L3 – Equivalent Circuits · L3 – Equivalent Circuits EIEN20 Design of Electrical machines, IEA,...

L3 – Equivalent Circuits

EIEN20 Design of Electrical machines, IEA, 2016 1

Industrial Electrical Engineering and AutomationLund University, Sweden

L3: Equivalent circuits

Formulation, implementation and experience

Avo R Design of Electrical machines 2

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Previous lecture

• Coil: L=0.02m, W=0.05m, J=2A/mm2, q=57600W/m3, amb=20°C ..

• Analytic: max=41.60°C (a rod) max=63.20°C (a plate)• Thermal EC: P=14.4W, Gconv=0.7W/K, surf=40.57°C,

Gcond=0.5+0.08W/K, max=65.4°C • Heat transfer FE: max=57.76°C surf=40.57°C


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Today’s goal

• Equivalent circuits• Calculation example in the first home assignment• Introduction to calculation methods• Equivalent circuit method• Magnetic equivalent equivalent circuit• Introduction to the second home assignment


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What did we learn …

• Trying to manage with 200 turns …

• Selecting diameter to fit into window …

– 1.0…0.8 mm

• Keeping the same current

– 5A – 6.3…9.4 A/mm2 0.44 0.48 0.52 0.56 0.6 0.64

70

80

90

100

110

120

pow

er lo

sses

P, [

W/m

]

fill factor, Kf [-]0.44 0.48 0.52 0.56 0.6 0.64

0

0.2

0.4

0.6

0.8

1

Tem

pera

ture

, [C

]

0.44 0.48 0.52 0.56 0.6 0.6460

80

100

120

pow

er lo

sses

P, [

W/m

]

fill factor, Kf [-]0.44 0.48 0.52 0.56 0.6 0.64

0

100

200

300

Tem

pera

ture

, [C

]

0.44 0.48 0.52 0.56 0.6 0.6460

80

100

120

pow

er lo

sses

P, [

W/m

]

fill factor, Kf [-]0.44 0.48 0.52 0.56 0.6 0.64

0

100

200

300

Tem

pera

ture

, [C

]

Changing the material between winding turns fromair (0.03W/mK to impregnation 0.3W/mK)


Home assignment A1

Analysis of heat transfer in a single-phase transformer


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Goal and Geometric modeller

• Thermal analysis of a single phase shell type of transformer

– Specify B find Pcore

– Specify J and Pcoils so that coil< limit

– Maximise I where IJAeand BAm

The proportions between the electric and magnetic circuit is changed: Ks=lslot/(lslot+lcore)




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nThermal EC and topology matrix

1

2 3

5

4

6

9

7

1011

12

8

1 2

3 4

5 6

7

8 9

10

11

12 13

1415

16

• th=[1 1 2 la(1)*Ath(1)/Lth(1);• 2 2 3 la(2)*Ath(2)/Lth(2);• 3 3 4 la(3)*Ath(3)/Lth(3);• 4 4 5 la(4)*Ath(4)/Lth(4);• 5 1 6 la(5)*Ath(5)/Lth(5);• 6 2 7 la(6)*Ath(6)/Lth(6);• 7 3 8 la(7)*Ath(7)/Lth(7); • 8 6 7 la(8)*Ath(8)/Lth(8); • 9 7 8 la(9)*Ath(9)/Lth(9);• 10 8 4 la(10)*Ath(10)/Lth(10);• 11 6 9 la(11)*Ath(11)/Lth(11);];• nelm = 11; % number of elements• ndof = 9; % number of nodes

Elements: thermal conductivity, la [W/mK],Area, Ath [m2] and length Lth [m]


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Thermal equivalent elements

• Elements defined by: – Thermal conductivity, la or λ [W/mK],– Area, Ath [m2] and length Lth [m]

• Homogeneous bidirectional flux flow is assumed in a single heat conductivity elementGth

• The cross section variation and convection can be included

cool

el

iee

i

ei

eeth

AA

cl

AG

1 1

1

y

x

z

h

wl


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Solving a TEC

• The node potential method is used to calculate the relations between the thermal conductivity (stiffness) matrix G, temperature (unknowns) vector and thermal flux input (load) vector Q of known losses

G=Q• Each and every thermal element Ge is assembled to a

global matrix G with a connection between elements and boundaries

• Unknowns (n) are calculated in respect to references (r)n= Gnn

-1(Qn - Knr r)


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Initial data from EC and FE models

75.22.25135.075.62.18134.890

77.72.47135.179.62.41134.880

80.52.73135.083.82.65134.870

83.23.06135.088.02.95134.860

86.23.53135.192.33.35134.850

89.14.21134.996.73.93134.840

92.45.33134.3101.34.87134.830

97.67.64135.7106.66.72134.820

103.216.06135.1112.913.59134.810

cFEM

[°C]JwFEM

[A/thth2]wFEM

[°C]cECM

[°C]JwECM

[A/thth2]wECM

[°C]Ks [%]




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nObservations

• The estimation error between the thermal EC and FE model is larger when the geometric proportionsbetween the sides of the coil (coil width/coil length) is bigger

• thermal EC shows a higher hotspot temperature than the thermal FE model for the same current loading

• It is inconvenient to use the same thermal converging conditions for a small un-proportional and a large proportional coil (max 25 iterations in FEM vs max 190 in TEC)


Equivalent Circuit Method

From physical understanding to mathematical formulation and

method implementation


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Physical understanding

Physical understanding

Mathematical formulation

Test and measurements

Model Real device

• Cause-effect relationship• mathematic description

and measurement of physical phenomena in order to get a good understanding

• material properties


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Mathematic formulation

• Basic formulation for electromagnetic devices

• Heat transfer is described by heat equation

• Electromagnetism by Maxwell’s equations

• Electro-mechanism by electromagnetic stress tensor or virtual work Change of system

energy

Magnetic stress per unit of area

Gauss’s Law, Electricity

Gauss’s Law, Electricity

Ampere’s circuital law

Faraday’s law

Gauss’s Law, Heat transfer

S m dstF

tBE

JH

D

q

0 B

mWF




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nNumerical modelling

• Model – description of system: process in a structure• Modelling – study of the behaviour of the model• Numerical modelling – handle the complexity of PDE

Method Finite element method (FEM)

Finite difference method (FDM)

Boundary element method (BEM)

Equivalent circuit method (ECM)

Point mirroring method (PMM)

Principle of discretisation

m1

m2

q

q*

Geometry approximation Extremely flexible Inflexible Extremely flexible Specific

geometries Simple

geometries

Non-linearity Possible Possible Troublesome Possible By constant factors

Computational cost High High High Very low Low


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Equivalent element

Length

Area

Flow

Potential difference• Definition of geometric

and medium properties• Obey to a physical laws

i.e. relations• Is equivalent to ‘physical

reality’ i.e. has similar or identical effect

areaklengthimpedance


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Formulation of a transfer problem

• Heat source Q(x) [J/sth] per unit of time and length

• The heat inflow H [J/s] at position x, and outflow H+dH at position x+dx

• Transfer problem is described by conservation equation and constitutive relation

xdx

L

H H+dH

Q(x)


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Conservation equation

• The conservation equation describes the balance of the time independent heat flow

QdxdHdHHdxQH




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nConstitutive relation

• the heat flux q [J/(m2s)] is specified as the flow through the cross-section area per unit time

• constitutive relation defines the heat flow inside the medium

AHq

dxdq


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Heat transfer

• Heat transfer problem according to potential and heat source/sink

• The stationary (i.e. time-independent) heat problem is described as a balance between heat supply to the body per unit of time and the amount of the heat leaving the body per unit of time.

0

Q

dxdA

dxd


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Solution

• The differential equation of the transfer balance is solved for the finite size of volume which boundaries specify the flow through them.

• In order to solve the heat transfer second-order differential equation, two boundary conditions need to be specified to the two ends of fin.

• At one of these ends either temperature or flux q is given.


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Constitutive relation –a conductivity element

• Flux tube• The scalar potential

difference (potential drop) – temperature– voltage– magnetomotive force

• The ratio of potential difference to flux is a function of flux tube geometry and medium properties

x A(x+dx)L

q

(L)

A(x)

(0)

q

l

xAxcdxuluR

0

0




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nSimilarities I

• The heat transfer problem describes the temperature distribution between a source Q and coolant

• The magnetostatic problem specifies the magnetic potential Vm according to the magnet flux Ψ.

• The displacement u is the unknown for elasticity problem with body forces b.

0

Q

dxdA

dxd

0

dx

dVA

dxd m

0

Q

dxdV

Adxd e

0

b

dxduAE

dxd


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Similarities II

• The ability to conduct flow in different media and physical problem are defied with thermal conductivity λ, magnetic permeability μ or elasticity E and the corresponding cross section A.

l

xAxdxR

0

l

m xAxdxR

0

l

xAxdxR

0

l

E xAxEdxR

0


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Equivalent circuit relations

=Q·RN·I=Φ·RU=I·ROhm’s Law

R=1/λ·l/AR=1/μ·l/AR=1/γ·l/AResistive element

Q=q·AΦ=B·AI=J·AFlow

=G·lN·I=H·lU=E·lPotential

Thermal circuit

Magnetic circuit

Electrical circuitRelation


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Thermal equivalent circuit

• Heat transfer in two different medium that is described by heat conductivity k1 (W/K) and k2 (W/K)

• elements are connected in series along x-axis• The positive direction of thermal flow Q (W) is chosen

to be in the direction of x-axis.

1 2 k1

Q1 Q2

2 3 k2

Q2 Q3x




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nThermal flux

• The thermal flux q within the element is described in accordance with a constitutive law i.e. a relation which describes how the material conducts heat.

• The heat flow is represented as the heat fluxes (including external) acting on the element nodes.

121 kq

1212 kQ 2111 kQ


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Matrix form of element

• According to balance equation, the sum of all the heat fluxes acting on the element nodes is equal to zero.

• The characterization of one element can be expressed in matrix forms.

• Each element independently in the system can be described in respect of the element relation.

2

1

2

1

11

11

QQ

kkkk

eee faK


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Expanded element relation

• The expanded element relation i.e. the first element relation to the whole system in accordance with this example is:

000000

12

11

3

2

1

11

11

QQ

kkkk

eeee11 faK


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The complete system

• The complete system equation of the entire thermal circuit is the sum of the expanded element stiffnessrelations of each element and load vector in accordance with equilibrium conditions for the nodal points.

3

22

12

1

3

2

1

22

2211

11

0

0

QQQ

Q

kkkkkk

kk

fKa




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nDefining a heat transfer problem

• The determinant of the assembled total stiffness matrix K is equal to zero.

• In order to obtain unique solution for the unknown temperature and at least one node point has to be prescribed a priori.

• The specification of the given temperature is an essential boundary condition, which prescribes the value of variable itself and is necessary in order to solve the system of equations.

• The heat flux is a natural boundary condition and this specifies thermal insulation, the heat flow into or out from the system.


MEC – Magnetic Equivalent Circuit

Example based on a electromagnetic circuit


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From Real circuitsto Equivalent circuits

i

Origin of magnetic flux• Current carrying coils• Permanent magnets

Medium carrying the magnetic flux• magnetic core• Air-gap

Force of origin• Interaction• Attraction

Problem solving domain• Symmetry• Boundary

A


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From Real circuitsto Equivalent circuits

i

Coil and Core• MMF source• Nonlinear reluctance of core Permanent magnet

• Remanence flux source• Inherent reluctance

Air-Gap• Parametric gap reluctance

Problem solving domain• Symmetry• Boundary

B

NiMMF ABl

)(

)()(

0 A

l

ABRR

lHAB

C

R




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nExample: electromagnetic circuit

• Electromagnet + a core with permanent magnet

• ‘magnetizing’ flux path through the core

• Ampere’s circuit Law

• Circuit consists of a soft magnetic core, a magnet and a coil.

iNlH

iNlHlHlH ggfefepmpm


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Example: parameterization

• Geometry and materials are parameterized

• magnetic field intensity H is replaced with flux density B

• The same flux slows through the cross-section areas A

ggfefefe

pmpmrpm

HBHB

HBB

00

0

gfepmggg

fefefepmpmpm

AB

ABAB


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Example: equivalent circuit

• After the first replacement

• After the second replacement

• Flux path in the same medium is summon up to a single element

iNlB

lB

lBB

gg

fefe

fepm

pm

rpm

000

pmpmpm

pmgfepm

fe

g

fefe

fe

pmpm

pm

Al

iN

Al

Al

Al

0

000


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Example: Finite element model

• FE model• Comparison

– Bc=0.73T froth previous formulation

– Bc=0.62T froth FE model

• Difference is due to a leakage flux path




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nFormulating a MEC

• Assume flux path (circular) and define elements• Three types of permeance elements Pm:

– nonlinear core elements Pm=f(B) – parametric gap elements Pm=f() – leakage permeances.

• The node potential method is used to calculate magnetic scalar potential (unknowns) vector Vm with respect to the permeance (stiffness) matrix G, and magnetic flux input (load) vector Ψ.

ΨGVm


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Magnetic equivalent circuit

• Relative simple to ‘add’leakage elements

• Circuit described in a topology matrix

• Comparison – Bc=0.73T from previous

formulation– Bc=0.62T from FE model– Bc=0.68T from MEC

model


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Sources

• The remanence BR of the permanent magnet and the cross-section area Apm of the magnet determine the value of the source magnetic flux.

• In order to consider the non-zero current in the armature coil the mmf source has to be added to the algebraic equations for the node points.

pmR AB pmC

pmRpm lH

ABP

12121221 PNiGVV mm


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Magnetic saturation

• The magnetic saturation is taken into account in nonlinear elements Gm,

• The permeability updatebases on the previous update and on the resent estimation from μ=f(B)

ababababba GFGuu

)(ψGu

eeeee AGuuB /21

ndof

e

ni

ndof

n

ni

ni

err

nelm

e

ei

nelm

e

ei

ei

err

u

uuu

B

BBB

1

2

1

2

1

1

2

1

2

1

e

ienl

ei

ei

enl

cB

11

3

3

10

10

err

err

u

B

Fpmmecl ,,, Mμ




Home assignment A2

Analysis of electromagnetism in a single-phase transformer


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Goal: Geometry vs Equivalent circuits

• Same transformer as in the first assignment

– Proportional core and slot for coils Ks=0.5

– Current driven magnetic circuit =f(I)

– Voltage driven electric circuit I=f(U)

– Comparison between equivalent circuit and finite element method


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Circuits & phasors

• Magnetisation– Resistance R1 and inductance L1

of primary coil– Leakage and mutual inductance

Lσ+Lm=L1

– Core losses Rm– Complex current Io and

magnetising flux Ψm• Magnetically coupled

– Secondary circuit– EMF E2=jωΨm

• Electrically loaded– Max P2 power I2=E2/(Z2+R)– Primary current I1=I2+Io

Io

IoR

IoLΨm

E2 I2

I1


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Equivalent circuit

• The resulting equivalent system according to equations• Corresponding components in phasor diagram

201 nIII

222111122222 jXRIjXRnInUjXRIEU

L3 – Equivalent Circuits · L3 – Equivalent Circuits EIEN20 Design of Electrical machines, IEA,...

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