Eindhoven University of Technology

MASTER

Multi-physical modeling and experimental verification of a respiratory system

Mennen, R.J.C.M.

Award date:2009

Link to publication

DisclaimerThis document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Studenttheses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the documentas presented in the repository. The required complexity or quality of research of student theses may vary by program, and the requiredminimum study period may vary in duration.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

Electromechanics & Power ElectronicsDepartment of Electrical Engineering

Den Dolech 2, 5612 AZ EindhovenP.O. Box 90159,5600 RM EindhovenThe Netherlandsw3.ele.tue.nl/epel

TU Technische UniversiteitEindhovenUniversity ofTechnology

-------..-------.-----.---f---------------.--

Author: R.J.C.M. Mennen

Supervisors:

E.A. Lomonova, EPE, TU/eJW. Jansen, EPE, TU/eG. van Dijk (Drager MedicalBV, Best)

Multi-Physical Modeling andExperimental Verification of a

Respiratory System

Master thesis report

Reference

EPE.2009.A.03

Date

19 me; 2009

Where innovation startsThe department of Electrical Engineering of the Eindhoven University of Technologydisclaims any responsibility for the contents of this report

Abstract

This report describes the dynamic and steady-state modeling of a respiratory device in the multi­physical domain. The multi-physical domain covers the electrical, the (electro)mechanical andpneumatic components of the system. The electromechanical parts are simulated using lumpedparameter analysis. These electromechanical parts consists of an inverter and a permanent mag­net synchronous motor. Several parameters can not be derived directly from the specifications.Therefore, several additional (semi-)analytical models are derived. The electromechanical devel­oped torque and the back-EMF constant are calculated by means of Fourier series. The self- andmutual inductance are derived by means of the Biot-Savart law. These models are verified bymeans of 2D FEM models and by means of measurements.

The dynamics of the pneumatic system are analyzed using the lumped parameter Greitzer­model. The model-parameters are derived by means of the measured surge-frequency. The steady­state performance of the pneumatic system is described by a compressor map. An analytical re­lation is derived to describe the compressor map, which is inverted to improve the implementedcontrol action.

One general model is obtained by coupling the models of the different parts of the multi­physical domain. Experiments are carried out to verify all the models, including the dynamicsand steady-state characteristics.

Using the models, the efficiency of the respiratory system with the implemented driving strat­egy is derived. To improve the calculated system efficiency an alternative strategy is proposed.

2

Acknowledgements

This research project was carried out within the company Drager Medical B.V., situated in Best.Many thanks to Drager for giving me an interesting multi-disciplinary assignment, and for givingme the opportunity to work within a high-tech medical environment. Furthermore, the supportand supervision from within Drager was excellent. Therefore, I want to thank especially mysupervisor Geert van Dijk for his efforts to guide me through my graduation, despite his busyagenda. From Daer I also want to thank Eugene Herben for his help with pneumatic modelingand John Bevers for his help in general with practical issues. From the TUIe I want to thankfirst of all Helm Jansen, my daily supervisor. Without his efforts and supply of knowledge, mypaper and project results wouldn't be of the same level. Furthermore, I also want to thank MarijnUyt De Willigen for all his help with the test setup and the measurements and for the interestingconversations about everything except electromechanics. Furthermore, for evaluating my work Iwant to thank Elena Lomonova. From the mechanics department, I want to thank Rick de Lange,for all his help and discussions about modeling of compressors. Also, many thanks to the (PhD)students, especially Timo Overboom, Frank van den Bergh, Koen Meessen, Bart Gysen, DavyKrop and Hans Rovers, for helping me with modeling and above all for fun. Finally, some specialthanks to Chantal Janzen for her daily presence during my graduation project, and enlighteningmy day every time I saw her.

3

List of symbols

Electromechanical Symbols

Quantity Unit DescriptionA Wb·m- 1 Magnetic vector potentialA m2 Enclosed area

B T Magnetic flux densityB rem T Remanent flux densityjj C·m-1 Electric flux densitydw m Diameter of the copper windings

E V -1 Electric field intensity·me Unit vector (en ee, ez )

E V Back-EMFF N (Lorentz) force vector

Fd N Force vector due to an electric field

Fm N Force vector due to a magnetic fieldII A-m-1 Magnetic field strengthI A Current

A CurrentJ A-m- 2 Current density vectorJm kg·m2 InertiaK N·m-A-1 Torque constantK c N·m Coulomb friction coefficientK N.m.rpm- 1 Viscous friction coefficientv

L H InductanceLax m Axial length of the windingsLm m Axial length of the rotor-magnetLm H Measured phase-phase inductanceL ss m Self inductanceMp

A -1 Remanent magnetization vector·mM ij H (Mutual) inductance between phase i and jN Number of turnsn Harmonic numberPin W Input powerPout W Output powerp Number of pole pairsq C Electric chargef m Radial position vectorR s n Electrical resistance phase-phaseT m Radial position

rs,m,mean,c,b m PMSM radii indicated in Fig. 3.1

4

5

Quantity Unit DescriptionS m~ SurfaceT N·m Torque vectorT N·m TorqueT °C TemperatureTe N·m Torque due to compression processT L N·m Total load torquet s Time'U V Phase voltage-I; m·s- 1 Velocity vectorV m3 VolumeV V VoltageVde V DC-voltagev m·s- 1 VelocityX Coefficienty Coefficientz m Axial positionQ rad·s- 2 Angular accelerationf3w rad Angle helical windings0 rad Rotation around z, stationary reference frameOr rad Rotation around z, (rotor-magnet)A Wb-turns Flux linkage

J.lo H -1 Permeability of vacuum (47l" x 10-7)·mJ.lr Relative permeability

P kg·m-3 Mass density

P C·m-3 Free charges per unit volume

PCu f1·m Electrical resistivity¢ Wb Magnetic fluxw rad's- 1 Motor speed

Pneumatic Symbols

DescriptionCompressor duct areaSonic velocitySpecific heat

Rate of strain tensor: 1 (EJUj + EJU,)2 BXt aXj

Resultant body force vector per unit massHelmholtz frequencyGravitational accelerationFluid elevationSpecific enthalpy, compression processTotal specific enthalpySpecific enthalpy, lossesInertiaConstant of the Bernoulli equationThermal conductivityFriction losses coefficientIncidence losses coefficientOutlet duct valve coefficientCompressor duct length(Compressor) mass flow

N·kg- 1

Hzm·s- 2

mJ·kg- 1

J·kg- 1

J·kg- 1

kg·m2

Unitm~

m·s- 1

J·kg-1 ·K- 1

Quantity

eij

FfH

9hhehe,thiass

JKb

kkfkins

kv

Le

In

6

Quantity Unitm v kg.s -1

nnPc J'S-l

P Pa

Po Pa

PI Pa

P2 Pa

Pp PaQ m3 ·s-1

1'1 m1'2 mS J·K- 1

T KTo KT c N·mT m N·mif m·s- 1

Ut,1 m·s- 1

Ut,2 m's- 1

Vp m3

X

(3

I6Jij

J-L Pa·sp kg.m-3

(J

<p

w rad·s- 1

Subscripts

DescriptionOutlet duct valve mass flowOutward normal, unit vectorPolytropic efficiency coefficientPower delivered to the fluidStatic pressureAmbient pressurePressure outside the outlet duct (Greitzer model)Instantaneous compressor pressure rise (Greitzer model)Plenum pressure (Greitzer model)Volumetric flowAverage impeller radiusExternal impeller radiusGain of entropy per unit massAbsolute temperatureAmbient temperatureTorque due to compression processTorque developed by the compressor motorLocal velocity of the fluidTangential gas velocity at radius 1'1

Tangential gas velocity at radius 1'2

Plenum volume (Greitzer model)Unit vector (x,y,z)

Coefficient of thermal expansion - *(~)p

Adiabatic index of airRate of expansion: \7. if = eii,

Kronecker delta function (see Appendix A)Viscosity of the fluidMass densitySlip factor2!"(e_ -e- - - 162 )P 'J 'J 3

Compressor motor speed

a Phase Ab Phase Bc Phase C

Component i of the vector or tensorj Component j of the vector or tensorI In region II I In region IIl' Radial component() Tangential componentz Axial componentn Harmonic number+, - Positive and negative winding phase-band respectively

Acronyms

DSPEMFFEMPMSMPSDPWMRMS

Digital signal processorElectro-motive forceFinite element modelingPermanent magnet synchronous motorPower spectral densityPulse-width modulationRoot mean square

7

Contents

List of symbols

1 Introduction1.1 Problem definition1.2 Project goals ...

2 Physics of the respiratory system2.1 Inverter .2.2 Permanent Magnet Synchronous Motor (PMSM)2.3 Pneumatics . . . . . . . . . . . . . . . . . . . . .2.4 Test setup . . . . . . . . . . . . . . . . . . . . . .2.5 Look-up tables implemented in the respiratory control-loop

3 Electromechanical models3.1 Steady state semi-analytical electromechanical model .3.2 Model describing the dynamics of the electromechanical subsystem3.3 Efficiency analysis of the electromechanical subsystem . . . . . . .

4 Pneumatic model4.1 Pneumatic model .4.2 Compressor map .4.3 Dynamic parameters derivation

5 General, multi-physical model5.1 Results of modeling the respiratory dynamics5.2 Steady-state results.5.3 Power efficiency .

6 Conclusions

Appendices

A Used mathematical operators and entities

B Six-step commutation with voltage control

C Navier-Stokes equationsC.1 Derivation of the Navier-Stokes equations

D Specifications Faulhaber PMSM

E Graduation paper

8

4

999

111212141616

17192629

31313539

42424445

49

52

53

56

6060

64

65

Chapter 1

Introduction

Medical appliances nowadays are more sophisticated, complex and demanding, and customers ex­pect more in terms of customization, innovation, quality and price than ever before. The lifetimesof products are short, due to fast changing technologies and new products from (new) competitors.The customers, therefore, have more choice, and good ideas are quickly copied by competitors.Drager Medical B.Y. in Best tries to meet these new requirements by the Compact YentilatorFamily (CYF) Concept project.

This project designs a concept-platform for the future small, transportable ventilator systems,which has to be the basis for the following 5 to 10 years. A platform is a set of subsystems, stan­dardized interfaces and concepts that form a common structure from which a stream of relatedproducts can be efficiently developed and produced. This approach has numerous advantages likeshortening product lead times, reduce design risks and cost, reduction of inventory, standard manu­facturing processes and tooling, more standard parts and faster response to changing market needs.

1.1 Problem definition

This new ventilator product is battery supplied, with an autonomous operation, without externalpower supply of approximately 60 min. This short autonomous operational time is due to alow system efficiency. Therefore efficiency and power management are important issues duringdevelopment of this device. Furthermore, to enable the required respiratory action, a whole rangeof control-strategies are used, which are not investigated during this project. However, thesecontrollers are based on a curve-fit on steady-state (pneumatic) measurements. The results areimplemented into a look-up table used by the control-loop. The drawbacks ofthis control approachare:

• Set-points used by the drive are known only for 1 particular set of ambient conditions.

• No information about the dynamics of the system is known.

1.2 Project goals

This project investigates the dynamics and steady-state characteristics of the PMSM and thepneumatic behavior of the entire blower, which operates in different physical domains. Thesedomains are the electrical, the electromechanical and pneumatic domain. Numerous techniquesare available to model the pneumatic [11], electromechanical [9] and electrical [16] componentsseparately, as part of the multi-physical domain covered by the respirators action. Earlier researchregarding respiratory subjects are mainly focussed on the biomechanical part of the system as

9

10 Chapter 1. Introduction

described in [21], [18], [8]. Several papers discuss briefly respiratory devices and control of thepneumatic output power, [6], [20], however, these models are also primarily based on the biome­chanical part of the system, as the electromechanical and pneumatic systems are left out of theanalysis. The results of this project will be used to carry out an efficiency and power-flow analysis,covering the entire respiratory action, therefore, the entire respiratory system has to be modeled.Using this analysis the drive of the respiratory device will be evaluated. Furthermore, the derivedmodels will be used to replace the empirically obtained look-up tables. The benefits which maybe accomplished, by implementing the models, are

• Better and more accurate control.

• Compensation of the control-action during change of the environmental conditions.

• More insight in power-flows, which makes it possible to derive the efficiency ofthe respiratoryaction.

• Determination of the system bottle-necks during respiration, and increasing the power­management.

In the next chapter the different components and functions of the respiratory system areexplained. Then, the focus will be on the analyzed respiratory system part, the blower unit.The blower operates in different physical sub-domains, namely the electrical, mechanical andpneumatic domain. In Chapter 3 the electromechanical models are presented, describing thedynamics and steady-state characteristics of this domain. Chapter 3 is split into a part in whichthe derived dynamical model is described, and a part in which the (static) semi-analytical modelsare discussed. Verification of the models is done by means of measurements. An efficiency analysis,regarding the electromechanical subsystem has been carried out at the end of this chapter. Thederived pneumatic models together with the results of the measurements are presented in Chapter4. The models describe the dynamics of the pneumatic subsystem, as well as the steady-statecharacteristics. In Chapter 5 the models derived for the different physical domains are combinedand the results are verified by means of measurements. An efficiency analysis, regarding the entirerespiratory action is carried out at the end of this chapter. In the final chapter, conclusions aredrawn regarding the models and measurements and recommendations for further research aregiven.

Chapter 2

Physics of the respiratory system

In this chapter, the physical properties of a respiratory system or ventilator are presented. Thestudied respiratory system is an update of a ventilator from the Carina™ family from the DragerMedical company. Schematically, a respiratory device is shown in Fig. 2.1 and consists of thefollowing functions and components. The gases required for (mechanical) respiration of a patientare air and oxygen. The ratio of these gases is adjustable by the oxygen dosage function. The airand oxygen are mixed in the blender. This mixture is forwarded by the pressure/flow generatorvia a system of hoses into the lungs of the patient during the inspiratory phase. At the startof the expiratory phase, the pressure/flow generator is operating at a lower power level. There­fore, the air is forced out of the patient due to the elasticity of the lungs and the pressure of theribcage and flows back into the environment. To keep the airway-pressure during expiration abovethe minimum allowable level, expiratory pressure control function together with the pressure/andflow generator are utilized. This minimum pressure-level is maintained to prevent collapsing ofthe lungs during expiration and is called Peak End-Expiratory Pressure (PEEP).

Pressure/FlowGenerator

Oxygeninlet

Air inlet

Patient airexhaust

Figure 2.1: Block-diagram of the respiratory system.

The system part analyzed for this project is the pressure/flow generator or blower unit. Theblower operates in two different physical domains, which are the electromechanical and pneumaticdomain as shown in Fig. 2.2. The different components of the blower are as follows. The electricalcircuit, or inverter is the power supply of the motor, which is a permanent magnet synchronous

11

12 Chapter 2. Physics of the respiratory system

Hallsi nals

Transistorswitching

yElectromechanical subsystem

Reference

1m lIer

•Pneumaticsubsystem

p

Q

Figure 2.2: Block-diagram of the blower unit.

motor (PMSM). An impeller is mounted on the shaft of the PMSM to deliver the pneumaticwork. The motor is commutated with a six-step commutation scheme. The rotor-position ismeasured with Hall-sensors. For more information about Hall sensors and six-step commutation,see Appendix B. Finally, to achieve the desired respiratory action, regarding the reference signal,the controller-block regulates the input-signal of the inverter. The symbols used in Fig. 2.2 are'

v = Voltage (V),

I = Current (A),

T = Torque (N·m),

w = Speed (rad's- 1),

p = Pressure (Pa),

Q = Volumetric flow (m3 ·s- 1).

2.1 Inverter

The power supply of the blower is a full-bridge inverter, which controls the voltages applied tothe three stator windings, as shown in Fig. 2.3. This is achieved by means of the six MOSFETsof the inverter, which are used as switches. The amplitude of the voltage delivered to the statorwindings is adjustable by means of the duty-cycle of the PWM signal. This control method iscalled voltage control. The required duty-cycle is dependent on the desired speed of the PMSMand the load torque. The commutation-algorithm and PWM-controller are embedded in a DSP.The inductances in the leads to the stator windings are included to reduce the current-peaksduring switching.

2.2 Permanent Magnet Synchronous Motor (PMSM)

The electric motor of the blower is a three-phase PMSM, which is designed especially for this(respiratory) application. It is capable of achieving high speeds up to 8.9.103 rad·s- 1 or 85000rpm, and accelerations up to 30000 rad·s- 2 • A cross-sectional view of the implemented motor isshown schematically in Fig. 2.4.

• New introduced symbols are explained under all equations. Equations introduced earlier in this thesis areconsidered familiar and not mentioned again. All symbols used in this thesis are listed in the "List of Symbols".

2. 2'~E:lr:I!l_~_ElIl~_MagnElt Synchr:()Il()ll~ lY!()t_()E(!'lY!~lY!) . 13

+

,---' TT T ,--:> Phase A

c,Phase B

Phase C

Figure 2.:3: Three-phase full-bridge inverter with additional inductances in the leads to the statorwindings.

Table 2.1: Magnetic properties of smeo material.

PaTameter Description ValueB re", Remanent magnetization 0.98 l'

Ilr Relative permeability 1.085 H.m ·1

x (m)

Laminated

back iron

Slalor Windings

Airgap

PM rotor

Shaft

0,015]

0.01

0.005!

~ .o.oo:~1-0.01

·0.015,0.01 ",>

0.005 '0,'",- ~~~----- 0.01

o~, --------.o.o~~/· /~o

y (m) -0.01 -0.01

Figure 2.4: Schematic of the three-phase Pl'vISM. Figure 2.5: Schematic of the helical winding

topology of the PMSM in 3D.

The rotor-magnet implemented in the PMSM is a two pole SmCo magnet with parallel magneti­zation, capable of retaining magnetization up to 130°C. The magnetic properties of this materialare given in Table 2.1.

The three stator windings of the P]'vISM are arranged in a Delta-connection, which gives alower EMF compared to Wye-connection. The EMF is explained in Chapter 3. The coils of thestator windings are wound in a helical configuration as shown in 3D in Fig. 2.5. A 2D view of thebelical winding configuration is shown in Fig. 2.6. This topology is treated earlier by [1] and [2].This winding topology has several advantages and drawbacks. It is implemented to obtain a rigidsloL!ess structure without the disturbing cogging forces. However, using the helical winding topol­ogy the torque capabilities of t.he motor are reduced, because the current is not perpendicular to

14 Chapter 2. Physics of the respiratory system-----"----------- ._ --",-----------------------",,----

Figure 2.6: Helical winding topology in 2D.

Table 2.2: Properties of the stator windings.

Number of windings per phase IV 32Pw in Fig. 2.6 (rad) 0.84Diameter of copper winding dw (Ill) 0.2998·10 .;;

PCu (0. m) at 20°C 1.72·1O-~

the rotor magnetic flux density. The influence is further iuvestigated in Chapter 3. The propertiesof the windings are depicted in Table 2.2.

The shaft and back-iron are made of the steel alloy X30CrMoN15-1. The material propertiesof these motor-parts are given in Table 2.:3. The curve, as given in Fig 2.7 is based on theBH-curve of steel often used in electromechanics, which resembles the particular steel used in thisapplication. Furthermore, see Appendix D or ?? for an overview of the electrical and mechanicalproperties of the P1\ISlVI used in this application, as indicated by the manufacturer Faulhaber.

2.3 Pneumatics

The pneumatic work in the compressor is done by a disc mounted on the shaft of the PlVISM,called the impeller. The type of compressor used in this application is a centrifugal compressor.Compression by centrifugal compressors is realized by transferring momentum of the blades of theimpeller to the fluid/gas and subsequently convert this kinetic energy into pressure by diffusion.This diffusion takes place in a channel of increasing radius around the impeller, named the diffuser.

Table 2.:3: 1\'faterial properties of the steel parts.

Part [ Material

Steel (X:30CrlVIoN15-1)lVIegaperm 40L

Fig. 2.79000

Saturation (1')

Fig. 2.71.48

2.3. Pneumatics 15

2,--------,------,----,-------,----,--------,------"

3.5

X 104

3°0·------0'5-----'·--------1'5-~- 2.5

Magnet field intensity (Am-1)

0.2

1.8

16~E 1.4~c !.~ 1.2~Q) I"0 Ix 1 ~

~ i:a; 0.8cCl ,

'" I2 0.6:

I0.4

Figure 2.7: Approximation of the BH-curve of the steel used for the shaft.

Figure 2.8: 3D Vlew of the 1111­

peller.Figure 2.9: Schematic of a centrifugal cornpressOI' [El].

The rise of the static pressure during the diffusion proces is a direct result of Bernoulli's theorem,which states that the sum of kinetic energy ~pv2

, the potential energy pgh and the static pressurep remains constant Kb during the process [19].

(2.1 )

where

[(b = Constant of the Bernoulli equation,

p = Fluid mass density (kg·m-3),

v = Fluid velocity (m·s-1),

9 = Gravitational acceleration (m.s- 2),

h = Fluid elevation (m),

J! = Static pressure rise (Pa).

16

This means that due to the increasing radius of the diffuser the speed of the fluid '1' decreases andtherefore the static pressure p rises, neglecting potential energy variations,

2.4 Test setup

The test setup used f()r the measurements during this project IS schematically represented byFigs. 2.10 and 2.11.

Tube Ball valve1.2 m

Outlet

Centrifugal compressor

Figure 2.10: Schematic of the test setup.

Figure 2.11: Picture of the test setup.

This setup consists of the analyzed compressor, which is the centrifugal compressor. However, thecommutation and motor control are realized in dSPACE with the 1104 control system, instead ofthe DSP-chip, used in the commercial application. At the outlet duct of the compressor. pressuremeasurements are carried out with the CTEM70:350GYO pressure gauge from SensorTechnics. Theoutlet duct is connected further to a tube with a length of 0.35 Ill. Flow measurements are carriedout at the end of this tube with a 1'81 model 4040 flow meter. A standard ventilation tube oflength 1.2 m is connected to the outlet of the flow meter. This pneumatic system is closed by astandard full port ball valve to adjust the outlet duct opening.

2.5 Look-up tables hnplelnented in the respiratory control­loop

As earlier mentioned in the introduction, the (feed-forward) respiratory control action is based ona curve-fit on steady-state measurements. The resulting expression is a relation between pressurep, volumetric flow Q and motor speed RPAt, as shown in (2.2).

RPAI = :15000 * (-O.00:n2 + 0.129 *p049:38 + 1.52· 1O-,5QU)o.5 + 0.0043G *p-04:ICJ ) (2.2)

As can be seen, the environmental conditions are not taken into account. This can be improvedby deriving an analytical expression, describing the steady-state characteristics of the analyzedcompressor. This expression has to include the following variables.

p = Pressure (Pa).

Tn = .Mass flow (kg·s- 1),

v.,' = Motor speed (mel.s - 1),

Po = Ambient pressure (Pa),

To = Ambient temperature (K).

Chapter 3

Electromechanical models

In this chapter, the different developed electromechanical models and the theory needed to derivethese models are explained. The operating principles of the used permanent magnet synchronousmotor (PMSM), accompanied with the used electrical and mechanical equations are presentedalso in this chapter. The derived models are verified by means of steady-state and dynamicmeasurements. At the end of this chapter, an efficiency analysis is carried out regarding theelectromechanical subsystem. A possible improvement on the actual driving strategy is proposed.The performance of the proposed driving strategy is compared with the actual system by meansof the efficiency analysis.

The theory of operation of any electromechanical system starts with the Maxwell equations asgiven in [9].

t jj. dl l (J+ a;}) . dB, Ampere's circuital law

i E.dB 0, Conservation of flux

t E.dl -1 813. dB Faraday's laws 8t '

i fJ. dB Ipdv, Gauss' law

where

(3.1)

(3.2)

(3.3)

(3.4)

jj = Magnetic field intensity (A.m- I),

J = Free current density (A-m-3),

fJ = Electric flux density (C.m- I),

13 = Magnetic flux density (T),

E = Electric field intensity (V.m- I),

p = Free charges per unit volume (C·m-3).

These equations describe the electromagnetic fields due to magnets and currents in detail, which isnecessary to predict the performance of an electromechanical device. The conditions investigatedin this thesis are relatively low-frequent and the dimensions of the regions of interest are smallrelative to the wavelength of the electromagnetic field that permeates it. The set of equations cantherefore be considered quasi-static and the term 8 fJ/8t in (3.1) is in this case negligible. As aresult, the coupled electromagnetic Maxwell equations (3.1) to (3.4) are partially uncoupled. This

17

18

is used to simplify the analysis of the machine.

Chapter 3. Electromechanical models

The force on a particle with charge q in a magnetic field can be expressed with the Lorentzforce P [15]

where

qv x ii,qE,Pm + Pd = q(E + V xii), (3.5)

Pm = Force vector experienced by a charged particle moving in a magnetic field (N),

Pd = Force vector experienced by a charge determined from the definition of E (N),

q = Electric charge (C),

V = Velocity (m.s- 1),

The derivation of the Lorentz force is based on Faraday's law (3.3) and the definition of the electricfield, E = P/q. For a more thorough discussion about the Lorentz force and its derivation fromthe Maxwell equations see [15]. For the electromechanical analysis, the force and torque producedby the currents and the ii-field are especially interesting. (3.5) is therefore written in terms ofcurrent density and magnetic flux density, as

Where

1Ix iidv,

1TX (I x ii)dv.

Lorentz force, [9]

Torque

(3.6)

(3.7)

P = Force (N),

T = Torque (N.m),

T = Vector about which the torque is computed (m).

Several other important physical quantities are defined to describe behavior of the electromechan­ical machines. They are listed below. The physical meaning of these quantities become clear whenthe model is developed in the remaining part of this section.

¢ l ii· ds, Flux

A l ii ·ds= N¢, Flux linkage

L8A

Inductance8i '

E8A

Electromotive force (EMF)at'

(3.8)

(3.9)

(3.10)

(3.11)

where

¢ = Magnetic flux (Wb),

A = Flux linkage (Wb-turns),

N = Number of turns of the coil,

L = Inductance (H),

E = EMF (V).

3.1. Steady state semi-analytical electromechanical model 19

Due to the time rate of change of magnetic flux through the circuit a voltage is induced, accordingto Faraday's law (3.3). This voltage is defined as EMF. The El'vlF term can be split into a termcaused by a change in currents (induced EMF) and a term caused by the rotational speed (motionalEMF or back-EMF), namely E = dA/dt = L (di / dt) + (aA/DO r ) w. For further reading and a morecomprehensive explanation of the above described thpory see [9].

3.1 Steady state semi-analytical electromechanical model

The torque of the PMSM is calculated with (:Ui). To dprivp the torque, EI'vlF and performance of

the PMSM, it is therefore important to have a description of the magnetic flux density (B) causedby the rotor-magnet inside the airgap at the radius of the windings. According to [23] and [22] themagnetic flux density imdde the airgap can be described by Fourier series after solving the Lapla­cian and/or Poissonian field equations. This is elaborated in the remaining part of this sectioll.

The rotor consists of a steel shaft and a magnet (2 poles) with parallel nwgnetization (Fig. :3.1).

Airgap Magnets(Region I) (Region II)

rmesn

rb

1111

"

Table :3.1: Model parameters.

I Geometric parameters (m) IT 1.5 x 10 -0

s

Tm 5.4 x 10 -0--

6.7 x 10 .;:5{"mean

::5T c 7.4 X 10 '.

rb 10.5 x 10 -0

Figure 3.1: Cross-section of the PMSM withthe two regions indicated as used f(lr mag­netic flux density calculations.

The following assumptions are done in order to apply the field equations.

• Length of the PMSM is considered infinite, consequently no end-effects and simplification ofthe model from three dimensions to two dimensions.

• Relative penneability (/1,.) oIthe iron parts is considered infinite, hence no satmation effects.

• Relative permeability of copper is considered 1, hence the coils will be modeled as air.

The shaft and back-iron are assumed to be made of iron with 111' = 00. In this case, only tworegions are of interest, namely the magnet-region (region II) ane! the airgap-region (region I). Thisis sho'vvn in Fig. 3.1. Furthermore, the radius of the copper windings is very small. Therefore, itis a,ssurned that the current flows through a sheet. of radius T me"".

20 Chapter 3. Electromechanical models

In general a function f (x) with period T can be described by Fourier series as follows

f(x) ao ~ [ (2mr ) . (2mr )]2 +~ an cos T x + bn sm T X ,

2 rT (2mr )T io f(x) cos T x dx,

2 rT. (2mr )T io f(x) sm T X dx,

(3.12)

(3.13)

(3.14)

where

1 j"ao = - f(x)dx.7r _"

(3.15)

The magnetization Mp of the rotor-magnet can be split into a part in the radial direction (er ) anda part in the tangential direction (eo):

The magnetization is visualized in Fig. 3.2. Both the radial and tangential components aresinusoidal waveforms in case of parallel magnetization.

Magnetization in radial direction,,~~~-~--'--c-~-~-~----,-----r-----.----=....,

i

~f 05r~ o~"iiiE I

~ ~j0------'0'2------'0'-4--O,Le--0'--,8-=-~,..--'---L----'-',.-----",8c-------",c-8 ---'

e (it rad)

Magnetization in tangential direction

-'Ol------c~~""--«;---:":-----',---c',':-2-c-,'-c,.-c-'-'c-,8--''''',8----.J

e (it rad)

Figure 3.2: Magnetization vector of the two-pole parallel magnetized rotor.

As can be concluded from Fig. 3.2 only the first harmonic is present. The description of the radialand tangential components of the magnetization vector can therefore be expressed in Fourier serieslike

n=1,3,5,

B remMr nCos (npB) = Mr 1 cos (B - Br ) = -- cos (B - Br ),, , /-La (3.16)

n=1,3,5,

(3.17)

Where (B - Br ) is the difference between angle of the rotor Br and the angle B as defined 1ll

3.1. Steady state semi-analytical electromechanical model 21

the stationary reference frame in Fig. 2.4. Furthermore, p is the number of pole-pairs of therotor-magnet, which is equal to one in this PMSM. To find an expression for the magnetic fluxdensity inside the airgap, the Maxwell equations are derived for the magneto-static case in vectornotation:*

VxH

V·BJ,o.

(3.18)

(3.19)

The constitutive relation is for the magnet and the airgap region respectively

BB

/-LO/-Lr H + /-LoMp,

/-Lo H .

(3.20)

(3.21)

Solving the magneto-static Maxwell equations is done with the vector potential. The vectorpotential A is defined as

B=VxA. Vector potential (3.22)

Substituting the expression for the vector potential into the constitutive relation (3.20) gives

v x A = /-LO/-LrH + /-LoAlp.

Taking the curl of both sides of the equation gives

- 2 - - -V(V . A) - V A = /-Lr/-LoJ + /-Lo V x Mp.

This results for V . A = 0 (Coulomb gauge) in the Poisson equation.

V2 A= -/-Lo (V x Mp + /-LrJ) .

(3.23)

(3.24)

(3.25)

The Poisson equation is valid within region II in Fig. 3.1, the magnets. In the case of a source-freemedium, e.g. the airgap (region I), the Poisson equation reduces to the Laplace equation.

(3.26)

For cylindrical coordinates the B-field as function of magnetic vector potential is

(3.29)

(3.28)

(3.27)Ber ree ez

VxA= a a aar ao azAr rAo Az

(! 8Az _ 8Ao) er+ (8Ar _ 8Az ) eo +! (8(rAo) _ DAr) ez.

r 88 8z 8z Dr r Dr 88

Furthermore, since that the magnetic flux density has only components in radial and tangentialdirection, the vector potential has only a component in the axial (z) direction.

B ~ ! 8Az __ 8Az -- r 88 er 8r eo·

(3.31 )

(3.30)

The Laplace and Poisson equation, (3.26) and (3.25) respectively, can further be elaborated bythe result of the Laplacian (V2 A) in cylindrical coordinates for both regions, as

(1 8 ( Az I) 1 8 2

A z I) _:;: 8r r 8; + r2~ ez = 0,

(1 8 ( A z II) 1 82

A z II) _ -:;: 8r r---a;:- + r2 88~ ez = -/-Lo V x Mp

_p.o (8(rMo) _ 8Mr ) __r 8r 88 ez - O.

*See App. A for a description of the V-operator

22 Chapter 3. Electromechanical models

Due to the specific magnetization of the rotor-magnet, the Poisson equation valid within themagnet-region is reduced to the Laplace equation. The (homogeneous) solutions for the Laplaceequations of the vector potential (A) in both regions are in general defined as a linear combinationof sin (np()) and cos (np()) and rnp and r-np . In this particular case p, the number of pole-pairsis one. The normal component of the magnetization vector contains only cosine-terms (3.16).Therefore, the normal component of the magnetic flux density Br also contains only cosine terms.Since B r = (l/r) (8A z I8()) Az contains only sine terms:

00

L (Xn,JTnp + Yn,Ir- np ) sin (np(() - ()r)) ezn=l

(XI,Ir l + YI,Ir- l) sin (() - ()r) ez ,

00

L (Xn,IIrnp + Yn,IIr- np ) sin (np(() - ()r))ezn=l

(3.32)

(XI,IIr l + YI,IIr- l) sin(() - ()r)ez. (3.33)

The final step is possible because only the first harmonic is present in case of parallel magnetizationand the number of pole-pairs is equal to one. The magnetic flux density can now be derivedby substituting (3.32) and (3.33) into (3.29). The results are formulated in four equations; onecomponent for the magnetic flux density in the radial direction and one component in the tangentialdirection for both regions.

00

n=1,3,5, ...

(3.34)00

n=I,3,5, ..

(3.35)00

n=1,3,5, ..

(3.36)00

L (npXn,IIrnp- 1- npYn,IIr-np- l

) sin (np(() - ()r))n=1,3,5, ...

(3.37)

(3.42)

(3.43)

(3.44)

(3.45)

The final step in the expressions for the magnetic flux densities can be explained by taking intoaccount just one harmonic and one pole-pair. The four unknowns (XI,I, YI,I, XI,II, YI,II) arefound by computing the following boundary conditions

He,Ilr=rc 0, (3.38)

He,II Ir=r, 0, (3.39)

Br,Ilr=rm Br,II Ir=rm , (3.40)

He,Ilr=rm He,II Ir=rm • (3.41)

Where the geometric parameters are as indicated in Fig. 3.1. The four boundary conditions arecomputed and the results are listed in the following equations.

XI,I - YI,Irc=-2 0,

XI,II - YI,IIr;2 -J.LoMe,l'

(XI,I - XI,II) + (Yi,I - Yi,II) r;;.,2 0,

(J.LrXi,I - Xi,II) + (-J.LrYi,I + Yi,II) r;;.,2 J.LoMe,l.

3.1. Steady state semi-analytical electromechanical model 23

Table 3.2: Axial length of the stator windings and magnet.

Length of the coil-axis Lax (Fig. 2.6) (m)Length of the magnet-axis L m (m)

This system of four equations with four unknowns can be solved straightforwardly using matrixcalculations with the mathematical software package used. The results of these calculations arecompared with FEM-simulations in Flux 2D [7] (Figs. 3.3 and 3.4).

Color Shade ResultsQuantity: iFlux densityi TeslaScale! Color4,228IE-6 92,25883£-392,25883£-3 I 184,5134£-3184,5134E-3 ,I 276,76803£-3276,76803E-3 369,02261E-3369,02261E-3 461,27719E-3461,27719E-3 553,53183E-3553,53183E-3 I 645,7864£-3645,7864£-3 738,04098£-3738,04098£-3 I 830,29556E-3830,29556E-3 922,55014E-3922,55014E-3 I 1,01481,0148 / 1,107061,10706 / 1,199311,19931 1,291571,29157 1.383821,38382 1,47608

Figure 3.3: Magnetic flux density throughout PMSM: FEM.

The waveforms for the radial and tangential components of the magnetic flux density agree forboth the FEM and semi-analytical simulations. The disagreement between the two different mod­els is less than 1 percent. This means that the assumptions done at the start of this section arevalid and there is no iron saturation.

The torque is calculated using (3.6). The magnetic flux density in the radial direction at radiusr mean is calculated with the above derived model. The current density J is calculated using theproperties of the windings indicated in Table 2.2. The axial length of the stator windings and themagnet as indicated in Table 3.2, are taken into account during the analysis. To calculate thecurrent density the following assumptions are used.

• The windings for every phase are divided in two phase-bands, one in the positive z-directionand one in the negative z-direction (Fig. 2.6).

• One phase-band covers 1200 or %1T' rad.

• The current flows through a sheet of radius r mean .

(3.47)

(3.46)21T'rmean ,

3Nia

21T'rmean ·

The resulting current density is an average over 32 windings covering 2/31T' rad on radius r mean .

The results for both the positive and negative bundle of phase A are expressed by

3Nia

-0.6

24 Chapter 3. Electromechanical models

0.6 ,-----------,-----,------,-----------,-----,-----~(

0.4

E 0.2

.i='·iii

~ 01

'i=

¥ -0.2Qlc01<l1

~ -0.4 ____ Bf

-_Beo B

fFEM

11< Be FEM-0.8

0L-------'------2.L----

3'---------4'------------'5====6=='--1

angle (rad)

Figure 3.4: Magnetic flux density (radial and tangential) at r = r mean.

Table 3.3: Integral limits used in (3.48).

Bf-----+-p-hTe

A--__+_p_h-,as,----e_B__-=J--+-Ph-as-e-c-_- J(}min 1 7r + ~z ~7r - ~z 17r + ~z li 7r - ~z 117r + ~z 117r - ~z

2 Lax 2 Lax 6 Lax 6 Lax 6 Lax 6 Lax

(}max 17r + ~z li 7r - ~'" 117r+ ~z 117r - ~z 11 7r + ~z ~7r- ~z6 Lax 6 Lax ""' 6 Lax 6 Lax 6 Lo,x 6 Lax

Zmin -~Lm -~Lm -~Lm -~Lm -~Lm -~Lm

Zmax ~Lm ~Lm ~Lm ~Lm ~Lm ~Lm

The same holds for the other 2 phases, with ib and i c instead of i a • The relevant component forthe current densities with torque calculations is the z-component. This axial component givestogether with the magnetic flux density in the radial direction after elaborating the cross-productin the Lorentz equation (3.6) the force in the (}-direction, or tangential direction. The currentdensity in the z-direetion Jz is simply obtained by multiplying the above discussed results forfabc,+- with sin (/3w) as defined in Fig. 2.6. To obtain the torque the following integral, withds = rd(}dz has to be evaluated:

(3.48)

The integral limits used to compute (3.48) for the positive (+) and negative (-) bundles for allthree phases are summarized in Table 3.3. To compute the torque produced by phase A, (3.48) issolved by integrating over the stator windings of this phase, arranged as shown by Fig. 2.6, and withthe phase-current ia equal to 1 A. The induced voltage over a phase, or the back-EMF constantK, can be derived from the torque produced by that phase as both are proportional to the change

3.1. Steady state semi-analytical electromechanical model 25

(X,y,z)

Figure 3.5: Current filament HS with distance vectors used in (3 ..50) to (:3.52).

in nux-linkage. The mathematical expression for the EMF voltage is then E = Kwsin((;I - er),where (;IT is tbe angle of the rotor magnet. The resulting value tc)r J{ is 1.9.10-3 Y·S·IIld- 1 . Thisvalue has a deviation of 7 percent from the value given in the specifications.

To complete the set of parameters to model the PMSM in (3.55), the (selt~)inductanceof thecoils and the mutual inductance between the coils are calculated. These parameters are derivedusing the vector potentiaL due to the current through the windings. Using Biot-Savart-Iaw, thevector potential is defined as [4]

If (.c') = p,oI j'~x I~ ~ I41r C X - x'

(:U9)

The vector potential in vacumn at any point in space, due to a line-current can, therefore, bederived directly from that line-current with current I flowing in the direction of rat :r' in space.The point at which the vector potential is calcula.ted is:r. Elaborating (3.49) for some arbitraryline-current in a Cartesian coordinate system the resulting expression for the vector potential isgiven in [4] by

(:3.00)

(:3.51)

(3.52)

where P R, PT and P S are the lengths of the vectors from the current filament RS to the pointl' at which the vector potential is calculated. This current filament and the relevant vectors areshown in Fig. :3.5. Subsequently, the self- and lIlutual inductances are expressed by

26 Chapter 3. Electromechanical models

Table 3.4: Calculated self and mutual inductance.

In vacuum With back-ironL S8 (IlH) 12 19M (IlH) 0.89 1.4

(3.54)

(3.53)L 1 J ~ ~- A· Jdv12 v '

1 J ~ ~-- A2 · J 1dv,hh VI

where 8 and 8 1 are the areas of the analyzed coils on the defined current sheet in Fig. 3.1.The windings of the three-phase coils in cylindrical coordinates shown in Fig. 2.6 are dividedinto a number of line-currents. Subsequently, the self and mutual inductances are calculated bynumerically integrating over the current sheet surface of that phase. Because the vector potentialsare derived in a Cartesian coordinate system, transformations from the Cartesian to the cylindricaland vice versa are carried out. The values calculated for the self inductance L BS and the mutualinductance M are shown in Table 3.4.However, instead of vacuum, the stator windings are surrounded by back-iron. To estimate theinfluence of the back-iron, simulations are done with a 2D FEM-model including and excluding theback-iron. The resulting increase in inductance because of the back-iron is approximately 60%.The inductance values using this approximation are shown in the second column of Table 3.4. Themeasured inductance L m in case of a Delta-connected three-phase winding configuration can beexpressed by the self and mutual inductance as Lm = (2/3) (L ss + M). This gives Lm = 14 IlH forthe derived inductance values. The disagreement is 18% in comparison with the 17 IlH obtainedfrom the specifications. The measured inductance is 18 IlH.The calculated values of the self- and mutual inductance show that the mutual inductance is only8% compared to the self inductance. Moreover, because of the additional inductances of 36 IlH inthe stator winding leads, as already indicated in Fig. 2.3, the mutual inductances are neglected inthe electromechanical model during further modeling.

3.2 Model describing the dynamics of the electromechanicalsubsystem

In this section, modeling of the dynamics of the PMSM is presented. Several simulations of thesemodels are shown together with the inverter bridge.

The dynamics of the PMSM are approximated by the stator currents ia,b,e of phase A, BandC respectively, the rotor angle B and speed w. The resulting (electro)mechanical model defined inthe three-phase stationary reference frame is [14]

L dia88 dt

L dibss dt

L diess dt

J d2 B(t)m dt2

dBdt

dib die-R i -E -M- -M- +u

S a a dt dt a,

dia die- R ib - E b - M- - M- + Ub

8 dt dt 'dia dib

-R i -E -M- -M- +usee dt dt e,

w. (3.55)

273.2. Model describing the dynamics of the electromechanical subsystem-------

Table 3.5: Relevant motor specifications.

Parameter Description ValueRs Phase-phase resistance 0.39 nKe Coulomb friction 0.3.10 -;5 N.m

Kv Viscous friction coefficient 0.5. 10 -15 N.m.rpm -1

K Torque constant 2.04·10 3 N-m.A 1

J m Inertia 1.74 . 10-r kg·m~

L m Phase-phase inductance 17 itH

Geometric motor dimensions (Fig. 3.1)rs Shaft radius 1.5 ·10 -;5 mrm Magnet radius 5.4 ·10 -;5 m

rmean Mean coil radius 6.7 ·10 -;5 m

rb Back-iron radius 7.4.10 -;5 m

Where Ea.,b,e are the back-EMF voltages of phase A, Band C respectively. Furthermore, R s isthe phase-resistance, L ss the self-inductance, M the mutual inductance and Ua.,b,e are the phasevoltages of phase A, Band C resp. The inertia and back-EMF constant are indicated by J m and K,respectively. The load torque TL is split into Coulomb friction, viscous friction and torque due tocompressor work Te . The influence of eddy-currents due to motion of the magnetic field is includedin the viscous friction coefficient. The compressor torque is further elaborated in Chapter 4. Theresulting expression for the load torque is

(3.56)

The relevant specifications regarding the model, as indicated by the manufacturer, are shown inTable 3.5. Because of the negligible influence of the mutual inductance on the performance, asderived in Section 3.1, these inductances are not included in further modeling. The reduced modelis

L ~ia88 dt

L dib88 dt

L die88 dt

J d20(t)m dt2

dOdt

= w. (3.57)

To simulate the described electromechanical model, (3.57) is implemented in Matlab and Simulinkand the inverter is modeled using the SimPowerSysterns toolbox. The (dynamic) performance ofthis electromechanical model is compared with the manufacturer specifications. The results areshown in Table 3.6 and deviate with a maximum of 6% from the specifications.

Due to the described six-step commutation and voltage control driving strategy, the voltagesapplied to the windings have a rectangular waveform. This waveform is in contrast with thesinusoidally shaped back-EMF, as derived in Section 3.1. The resulting (line-)current is shownin Fig. 3.6, together with the EMF-waveform and the applied voltage of the same phase. As isshown, the difference in waveform in applied voltage and EMF, by using the six-step commu­tation, results in higher harmonics in the current. These higher harmonics give rise to torque

28 Chapter 3. Electromechanical models

Table 3.6: Specifications of PMSM.

Manufacturer ModelMechanicaltime-constant 16 ms 16 msNo load speed 6.73.104 rpm 6.96.104 rpmLoad current 0.49. 10 -0 N'm.A ·1 0.46.10 -0 N'm.A ·1

EMF voltage

~ l\f\I\l\5--~;:~Q-100.735 0.736 0.737 0.738 0.739 0.74 0.741 0.742

Applied voltage

~c~:;() -2

0.735

0.736

0.736

0.737

0.737

0.738 0.739Current

0.738 0.739Time (5)

0.74

0.74

0.741

0.741

0.742

0742

Figure 3.6: Simulated EMF waveform, applied voltage and resulting line-current using six-stepcommutation and sinusoidal current control. Open-loop operation with Vdc = 10V.

3.3. Efficiency analysis of the electromechanical subsystem 29

ripple, as indicated in Fig. 3.7, and extra copper losses. The torque ripple with a peak-peakamplitude of approximately 1.5 mN·m seems relevant, because the rated output torque is only5 mN·m. However, the resulting speed ripple is negligible, due to a sufficiently high inertia, asshown in Fig. 3.8. Six-step commutation and voltage control is further elaborated in App. B.

--Six-step commutation---- Sinusoidal current control

S.2f J94

5.15:-

105.1

0.362 0.363Time(s)

036503640362 0363Time (s)

0.361

Ezi 5FH1IH1itH'Hi1t1i#+HH+tHH,I#~H'H11f+HH+I+HI

! I

Io!-

J~-~~----~----,036

Figure 3.7: Simulated torque ripple with six­step commutation and ideal sinusoidal cur­rent control at 5.0.104 rpm.

Figure 3.8: Simulated speed ripple with six­step commutation at 5.0.104 rpm.

However, generally a PMSM with a sinusoidal shaped back-EMF is driven by a current having thesame sinusoidal waveform as this back-EMF [12]. This is accomplished by a three-phase sinusoidalcurrent control drive. The resulting waveforms using sinusoidal current control are shown in Fig.3.6. The two driving strategies, six-step commutation with voltage control and sinusoidal currentcontrol, are compared by means of an efficiency analysis in the following section.

3.3 Efficiency analysis of the electromechanical subsystem

The efficiency analysis of the electromechanical subsystem, to compare six-step commutation andvoltage control with sinusoidal current control, is carried out in this section. The power flowmodel of the electromechanical subsystem is shown in Fig. 3.9. The losses as indicated in the

Figure 3.9: Power flow model of the electromechanical subsystem.

power flow model, shown in Fig. 3.9 are summarized in Table 3.7. Using the ratio between outputand input power, the two resulting efficiency maps for six-step commutation with voltage controland sinusoidal current control are shown in Figs. 3.10 and 3.11 respectively. The input poweris Pin = V I at the DC voltage side of the used full-bridge inverter (Fig. 2.3). The mechanicaloutput power is defined as Pou.t = Tw.

30 Chapter 3. Electromechanical models

Table 3.7: Losses taken into account with analysis.

Electrical subsystemI'2R I Copper losses (W)

Mechanical subsystemKcw I Coulomb friction (W)Kvw" I Viscous friction including eddy current losses (W)

0.'

0.'

4 5 6Speed (rpm)

Figure 3.10: Efficiency map of the elec­tromechanical subsystem using six-stepcommutation with voltage control.

Figure 3.11: Efficiency map of the elec­tromechanical subsystem using sinusoidalcurrent control.

In the normal working region of the PMSM for the respiratory application, the efficiency is between50% and 65% (Fig. 3.10). However, by implementing sinusoidal current control, the efficiency ofthe drive in the normal working area of the PMSM is between 70% and 78% (Fig. 3.11). Therefore,the efficiency of the drive can be significantly improved by changing the driving strategy fromsix-step commutation to sinusoidal current control. The influence of the inverter on the lossesis taken into account with the efficiency analysis regarding six-step commutation with voltagecontrol. However, no specific amplifier topology is taken into account by the analysis of sinusoidalcurrent control. This should be investigated further. Moreover, to complete the efficiency analysisfor this project, the pneumatic subsystem has to be included. The derivation of the models for thepneumatic subsystem and the efficiency analysis regarding the entire respiratory action is carriedin the rest of this thesis.

Chapter 4

Pneumatic model

In this chapter the pneumatic equations to describe the behavior of a fluid inside a centrifugalcompressor are derived. In general, the behavior of a fluid is described in detail by the Navier­Stokes equations, as derived in Appendix C. However, solving this system of equations for acomplex system like the blower is computationally very hard, and therefore, takes a very longtime. The three dimensional Navier-Stokes equations are therefore simplified to a one dimensionalmodel with only a few parameters, namely the Greitzer model. The derivation and performanceof this model is described in this chapter, together with the required assumptions to enable thesimplifications from the Navies-Stokes equations to the Greitzer model.

4.1 Pneumatic model

The model derived in this section describes the pneumatic behavior of the blower unit. It takesinto account the different thermodynamical quantities, pressure p, mass flow m, temperature Tand compressor speed w.

A blower, or compressor, is a device that increases the pressure of a medium. It accomplishesthis by adding kinetic energy to the medium and subsequently increasing the potential energy bydecelerating the fluid. There are several different types of compressors, e.g. axial and centrifu­gal compress. The type used in this application is the centrifugal compressor. See Fig. 4.1 fora schematic overview of the different parts of the compressor. The different components of thecompressor indicated in this figure are now discussed briefly. The base-component of a centrifugalcompressor is a rotating disc with a number of blades on it, called the impeller. The impeller,with radius "'2 in Fig. 4.1, is mounted on the shaft of the PMSM. The fluid is sucked into themiddle-point of the wheel, the eye of the impeller (inducer) and whirled round at high velocity.The accelerated fluid is decelerated inside the one or more diverging passages of the diffuser. Bydecelerating the fluid with more passages separated by the diffuser vanes, a smaller length is nec­essary to obtain the deceleration. The used compressor however has a vaneless diffuser. The fluidis guided to the exit of the compressor by a curled, diverging channel, called the volute (Fig. 4.2).The function is to collect and guide the flow of the medium as efficiently as possible to the exit ofthe compressor.

The equations used to model a fluid flow in general are, as mentioned, the Navier-Stokesequations, which are derived from the conservation law of momentum for some element of fluid,the conservation law of mass and the conservation law of energy. To obtain a complete set ofequations to fully describe the system, some additional thermodynamic equations of state haveto be included. With this set of equations the behavior of the flow is described in detail in three

31

32 Chapter 4. Pneumatic model••••••• •• - - 0 •• _. ••__ ••__ ._••" •••••• ••••_. ._••••• "_•••_ ••, _."._, • _

diffuser, (vaned)"-,

Iimpeller eye (inducer)

Figure 4.1: Schematic of a centrifugal compressor with vaned diffuser [.5].

Inlet

Impeller

Figure 4.2: Schematic of a centrifugal compressor with volute [5].

4.1. Pneumatic model

dimensions. The full set of equations governing the motion of a fluid is

33

where

~ Dp + \7. itpDt

0,

pF - ap +~ {2/1 (e. - ~6.b)}taXi aXj tJ 3 tJ '

DT {3T Dp 1 a ( aT)c -----<l>+-- k-p Dt p Dt - p aXi aXi '

Conservation of mass

Equation of motion

Energy balance

(4.1)

(4.2)

(4.3)

DDt = Material derivative, see Appendix A,

p = Mass density (kg·m- 3 ),

it = Local velocity of the fluid (m·s- I),

F= Resultant body force vector per unit mass (N·kg- I),

p = Static fluid pressure (Pa),

if = Unit vector, e.g. (x,y,z),

/1 = Viscosity of the fluid (Pa·s),

1 (au aUi)eij = Rate of strain tensor: - ~ +~ ,2 VXi VXj

6. = Rate of expansion: \7 . it = eii"

bij = Kronecker delta tensor, see Appendix A,

T = Absolute temperature (K),

S = Gain of entropy per unit mass (J.K- I),

Cp = Specific heat: T (~~) p (J.kg-I.K- I),

. . 1 (ap )(3 = Coefficient of thermal expansion - - aT 'p p

2/1 1 2<l> = r;(eijeij - 36. ),

k = Thermal conductivity (W.m-I.K- I).

For a full description of all the symbols and variables involving the flow equations and the deriva­tion of them from the basic laws of physics see [3]. A summary of the most important steps toobtain this set of equations is given in Appendix C. These general equations can only be solvedanalytically in a few very simple cases, and numerically with the help of computationally power­ful computers and Finite-Element-Method (FEM) software packages. However, solving the flowequations numerically, even in simple cases takes a large amount of time.

To obtain a model suitable for control or simulation purposes, the same equations or laws areused. However, instead of computing this set of equations for every point of the fluid in space, theproperties of the fluid in motion are lumped together into a few model parts. The resulting modelof the system contains the fluid flow through and pressure rise over the compressor, a subsequentvolume (plenum) and a throttle in the outlet duct, which is known as the Greitzer model andpresented in Fig. 4.3.

For every model part, the equations governing the state of the fluid will be derived as describedin [19]. Inside the compressor and throttle/outlet ducts, the fluid is assumed to be incompressible.The (material) derivative of the density is therefore equal to zero, thus p-I Dp/Dt = 0. The

34 Chapter 4. Pneumatic model

Plenum

P2

DuctIIm

Llpv

mv

Valve

P1

Figure 4.3: Schematic representation of the Greitzer model.

equation of mass in (4.1 ) is then reduced to \7 . ii = O. If the fluid is air under normal conditions,and fluctuations in pressure and temperature are not too large, the gravitational and viscouseffects in (4.2) are negligible. This is valid in case of the respiratory application. The reduced lawof conservation of momentum (4.2) is then

(4.4)

Which is for the 1 dimensional case

and with applying \7 . ii = 0 for the incompressible case this is described by

(4.5)

Moreover, due to the assumption of incompressibility the mass flow through the inlet duct andcompressor can be expressed as m = puAc • After separating op and ax and introducing thecompressor action !:lpc(m) the conservation of momentum is given by

Lc dm )-- = !:lp (m -!:lpA

cdt c ,

(4.6)

where Lc is the length of the compressor duct and Ac the area at the impeller inlet. The pressuredifference between the environment and the plenum is !:lp = Pp - Po. !:lpc(m) is the instantaneouspressure rise due to the compressor action, which is denoted as P2 - Po in Fig. 4.3. The sameholds for the throttle or outlet duct. Therefore, the assumptions made with the compressor partare also adopted for this part of the system.

(4.7)

The two nonlinear functions !:lpc(m) and !:lpv(mv) describe the pressure rise over the compressorand pressure drop over throttle respectively. However, the length of the throttle duct is considerednegligible. (4.7) is therefore omitted during the rest of this study.

The final part of the Greitzer model to be discussed is the plenum volume. The followingassumptions regarding the properties of the fluid inside the plenum volume can be made:

• The pressure is uniformly distributed throughout the plenum volume.

4.2. Compressor map

• Gravitational effects are negligible.

35

• Viscous effects are negligible.

The conservation of momentum can then be omitted. With the help of the divergence theorem(Appendix A) the integral for the conservation of mass can be expressed as

{ ~edV = - ( p(ii· il)dA.1v at JA (4.8)

The right-hand side of the above equation can be expressed like - L mi for a control volume withi ports. The expression for the speed of sound is given by

2 _ (up)ao - up s(4.9)

For the full derivation of (4.8) and (4.9) see Appendix A.l and A.2 of [19] respectively. Assumingthat the process inside the plenum volume is isentropic' and by using (4.9) the resulting equationdescribing the dynamics of the pressure fluctuations inside the plenum is

upat

1 upa2 at·o

(4.10)

The resulting equation for the conservation of mass inside the plenum, using (4.8) and (4.9) is

(4.11 )

With all the symbols as indicated in Fig. 4.3. The mass flows m and m v are determined by (4.6)and (4.7) respectively. As indicated, the length of the throttle duct is often much smaller thanthe length of the compressor duct. Therefore dmv/dt is neglected during this study. However, themass flow through the outlet duct valve is generally modeled as

(4.12)

where kv is the outlet valve coefficient, and which is a measure for the outlet duct opening.

The resulting model describing the dynamics of the pneumatic subsystem is as given in [5]

dpp a2

dt; (m-mv ), (4.13)

p

dm Ac (4.14)dt L

c(P2 - Po) ,

dJJJ~ (Tm -7~). (4.15)

dt

The third equation, (4.15) is added to include the dynamics of the compressor motor. The variablesm v are as given in (4.12), Tm is the torque delivered by the motor and Tc is the load-torque due tocompressor action. The instantaneous pressure rise because of compressor action P2 is defined bythe steady-state characteristics of the compressor. The variables Tc and P2 are further elaboratedin the following section.

4.2 Compressor map

Steady-state behavior of the compressor is described by the compressor map. This compressor mapgives the relation between stagnation pressure and (mass) flow for several speeds of the PMSM.The map derived for this compression system is shown in Fig. 4.4. The meaning of the surge line

36 Chapter 4. Pneumatic model

-ModelMeasurements*

--- jsurge linel-

*

1.07 f--*':":_,.;."._.._.-_*_-~ __._....,'.',-*_<_-'__.....;.'..;;.*.:..~_--~-- - _. -~X60000 -r~~' r····­

. * .

1.0!1LL.:..:..:..:..::....:..!:.:..:..:..J.----L1-------.J2~----L3---4.L-----'---':.==-=-=-~=~6

massflow (kg/s) x 10-3

1.08

1.09 *

.--.. 1.06coa..-Q) -_:¥L.. 1.05:::Jenen *~

a.. 1.04

*1.03

Figure 4.4: Compressor map.

4.2. Compressor map

indicated in this figure is explained in the next section.

37

Measurements to derive the compressor map are carried out by controlling the PMSM to arange of speeds and adjusting the outlet duct valve into a set of positions between fully closed andfully open. For every set-point of motor-speed and valve position the steady-state pressure andflow are measured. The results are shown in Fig. 4.4.

An analytical relation is derived to model the steady-state performance, or compressor mapof the used compressor. This relation is derived as follows. Generally, the pressure rise due tocompressor work can be described as an isentropic process, which gives an increase in enthalpyt!..hc for constant entropy. In practice, however, the pressure increase in the compressor is notisentropic, because there will be an entropy increase due to inevitable losses t!..h1oss' These lossescan be described by an isobaric process in series with the previously described isentropic process.The compression process is shown in Fig. 4.5 and can be represented by

(4.16)

where t!..hc,t is the total specific enthalpy delivered to the fluid [5]. The enthalpy delivered to the

he,13

hc:;

~j P2-I>.1

B-1lU,..c::1

ho..... ~

1:1W'

Po

Entropy

Figure 4.5: Total compression process.

fluid is calculated by the power delivered to the fluid. The power can be expressed by

(4.17)

The torque applied to the fluid Tc is defined as the change in angular momentum given in [10].

(4.18)

where Ut,l,Z are the tangential gas velocities at the according radii (Tl,Z)' The righthand side ofthis equation is obtained by assuming Tl Ut,l = O. The slip factor (j is dependent on the number

*An isentropic process is a process during which the entropy remains the same. It is therefore a lossless andreversible process.

38 Chapter 4. Pneumatic model

Table 4.1: Parameters compressor map.

Parameter Description Valueu Slip factor 0.721'2 External impeller radius 15.10 -0 m

1'1 Average impeller radius 7.6.10 -0 m

kins Incidence losses coefficient 0.054.10 7

kf Friction losses coefficient 8.0.10 -7

n Polytropic efficiency coefficient 1.55

of blades and the angle between the blades and tangent of the impeller. Ideally, the flow leavesthe impeller only in the tangential direction. However, in practice the direction of the flow has anangle relative to the tangential, which is accounted for by u. The slip factor is approximated to be0.72, which is obtained by means of a curve-fit on the above described steady-state measurements.Subsequently, the 1:1hc,t is derived as UW21'~. From the standard isentropic relations it is knownthat

P2

Po (1:1h) ~"1l+ __c

cp1'o(4.19)

where 1:1hc is as in (4.16) and I is a physical parameter, known as the adiabatic index, and equalto 1.40. The same notations as shown in Fig. 4.3 are used. The main compressor losses 1:1h1ossare split into incidence 1:1hi and frictional1:1h f losses. These can respectively be represented by

1'i 22 (w - kinsm) ,

kf m 2•

(4.20)

(4.21)

The incidence losses occur due to instantaneous change of the gas velocity as it hits the impellerblades. The constant kins is determined by means of a curve-fit on the measurements. The secondtype of compressor losses are due to the friction of a fluid flow through a pipe. The constant kfis also determined by means of a curve-fit on the steady-state measurements. Other parametersmentioned are the absolute ambient temperature To and specific heat at constant pressure cpo Thevalues of these parameters are 293 K and 1012 J·kg- 1 ·K-1 respectively. The isentropic process,(4.19) can be represented by

P2(

2 2 ,,2 2 2) n~ 1uw l' _:...l. (w - k m) - kfm1 + 2 2 tnS Po.

Tocp

(4.22)

The above discussed losses (4.20) and (4.21) are taken into account. The adiabatic constant Iis replaced by n, which is slightly higher than,. This a correction regarding the real compres­sion process in contrast with an isentropic process. To take negative mass flow into account,(w - kins m)2 has to be written as (w - kinslml)· (w - kin.m) and kf m 2 as kfm ·Iml. The resultsare visualized in Fig. 4.4, together with the steady-state measurements.

The model of the compressor map is valid with a maximum deviation of 5% for speeds up to7.0· 104 rpm, which includes the normal operating range during ventilation of 1.7· 104 to 4.6· 104

rpm. The derived parameters describing the compressor map for this system used in (4.22) aresummarized in Table 4.1. The above derived analytical relation of the compressor map (4.22) isinserted into the Greitzer model in (4.14) for P2.

4.3. Dynamic parameters derivation 39

Information derived from the compressor map can be applied into the control-loop. Afterinverting (4.22) the required motor-speed w is known for every combination of pressure P2 andflow m. The resulting relation between speed, pressure and flow is shown in (4.23).

w (4.23)

Other parameters which influence the performance of the compressor are the ambient temperatureTo and pressure Po. This influence is accounted for by the derived compressor map (4.22) and,therefore, also by the relation 4.23.

Another method of deriving the compressor map is by means of a curve-fit of a cubic polynomialwith the measurements [17], [19]. However, in this study the above elaborated method is preferred,because of the presence of the ambient parameters in the expressions, which gives more flexibilityregarding controller design.

4.3 Dynamic parameters derivation

To complete the Greitzer model, (4.13) to (4.15), the parameters v;, , Ac and Lc are derived inthis section. These three parameters determine the dynamic behavior of the pneumatic sub-system.

These parameters are the geometrical parameters of the compressor. However, it is not alwayspossible to derive these parameters directly from the physical compressor dimensions. They candiffer up to 9 times from the physical based values [17]. Therefore, Ac , Lc and Vp are obtainedby means of pressure measurements with closed outlet duct, which forces the fluid in the plenumto go into an oscillatory behavior, known as surge. Surge can be explained by the compressormap derived in the previous section. A stable and unstable operating region in this compressormap can be distinguished. The unstable region is apparent for small positive mass flows and highspeeds, which is to the left of the surge line in Fig. 4.4. However, due to the small gradient of thecompressor characteristic in the unstable region, for the used compressor, these surge oscillationsare small. Moreover, this effect is not measured for speeds below 5.0· 104 rpm. A drawing of asurge cycle is shown in Fig. 4.6 together with the compressor characteristic for some particularmotor speed. Surge is investigated in [11], [17], and can be explained by means of Fig. 4.6. Thecycle starts where the flow becomes unstable at (1). Then it jumps to the reversed flow char­acteristic of the compressor at (2). The surge cycle follows the reversed flow characteristic untilapproximately zero mass flow at (3). At this point it jumps to the compressor characteristic forpositive mass flows at (4) and follows this branch back to (1) again. The mechanism behind surgecan be explained as follows. In the unstable region, left to the surge line, a drop in mass flowresults in a drop of the compressor delivery pressure. Therefore, the compressor is not able towork-up to the pressure of the plenum volume earlier produced, which results in a deceleration oreven reversal of the flow. Due to the reduction of mass flow, the plenum pressure will decrease,until the compressor is able again to work-up to the pressure in this volume and restoring the massflow. The frequency of the pressure oscillations during surge is dependent on the above mentionedparameters Ac , Lc and Vp . However, no analytical expression exists to give the relation betweenthe three parameters and the surge frequency.

According to [11] the surge frequency is around, but below the resonance frequency of thesystem. The resonance frequency of the system is generally known as the Helmholtz frequency

~4..~OI... __. _._.._ __ . _ ~I~~[:>~~~r ..~4b:. JP~.neuIlla tic Illodel

;..- ...".'" ,'" .'" ,'" .'",,,,,,,

'"'"

Pressure rise . Surge line

.,' (1)

(4)1------ --1_,,,,Compressor"characteristic

o Flow rate

Figure 4.6: Example of a surge cycle.

iIi, and can be expressed, analytically, by the three pneumatic parameters as

(4.24)

T'he values for the pneumatic parameters Vp , A c and L c are estimated by taking into accountthe physical dimensions of the system and the Helmholtz frequency. The parameters are furtherrefined by means of fitting the modeling results on the surge measurements. To obtain the bestmeasurement results of the;;e pressure fluctuations, the pres;;ure gauge placed at the outlet ductof the compressor in Fig. 2.10 i;; moved to the ball valve at the end. As mentioned earlier, surgebehavior is only measurable at high speeds and low mass flows. Therefore, the outlet duct valveis closed and the PMSM is driven in open-loop operation with a high DC-voltage of 20 V. Thisgives a speed of approximately 8.0.104 rpm. '1'he;;e measurements are carried out with differentlengths of connected tubes, hence different plenum volumes. The lengths of the used tubes axe0.85, 1.2 and 1.6 m. The power spectral densitie;; (PSD's) calculated from the measured pressurefluctuations are shown in Fig. 4.7. As can be seen in Fig. 4.7, the measured frequency decreaseswith increasing plenum volume, as expected. These results verify that the measured frequencyis the surge frequency. The connected tubes of the te;;t-setup as shown in Fig. 2.10 have a totallength of ] .6 n1. The frequency of the pressure oscillations found for this Ci:hge is 4.'i Hz.

The modeling results using (4.13) to (4.15) for all three plenum volumes are indicated inFig. 4.7, and these results are also summarized in Table 4.2. The derived vahles for v~" A c andL c deviate from the physical systern dimensions between I and 6 times, as already assumed in thefirst paragraph of this section. The derived values, as well as the phy;;ical sy;;tem dimension;; are;;hown in Table 4.3. Finally, the Helmholtz frequency calculated by means of the three pneumaticparameters is also presented in Table 4.:3. This value is of the same magnitude, but above thesurge ti·equency.

4.3. Dynamic parameters derivation

Tubing: 0.85 m

¥ 1000 r, ~ Measurements

~ 1 ,500 1 , ==Slmulatlons, .

0 ,(f) 1 ,U. 0

, ,0 50 100 150 200

Frequency (Hz)TUbing: 1.2 m

f 1000l ..1

1--Measurements I1 ---- Simulations

~ 5001

"

I' .~.,., ',

IAJ \U. 0

0 50 100 150 200Frequency (Hz)Tubing: 1.6 m

f 1000

~ "500 .. . i\ ..

0 \,(f) \U. 0

0 50 100 150 200Frequency (Hz)

Figure 4.7: PSD of pressure measurements with 20 V.

41

Table 4.2: Surge frequency and PSD amplitude with frequency in Hz and PSD amplitude inW·Hz- 1 .

I Tube length (m) I Measured frequency I Simulated frequency I

~r Jr---~-;-:~;---------1Rf I[11.l§:e length (m) ~ured amplitude I Simulated amplitude I

0.85 7.2.102 9.3·10~

1.2 1.1.103 8.8.102

1.6 5.5·1O~ 7.1·1O~~-

Table 4.3: Pneumatic parameters for setup 1.6 m without mass flow.

Parameter Matched Physical dimensions IA c 1.5.10 -4 m~ 1.5.10 -4 m~

L c 4.5· 10 -~ m 1.0.10 ·1 m

Vp 3.2.10 -o m0 5.4.10 -4 m0

fH 55 Hz 90 Hz

Chapter 5

General, multi-physical model

To validate the different earlier derived models, measurements on all multi-physical sub-domainsare carried out. The measurements on the system, shown in Fig. 2.10, are carried out duringopen-loop operation. Open-loop operation is done by establishing a DC-voltage step from 5.0 to10 V at the inverter, and by keeping the duty cycle of the PWM on unity. This has been done tominimize the influence of the switching actions during PWM on measurements. This open-loopoperation is also implemented in the models during simulations. Therefore, the conditions duringsimulations and measurements are the same and by neglecting the high frequent PWM-switchingthe computational effort is reduced. The applied voltage step from 5.0 to 10 V agrees well with thenormal operating conditions, which are between 1.7.104 rpm and 4.6 . 104 rpm. These open-loopmeasurements have been reiterated for several positions of the outlet duct valve. The voltage-stepapplied to the stator windings in the test setup is not ideal due to the limited slew-rate of the usedvoltage source. Therefore, this voltage step is measured and also applied to the models duringsimulation. The following variables on all parts of the multi-physical domain have been measured:current, speed, pressure and (volumetric) flow.

5.1 Results of modeling the respiratory dynamics

The models described in the preceding chapters for the electromechanical domain (3.57), and thepneumatic domain (4.13) to (4.15) are integrated and the simulation results are compared withthe measurements on all physical domains in this section. The inverter is modeled with the Sim­PowerSystems toolbox of Simulink.

During normal operation of the respiratory system the compressor is delivering flow. Thereforethe measurements described in this section are carried out with an open outlet duct valve. The line­currents measured during steady-state operation with Vdc = 10 V are shown in Fig. 5.1, togetherwith the simulation results. The current amplitude and waveform of the measurements and thesimulations are in good agreement. This holds also for the rms values of the current at the usedvoltages as represented in Table 5.1. The response of the speed on the applied step voltage from5.0 V to 10 V is shown in Fig. 5.2. The simulation results during the same configuration are alsoincluded in Fig. 5.2. The steady-state values and dynamics of the simulations and measurements

Applied voltage

5V10 V

Measurements I Simulations

0.72 A 10.66 A1.4 A f--1.-5~A---

~-----

Table 5.1: Current-rms values measured and simulated.

42

5.1. Results of modeling the respiratory dynamics 43

3 - •.

?....c 0~:Jo

-1

-2

-3

0.4170.4160.415-4~ -'---__--'- -'---__--'- ...L-__--'-__-----.J

0.41 0.411 0.412 0.413 0.414time (s)

Figure 5.1: Measured and simulated current waveform (Vdc0.87-10-4 ).

10 V, w 4.1.104 rpm, kv

44.5 x 10

- Measurements4 ---- Simulations

3.5

3

'[ 2 5.....

Speed

2

1.5

0.5

8.9 1.1 1.2 1.3 1.4Time (s)

Figure 5.2: Measured and simulated response of the speed on a step-voltage from 5.0 to 10 V.

44 Chapter 5. General, multi-physical model

are in good agreement. The measurements and simulation results for the response of the pneumaticquantities mass flow and pressure on the voltage step from 5.0 to 10 V are shown in Fig. 5.3. The

5 Pressure1.04

x1O

-- Measurements~ 1.035 ---- Simulationsell

~ _....- -- -1.03~::l 1.025 . .. ........•..<f)<f)CI)

It 1.02 ---------- ----1.015

0.9 1.1 1.2 1.3 1.4Time (s)

x 10.3 Flow

- Measurementj~3 ---- Simulations ..~

~ 2 ---------------l;:<f)<f)

ell:::aE

I8.9 1.1 1.2 1.3 1.4Time (s)

Figure 5.3: Measured and simulated response of the pressure and mass flow on a step-voltage from5.0 to 10 V.

time-constants and steady-state values of the measurements and simulations derived from theFigs. 5.2 and 5.3 are summarized in Table 5.2. In this table, the pressure is expressed in pressurerise above ambient pressure Po. The largest deviation between measurements and simulationresults of the dynamics is the time-constant of the flow response and is 33%. This error occursdue to changing pneumatic parameters as function of the working point of the compressor. Thesteady-state values are all within a deviation of 13%, as can be derived from Table 5.2.

5.2 Steady-state results

Measurements on the system during steady-state operation are also carried out to obtain the per­formance of the model over the whole operating range of the respiratory system, hence for differentoutlet duct valve openings. The results of these measurements and simulations are shown in Figs.5.4 and 5.5.

5.3. Power efficiency

Time-constants and steady-state values

45

Voltage Time-constant speed Steady-state speedin ms in 103 rpm

Measured Model Measured Model5V

79 8522 23

lOV 42 42

Time-constant pressure Steady-state pressurein illS in Pa above Po

Measured Model Measured Model~ 84 1.0.102 4.6·10~ 5.2·10~

lOV 1.8.103 1.8.103

Time-constant flow Steady-state flowin ms in 10-3 kg·s- 1

Measured Model Measured Model5V

70 931.8 2.0

lOV 3.7 3.7

Table 5.2: Measurements and model-based results for a voltage step from 5.0 to 10 V with ku is0.87.10-4 .

6 X 104

i Speed

c '"'~ ! - MeasurementsI ---..- ........---- ..._......__.. 12 v 1---- Si~~lations

5 ---------c--- • _

~ 4f

--------------------..---- 10 V j- -----..._---------""---- ..._-----_ ..

aLi-----;';---;;-------;4,-------;:-5----c6;;------::---'6Valve opening coefficient k" x 10·~

Figure 5.4: Steady-state speed values.

Pressure

5V

-~-.-..4 5 678

Valve opening coefficient k... x 10.5

Figure 5.5: Steady-state pressure and mass flowvalues.

These figures indicate that the steady-state modeling results are more accurate for higher outletvalve coefficient, hence for higher mass flows. This is because the torque due to mass flow, de­scribed by (4.18), is well defined and becomes dominant over the other losses for higher massflows.

5.3 Power efficiency

Power efficiency calculations for the respiratory system are carried in this section to evaluatethe implemented six-step commutation driving strategy. Because of the difference between EMFwaveform and current waveform already discussed in Sections 3.2 and 3.3, the system efficiency isnot optimal. To derive the advantage of the suggested sinusoidal current control driving strategyin Section 3.2, the efficiency of the used six-step commutation strategy is compared with the the­oretical efficiency using the proposed driving strategy for the used PMSM. The losses taken into

46 Chapter 5. General, multi-physical model

Electrical subsystemPR I Copper losses (W)

Mechanical subsystemKcw I Coulomb friction (W)K vw2 I Viscous friction including eddy current losses (W)

Pneumatic subsystem

q. (w - kinsm)2 m I Incidence losses (W)k f m:5 I Fluid friction losses (W)

Table 5.3: Losses taken into account with analysis.

account with the analysis are summarized in Table 5.3. The total power flow mode, regardingthe whole respiratory action is shown in Fig. 5.6. The losses of the pneumatic subsystem are

Figure 5.6: Total power flow model of the entire respiratory action.included compared with the analysis done in Sect. 3.3. For reasons of convenience, the resultsof the analysis carried out earlier are shown again in Figs. 5.7 and 5.8. The results shown inthese figures includes only the electromechanical subsystem and losses. The blank area in the topright corner of Fig. 5.7 is due to the voltage limit of 20 V. The efficiency map for the proposedsinusoidal current control strategy is shown in Fig. 5.8.

The results of the efficiency analysis for the whole respiratory system, including the pneumaticsubsystem are shown in Figs. 5.9 and 5.10, for the six-step commutation drive and sinusoidalcurrent control drive respectively. In the normal working area the power efficiency of the six-stepcommutation drive, without pneumatics, is between 50% and 65% (Fig. 5.7). The drive efficiencyby applying sinusoidal current control increases up to 70% and 78% (Fig. 5.8). Therefore, theimprovement in power efficiency of the drive, without pneumatic sub-system, using sinusoidal cur­rent control is approximately between 10% and 20%, as can be concluded by comparing Figs. 5.7and 5.8. However, the power efficiency is also derived for the total respiration action for the imple­mented, six-step commutation drive and sinusoidal current control drive as shown in Figs. 5.9 and5.10 respectively. The power efficiency of the whole respiratory system is between 15% and 20%for the six-step commutation drive. These values are valid within the normal operating region ofthe respiratory system, as indicated in Fig. 5.9. The resulting efficiency increase by implementingthe sinusoidal current control drive is reduced to a maximum of 5%.

Possible disadvantages of the sinusoidal current control strategy are higher losses iN. the elec­tronics due to a higher switching frequency. This reduces the calculated improvement in drive­efficiency of 5%. However, because no particular amplifier topology is chosen for the analysis,these losses are not included. Therefore, further research is necessary to derive the influence ofthe amplifier on the performance. Finally, to enable sinusoidal current control, a more accurateand expensive speed and position sensor is needed, instead of the implemented Hall sensors.

5.3. Power efficiency

~ 10Ez~

Q):::J

l!.8"Croo

...J

4

2

o 2 4 6Speed (rpm)

8

47

Figure 5.7: Simulated efficiency map of the six-step commutated drive with PMSM with voltagelimit of 20 V.

14

12

~ 10E~Q):::J

l!.8"C

~ 6...J

4

2

2 3 456Speed (rpm)

7 8

Figure 5.8: Simulated efficiency map of the sinusoidal current controlled drive with PMSM.

48 Chapter 5. General, multi-physical model__• """••••• 0 _ •••_._. ..••_ _ ••••••••_ ••_._. _

X 105

1.09

1.08

1.07

ro 1.06e:..~~ 1.05f!!0.

1.04

1.03

1.02

o 234Mass flow (kg/s)

OA

0.35

0.3

0.25

0.2

0.15

0.1

0.05

05 6

X 10-3

Figure 5.9: Simulated efficiency map of the respirator system with trapezoidal voltage controlleddrive.

1.08

1.07

ro 1.06e:..~

~ 1.05~0.

1.04

1.03

1.02

o 234Mass flow (kg/s)

5

Figure 5.10: Simulated efficiency map of the respirator system with sinusoidal current controlleddrive.

Chapter 6

Conclusions

• In this thesis, electrical, electromechanical and pneumatic models are derived and experi­mentally verified to simulate the dynamics of a respiratory system. The respiratory systemconsists of a voltage driven high-speed PMSM with helical stator windings, with an impellermounted on the shaft.

• The parameters of the model simulating the dynamics of the voltage driven PMSM areobtained from measurements, specifications and two (semi-)analytical models.

- The first analytical model gives the magnetic flux density inside the airgap of thePMSM using analytical Fourier series. The magnetic flux density is used to determinethe electromechanically developed torque, the EMF-constant and the EMF-waveform.

- The second model gives the self inductance and mutual inductance of the helical statorwindings using Biot-Savart. The semi-analytically derived parameters are verified with2D FEM-simulations.

• The electromechanical models show a good agreement with the specifications and measure­ments.

• The steady-state characteristics of the pneumatics of the respiratory system are determinedby the compressor map. A model is derived to approximate the compressor map obtainedby measurements.

- This model is partially derived from the physical dimensions of the respiratory system.

- Some model parameters can not be determined directly from the physics of the pneu-matic system. These parameters are derived by fitting the model on the measurementresults.

• The maximum error between the modeled compressor map and the measured compressormap is within 5%. This is valid for speeds up to 7.0.104 rpm, which include the normalworking area of the compressor.

• By inverting the derived compressor map, an analytical relation is obtained between therequired motor speed for any combination of pressure and flow. This relation can be imple­mented into the respiratory control-loop.

• The dynamical behavior of the pneumatic sub-system is modeled by means of the Greitzermodel.

• The parameters of the Greitzer model are derived using the surge frequency. Surge is initiatedby closing the pneumatic sub-system, hence the Greitzer model is derived in case of no flow.

49

50 Chapter 6. Conclusions

• The accuracy of the model reduces with increasing flow, which indicate shifting of the modelparameters with changing compressor working-point.

• Using the derived models, the power efficiency of the respiratory system is calculated. Theseefficiency calculations are carried out for two different driving strategies, i.e. six-step com­mutation and sinusoidal current control. This is done to veritY whether the respiratorysystem efficiency improves by changing the driving strategy from six-step commutation tosinusoidal current control, because the driving strategy for a PMSM should generally besinusoidal current control.

The efficiency is first calculated for the drive only, hence without impeller and pneu­matic sub-system. The resulting power efficiency of the six-step commutated drive, inthe normal operating region, is between 50% and 65%. The efficiency is also calculatedin the working region for sinusoidal current control. The resulting power efficiency in­creases up to 70% and 78%, hence the proposed driving strategy increases the efficiencysignificantly.

The same efficiency calculations are carried out for the entire respiratory system, henceincluding the impeller and pneumatic sub-system. The results obtained for the actualdriving strategy indicate a efficiency between 15% and 20%. Since the losses in thepneumatic sub-system are unaffected by the drive strategy, the efficiency gain of thetotal respiratory system is reduced to a maximum of 5%, when using sinusoidal currentcontrol instead of six-step commutation.

Recommendations for further work are

• to implement sinusoidal current control to veritY the calculated gain in efficiency and todecide if it is worthwhile realizing this driving strategy.

• to investigate the performance of the derived compressor map for changing ambient condi­tions to analyze whether it is possible to implement it in the control-loop.

Bibliography

[1] A.F. Anderson, J.R. Bumby, and B.!. Hassal!. Analysis of helical armature windings withparticular reference to superconducting a.c. generators. Generation, Transmission and Dis­tribution, lEE Proceedings, 1980.

[2] M.R. Bailey, J.R. Bumby, B.!. Hassall, and A.F. Anderson. Magnetic fields and inductancesof helical windings with 1200 phasebands. Electric Power Components and Systems, 1981.

[3] G.K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, 1967.

[4] K.J. Binns, P.J. Lawrenson, and C.W. Trowbridge. The Analytical and Numerical Solutionof Electric and Magnetic Fields. John Wiley & Sons, 1992.

[5] Konstantin Boinov. Efficiency and Time-Optimal Control of Fuel Cell-Compressor-ElectricalDrive Systems. PhD thesis, University of Technology Eindhoven, 2008.

[6] M. Borrello. Modeling and control of systems for critical care ventilation. 2005.

[7] Cedrat. Flux 9.3 User's Guide, 2006.

[8] F. Eugenio, P. Cappa, S.A., and S. Silvestri. Linear model and algorithm to automaticallyestimate the pressure limit of pressure controlled ventilation for delivering a target tidalvolume. Journal of Clinical Monitoring and Computing.

[9] Edward P. Furlani. Permanent Magnet and Electromechanical Devices. Academic Press, 200l.

[10] Jan Tommy Gravdahl and Olav Egeland. Speed and surge control for a low order centrifugalcompressor mode!. Proceedings of the IEEE International Conference on Control Applications,1997.

[11] J.T. Gravdahl and O. Egeland. Compressor Surge and Rotating Stall: Modeling and Control.Springer Verlag, 1999.

[12] D.C. Hanselman. Brushless permanent-magnet motor design. London: McGraw-hill, 1994.

[13] J.H. Heinbocke!. Introduction to Tensor Calculus and Continuum Mechanics. Trafford, 1996.

[14] J.R. Hendershot and T.J.E. Miller. Design of Brushless Permanent-Magnet Motors. MagnaPhysics Publishing and Oxford University Press, 1994.

[15] William P. Houser. Deriving the lorentz force equation from maxwell's equations. ProceedingsIEEE SoutheastCon 2002, pages 422-425, 2002.

[16] M.P. Kazmierkowski, R. Krishnan, and F. Blaabjerg. Control in Power Electronics: SelectedProblems. Elsevier Science, 2002.

[17] Corina H.J. Meuleman. Measurement and Unsteady Flow Modelling of Centrifugal Compres­sor Surge. PhD thesis, University of Technology Eindhoven, 2002.

51

52 BIBLIOGRAPHY

[18] V. Tamburrelli, S. Silvestri, and S.A. Sciuto. Reliable procedure to switch from volumecontrolled to pressure controlled ventilation assuring tidal volume during anesthesia. 2005.

[19] Jan van Helvoirt. Centrifugal Compressor Surge: Modeling and Identification for Control.PhD thesis, University of Technology Eindhoven, 2007.

[20] Alvin A. Wald, Terence W. Murphy, and Valentino D.E. Mazzia. A theoretical study ofcontrolled ventilation. IEEE Transactions on Bio-Medical Engineering, 1968.

[21] T. Yuta. Minimal Model of Lung Mechanics for Optimising Vent'ilator Therapy in CriticalCare. PhD thesis, University of Canterbury, 2007.

[22] Z.Q. Zhu, David Howe, E. Bolte, and B Ackermann. Instantaneous magnetic field distributionin brushless permanent magnet dc motors, part i: Open-circuit field. IEEE Transactions onMagnetics, 29(1), January 1993.

[23] Z.Q. Zhu, David Howe, and C.C. Chan. Improved analytical model for predicting the magneticfield distribution in brushless permanent-magnet machines. IEEE 7ransactions on Magnetics,38(1), January 2002.

Appendix A

Used mathematical operators andentities

In this work the following mathematical assumptions and operators are used. For 1110re informa­tion about mathematical hypotheses and in particular tensor calculus see [1:3].

i, ] and k are the unit vectors of the used Cartesian coordinate system for the .1:-, y- andz-direction respectively, as shown in Fig. A.l. The base vector is therefore defined as e= i+:7+k.or e= ei +ej +ek. The cylindrical coordinate system (F,O,:3) is also used often in this thesis. Thiscoordinate system is defined as indicated in Fig. A.2.

IIIIIII

-~------

---z

r

Figure A.I: Cartesian coordinate system Figure A.2: Cylindrical coordinate sys­tem

Several vector and scalar operators are used throughout the text which are summarized below.The dot-product of 2, 3 dimensional vectors il = ali + 02] + a8k = a;e; = (lie;* and [} =

hi+ b2.7 + b8k= b;e; in a orthogonal coordinate system is defined as:

(A.I)

*Einstein's summation convention. Summation over the components always takes place over a repeated pairof indices, 1 covariant (subscript) and onc contravariant (supersnipt) component. In Cartesia.n coordinaLes thecovariant and contravariant components are the same.

54 Chapter A. Used mathematical operators and entities

The cross-product of 2, 3 dimensional vectors, A and fj in a orthogonal, Cartesian coordinatesystem is defined as:

(A.2)

The del operator \7 in a 3 dimensional Cartesian coordinate system R3 with coordinates (x, y, z)is defined as:

~B ~B ~B\7=i-+j-+k-

Bx By Bz(A.3)

The del operator is used for several operations on scalar and vector functions. The first mentionedis the vector gradient of a scalar function ¢ in a Cartesian coordinate system and is defined as:

The divergence of a vector function A in a Cartesian coordinate system is defined as:

. ~ ~ ~ BAI BA2 BA3dwA = \7 . A = Ai i = -- + -- +--, Bx By Bz

with:

Next, the curl of a vector is defined as:

~ j kcurlA= \7 x A= 0 0 0

ox oy OZAl A2 A 3

The components of \72 A may be calculated with the following entity:

\72A= \7 (\7 .A) - \7 x (\7 x A)

(A.4)

(A.5)

(A.6)

(A.7)

An operator frequently used in fluid dynamics is the material derivative J]t' This operator isa derivative taken along a path moving with a velocity il. It describes the rate of change of somequantity that is being transported by a fluid current and is defined by:

!2=~+il.\7Dt Bt

This operation can be elaborated on a scalar field as well as a vector field.

(A.8)

The ~-symbol is defined in fluid dynamics as eii, = \7 . il, and is known as the rate of straintensor.

55

The 6ij symbol is generally known as the Kronecker delta tensor (formally 6I). The compo­nents of this tensor are defined by 6ij = 1 for i = j and 6ij = 0 for i i- j.

Several used theorems regarding integrals of vector functions are the divergence theorem andStokes' theorem:

fff (V. X) dV

v

hV x A. d~

ff 1.ndSav

1 X.diJaz~

Divergence theorem

Stokes' theorem

(A.9)

(A.IO)

The divergence theorem relates the flow of a vector field X through the closed surface S of avolume to the behavior of the vector field inside the specified volume V. In (A.9) n is the outwardunit vector normal to the surface S.The Stokes' theorem relates the surface integral of the curl of a vector field over a surface 1:: to theline integral of the vector field over its boundary. The curve of the line integral 81:: has positiveorientation (right-hand rule with the surface-normal). With i is the unit vector in the directionof the curve.

Appendix B

Six-step commutation withvoltage control

The commutation and control of the blower unit is based six-step modulation with voltage control.This method utilizes the signals of Hall sensors to arrange the commutation, and a PWM signalto obtain a net DC-voltage to achieve the required angular velocity and torque.

The operation of the Hall sensors is based on the Hall effect. This effect makes use of thedeflection of a current in the presence of a changing magnetic field normal to the current becauseof the Lorentz force. This gives a net voltage in the direction perpendicular to the other two axes.See Fig. B.l for a visualization of the operation of a Hall sensor.

Figure B.l: Hall effect

Three Hall sensors are placed in the stator part of the PMSM, evenly distributed over 360°. Duringone rotation of the rotor, six different signal combinations of the three Hall sensors occur. Thesesix combinations are used to operate the MOSFETs or switches of the inverter, as shown in Fig.B.2, to obtain the required commutation. The schematic of the inverter is the same as mentionedin Chapter 2, but re-inserted for reasons of convenience. With this technique it is possible todetect the rotor-position every 60°, which is sufficient for rotation of the PMSM. The switchingscheme, as function of the Hall signals is summarized in Table B.l and graphically represented in

56

57

+

,.-----F"yy'---_-, Phase A

~----___+============t==;--------L T T T'------_~ Phase B

Ts

Figure B.2: Full-bridge inverter with additional inductances in the leads to the windings

Table 13.1: Comnllltation sequence for clockwise rotation, with NC is 'not connected' and 'l'x theactivated transistor

I Hall sellsor A [Hall sensorB[JIall ser~haseA I Phase 13 IPh~0 0 1 '1'1 NC '1;,() 1 1 1'1 15 NC0 1 0 NC 15 1~

I 1 0 14--

NC 1:,I 0 0 14 12 NC1 () 1 lYC 12 :16

Figs. 13.3 to 13.5. The speed is controlled by adjusting the amplitude (net DC voltage) of thespace vector with a PWM signal. The duty cycle of this signal is a function of the error betweenthe actual and the desired angulaT velocity. The six space vectors together with the six Hall signalcombinations can be seen in Fig. B.G.

Chapter B. Six-step commutation with voltage control58----------~----~----------"'-------

Normalized EMF waveform phase A-B

~:L~~o 1 Z 3 4 5 6

Normalized EMF waveform phase B-C

t=Z=~o 1 2 3 4 5 6

Normalized EMF waveform phase C-A

fS~~la 1 2 3 4 5 6

Figure B.3: Normalized EMF waveforms.

Signal Hall sensor A

~l. .1: : J0' 3 4 6

Signal Hall sensor 8

::rJ 'J .1::l . j00 1 --:-----~-- ----"-----------'--,---J

Signal Hall sensor C

iJ • .1' .1 ,jo-----'-,-----"-L..------"----....L.--------""' ,

Figure B.4: Normalized Hall signals.

Normalized voltage phase A

f------"-----------:'-------:[ --!---1--,I ~:[ : jo:F ]

-0.5

-'0 - ,'--__-'- ....1.-L -'---__--'- ----'--

'r0.5 •

-0::1"----,O'---------'J'---------L------L-----'----------'--

Figure B.5: Normalized voltages applied to the stator windings.

ABC - Hall Sen:sors

Figure B.6: Space vectors with Hall combinations

59

Appendix C

N avier-Stokes equations

In this chapter, the complete set of equations to describe the motion of a fluid are explainedtogether with the appropriate simplifications to obtain the generally used model for centrifugalcompressors, namely the Greitzer-model.

C.l Derivation of the N avier-Stokes equations

The complete set of equations to describe the motion of a fluid is based on the conservation lawsof momentum, mass and energy and some equation of state or boundary conditions, e.g. equationof state of a perfect gas p = RpT. The equation of state of a perfect gas relates the pressure p,mass density p and absolute temperature T with the gas constant R. In the following text, thementioned Navier-Stokes set of equations is derived from the conservation laws and the applica­bility for the project is discussed.

The general law of conservation of mass of a volume V enclosed by a surface A, and of which theposition is fixed relative to the coordinate axes and entirely occupied by a fluid can be expressedas follows.

~JpdV = - Jpu· ndA, Conservation of mass, [3] (C.I)

where

(C.2)Conservation of mass, differential form

p = Mass density (kg·m-3),

u= Local velocity of the fluid (ms- 1),

n= Unit vector normal to the surface of the volume, directed outwards.

This relation states that the time-rate of change of mass in a volume V of fluid is equal to the fluxof mass flowing through the surface A of this volume. In differential form (C.I) is written like

~ Dp + ~. u= 0pDt '

where 17 is the velocity of the element of fluid under study and D / Dt is the material derivative (seeAppendix A). In this differential form, the equation may be interpreted in terms of changes in thevolume of a given mass of fluid. For an incompressible fluid, with Dp/Dt = 0, the conservation ofmass reduces to ~ . 'il = O.

(C.3)

The next law of conservation is the conservation of momentum. The rate of change of momen­tum in a volume V of fluid enclosed by the material surface S is given by

JDu dV.DiP

60

C.l. Derivation of the Navier-Stokes equations 61

The forces acting on this piece of volume are divided into body forces and surface forces. Theseforces (per unit mass) are represented by

Fsurface

(C.4)

(C.5)

where

l = Resultant force per unit mass of fluid (N·kg-I),

aij = 2 dimensional symmetric stress-tensor (N·m- 2).

(C.6)Conservation of momentum, differential form

In which aij represents the surface force components on the volume of fluid. The diagonal com­ponents represent the normal stresses and the off-diagonal components the tangential, or shearingstresses. The stress tensor aij can be subdivided further into several components, namely staticand dynamic parts, which will become clear in the following part. For now, the momentum balancecan be represented in differential form as follows, and can be explained as the acceleration of thefluid in terms of the local body and surface forces.

DUi aaijP-

D=pFi +-a .

t Xj

In a fluid at rest there is no deformation of the volume of the fluid. The off-diagonal, shearingcomponents of the stress tensor are therefore all zero. The only components which may act arethe normal components. This statement describes the well known property of static fluid pressure,in which the pressure acts equally in all directions. It is possible to write the stress tensor in thestatic case like aij = -POi/' The p in the equation is the so-called static fluid pressure. In a fluidin motion these results are generally not valid. It is however possible to define a scalar quantity,like static fluid pressure, to describe partly a fluid in motion. The other part of the stress tensordescribed by (C.7), is entirely due to the motion of the fluid, and is called the deviatoric stresstensor d ij . The stress tensor may be consequently written like

aij = -POij + d ij . (C.7)

If the fluids under study belong to the Newtonian fluids, the deviatoric stress tensor can beexpressed like

(C.8)

where

J-L = viscosity of the fluid (Pa·s),

1 (au au)eij = rate of strain tensor: - -aJ + -aZ ,

2 Xi Xj

6. = rate of expansion: 'V. u= eii,'

The Newtonian fluids are defined by a linear relation between rate of strain and the resulting(deviatoric) stress. However, this simple expression holds for a large range of liquids and gasesunder normal conditions for temperature and pressure. (C.8) holds also for air and water [3].Using (C.8) the total stress tensor for Newtonian fluids becomes:

(C.g)

'The Oij is known as the Kronecker delta tensor (formally oIl. The components of this tensor are defined byOij = 1 for i = j and Oij = 0 for i i j.

62 Chapter C. Navier-Stokes equations

Navier-Stokes equation of motion (C.IO)

Substituting C.g into the equation of motion, described in (C.6), the following expression holds.

p~~i =pFi - ::i + a~j {2Jt(eij-~~Oij)}'

This is known as the Navier-Stokes equation of motion. (C.IO) may be reduced in the case ofincompressible fluid and Jt uniform over the fluid into

(C.lI)

Furthermore, to describe a fluid in motion an expression for the energy balance is necessary.Body and surface forces are doing work on the mass of fluid, and due to transfer across theboundaries, heat may be gained. According to the first law of thermodynamics (C.12), some ofthis total gain of energy is manifested in an increase in kinetic energy and the remainder in anincrease in internal energy.

~E

~E

where

Q+W,

T~S-p~V.

First law of thermodynamics

Fundamental thermodynamic relation

(C.12)

(C.13)

~E = Change of internal energy per unit mass (J·kg- 1),

Q = Gain of heat per unit mass (J.kg- 1),

W = Work done per unit mass (J.kg- 1),

~S = Increase in entropy (J.K- 1),

~V = Change in volume (m3).

The fundamental thermodynamic relation is a combination of the first and second law of thermo­dynamics. t This relation between the internal energy E and is used by deriving the equation forthe conservation of energy. The amount of work per unit mass W done by the body and surfaceforces can be expressed by

~ 'Ui aGij Gij aUiW = UiFi + --- + ---.

p aXj p aXj(C.14)

The part which is associated with the increase in internal energy is the component (Gij/ p) (aui/axj).Moreover, the rate of gain of heat being transferred by molecular conduction is expressed as(l/p) (a/axi)(k(aT/axi))' This equation is obtained by J f'11dA, with f = -k"VT and thedivergence theorem [3], with k is the thermal conductivity of the medium. Concluding, the rateof change of internal energy per unit mass of a material element of fluid is given by

(C.15)

After substituting (C.g), the equation for the rate of change of internal energy per unit mass offluid becomes

DE -p~ 2Jt I 2 I a ( aT)- = -- + -(eiei' - - ~ ) + - - k- .Dt p p J J 3 p aXi aXi

(C.16)

t The second law of thermodynamics states that the entropy of an isolated system cannot decrease. The entropyof a system is a measure for chaos

C.l. Derivation of the Navier-Stokes equations 63

The rate of increase in heat per unit mass can be expressed as an increase in entropy, using (C.13)as given in [3].

(C.17)

where

S = Gain of entropy per unit mass (J·K- 1),

cp = Specific heat: T (~~) (J·kg-1·K- 1),P

(3 = Coefficient of thermal expansion: - ~ (~;) p'

This is equal to the following expression, after substituting (C.16).

(C.18)

where

The above equation consists only of molecular transport effects. However, these effects are inmany flow fields negligible, ergo T ~~ ::::: O.

After deriving the conservation of energy, the set of equations to describe a (Newtonian) flowfield is complete. Using (C.2), (C.lO) and (C.18) the flow dynamics can be described. However,some additional equations of state are necessary to be able to solve this set of equations. Generallyf (p, p, T) = 0 is used, as given in [3]. The resulting system of equations is

1 Dp ~--+\7·upDt

0, Conservation of mass

pF - ap +!!..- {21t (e - ~t.8)} Equation of motiontaxi ax j tJ 3 tJ ,

CDT _ (31' Dp __ ..... + _1~ (k aT )

'l' Energy balancep Dt p Dt p aXi aXi .

(C.19)

(C.20)

(C.2l)

Appendix D

Specifications Faulhaber PMSM~FAULHABER

Bi.irstenloser De-ServomotorElektronische Kommutierung

5mNm

Serie 2446 ••• B

22,·24, Wellenlagerung, Wellenbelasillng, Wellen.piel

OO2סס.2446

1. Nennsoannuna2, AnschluBwidersland, Phese-Pha••3, Abgal»leistung, """,,4, Wiri<ungsgrad, max,

5, Loorlaufdrehzahl6, L""riaufstrom7. AnhaJtemoment8. Raibunasdrehmoment9. Dvnamisch4;u Reibungskooffizient

10, Drohwhlkonstarrle11. Generntofspannungskonslan'te12. Drahmomentkonstante13, Stromkonstanta

14, Steigung dor !>-MoKennlin"15, Induklivlill zwischen dan Phasan16, Mech, Anloulzailkonslanfa17, RolortrSgheilsmomenl18. Wink-elbaschleunigung, max.

19, WSl111owidersUlr<le:20. Thormls.-chs ZsitkonstantQft:

21, Batriebslemperalul'b9reicll

25, Gehausemalerial26, Gewichl'D, Dreh'ichtung

Empfohlel1El Welte

28, Drehzahl bis29, Dauerdrehmoment bis30. Thermis.ch zulassiger Oauerstrom

Bemerkung

Rthl! Rlh2\wI! 1\\'2

2446S0tsa MKD26lI4 06 3500

15 V0,39 Ohm

1) 29,7 W82,5 %

Siohe unten rpmSiohe umen A78,1 mNm0,3 mNm0,5 10E·5 mNmlrpm

4681 rpmN0,214 mY/rpm2,04 mNm/A0,49 NmNm

895 rpmImNm17 uH16 ms1,74 gcm:2450 1lJ3 radlSZ

2/12 KIW3/ aoo

·30 125 'C

on!spreche,u MaBblatt

Edelstahl170 9ans.t9UBfUngsbedingt

2) rpm1)2) 4,1 mNm1) 2) 2,4 A

Daten des Motors Dl'!n9zusa.tzliche KOhlullg, d.h.· I] bet 600Cl0 rpm 2) W!:1nTlewiderstand Rlh2 nicht reduzlertZu S.: Leerlautdrehzahl nach 15; 66800 rpm

Leerlaufdrehzahl"ach 600s: 67300 rpmZu 6.: Leerlautetrom nach 1 t.. 0,59 A

Leerlautalrom nach €lOOs: 0,31 AZusatztiche Vereinbarung: Mot0f3tmmdiffelenz zwm.chen Unks- Wld Rechtslauf <.1 0% yom gemes5enen Motorslrom.zu 28.: 73000 Ipl"n tOr 213 dar maxlrnal zulAssigen Molort&rnp6ratuf"

B5000 rpm kurzfristiglul:'1sslge Drehzahl

Ind&x CDatum 10,03.2008

64

Appendix E

Graduation paper

65

GRADUATION SYMPOSIUM, 28 APRIL 2009

Multi-Physical Modeling and ExperimentalVerification of a Respiratory System

RJ.C.M. Mennen,Electromechanics and Power Electronics group,

E-mail: RJ.C.M.Mennen@student.tue.nl

Fig. I. Block diagram of a respiratory device with patient.

Pressure/FlowGenerator

II. RESPIRATORY SYSTEM OVERVIEW

The studied respiratory system is an update of a ventilatorfrom the Carina™ family from the Drager Medical company.This Drager respiratory device, as shown in Fig. 1, consistsschematically of the following parts. The required gases streaminto the system via an air- and oxygen-inlet. The ratio of thesegases is adjusted by the oxygen dosage function and mixedtogether in the blender. Pressure and flow of this mixture iscontrollable and forwarded by the pressurelftow generator viahoses and valves into patient lungs. This action occurs duringthe inspiration phase. The expiration phase, is the phase during

Patient airexhaustAir inlet

Oxygen-4­inlet ~

This paper describes the models covering the entire multi­physical domain of the respiratory action. After analyzingthe physical system, the models are derived for the differentdomains; electrical, electromechanical and pneumatic. To ob­tain all the necessary parameters for completing the dynamicmulti-physical model, several magneto-static semi-analyticalmodels are presented. The remaining parameters in the modelspredicting the dynamics of the respiratory process, and inparticular the pneumatic parameters, are derived by meansof measurements. Subsequently, the models are integrated toobtain one general model to simulate the entire respiratorydevice. The results acquired by this general model are ex­perimentally verified. Furthermore, the results obtained fromthe derived models can be used to improve the respiratorycontrol-loop in case of changing environmental conditionsduring operation, since the current control strategy is based onsteady-state measurements for only one set of environmentalconditions. Finally, the efficiency of the respiratory system isderived and used to evaluate the actual drive.

I. INTRODUCTION

Respiratory devices or respirators are usually battery sup­plied. This increases the usability, reliability and freedom ofmovement of the devices. These qualities become especiallyemphasized with the usage of this ventilation equipment formedical care in autonomous applications, like ambulances andrescue planes or emergency helicopters. Due to the limitedavailability of energy stored in batteries and the conceivablescarcity of recharge points, the working respirator shouldoperate efficiently. Detailed insight into the working principlesof the respirator is necessary to derive the efficiency of thesystem as a whole. Furthermore, because the respiratory sys­tem is subject to changing ambient conditions and numerousrespiratory disorders, these devices have to be able to copewith different situations.

Numerous techniques are available to model the pneumatic[1], electromechanical [2] and electrical [3] components sepa­rately, as part of the multi-physical domain covered by therespirators action. Models discussed in literature, regardingrespiratory subjects are mainly focussed on the biomechanicalpart of the system as described in [4], [5], [6]. Severalpapers discuss briefly respiratory devices and control of thepneumatic output power, [7], [8], however, these models arealso primarily based on the biomechanical part of the system,as the electromechanical and pneumatic systems are left outof the analysis. Furthermore, modeling of compressor systems,axial or centrifugal is done with a variety of versions of thelumped parameter Greitzer-model. Several alternatives of theGreitzer model for different compressor types and systems arediscussed in [1], [9], [10]. However, the drive used in thisapplication differs in power-range and used compressor-type.Modeling techniques to simulate the dynamics of the differentparts of respiratory devices are all discussed separately in lit­erature, however to derive an efficiency map of the respirationaction all models comprising the multi-physical domain haveto be integrated.

Abstract-This paper describes the dynamic and steady­state modeling of a respiratory device in the multi-physicaldomain. The multi-physical domain covers the electrical,electromechanical and pneumatic components of the system.The electromechanical parts are simulated using lumpedparameter analysis. Analytical Fourier series are derived toobtain the necessary model parameters. The dynamics of thepneumatic system are analyzed using the lumped parameterGreitzer-model. One general model is obtained by couplingthe models of the different parts of the multi-physical domain.Subsequently, efficiency of the system is derived using themodels. The simulation results are verified by means ofmeasurements on the physical respiratory device.

GRADUATION SYMPOSIUM. 28 APRJL 2009

Shaft

Airgap

PM rotor

Laminated

back iron

Stator windings

Fig. S. Schematic of the helical winding configuration.

Fig. 6. 3D view of the conical shaped impeller.

i7~T/

ch-to the correct stator windings. The windings are arranged ina helical winding configuration with a 120° phaseband asrepresented in Fig. 5 and treated by [Il] and [12]. For moreinfonnation about the type of motor and the winding topologysee [13].

TIle PMSM drives the centrifugal compressor by a conicalshaped impeller as shown in Fig. 6. A schematic of the entirepneumatic system of the respirator is shown in Fig. 7.

The test setup used for the measurements is schematicallyrepresented by Fig. 8. The commutation and motor controlare realized by means of dSPACE, instead of the DSP-chip,used in the commercial application. At the outlet duct of thecompressor pressure measurements are carried out with theCTEM70350GYO pressure gauge from SensorTechnics. Thisoutlet duct is connected to a tube with a length of 0.35 m. Howmeasurements are carried out at the end of this tube with aTSI model 4040 flow meter. A standard ventilation tube of

Lungs

signal

PWM 51 nal

Transistor

~~". "...,. ,."'~j~ commutation

l FUll-bridge ~: Penmanenl Magnet i i, . t I ' Synchronous motor ~ Impeller Q

Inver er ~:, p(MSM) ~ ....

Fig. 2. Block diagram of the blower and control blocks with patient.

Fig. 3. Three-phase full-bridge invetter with extra inductances in the leadsto the stator windings.

t:- --'E======:f=:;:---""''"'-~.Phase B Fig. 4. Cross-section of the PMSM.

'---~"""""'-.PhaseC

which the gases flow from the patient through the exhaust intothe ambient atmosphere. Expiratory pressure control duringthis phase keeps the pressure above the minimum allowedpositive end-expiratory pressure (PEEP). This is necessary toprevent collapsing of the lungs.

The analyzed system part is the pressure/flow generator orblower as is schematically shown in Fig. 2. A full-bridgeinverter is the power supply for the high-speed permanentmagnet synchronous motor (PMSM). An impeller is mountedonto the shaft of the PMSM to enable pneumatic work. Theoutput of the pneumatic system is connected to the patientlungs. Furthennore, a controller is needed to achieve the de­sired vcntilation strategy. Finally, using the signals from thrceHall sensors together with a six-step commutation-schemethe order at which the voltage is applied to the three statorwindings is regulated. The relevant variables in the electricaLelectromechanical and pneumatic domain are also indicated inFig. 2. These are voltage V and current I (electrical domain),torque T and speed iJJ (mechanical domain), and pressure pand volumetric flow Q (pneumatic domain).

The used inverter is shown in Fig. 3. This inverter consistsof six MOSFETs which are used as switches to control thevoltages applied to the three stator windings of the PMSM.The speed of the motor is controlled by means of PWMvoltage control. The extra inductances in the leads of the statorwindings in Fig. 3 are added to reduce the current peaks duringthe switching of the transistors.

The motor of this respirator is a three-phase pennanentmagnet synchronous motor (PMSM). To enable a satisfactorilyventilation action, this PMSM has to achieve high speeds up to8.9'10,3 rad·s- 1 or 8.5.104 rpm and accelerations up to 3.0.104

rad·s- 2 • Figure 4 shows a schematic representation of thePMSM. As already indicated, the electronic commutation isenabled by three Hall-sensors placed around the rotor magnet.Every 60 electrical degrees the rotor position is known as afunction of the three Hall signals. This infoffilation is used,together with a six-step switching scheme, to apply the voltage

GRADUATION SYMPOSIUM. 28 APRIL 2009

Centrifugal compressor

Afr(O)B,.

(3)- cos (0) ,110

Alo(e)BT •

(4)--'- sm (0),110

Parameter Description ValueRs Phase-phase resistance 0.39 nKc Coulomb friction 0.3·10' NmKv Viscous fdction coeffiCIent 0.5· 1O-~ Nm·rpm··'· i

J( Torque constant 2.04· 10 .J Nm·A ~Jrn Inenia 1.74·10 kg·m~

L m Phase-phase inductance 17 ItHGeometdc motor dimensions (Fig. 9)

1'8 Shaft radius 1.5 . 10 ml'm Magnet radius 5.4 . 10 ' m1'rneall Mean coil radius 6.7.10 -0 III

Tb Back-iron radius 7.4.10 -J m

TABLE JRELEVANT MOTOR SPECIFICATIONS.

where Br is the remanent flux density and 110 is the permeabil­ity in free spac.:e. Evidently, no higher harmonics are present

A. Back-EMF calculations

The rotor magnetic lleId in the airgap B is described withFourier-series in [15], and is derived using the vector potentialA in cylindrical coordinates as

~ ~ 1 aA, aA z ~B=\7xA=:;: ao':.e'r- aT eO· (2)

The following assumptions have been made in order to applythe field-equations:

• Length of the PMSM is considered infinite, consequentlyno end-etTeets and simplification of the model from 3­dimensional to 2-dimensional.Relative permeability (Pr) of back-iron and shaft isconsidered infinite.

• Relative permeability of copper is considered one, hencethe coils are modeled as air.

In this case, only two regions are of interest, namely themagnets (region II) and the airgap (region I). In Fig. 9 theresulting model with the two relevant regions and the ironparts are represented. The radius of the copper windings is verysmall, therefore it is assumed that the current flows through asheet at radius 'fme",.". The rotor consists of I SmCo magnet,hence I pole-pair, with a remanent /lux density of 0.98 Tand capable of retaining magnetization at temperatures over130°C. The magnet (region II) has parallel magnetization,for which the remanent magnetization vector in cylindricalcoordinates is given by

of the magnetic field is included in the viscous frictioncoefficient. The compressor torque is further elaborated inSection IV. The relevant specifications regarding the model.as indicated by the manufacturer, are shown in Table I. Tocalculate the efficiency of the used drive together with thePMSM the waveforms of the back-EMF voltages E",b,c arederived. Furthermore, to complete the model, the phase-phaseinductance L m is fragmented into the self-inductance L S8 andmutual inductance 111 by semi-analytical lllodeling.

(1)

Outlet

'inK sin (0) + ibJ( sin (0 - 2/31T)

+icK sin (e + 2j:37T) - 7[",

w,dedt

III. ELECTROMECHAN1CS

In this section, modeling of the PMSM is presented. Severalsimulation results of these models, together with the modeledinverter, are presented. Due to insufficient motor speciticationdata regarding the dynamic behavior of the PMSM, severalsemi-analytical models are used to derive the required param­eters.

The dynamics of the PMSM are approximated by theelectromechanical model, with the stator currents i".b.c ofphase A, Band C respectively, the rotor angle 0 and the speedw, as defined in the three-phase stationary reference frame[14]:

Fig. 8. Schematic of the test setup.

where E",b,c are the back-EMF voltages of phase A, Band C respectively. Its is the phase-resistance, Lss the self­inductance, ]\if the mutual inductance and V",b,c are the phasevoltages of phase A, Band C respectively. The inertia andback-EMF constant are defined by Jm and K, consequently.The load torque 7£ is split into Coulomb friction Ie" viscousfriction with coefficient J<" and torque due to compressor work7~,. The influence of eddy currents induced due to variation

length 1.2 m is connected to the outlet of the flow meter. Thispneumatic system is closed by a standard full port ball valveto adjust the outlet duct opening.

Fig. 7. Schematic of the pneumatic system [10].

Tube Ball valve1.2 m

GRADlJATJON SYMPOSJUM. 28 APRlL 2009

Airgap Magnets(Region I) (Region II)

ZI II r, I Ir-' II rm II -.I rmeM

3angle (rad)

Fig. 10. Magnetic flux density (radial and tangential) at r = r mean .

Fig. 9. Cross-section of the PMSM with the IWO regions indicated as usedfor magnetic nux density calculations.

To solve these four equations with four unknowns (X J,I, Y1,l,Xu1 , Yt.lI) the following boundary conditions are defined:

He,Ilr=n, 0, (15)

He,I111'=1', 0, (16)

BrJ 11'=1'"n B,.,IIII1'=1'm' (17)

He,Ilr=1',,, He,! 11 1'= I'm . (18)

AI(T,O) (Xl,lr 1 +Y1,lr-1 )sin(O)ez, (9)

AI1 (1',O) (XLlI1' 1 + }iII1'- 1) sin(O)ez . (10)

Using (9) and (10) with (2), the magnetic !lux density in bothregions is

(20)

B. Inductance calculations

To complete the set of parameters to model the PMSM (1),the (self-)inductance of the coils and the mutual inductancebetween the coils are calculated. These parameters are derivedusing the Biot-Savart-law [16]:

~ ~ flolf dfit (;r) = -1 -I~.c~/1

'IT C x-.r

r Jzm.a xl emnr

T = Is rx ']zB,.ds = Zm;n em", T;"canJzBrdOdz. (19)

This equation is solved by integrating over the stator windingsof one phase, arranged as shown by Fig. 5. For the torquecalculations only the z-component of the current-density isimportant. Consequently, the angle /j", in Fig. 5 due to thehelical winding configuration has to be taken into account.The induced voltage in a phase winding can be delived fromthe torque produced by that phase as both are proportional tothe change in !lux-linkage. The mathematical expression forthe EMF voltage is given by E = J(w sin (0 -- Or), where Oris the angle of the rotor magnet. The resulting value for Kis 1.9 . 10-:3 V·s-rad- 1 . This gives an enor of 7% with thespecifications. The wavefonn of the backeEMF is, due to thesinusoidal magnetic flux density in the airgap (Fig. 10), alsosinusoidal.

The resulting magnetic flux density in the airgap obtained withthe semi-analytical model is validated with FEM-simulations,as shown in Fig. 10. The saturation curve of the material usedfor the back-iron and shaft are included in the FEM-model.The semi-analytical model, just as the FEM-simulations pre­dict a maximum value for the magnetic flux density in theradial direction of 0.55 T at radius "mean' As shown in Fig. 10the FEM and semi-analytical results are in good agreement.Furthennore, the FEM results indicate that the assumptionsof Vr = (Xl for the back-iron and shaft-material is legitimate,because the material is not saturated.

From the magnetic flux density in the airgap, the electrome­chanically developed torque is calculated with the Lorentzforce:

(11)

(12)

(13)

(14)

(xu + YI.j1·-2) cos (fJ),

- (X1,l- Y1,I1'-2

) sin (0),

(XUI + YUI,,-2) cos(fJ),

- (X1,ll - y1.TI ,,-2) sin (0).

3 r,I(1',0)

Be,I(1', 0)

B",lI(1', fJ)

Be.lI (l"O)

because of the parallel magnetization. To find an expressionfor magnetic flux density in the airgap, the 111~gneto-statis

Maxwell equations are solved by substituting B = \7 x Ainto the constitutive relation for the magnet and airgap regionrespectively, which is stated below:

jj ILo/l,Ji + II'oi11" (5)

iJ /loti, (6)

where the remanent magnetization vector is i11' fi,fri!r +Alei!e. This results in the Laplace equation for the airgapregion and the Poisson equation for the magnet region

\72 Al 0, (7)

\72 All -ILo \7 x A/p = O. (8)

For this particular magnetization, the Poisson equation re­duces to the Laplace equation. The homogeneous solution forLaplace equations with only the fundamental harmonic is alinear combination of the functions sin (0) and cos (0) and 1'1and 1'-1. The vector potential in both regions can be expressedas

GRADUATION SYMPOSIUM. 28 APRIL 2009

(K.Y.z)

TABLE nCALCULATED SELF AND MUTUAL INDUCTANCE.

Fig. 11. Current filament RS with distance vectors used in (21) to (23).

TABLE HISl'l:ClFlCATIONS OF PMSM.

Manut~~ctllrer ModelMechanicaltime-constant 16 rns 16 illS

No load speed 6.73.104 rpm 6.96·10'" rpmLoad ellrrent 0.49· 10 Nm·A 0.46. 10 -3 Nm.A T

where P R, PT and P 8 are the lengths of the vectors fromthe current filament RS to the point T at which the vectorpotential is calculated. This current filament and the relevantvectors are shown in Fig. II. The self- and mutual inductancesare expressed by

The vector potential in vacuum at any point in space, due to aline-current can, therefore, be derived directly from that line­current with current] flowing in the direction of rat ;7!' inspace. The point at which the vector potential is calculatedis ;'/:. Elaborating (20) for some arbitraJy line-current in aCartesian coordinate system the resulting expression for thevector potential is given in [16] by

]~to (. -I (PR) . ~I (PS))Ax ~ smh PT - smh PT

x 1;1;2 - ;EIIIR8, (21)

Ay ~~ (sinh-1 (~~:) - sinh-

I (~~,))X IY2 - YIII RS, (22)

Az ~~o (Sinh-I (~~:) - sinh~1 (~~))x IZ2 -'IIIRS, (23)

C. Specifications and simulations

To simulate the described electromechanical model, (I)is implemented in Matlab and Simulink and the inverter ismodeled using the SimPowerSystems toolbox. The (dynamic)performance of this electromechanical model is compared withthe manufacturer specifications. The simulation results areshown in Table III and deviate with a maximum of 6% fromthe specifications.

Due to the used six-step commutation and driving strat­egy, the voltages applied to the windings have a rectangularwaveform. This waveform is in contrast with the sinusoidallyshaped back-EMF, derived in Section III-A. The resulting(line-)current is shown in Fig. 12, together with the EMF­waveform and the applied voltage of the same phase. Ingeneral, a PMSM with a sinusoidal shaped back-EMF has tobe driven by a current having the same sinusoidal waveformas the back-EMF [17]. This is accomplished by a three-phasesinusoidal current control drive. However, using the six-stepcommutation to control the PMSM, higher harmonics arcpresent in the current waveform, as shown in Fig. 12. Thesehigher hannonics give rise to torque ripple, as indicated inFig. 13, and extra copper losses. The perfonnance of theproposed sinusoidal current control drive is further investigatedin Section V.

IV. PNEUMATICS

The behavior of a pneumatic system is governed by theNavier-Stokes equations, which arc derived from the con-

using this approximation are shown in the second column ofTable II. The measured inductance L m in case of a Delta­connected three-phase winding configuration can be expressedby the self and mutual inductance as Lm = ~ (L'8 + M). Thisgives L rn = 14 pH for the derived inductance values. Thedisagreement is 18% in comparison with the 17 pH obtainedfrom the specifications. The measured inductance is 18 fiH.

The calculated values of the self- and mutual inductanceshow that the mutual inductance is only 8% compared tothe self inductance. Moreover, because of the additionalinductances of 36 /tH in the stator winding leads, as alreadyindicated in Fig. 3, the mutual inductances are neglected inthe electromechanical model (I).

(24)

(25)

L1 r ~ ~

]2 is A· ,Ids,

1 1 ~ ~IT A2 · ,lIds.I 2 8,

Where 8 and SI are the areas of the analyzed coils on thedefined current sheet in Fig. 9. The windings of the three-phasecoils in cylindIical coordinates shown in Fig. 5 are dividedinto a number of line-currents. Subsequently, the self andmutual inductances are calculated by numerically integratingover the current sheet surface of that phase. Because thevector potentials are derived in a Cartesian coordinate system,transformations from the Cartesian to the cylindrical andvice versa are carried out. The values calculated for the selfinductance L.,s and the mutual inductance lvI are shown inTable Il.

Instead of vacuum, the stator windings are surrounded byback-iron. To estimate the inlluence of the back-iron, simula­tions are done with a 2D FEM-model including and excludingthe back-iron. The resulting increase in inductance because ofthe back-iron is approximately 60%. The inductance values

GRADUATION SYMPOSIUM, 28 APRIL 2009 6

Lumped parameter Greitzer model.

TABLE IVPARAMETERS GREITZER MODEL.

ao Sonic velocity (m.s 'i)

V Plenum volume (m')p

m v Outlet duct mass flow (kg's -i)

Po Ambient pressure (Pa)A c Compressor duct length (m)Lc Compressor duct area (m")(J Slip factorT2 Outer radius of the impeller (m)w Motor speed (rad.s . i)J Inertia (kg·m~)

Tm Motor torque (Nm)k v Outlet duct valve coefficient

Duct

P1mv

Valve

Plenum

ll.pc-----'

Compressor pz -Po ;::===m~

Fig. 14.

10

JJ\/\I\lV\J6.735 0.736 0.737 0.738 0.739 074 0.741 0.742

Time (s)Applied voltage

2 10lb .,::'.1: .•••......~,~~~ -'-"-'-

0.735 0.736 0.737 0.738 0.739 0.74 0.741 0742Time (s)Current

i-~0.735 0.736 0737 0738 0.739 0.74 0.741 0.742

Time (s)

X 10-4151-- '~----'-~---,--Six-ste~'To;;;mutatl~~

1 ..--- Sinusoidal currentCO~~ II

EMF voltage

Fig. 12. Simulated EMF wavefonn, applied voltage and resulting line-currentof the used drive during open-loop operation of the compressor.

Fig. 13. Simulated electromagnetical developed torque ripple with six-stepcommutation and ideal sinusoidal current control at 5.0.104 rpm.

servation laws of momentum, mass and energy. Principally,these equations describe the fluid dynamics in detail of thesystem under study [18]. Solving this system of equationscan be accomplished in two ways, namely analytically withFEM-software or by simplifying the system with the Greitzermodel. In this study, the Greitzer model is preferred, becauseof the computational gain in simplifying the pneumatic system.The Greitzer model reduces a pneumatic system including thecompressor and connected tubes or volumes into three maincomponents, namely the compressor, a (plenum) volume andoutlet duct as shown schematically in Fig. 14. The dynamicsof the pneumatic variables of the plenum, namely pressurePp and compressor mass flow m are described by the Greitzermodel. The used test setup shown in Fig. 8 is physically linkedto this model as follows. The pressure rise due to compressoraction is described by !1pc = P2 - Po in the Greitzer model inFig. 14. The compressor duct length and area are L c and A c ,

respectively. The ventilation tubes connected at the outlet ofthe compressor in the test setup are represented by the plenum

(29)

where

The used parameters in (26) to (29) are given in Table IV. Theexpression (29) is used generally to model the flow through theoutlet duct [10]. The ambient pressure Po is equal to 101325Pa and the sonic velocity ao is 340 m·s-1 . The outlet duct

volume Vp in the model. Vp , L c and A c cannot always bederived straightforward from the compressor dimensions andcan differ up to 9 times the expected values [19]. Finally,the ball valve is modeled with the valve described by theGreitzer model, with m v the mass flow out of the plenum. Thepressure PI outside the plenum is during testing equal to theambient pressure. However, during respiration of a patient PI

is equal to the airway pressure. To obtain the lumped parameterGreitzer model from the Navier-Stokes equations the followingassumptions have been made.

• Relevant pressure and flow behavior is one dimensional,• The fluid in the inlet and outlet ducts is assumed to be

incompressible,• The length of the outlet duct is negligible.• The pressure inside the plenum volume is uniformly

distributed,• Gravitational and viscous effects are negligible.

The reduced model which describes the pneumatic quantitiesplenum pressure Pp and compressor mass flow m is [10]:

dpp a6dt Vp (m - m v ), (26)

dm AL

c(P2 - po) , (27)

dt c

~ ~ (Tm - mo'W1-~) , (28)

0.3650.3640.362 0.363Time (s)

0.361

GRADUATION SYMPOSIUM, 28 APRIL 2009 7

TABLE VPARAMETERS COMPRESSOR MAP.

flow has an angle relative to the tangential, which is accountedfor by a. The slip factor is approximated to be 0.72, whichis obtained by means of a curve-fit on the above describedsteady-state measurements. Subsequently, the tlhc,t is derivedas aw2r~. From the standard isentropic relations it is knownthat

(34)

(35)

(33)

ri 2'2 (w - kinsm) ,

kf m 2.

a 0.72r2 15 . 10 -j rn

rl 7.6· 10 -0 rn

kins 0.054·10kf 8.0·10n 1.55

tlhi

The incidence losses occur due to instantaneous change ofthe gas velocity as it hits the impeller blades. The constantk ins is detennined by means of a curve-fit regarding themeasurements. The second type of compressor losses are dueto the friction of a fluid flow through a pipe. The constantkf is also determined by means of a curve-fit regarding thesteady-state measurements. Other parameters mentioned arethe absolute ambient temperature To and specific heat atconstant pressure cpo The values of these parameters are 293K and 1012 ]·kg-1 .K-1 respectively. The isentropic process,(33) can be represented by

P2 = (1 + tlhc ) ~Po cpTo

where tlhc is as in (30) and, is a physical parameter, knownas the adiabatic index, and equal to 1.40. The same notationsas shown in Fig. 14 are used. The main compressor lossestlh10ss are split into incidence tlhi and frictional tlhflosses.These can respectively be represented by

P2(30)

A. Compressor map

Steady-state behavior of the compressor is described bythe compressor map. This compressor map gives the relationbetween stagnation pressure and (mass) flow for several speedsof the PMSM. The map derived for this compression system isshown in Fig. IS. A stable and unstable operating region in thecompressor map can be distinguished. The unstable region isapparent for small positive mass flows and high speeds, whichis to the left of the surge line. The unstable behavior is knownas surge [lJ, [19]. However, due to the small gradient of thecompressor characteristic in the unstable region, these surgeoscillations are small for this compressor. Moreover, this effectis not measured for speeds below 5.0 . 104 "-t'ln. The stableregion is to the right of the surge line.

Measurements to derive the compressor map have beencarried out by controlling the PMSM to a range of speedsand adjusting the outlet duct valve into a set of positionsbetween fully closed and fully open. For every set-point ofmotor-speed and valve position the steady-state pressure andflow are measured, and shown in Fig. IS.

The compressor map is derived as follows. Generally, thepressure rise due to compressor work can be described asan isentropic process, which gives an increase in enthalpytlhc for constant entropy. In practice, however, the pressureincrease in the compressor is not isentropic, because there willbe an entropy increase due to inevitable losses tlh1oss. Theselosses can be described by an isobaric process in series withthe previously described isentropic process. The compressionprocess is then

valve coefficient kv is a coefficient indicating the openingof the valve and calculated by means of measurements. Theremaining parameters Vp , Ac and Lc are defined fonnerly,and detennined by means of measurements. The results arepresented in Section IV-B. The load torque due to compressoraction is defined by mawr~ in (28) and is elaborated inSection IV-A. The steady-state pressure increase caused by thecompressor is indicated by P2 in (27). This pressure increase isa function of mass flow m and motor speed w, and is derivedin Section IV-A.

where tlhc,t is the total specific enthalpy delivered to the fluid[10]. The enthalpy delivered to the fluid is calculated by thepower delivered to the fluid. The power can be expressed by

Pc = wTc = w2mar~ = mtlhc,t. (31)

The torque applied to the fluid 1~ is defined as the change inangular momentum given in [9]:

T c = m(r2Ut,2 - rlUt,l) = mawr~, (32)

where Ut,I,2 are the tangential gas velocities at the accordingradii (rl,2). The righthand side of this equation is obtained byassuming rl Ut,1 = O. The slip factor a is dependent on thenumber of blades and the angle between the blades and tangentof the impeller. Ideally, the flow leaves the impeller only in thetangential direction. However, in practice the direction of the

(36)

The above discussed losses (34) and (35) are taken intoaccount. The adiabatic constant , is replaced by n, whichis slightly higher than ,. This a correction regarding the realcompression process in contrast with an isentropic process.To take negative mass flow into account, (w - k ins m)2 hasto be written as (w - kinslml) . (w - kinsm) and k f m 2 askfm ·Iml. The results are visualized in Fig. 15, together withthe steady-state measurements. The model of the compressormap is valid with a maximum deviation of 5% for speeds up to7.0.104 rpm, which includes the nonnal operating range duringventilation of 1.7 .104 to 4.6.104 rpm. The derived parametersdescribing the compressor map for this system specifically in(36) are summarized in Table V. The above derived expressionfor P2 (36) is inserted into the Greitzer model in (27).

GRADUATION SYMPOSIUM, 28 APRIL 2009

w 1 (pPo2) -;,1 (k;ns 1Ttpo (pPo2) *rr (37)Po (rt - 2r'§CT)

. /2. "" (;:) l (-<;,P,10('1 - 2,1.)+ Po (;:) ~ (,i (-kim' + <;,To) + ,1 (m' (2k, + k;no'll - 2" TO).)) )

150 200

Tubing: 0.85 m

~uretnents ~

==~¥ 1000f ' ,

~ 500f f\~~ O~'------5~;O"----Li'k.., "'----,~OO'

Frequency (Hz)Tubing: 1.2 m

~ 1050000":'~- ------T~--- ------,._----------r-= ~eas"U~~~n;~

~, l , • l=::SimulationsJj~ 01 /_\ . ---1--

1

o 50 100 150 200Frequency (HZ)Tubing: 1.6 m

I 1OoofL---~----'---flL:=__=_C;_~"',~':'~O='~~=~=n:=nt::'iJs i

g 500r ' fl\, ~~-'--__10. 00'-----"---c5J>..0.:>.......---,~00,--- 150 200

Frequency (Hz)

~ 1.06

§ 1.05!---"-'-- ...:L..-t"'_-'L.:.....;j

~a.. 1.04

Fig. 16. PSD of pressure measurements with 20 V.

Fig. 15. Compressor map.

asInfonnation derived from the compressor map can be ap­

plied into the control-loop. After inverting the model therequired motor-speed is known for every combination ofpressure and flow. The resulting relation between speed,pressure and flow is shown in (37). Other parameters whichhave influence on the perfonnance of the compressor are theambient temperature To and pressure Po. This influence isaccounted for by the derived compressor map in (36) and,therefore, also by the relation (37).

Another method of deriving the compressor map is bymeans of a curve-fit of a cubic polynomial with the measure­ments [19], [20]. However, in this study the above elaboratedmethod is preferred. This is because of the presence ofthe ambient parameters in the expressions, which give moreflexibility regarding controller design.

B. Dynamic parameters derivation

To complete the Greitzer model (26) to (28), the parametersv;" Ac and Lc are derived in this section. These three param­eters detennine the dynamical behavior of the pneumatic sub­system. They are obtained by means of pressure measurementswith closed outlet duct, which forces the fluid in the plenumto go into surge. The frequency of the pressure oscillationsduring surge is dependent on the above mentioned parameters.However, no analytical expression exists to give the relationbetween the three parameters and the surge frequency. Ac­cording to [1] the surge frequency is around, but below theresonance frequency of the system. The resonance frequencyof the system is generally known as the Helmholtz frequencyfH, and can be expressed by the three pneumatic parameters

fH = aD VA c (38)211' VpL c

Taking into account the physical dimensions of the system andthe Helmholtz frequency, the values for the pneumatic param­eters Vp , Ac and Lc are derived. The parameters are refined bymeans of fitting the modeling results on the measured surgefrequency.

To obtain the best measurement results of the pressurefluctuations, the pressure gauge placed at the outlet duct ofthe compressor in Fig. 8 is moved to the ball valve at theend. As mentioned in IV-A, surge behavior is only measurableat high speeds and low mass flows. Therefore, the outletduct valve is closed and the PMSM is driven in open-loopoperation with a high DC-voltage of 20 V. This gives a speedof approximately 8.0.104 rpm. These measurements are carriedout with different lengths of connected tubes, hence differentplenum volumes. The lengths of the used tubes are 0.85, 1.2and 1.6 m. The power spectral densities (PSD's) resulting fromthe measured pressure fluctuations are shown in Fig. 16. Ascan be seen in Fig. 16, the measured frequency decreases withincreasing plenum volume, which is as expected. These resultsverify, that the measured frequency is the surge frequency. Theconnected tubes of the test-setup as shown in Fig. 8 have atotal length of 1.6 m. The frequency of the pressure oscillationsfound for this case is 45 Hz.

The modeling results for all three plenum volumes areindicated in Fig. 16, and these results are also summarizedin Table VI. The derived values for Vp , Ac and Lc deviatefrom the physical system dimensions between 1 and 6 times,as already assumed in the first paragraph of Section IV. The

GRADUATION SYMPOSIUM, 28 APRIL 2009 9

TABLE VIIPNEUMATIC PARAMETERS FOR SETUP 1.6 M WITHOUT MASS FLOW.

TABLE VISURGE FREQUENCY AND PSD AMPLITUDE WITH FREQUENCY IN Hz AND

PSD AMPLITUDE IN W·Hz- 1 .

1-Measur~mentSl1_.......... SlmulatlC~~

-3

-40,41--~O~.41~1-·0.4~1·2-oA13---~0.414 0.415 0.416time (5)

Simulated amplitude I

Simulated frequency ITube length (m) I Measured frequency

Tube length (m) I Measured amplitude I

_0.i:,,~_5 §E _0.85 7.2·1O~ 9.3·10"1.2 1.1·10' 8.8·10"1.6 5.5·10" 7.1·10"

Parameter Matched Physical dimensionsAc 1.5 . 10 m" 1.5.10-4 m~

L c 4.5· 10 -~ m 1.0.10 -L m

Vp 3.2·10 -, m' 5.4 . 10 -4 m'

fH 55 Hz 90 Hz

Fig. 17. Measured and simulated current waveform (VDC = 10 V, w =4.1.104 rpm, kt = 0.87·10 - 4).

TABLE VIIICURRENT-RMS VALUES MEASURED AND SIMULATED.

Time(s)

Applied voltage I Measurements Simulations

Fig. 18. Measured and simulated response of the speed on a step-voltagefrom 5.0 to 10 V.

~......._--«.~-------------Speed

3.5

4.5~~104- Measurements........ Simulations

5 V [~0c..;.7.=27-A-=----__h0-';i.6,.:.6.A__IOV [IAA 1.5 A

-----~----

represented in Table VIII. The response of the speed on theapplied step voltage is shown in Fig. 18. The simulation resultswith the same configuration are also included in Fig. 18.The steady-state values and dynamics of the simulations andmeasurements are in good agreement. The measurements andsimulation results for the response of the pneumatic quantitiesmass flow and pressure on the voltage step from 5 to 10 V areshown in Fig. 19. The calculated time-constants and steady­state values of the measurements and simulations are summa­rized in Table IX. In this table, the pressure is expressed inpressure rise above ambient pressure Po. The largest deviationbetween measurements and simulation results of the dynamicsis the time-constant of the flow response and is 33%. This erroroccurs due to changing pneumatic parameters as function ofthe working point of the compressor. The steady-state values

~ 2.5 ,;, ...../~

1:1 •.. . "~.1------:";;1.2-------=-C

1

.3:;----J

A. Results of modeling the respiratory dynamics

The models described in the preceding sections are con­nected and compared with the measurements on all physicaldomains in this section. During normal operation of therespiratory system the compressor is delivering flow. The line­currents for this operation mode are measured. These results,together with the simulation results are shown in Fig. 17.The current amplitude and waveform of the measurementsand the simulations are in good agreement. This holds alsofor the rms values of the current at the used voltages as

V. GENERAL, MULTI-PHYSICAL MODEL

To validate the different derived models, measurements onall multi-physical sub-domains are carried out. To obtain thedynamical behavior, the measurements on the system havebeen done during open-loop operation. Open-loop operationis accomplished by establishing a DC-voltage step from 5.0to 10 V at the inverter, and by keeping the duty cycle of thePWM on unity. This has been done to minimize the influenceof the switching actions during PWM on measurements. Thisopen-loop operation is also implemented in the models, whichreduces the computation time. The applied voltage step from5.0 to 10 V agrees well with the normal operating conditions,which are between 1.7.104 rpm and 4.6.104 rpm. This voltagestep has been reiterated for several positions of the outlet ductvalve. The step applied to the stator-windings is not ideal, dueto the limited slew-rate of the used voltage source. Therefore,the voltage step is measured and also applied to the modelsduring simulation. The following quantities on all parts of themulti-physical domain have been measured: current, speed,pressure and (volumetric) flow.

derived values, as well as the physical system dimensions areshown in Table VII. The Helmholtz frequency calculated withthe three pneumatic parameters is also presented in Table VII.This value is as predicted of the same magnitude, but abovethe surge frequency.

GRADUATION SYMPOSIUM. 28 APRIL 2009 10

Pressure

-----.., .._...--------'--........ -___ 10 V

X 104 Speed6, ..= I

..........__'..... - Measurements "I

..--..- ... ------_._..._... __~;_v .__~.--~:~ ..simula~J"'''--------'''--''-'1

.:~--~--------~---------:-~'-'---~2345678

Valve opening coefficient k" x 10-~

1.04 X 10';- Measurements

{ij 1.035 - ....- Simulations

~ 103~~ 1,025~ 1 02 _

101~':c.9---~--~1.'c-l---~1.c:-2--~1.c:-3--~14

Time (s)Flow

~ 2;--------------

ilL89 -1;-------:-1':-1-------c1.';;-2-----,-1'::-3------:"1

1

4

Time (5)

Fig. 19. Measured and simulated response of the pressure and mass flow ona step-voltage from 5.0 to 10 V.

Fig. 20. Steady-state speed values.

TABLE IXMEASUREMENTS AND MODEL-BASED RESULTS POR A VOLTAGE STEP

PROM 5 TO 10 V WITH kt [S 0.87.10-4 .

c=:::: Time-constants and steady-state values

Voltage Time-constant speed Steady-state speedin ms in 103 rpm

Measured I Model Measured I Model5V

79 I 8522 I 23

IOV 42 42

Time-constant pressure Steady-state pressurein ms in Pa above Po

Measured I Model Measured I Model5V

84 11.0.102 4.6·10~ I 5.2·10"10V 1.8.103 1.8.103

Time-constant flow Steady-state flowin ms in 10-3 kg'C 1

Measured I Model Measured I Model5V

70 I 931.8 I 2.0

10 V 3.7 3.7

5V

OL--~----L__~_--L-_~__o____-~----.J

4 5 6 8Valve opening coeffIcient k" x 10·s

Fig. 21. Steady-state pressure and maSS flow values.

are all within a deviation of 13%, as can be derived fromTable IX.

B. Steady-state results

Measurements on the system during steady-state operationare also carried out to obtain the performance of the modelover the whole operating range of the respiratory system.Figs. 20 and 21 indicate that the steady-state model is moreaccurate for higher outlet valve coefficient, hence for highermass flows. This is because the torque due to mass flow (32)is well defined and becomes dominant over the other lossesfor higher mass flows.

theoretical efficiency using the proposed driving strategy forthe used PMSM. The losses taken into account with theanalysis are summarized in Table X. The resulting simulatedefficiency map for the six-step commutation drive togetherwith the PMSM is shown in Fig. 22. The blank area in thetop right comer is due to the voltage limit of 20 V. Theefficiency map for the proposed sinusoidal current controlstrategy is shown in Fig. 23. In the normal working area thepower efficiency of the six-step commutation drive, withoutpneumatics, is between 50% and 65%. The drive efficiencyby applying sinusoidal current control increases up to 70%

C. Power efficiency

Power efficiency calculations for the respiratory systemare carried in this section to evaluate the implemented six­step commutation driving strategy. Because of the differencebetween EMF waveform and current waveform discussed inSections III-A and III-C, efficiency gain in drive operationcan be obtained. To derive the advantage of the suggestedsinusoidal current control driving strategy, the efficiency ofthe used six-step commutation strategy is compared with the

TABLE XLOSSES TAKEN INTO ACCOUNT WITH ANALYSIS.

Electrical subsystemI"R I Copper losses (W)

Mechanical subsystemKcw I Coulomb friction (W)Kvw" I Viscous friction including eddy current losses (W)

Pneumatic subsystem

"-j (w - kinsm)2 m I Incidence losses (W)

kr m6 I Fluid friction losses (W)

GRADUATION SYMPOSIUM. 28 APRIL 2009 11

14

12_10E~

~ BI!£

!

a.5

a.4

0.3

a.2

a,

108

107

,,1.06~

l!!~ 1.05l!!D.

104

103

1.02

IiiIiIIiIIIIII3

Mass flow (kg/s)

Fig. 22. Simulated efficiency lllap of the six-step commutated drive withPMSM with voltage limit of 20 V.

Fig. 23. Simulated efficiency map of the sinusoidal current controlled dlivewith PMSM.

Fig. 24. Simulated efficiency Illap of the respirator system with trapezoidalvoltage controlled drive.

14

12_10E~

~ 8I!£

~

1.08

107

104

1.03

1.02~!!!ii.ii

4 6Speed (rpm)

4 5 6Speed (rpm)

3Mass flow (kg/sJ

10)( lO

A

a.6

a.7

0.5

a.4

0.3

a2

0'

Fig. 25. Simulated efficiency map of the respitator system with sinusoidalcurrent controlled drive.

and 78%. Therefore, the gain in power efficiency of thedrive, without pneumatic sub-system, using sinusoidal currentcontrol is approximately between 10% and 20%, as can beconcluded by comparing Figs. 22 and 23. Disadvantages of thesinusoidal current control strategy are increasing losses in theelectronics due to a higher switching frequency. This reducesthe calculated efficiency gain of 10% and 20%. These lossesare not included in the models. Moreover, to enable sinusoidalcurrent control, a more accurate and expensive speed andposition sensor is needed, instead of the Hall sensors.

The power efficiency of the total respiration action is derivedfor the actual, six-step commutation drive and sinusoidalcurrent control drive as shown in Figs. 24 and 25 respectively.The power efficiency of the whole respiratory system isbetween 15% and 20% for the six-step commutation drive.These values are valid within the nonnal operating region ofthe respiratory system, as indicated in Fig. 24. The resultingefficiency increase by implementing the sinusoidal currentcontrol drive is reduced to a maximum of 5%.

VI. CONCLUSIONS

Tn this paper, electrical, electromechanical and pneumaticmodels are derived and experimentally verified to simulate thedynamics of a respiratory system. The respiratory system con­sists of a voltage driven high-speed PMSM with helical statorwindings, with an impeller mounted on the shaft. The pa­rameters of the model simulating the dynamics of the voltagedriven PMSM are obtained from measurements, specificationsand two (semi-)analytical models. The first analytical modelgives the magnetic /lux density inside the airgap of the PMSMusing analytical Fourier series. The magnetic /lux density isused to detennine the electromechanically developed torque,the EMF-constant and the EMF-wavefoTIll. The second modelgives the self inductance and mutual inductance of the helicalstator windings using Biot-Savar!. The semi-analytically de­rived parameters are verified with 2D FEM-simulations. Theelectromechanical models show a good agreement with thespecifications and measurements.

The steady-state characteristics of the pneumatics of therespiratory system are detennined by the compressor map.

GRADUATION SYMPOSIUM, 28 APRIL 2009

A model is derived to approximate the compressor map ob­tained by measurements. This model is partially derived fromthe physical dimensions of the respiratory system. However,some model parameters can not be determined directly fromthe physics of the pneumatic system. These parameters arederived by fitting the model on the measurement results. Themaximum error between the modeled compressor map andthe measured compressor map is within 5%. This is valid forspeeds up to 7.0.104 rpm, which include the normal workingarea of the compressor. By inverting the derived compressormap, an analytical relation is obtained between the requiredmotor speed for any combination of pressure and flow. Thisrelation can be implemented into the respiratory control-loop.

The dynamical behavior of the pneumatic sub-system ismodeled by means of the Greitzer model. The parametersof the Greitzer model are derived using the surge frequency.Surge is initiated by closing the pneumatic sub-system, hencethe Greitzer model is derived in case of no flow. The accuracyof the model reduces with increasing flow, which indicateshifting of the model parameters with changing compressorworking-point.

Using the derived models, the power efficiency of therespiratory system is calculated. These efficiency calculationsare carried out for two different driving strategies, i.e. six­step commutation and sinusoidal current control. This is doneto verify whether the respiratory system efficiency improvesby changing the driving strategy from six-step commutationto sinusoidal current control, because the driving strategy fora PMSM should generally be sinusoidal current control. Theefficiency is first calculated for the drive only, hence with­out impeller and pneumatic sub-system. The resulting powerefficiency of the six-step commutated drive, in the normaloperating region, is between 50% and 65%. The efficiencyis also calculated in the working region for sinusoidal currentcontrol. The resulting power efficiency increases up to 70%and 78%, hence the proposed driving strategy increases theefficiency significantly. However, the same efficiency calcula­tions are carried out for the entire respiratory system, henceincluding the impeller and pneumatic sub-system. The resultsobtained for the actual driving strategy indicate a efficiencybetween 15% and 20%. Since the losses in the pneumatic sub­system are unaffected by the drive strategy, the efficiency gainof the total respiratory system is reduced to a maximum of5%, when using sinusoidal current control instead of six-stepcommutation.

Recommendations for further work will be to implementsinusoidal current control to verify the calculated gain inefficiency and to decide if it is worthwhile realizing thisdriving strategy. Furthermore, investigating the performance ofthe derived compressor map for changing ambient conditionsis necessary to analyze whether it is possible to implement itin the control-loop.

REFERENCES

[I] J. Gravdahl and O. Egeland, Compressor Surge and Rotating Stall:Modeling and Control. Springer Verlag, 1999.

[2] E. P. Furlani. Permanent Magnet and Electromechanical Devices. Aca­demic Press. 200 l.

12

[3] M. Kazmierkowski, R. Krishnan, and F. Blaabjerg, Control in PowerElectronics: Selected Problems. Elsevier Science, 2002.

[4] T. Yuta, Minimal Model of Lung Mechanics for Optimising VelUi[atorTherapy in Critical Care. PhD thesis, University of Canterbury, 2007.

[5] V. Tamburrelli, S. Silvestri, and S. Sciuto, "Reliable procedure to switchfrom volume controlled to pressure controlled ventilation assuring tidalvolume during anesthesia," 2005.

[6] F. Eugenio. P. Cappa, SA, and S. Silvestri, "Linear model and algorithmto automatically estimate the pressure limit of pressure controlledventilation for delivering a target tidal volume," foumal (It' ClinicalMonitoring and Computing.

[7] M. Borrello, "Modeling and control of systems for critical care ventila­tion," 2005.

[8] A. A. Wald, T. W. Murphy, and V. D. Mazzia, "A theoretical study ofcontrolled ventilation," IEEE Transactions on Bio-Medical Engineering,1968.

[9] J. T. Gravdahl and O. Egeland, "Speed and surge control for a low ordercentrifugal compressor model," Proceedings 0/ the IEEE ImemationalConference on Colttrol Applications, 1997.

[10] K. Boinov, Efficiency and Time-Optimal Control of Fuel Cell­Compressor-Electrical Drive Systems. PhD thesis, University of Tech­nology Eindhoven, 2008.

[11] A. Anderson, J. Bumby, and B. Hassall, "Analysis of helical armaturewindings with particular reference to superconducting a.c. generators,"Generation, Transmission and Distribution, lEE Proceedings, 1980.

[12] M. Bailey, J. Bumby, B. Hassall, and A. Anderson, "Magnetic fields andinductances of helical windings with 120 0 phasebands," Electric PowerComponellts and Systems, 1981.

[13] T. Bertolini, R. Bessey, R. Keller, A. Seegen, and H. Wallner, Glocke­nankermotoren: AUfbau, Betriebsverhalten, Anwendungen. sv corporatemedia, 2006.

[14] J. Hendershot and T. Miller, Design of Brushless Permanent-MagnetMotors. Magna Physics Publishing and Oxford University Press, 1994.

[15] Z. Zhu, D. Howe, E. Bolte, and B. Ackermann, "Instantaneous magneticfield distribution in brushless permanent magnet dc motors, part i: Open­circuit field," IEEE Transactions on Magnetics, vol. 29, January 1993.

[16] K. Binns, P. Lawrenson, and C. Trowbridge, The Analytical and Nu­merical Solution of Electric and Magnetic Fields. John Wiley & Sons,1992.

[17] D. Hanselman, Brushless permanent-magnet motor design. London:McGraw-hill, 1994.

[18] G. Batchelor, An Introduction to Fluid Dynamics. Cambridge UniversityPress, 1967.

[19] C. H. Meuleman, Measuremellt alld Unsteady Flow Modelling ofCentrifugal Compressor Surge. PhD thesis, University of TechnologyEindhoven, 2002.

[20] J. van Helvoirt, Celttrijugal Compressor Surge: Modeling and Identifi­cation for Control. PhD thesis, University of Technology Eindhoven,2007.