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Practical Feedback-loop Design of Bus Converters Supplying Regulated Voltage to DC-Input-Port Converters 1

http://gsep.uc3m.esGSEPU C 3 M

Practical Feedback-loop Design of Bus Converters Supplying Regulated Voltage to DC-Input-Port Converters

Power Electronics Systems Group (GSEP)

Avda. Universidad, 30 - 28911- Leganés – Madrid - SPAIN

http://gsep.uc3m.es

E-mail: [email protected]

Dr. Marina Sanz IEEE member (PELS, IES and IEd)Associate Professor



Madrid

Leganes

Polytechnic SchoolLeganes Campus

GSEP is a research group of Carlos III University of Madrid



Outline

Lesson 1: Stability of Dc Power

Distribution Systems

Lesson 2: Feedback-loop design considerations

for the BUS converter

Lesson 3: Complete characterization of the

input impedance of a DC-input-Port

Converter

Lesson 4: Input impedance estimation of

commercial DC-input-Port Converters



Outline

Trends in Dc Power distribution systems

Arquitectures of main applicationsMain challenges

Interaction between cascade converters

Constant power load effectEquivalent small-signal circuit



Outline







Centralized architecture

Before1990

1990Since2000

Single power converter delivers power to the loads through multiple outputs

Telecom application

Power processing technology, thermal management, control, protection, etc. is integrated into a single unit

Can be purchased or manufactured as a stand-alone system

Customized design, meaning long time-to-market and lack of flexibility

Failure of the converter means failure of the whole system.

Static and dynamic regulation of DC voltage is poor.

PROS CONS



Distributed architecture

Before1990

1990Since2000

Telecom application

The loads are supplied by a number of small power converters which are distributed throughout the system to

perform the power processing close to the load

Standardizatio n (commercial off-the-self (COTS) converters) Reduce time-to-market and development cost

Redundancy Improve reiliability On-line replacement (hot-swapping ) Maintenance in non-interrupting way Decoupling between load and source dynamics Load can be supplied with high dynamic

response

AC Input AC/DCconverter

Bus -48 V

Load

Load

Load....

+ 5 V

+ 3.3 V

+ 1.8 V

IsolatedDC/DC

Battery

IsolatedDC/DC

IsolatedDC/DC



Distributed architecture

Before1990

1990Since2000

Telecom application

Isolation is provided by the bus converter

Non-isolated converters are tied to the intermediate bus and supply each load (Point-of-load converters)

An isolated bus converter creates an intermediate regulated or unregulated (12 V) bus from the main (48 V) bus



Small-Scale Power System for Telecommunication Application

AC input

AC/DCFront-endconverter

Bus -48 VDC/DC

Isolated busconverter

Bus 12 V

Load

Load

Load....

+ 2.5 V

-3.3 V

+ 5 V

DC/DCPoint of Load

DC/DCPoint of Load

DC/DCPoint of Load



Small-Scale Power System for Telecommunication Application

AC input


Bus -48 VDC/DC


Bus 12 V

Load

Load

Load....

+ 2.5 V

-3.3 V

+ 5 V

DC/DCPoint of Load

DC/DCPoint of Load

DC/DCPoint of Load

Development of behavioral models that only model the input

and the output suitable not for converter analysis but for

system analysis

0.5 1 1.5 2 2.5 3 3.540

45

50

55

Time (ms)Time (ms)

ModelsimulationMeasured

Vol

tage

(V)

1 2 3

4

6

8

10

Cur

rent

(A)

Time (ms)

Predict system instability

ModelsimulationMeasured

+-

+

vo

-

io

io·HiY i

ii

vi

+

-

vi·Go

Zo



Power distribution in High-Power Energy Harvesting System

http://www.wedgeglobal.com/en/waveenergy

Wave Energy

Canary Islands (SPAIN)



Wave Energy Power Distribution System

Wave power

absorber

Wave Energy is used for grid and self-consumption since autonomus system must fed auxiliary systems

Sea level

Floating buoy

Upcoming wave

generator and drive

DC Bus

Grid converter

DC - DC

Bidirectional converter

1000 V

320 V

Energy storage

(batteries / supercap)

DC - DC

Auxiliary DC Bus

400 V

Auxiliary

services

Auxiliary boost

converter

DC – DCor

DC - AC

Cbus

DC - DC

DC - AC

DC or AC grid

Constant power

“Instrumentation”“Communications”

PumpCompressor



More Electric “Transport”

optimize performance

decrease operating and maintenance costs

increase dispatch reliability

Reduce environmental impact

Multiple primary energy sources types and multiple electric loads

Network of power electronics converter is required

28VDC

270 VDC

3Ø AC Loads115 VAC

400Hz

High-Voltage

Battery

DC /DC Converter

Fuel Cell

Battery

DC /DC

High-Voltage DC Loads

3Ø

PLMU

DC LoadsAuxiliary Power Unit

EngineStarter /Generator

DC/ACconverter

AC/DCBidirectional

DC/DC

DC/DC



Bidirectional converter

DC - DC

New challenges in stability of Power Distribution Systems

AC input


Bus -48 VDC/DC


Bus 12 V

Load

Load

Load....

+ 2.5 V

-3.3 V

+ 5 V

DC/DCPoint of Load

DC/DCPoint of Load

DC/DCPoint of Load

generator and drive

DC Bus

Grid converter

320 V

Energy storage (batteries / supercap)

DC - DC

Auxiliary DC Bus

400 V

Auxiliary boost converter

DC – DC

or

DC - AC

CbusDC - DC

DC - AC

DC or AC grid Multiple

converters

interacting

↓COMPLEX

DYNAMIC

BEHAVIOR

28VDC

270 VDC

3Ø AC Loads115 VAC

400Hz

High-Voltage

Battery

DC /DC Converter

Fuel Cell

Battery

DC /DC

High-Voltage DC Loads

3Ø

PLMU

DC LoadsAuxiliaryPowerUnit

Engine Starter /Generator

DC/ACconverter

AC/DCBidirectional

DC/DC

DC/DC



New challenges in stability of Power Distribution Systems

AC input


Bus -48 VDC/DC


Bus 12 V

Load

Load

Load....

+ 2.5 V

-3.3 V

+ 5 V

DC/DCPoint of Load

DC/DCPoint of Load

DC/DCPoint of Load

Use of commercial off-the-self (COTS) converters Not enough manufacturer data

to parameterize a model

AC/DC

Converter

DC/DCIsolated

bus converter



Outline







AC input


Bus -48 VDC/DC

Isolated BUSconverter

Bus 12 V

Load

Load

Load....

+ 2.5 V

-3.3 V

+ 5 V

DC/DCPoint of Load

DC/DCPoint of Load

DC/DCPoint of Load

Main goal for BUS converter control design

Feedback-loop design for Small-signal Stability of DC bus

DC BUS VoltageDC BUS Voltage

Load currentLoad current

Load voltageLoad voltage

DC bus current DC bus current



Behavior at input port of unregulated DC/DC power converters

A DC/DC converter is an electronic circuit that consists on a chopper and a low-pass

filter that supplies DC voltage to the load

VGV0

+

-

L

CR0

Switching converter (continuous conduction mode)

iG

T

T

TT

Td ON

OFFON

ON =+

=TON TOFF

T

ON OFF ON OFF

drivingsignal

Filter

T·D

T

VG

1

fs kHz

Filter Gain

0Hz T·D

T

vo

Vi



Behavior at input port of unregulated DC/DC power converters

VGV0

+

-

L

CR0


iG

vG

1 : dIG

R V0

+

-

L

C

Filter

At low frequency, un unregulated converter behaves as a positive resistor at its input port

Low frequency

Zi 1 : d

T

T

TT

Td ON

OFFON

ON =+

=TON TOFF

T

ON OFF ON OFF

drivingsignal



Behavior at input port of regulated DC/DC power converters

At low frequency, a regulated DC/DC converter behaves as a constant power load (CPL)

Switching converter

At frequencies lower than the bandwidth of the feedback-loop, due to the action of the control (perfect tracking of the reference) Vo=cte→ Po=cte Pi=cte (power balance)

If the input voltage increases, the input current decreases and vice versa

Gv

Gi

GI

GV

iG

VGV0

+

-

L

CR0

Vref+

- Vfb=β·V0

+

-Modulator

Compensator

Sensor



Other DC-input port converters: Inverter

S3

S4

D3

D4

S1

S2

D1

D2

S5

S6

D5

D6

+

VG

-

AiA B

iB CiC

Switching block

+

VG

-

Three-phase DC/AC converter



Behavior as CPL of other DC-input port converters

Control Reference generator

Compensator

a

b

c

d

q

Modulator

pref

qref

Id_ref

Iq_ref

+

+

-

-

id

iq

a

b

c

d

q

VG

Switching block

Three-phase DC/AC converterwith dq control (grid-tied inverter)

Compensator

Sen

sor

Sen

sor

Sen

sor

Reference Active power

Reactive power

RFT

RFT

RFT=Reference FrameTransformation

At frequencies lower than the bandwidth of the

feedback-loop, due to the action of the control

(perfect tracking of the reference)

Po=cte Pi=cte

At low frequency, a regulated DC/AC converter behaves as a CPL



The small-signal model of a constant power load

In small-signal variations the constant power load behavior can be modeled as a negative resistor that

depends on the operating point

The worst case regarding system stability should be considered as operating point for the design

iv

ii

1iI

1iV

+

-inv

ii

1cplR−Q1

iv

ii

2iI

2iV

Q2+

-inv

ii

2cplR−



Effect of negative load resistor

A negative resistor load is unstable!!!

A simple dynamic system such as LC filter is unstable with negative resistor since the filter damping is negative

Resistor voltage

Resistor voltage

Ω

Ω



Experimental validation of the small-signal behavior of DC/DC converter at the input port

Setup for time domain

Setup for frequency sweep

C=100mF

0.181Ω/40.6 µH

zo

Vdc

Power

supply

dc-dc

Converter

Zi

+

vbus

-

i i

Rload

Frequency response analyzer

Oscilloscope

zo

Vdc

Power

supplydc-dc

Converter

R1load

R2load

CH2

+

-

CH1

CH4

+

-

CH3

C=100mF

0.181Ω/40.6 µH

Long-wire effects



COTS: Unregulated DC/DC converter

-40

-30

-20

-10

0

10

20

30

40

10 100 1000 10000 100000

-100

-80

-60

-40

-20

0

20

40

10 100 1000 10000 100000

Frequency (Hz)

Ma

gn

itu

de

(Ω

)

2A

7A

7A

2A

Frequency (Hz)

Ph

ase

(˚)

IB050E120T32N1-00

INPUT CURRENT

OUTPUT CURRENT

INPUT VOLTAGE

OUTPUT VOLTAGE

Positive resistor at low

frequency



COTS: Regulated DC/DC converter

-250

-200

-150

-100

-50

0

50

100

10 100 1000 10000 100000

Frequency (Hz)

7A

Ph

ase

(˚)

2A

-40

-30

-20

-10

0

10

20

30

40

10 100 1000 10000 100000

Frequency (Hz)

Ma

gn

itu

de

(Ω

)

2A

7A

HPQ-12/25-D48

Negative resistor at low

frequency

INPUT CURRENT

OUTPUT CURRENT

INPUT VOLTAGE

OUTPUT VOLTAGE

Play video



AC input


Bus -48 VDC/DC

Isolated BUSconverter

Bus 12 V

Load

Load

Load....

+ 2.5 V

-3.3 V

+ 5 V

DC/DCPoint of Load

DC/DCPoint of Load

DC/DCPoint of Load

Analysis of power system stability

Vi R

+Vo-

+Vbus

-Sourceconverter

LOAD converter

R

+Vo-LOAD

converter

A simple system consisting on cascade converters should be analyzed



Equivalent model for small-signal stability analysis

Small-signal system-level stability can be well explained by making use of small-signal impedance-

based models

Closed-Loop output impedance, Zo(s) Closed-Loop input impedance, Zi(s)

Vi R

+Vo-

+Vbus

-Sourceconverter

LOAD converter

Vi Zi(s)

+Vbus

-

Zo(s)SourceConverter

LoadConverter



X re (s)

Loop Gain =200Loop Gain = 10

Stable Oscillation

Gain increase

Stability of linear dynamic systems

G (s)

H(s)

-+X i (s) X O (s)X e (s)

Control to output transfer loop gainor Loop Gain

“The stability of the closed loop system is determined from the control to output transfer function, T(s) = G(s) ⋅⋅⋅⋅H(s) or open loop gain or simply loop gain ”

)(1)(

)()(1)(

)()(

sT

sG

sHsG

sG

sX

sX

i

O

+=

⋅+=

It is supposed to be a negative feedback structure but there is a“dark side”



The “dark side” of the negative feedback

• G and H blocks depend on frequency.

• At the frequency f-180 , the phase of loop gain is -180º.⇒ Positive feedback!!!!

• If at this frequency (f-180 ) the module of the loop gain is equal to 1, the oscillation conditions are finally complied: 1)2()2( 180180 −=⋅⋅⋅ −− fjHfjG ππ

∞→−

=⋅+

==−−

− 111180

180

180

G

HG

G

V

vCL

ffref

Of

Closed –loop transfer function at f-180

Infinite gain means that without any input, an output voltage is obtained

Negative feedback The oscillation amplitude progressively decreases.

1)()( º180º180 <⋅ ∠∠ ωω jHjG

Positive feedbackExponential growth of the oscillation

1)()( º180º180 >⋅ ∠∠ ωω jHjG

Stable

UnstableG(s)

H(s)

-+

G(s)

H(s)

-+



General Criterion for stability of linear dynamic systems

Nyquist Criterion

G (s)

H(s)

-+X i (s) X O (s)X e (s)

Re T(s)

Im T(s)

Stable

(–1 + 0j)

Re T(s)

Im T(s)

(–1 + 0j)

Unstable

X re (s)

)(sT

)(1)(

)()(1)(

)()(

sT

sG

sHsG

sG

sX

sX

i

O

+=

⋅+=

The system is stable if T(s) does not encircled point -1+0j

Control to output transfer loop gain

or Loop Gain



Nyquist Criterion for small-signal stability of the converter interaction

Nyquist Criterion

G (s)

H(s)

-+X i (s) X O (s)X e (s)

X re (s)

)(

)()(

sZi

sZosT =Vi Zi(s)

+Vbus

-

Zo(s)Source Load

Minor Loop Gain

Nyquist criterion should be satisfied by small-signal impedance ratio T(s)

)(1)(

)()(1)(

)()(

sT

sG

sHsG

sG

sX

sX

i

O

+=

⋅+=

)()(

1

1

)()(

)(

)(

)(

sZi

sZosZosZi

sZi

sVi

sVbus

+=

+=

Control to output transfer loop gain

or Loop Gain



Stability Criteria for Dc Power Distribution systems

)()(

1

1

)()(

)(

)(

)(

sZi

sZosZosZi

sZi

sVi

sVbus

+=

+=

)(

)()(

sZi

sZosT =Loop Gain

Nyquist Criterion

Re T(s)

Im T(s)

Stable

(–1 + 0j)

Middlebrook Criterion is the most conservative but is the simplest approach since only it only

takes into account the magnitude of both impedances

1)(

)()( <=

sZi

sZosT

Stable if



System 1:

UNREGULATED SOURCE CONVERTER

UNREGULATED LOAD CONVERTER

Example of system stability analysis: Main diagram blockSystem 2:


REGULATED LOAD CONVERTER

DC/DC Converter

Unregulated(Buck 1)

Voltage-ModeControl

DC/DC Converter

Unregulated(Buck 2)

DC/DC Converter

Unregulated(Buck 1)

DC/DC ConverterRegulated(Buck 2)



s

Vbus

+

-

L

C

vgs

ViV0

+

-

L

CR0

Vref+

- Vfb=β·V0

+

-Modulator

Compensator

d

s

Vbus

+

-

L

C

vgs

ViV0

+

-

L

CR0

System 1:



System 2:



Example of system stability analysis: The converters



Behavior of an unregulated converter at its output port

ViV0

+

-

L

C


ii

Zo

The small-signal model at the output port is the output filter of the converter

ioR0

vi

1 : dIi

V0

+

-

L

CR0

No variations of the input voltage

vi

1 : dIi

V0

+

-

L

CR0



Output impedance: source converter Input impedance: load converter

Unregulated Regulated

ZicplR−Zo

Zi

0

-20

-40

20

MagnitudZof

1 10 100 1000 10000Frequency (Hz)

0

-50

-100

50

100FaseZof

0

5

10

15

20

25

MagnitudZic

1 10 100 1000 10000Frequency (Hz)

0

-50

-100

50

100FaseZic

0

-20

-40

20

MagnitudZic

1 10 100 1000 10000Frequency (Hz)

0

-50

-100

-150

-200

50

100FaseZic

90º-90º

Magnitude_Zo Magnitude_Zi Magnitude_Zi

increasing

decreasing

0º

-90º

cte

decreasing

decreasing

-90º

Phase_Zo Phase_Zi Phase_Zi

-180º

cte

Input and output impedances

Buck 1(Source) Buck 2

(Load)

Buck 2(Load)

2d

R



(–1 + 0j)(–1 + 0j)

Constant Power Load Behaviour makes possible T encircles the point (-1+0j) giving as a result system instability

0

-20

-40

20

MagnitudZof FaseZic_9A MagnitudZic_1A

1 10 100 1000 10000Frequency (Hz)

0

-50

-100

50

100

FaseZof FaseZic_9A FaseZic_1A

↑↑↑↑Io

↑↑↑↑Io

↑↑↑↑Io

0

-20

-40

20

MagnitudZof MagnitudZic_1A MagnitudZic_9A

1 10 100 1000 10000Frequency (Hz)

0

-50

-100

-150

-200

50

100

FaseZof FaseZic_1A FaseZic_9A

↑↑↑↑IoZo

Zi1

Zi2

ZoZi2

Zi1

0

30

60

90

120

150

180

210

240

270

300

330

0.5

1

1.5

0

30

60

90

120

150

180

210

240

270

300

330

0.5

1

1.5

)(

)()(

sZi

sZosT =

0º

90º

-90º

0º

-180º

90º

-90º

Stability CriterionSystem 1:



System 2:



UNSTABLE STABLE



0

2

4

6

8

10io

0

0.5

1

1.5

2

2.5

3Vo

0.02 0.025 0.03 0.035 0.04 0.045 0.05Time (s)

0

2

4

6

8

10

12Vbus

UNSTABLE

0

2

4

6

8

10Io

0

0.5

1

1.5

2

2.5

3Vo

0.02 0.025 0.03 0.035 0.04 0.045 0.05Time (s)

0

2

4

6

8

10

12Vbus

Current Load of Buck 2 (Io)

Output Voltage (Vo)

Bus Voltage (Vbus)

STABLE

System 1:



Transient responseSystem 2:



Current Load of Buck 2 (Io)

Output Voltage (Vo)

Bus Voltage (Vbus)

The system is unstable when the converter load is regulated, and so behaves as constant power load



Summary

The interaction of two cascade converters should be analyzed

Middelbrook Criterion is the simplest approach

DISTRIBUTEDARCHITECTURES

USE OF COMMERCIALCONVERTERSWITH UNKNOWNPARAMETERS

Vi R

+Vo-

+Vbus

-

LOAD converter

SourceconverterSourceconverter

Design close-loop output impedance of source converterCalculation of the close-loop input impedance of the load converter

AC inputAC/DC

Front-endconverter

Bus -48 VDC/DC

Isolatedbusconverter

Bus 12 V

Load

Load

Load....

+ 2.5 V

-3.3 V

+ 5 V

DC/DCPointofLoad

DC/DCPointofLoad

DC/DCPointofLoad



Lesson 2: Feedback-loop design considerations for the BUS converter

Outline

Refreshing basic concepts

Complete characterization of the output port of the BUS converter

Output impedance shaping below the input impedance envelope




Outline






vinThe input voltage can change

The output current can change

iO

Input Perturbation

Output Perturbation

+-

LOAD

Power converter

Measured magnitude

Reference

control magnitude -

+Control

Zero error in steady state (perfect tracking of the reference)

Feedback loop objective



Output Voltage Under Load Current & Input Voltage Steps

vin v0

s

+

-

L

C R0

vgs

Input PerturbationOutput Perturbation

The input voltage can change


iO

vin

v0

iO

Input voltage step (positive increment)

Output current step (positive increment)

dDuty cycle

(control variable)

VO VO VO

Overshoot

Undershoot

Audio-susceptibilityVariation of the output voltage due to the input voltage change

Output impedanceVariation of the output voltage due to the output current change



The nature of DC-DC converters and stability theory

vG

+

-Va

VREF

+- +

-

LINEAR

LINEAR

NON LINEAR

NON LINEARSwitching converters are NON LINEAR SYSTEMS. However,…

Q

Considering the average value

And small variations around the operating point

iL

iL

Then, the converter can be considered linear

Nyquist stability criterion to design the feedback loop

12



Equivalent Linear System Dynamic modelling of the switching converter:

CompensatorPlantGVd

Sensor

+

-Modulator OvREFv d

FBv

ERRv

Averaging

Linearization & perturbation

1

2

Block diagram of the equivalent linear system

A linear model of the converter is obtained d

vG O

vd ˆˆ

=

G(s)

H(s)

J. A. Oliver, J. A. Cobos, J. Uceda, M. Rascon, C. Quinones “ Systematic approach for developing large-signal averaged models of multi-output PWM converters”. IEEE Proc. PESC, vol.2, pp. 696 - 701, 2000

A. Kislovski, R. Redl, and N. Sokal, Dynamic Analysis of Switching-Mode DC/DC Converters,New York: Van Nostrand Reinhold, 1996.

VG V0

+

-

L

CR0

Vref+

- Vfb=β·V0

+

-

ModulatorCompensator

Switching converter

d

Sensor



Nyquist Stability Criterion

Compensator Plant

Sensor

+

-Modulator OvREFv d

FBv

ERRv

R(s) Gvd(s)M(s)

G(s)

Control to output transfer function ( loop gain ):

H(s)(s)GM(s)R(s)(s)V

(s)V)()·(GT(s) vd

err

FB ⋅⋅⋅=== sHs

The converter is stable if T(s) does not encircled point -1+0j

)()(1

)(

)(

)()(

sHsG

sG

sV

sVsCL

ref

O

⋅+==

Re T(s)

Im T(s)

Stable

(–1 + 0j)

H(s)Closed loop transfer function

Design the compensator R(s) to T(s) meet Nyquist criterion



Nyquist and Bode Diagrams

If the converter is stable, the loop gain T(s) should be far enough of point -1+0j

Or, the module of the loop gain T(s) should be enough lower than 1 (0dB) when the phase of the loop gain T(s) is -180º

dBsT )(

Re T(s)

Im T(s)

)(sT

Nyquist diagramBode diagram

1)()( º180º180 <⋅ ∠∠ ωω jHjG



Design criteria for Stability

0dBdB

sT )(dB

sT )(

-180º

fC

Crossover frequency, fC

Phase Margin, PM

@ fc, |G(s)⋅H(s)|=1

The phase margin determines the stability of the

regulated converter

-180º+PM = T(jωc)

0 º)(sT

Bode diagram

)(sT

Bandwidth

fc=2kHz

PM=45º



fc=2kHz

PM=30º PM=45º

PM=10º

PM=1º

PM=45º

PM=30º

PM=10º

PM=1º

Reducing the PHASE MARGIN provokes a closed loop transfer function with complex poles, which causes a bigger oscillation in the transient response.

Increasing the PHASE MARGIN provokes a closed loop transfer function with real poles, which turns into a smaller oscillation in the transient response.

Phase Margin and Transient Response

The phase margin has

influence on the transient

response of the regulated

converter



Complete Block diagram of the equivalent linear system

Compensator

Sensor

+

-Modulator

OvREFv d

FBv

ERRv ++

G

Ovg v

vG

ˆ

ˆ=

Gv

load

O

i

vZ =O

loadi

Audio susceptibility : relationship of the input voltage variations to the output voltage variations).

Output impedance : relationship of the output current variations to the output voltage variations).

Perturbations

Input voltage

Load Current

Switching converter

VG V0

+

-

L

CR0

Vref+

- Vfb=β·V0

+

-Modulator

Compensator

d

Sensor

+

iload(s)

PlantGVd




Outline






Output impedance of the source converter

VG R

+Vo-

+Vbus

-

DC/DC

Converter

(SOURCE)

DC/DC

converter

(LOAD)

Closed-Loop output impedance, Zo(s) Closed-Loop input impedance, Zi(s)

Sourceconverter

Zo (s), terminated

Zo(s), unterminated

Load Converter

Zo(s), unterminated

• The unterminated small signal impedance takes into account the steady state, but leaves outside the small signal effect of the load resistor because in a cascaded connection it will not appears.

• The unterminated impedance considers the output capacitor, because it belongs to the DC-DC converter

• The converter should be at the operating point (consider dc current source at the output )

Zo (s), terminated

To analyze the system stability, the unterminated output impedance must be used

• The terminated impedance takes into account the load resistor that imposes the steady estate

VG



Unterminated versus Terminated output impedance (Open-Loop)

Sourceconverter

Zo (s), terminated

Zo(s), unterminated

Load ConverterVG

0

-10

-20

-30

-40

10

20

amp(Vo_terminated) amp(Vo_terminated)_(Open_Loop_Output_impedance_source_converter_unterminated)

100 500 1000 5000 10000

Frequency (Hz)

0K

-0.05K

-0.1K

-0.15K

0.05K

0.1K

phase(Vo_terminated) phase(Vo_terminated)_(Open_Loop_Output_impedance_source_converter_unterminated)

The unterminated Zo

magnitude increases

significantly

Zo terminated

Zo unterminated



Equivalent linear circuit for analytical calculationsDynamic modelling of the switching converter:

Averaging

Linearization & perturbation

1

2

A linear model of the converter is obtained

J. A. Oliver, J. A. Cobos, J. Uceda, M. Rascon, C. Quinones “ Systematic approach for developing large-signal averaged models of multi-output PWM converters”. IEEE Proc. PESC, vol.2, pp. 696 -701, 2000

A. Kislovski, R. Redl, and N. Sokal, Dynamic Analysis of Switching-Mode DC/DC Converters,New York: Van Nostrand Reinhold, 1996.

Injected-Absorbed Current Equivalent linear Circuit

6 coefficients A and B

which value depends

on topology and

conduction mode

VG V0

+

-

L

CR0

Switching converter

T

T

TT

Td ON

OFFON

ON =+

=TON TOFF

T

ON OFF ON OFF

drivingsignal

)(ˆ)()(ˆ)()(ˆ)()(ˆ svsCsvsBsdsAsi Gioiii ⋅+⋅−⋅=

)(ˆ)()(ˆ)()(ˆ)()(ˆ svsCsvsBsdsAsi Gooooo ⋅+⋅−⋅=

+-

CiAi Bi

Co

Ao Bo Rc

C

d(s)

)(svo

)(svi

ZLo

ad(s

)

i i(s)

iload(s)

Zc(s)

io(s)



Application example: Buck converter in CCM (I)

Vin V0

+

-

L

CR0

1 Input current and output current as function of inductance current

2 Inductor current as a function of the rest of specified values (Vin and vo)

i iio

3 Relationships



Application example: Buck converter in CCM (II)

Vin V0

+

-

L

CR0

i iio

+-

CiAi Bi

Co

Ao Bo Rc

C

d(s)

)(svo

)(svi

ZLo

ad(s

)

i i(s)

iload(s)

Zc(s)

io(s)



Analytical calculation of the open-loop output impedance

CIRCUIT

BLOCK DIAGRAM

+-

CiAi Bi

Co

Ao Bo Rc

C

d(s)

)(svo

)(svi

ZLo

ad(s

)

i i(s)

iload(s)

Zc(s)

io(s)

)(sAOd + )(sZL

-Ov

)(sBO

Oi ++

loadi

)()(1

)(ˆˆ

)(,sZsB

sZ

i

vsOLZo

LO

L

load

O

⋅+==

)(sZL Ov

)(sBO

Oi ++

loadi

-1



Unterminated & Terminated open-loop output impedance

+-

CiAi Bi

Co

Ao Bo Rc

C

d(s)

)(svo

)(svi

Zou_ol

ZLo

ad(s

)

i i(s)

iload(s)

Zc(s)

Zo_ol

Zou,ol = Open-loop “unterminated” output impedance

Zot,ol = Open-loop “terminated” output impedance

Open-loop & terminated

Open-loop & unterminated

)()()( , sZcsZsZ OLOuL ==

)()(

)()()()( , sZsZ

sZsZsZsZ

Loadc

LoadcOLOtL +

⋅==

• The terminated impedance takes into account the load resistor that imposes the steady estate.

• The unterminated small signal impedance takes into account this steady state, but leaves outside the small signal effect of the load resistor because in a cascade connection it will not appears.

• The unterminated impedance considers the output capacitor, because it belongs to the DC-DC converter

• To analyze the stability, the unterminated output i mpedance must be used

)(sAOd + )(sZL

-Ov

)(sBO

Oi ++

loadiOutput impedance

)()(1

)(ˆˆ

)(,sZsB

sZ

i

vsOLZoG

LO

L

load

Ovi ⋅+

===



Basic calculation of the close-loop “unterminated” output impedance using single control loop: Voltage-Mode Control (I)

Switching converter (buck converter in CCM & Volta ge Mode control)

Compensator

Sensor

+

-Modulator

OvREFv d

FBv

ERRv ++

G

Ovg v

vG

ˆ

ˆ=

Gv

load

Ovi

i

vZG == O

loadi

G(s)

H(s)

+PlantGVd

0

0

CoCi

loadi

VGIo



Basic calculation of the close-loop “unterminated” output impedance using single control loop: Voltage-Mode Control (III)

)(1

)(

)()()()(1

)(ˆˆ

)()( __ sT

sGvi

sHsRsMsGvd

sGvi

i

vsZsG

load

Ocloclvi +

=⋅⋅⋅+

===

Compensator

Sensor

ModulatorOvdERRv +

load

Ovi

i

vZG == O

loadi

G(s)

H(s)

+PlantGVd

-1R(s) M(s)

(s)

Where Gvi is the open-loop output impedance.To calculate the unterminated closed-loop output impedance, Gvi must be computed accordingly.

To determine the stability, the unterminated closed loop output impedance (without load resistor)

must me used

)()(1

)(ˆˆ

)()( _ sZsB

sZ

i

vsZsG

LO

L

load

Oolovi ⋅+

===



Open-Loop versus Closed-Loop unterminated output Impedance

The output impedance of the source converter can be properly designed by means of the loop gain T(s)

Zou,OL

T

Zou,CL

f<<f cZou,CL≅≅≅≅Zo,OL /T

f>>fCZou,CL≅≅≅≅Zou,OL

Zo

(Buck converter)

)(, sZ OLouViV0

+

-

L

C

ii

R0

)(1

)()( ,

, sT

sZsZ OLou

CLou +=

fc

0dB




Outline






Design of the feed-back loop of source converter

Vi R

+Vo-

+Vbus

-

dc-dc

Converter

(SOURCE)

dc-dc

converter

(LOAD)

Source converter Load Converter



Application design example

Narrow bandwidth converter(slow converter)

Converter with fast compensation(fast converter)

Vi R

+Vo-

+Vbus

-

dc-dc

converter

(Buck 1)

dc-dc

converter

(Buck 2)

Magnitude Buck 1 (Source) Buck 2 (Load)

Input voltage 12 V 5V

Output voltage 5 V 1.5 V

Switching

frequency20 kHz 100 kHz

Power 20 W 20 W

Input capacitance 100 µF / 5 mΩ 100 µF / 5 mΩ

Output capacitance 600 µF / 10 mΩ 150 µF / 5 mΩ

Inductance 150 µH / 10 mΩ 30 µH / 1 mΩ

Bandwidth 100 Hz 20 kHz



Regulator design of source converter

Gvd

Magnitude (dB)

Phase (degrees)

Open Loop

Voltage Regulator

Phase Margin= 100 degrees

Cross frequency= 100Hz

R11 = 100 kOhmR1 = 345.792 kOhmR2 = 158.534 kOhm C1 = 4.05366 nF C2 = 11.3987 nFC3 = 39.4159 nF Ra = 500 Ohm Rb = 750 OhmVref = 3 V



Regulator design of load converter

Gvd

Magnitude (dB)

Phase (degrees)

Open Loop

Voltage Regulator

Phase Margin=45 degrees

Cross frequency=20 kHz

R11 = 10 kOhmR1 = 1.30503 kOhmR2 = 277.27 kOhm C1 = 2.07178 nF C2 = 84.4722 pFC3 = 11.0239 pFRa = 33.4111 OhmRb = 66.8224 Ohm Vref = 1 V



Both converters are individually stable ... ... but the system is unstable

Transient response

Vi

R

+Vo-

+Vbus

-

dc-dc

converter

(Buck 1)

fsw=20 kHz

dc-dc

converter

(Buck 2)

fsw=100kHz

Output voltage source converter (fsw=20kHz)

Output voltage load converter (fsw=100kHz)

Dc bus voltage

Output voltage

Vi

R1

+Vo-

dc-dc

converter

(Buck 1)

fsw=20 kHz

Vi R

+Vo-

dc-dc

converter

(Buck 2)

fsw=100kHz



Stability criterion of the power system

Vi R

+Vo-

+Vbus

-

dc-dc

converter

(Buck 1)

dc-dc

converter

(Buck 2)

Unterminated close-loop output impedance(source converter)

close-loop input impedance(load converter)

+

-busv

ii

cplR−

Close-loop input impedance of load converterbelow crossover frequency

Magnitude Buck 1 (Source) Buck 2 (Load)

Input voltage 12 V 5V

Output voltage 5 V 1.5 V

Switching

frequency20 kHz 100 kHz

Power 20 W 20 W

Input capacitance 100 µF / 5 mΩ 100 µF / 5 mΩ

Output capacitance 600 µF / 10 mΩ 150 µF / 5 mΩ

Inductance 150 µH / 10 mΩ 30 µH / 1 mΩ

Bandwidth 100 Hz 20 kHz

System is unstable synce Nyquist criterion is not satisified )()( sZisZo >



Vi(s) Zi(s)

+

Vbus(s)-

Zo(s)Source Converter(Buck 1)

Load Converter(Buck 2)

)()(

1

1)()(

)()()(

sZi

sZosZosZi

sZi

sVin

sVbus

+=

+=

)()( sZisZo <

Design the regulator of the source converter not only to satisfy stability criterion of the regulator but

also to satisfy the impedance criterion

converter load i

converter load i

P

VR

2

=

Design the Compensator Considering the Impedance Criterion

Compensator



New compensator design of the source converter (Bus converter)

Gvd

Magnitude (dB)

Phase (degrees)

Open Loop

Voltage Regulator

Phase Margin=45 degrees

Cross frequency= 2.5 kHz

R11 = 100 kOhmR1 = 6.55793 kOhmR2 = 235.502 kOhm C1 = 2.40826 nF C2 = 1.08967 nFC3 = 71.4598 pF Ra = 500 Ohm Rb = 750 OhmVref = 3 V



Stability criterion of both power systems

Use the input impedance of the load converter as an envelope or mask

Set the design point to shape the output impedance of the source converter below the mask at any

frequency

Compensator design 1 Compensator design 2

)()( sZisZo <)()( sZisZo >


Cross frequency= 100Hz


Cross frequency= 2.5Hz

Stable stand-aloneconverter

Unstablesystem

Stable stand-aloneconverter

Stablesystem



Proper control design of the regulator assures system stability

Vi

R

+Vo-

+Vbus

-

dc-dc

converter

(Buck 1)

fsw=20 kHz

dc-dc

converter

(Buck 2)

fsw=100kHz

Vi

R1

+Vo-

dc-dc

converter

(Buck 1)

fsw=20 kHz

Vi R

+Vo-

dc-dc

converter

(Buck 2)

fsw=100kHz

Both converters are individually stable ... ... and the system is stable!!!

Output voltage source converter (fsw=20kHz)

Output voltage load converter (fsw=100kHz)

Dc bus voltage

Output voltage



Using software tool for automatic design of the compensator

Bus converter plant (source converter), Gvd

Bus converter loop gain, T

Load converter Input

impedance, Zi

Source converter

Unterminatted output

impedance, Zo

Solution MAP with feasible

designs for compensator

Feasible solutions

Analytical Input and output impedances of

many different DC/DC converters




Summary

Compensator

Sensor

Modulator +

G(s)

H(s)

+PlantGVd

R(s) M(s)(s)

Gvi=Zo,OL

-1

vo

iload

Calculate open-loop unterminated output impedance (injected-absorbed current equivalent linear model)

1

Calculate close-loop unterminated output impedance

2

Calculate compensator to MEET NYQUIST CRITERION IN STAND ALONE OPERATION AND IN THE CONVERTER INTERACTION

3

)(sAOd+ )(sZL

-Ov

)(sBO

Oi+

+loadi

)()(1

)(ˆˆ

)(,)(_sZsB

sZ

i

vsOLZosOLG

LO

L

load

Ovi ⋅+

===

Zin

Zo

)(1

)(_)()(_ _ sT

sOLGvisZsCLG clovi +

==

-180º-180º+PM = T(jωc)

0dB

)(sT

)()( sZcsZL = Only output capacitor as output load impedance

fc, crossover frequency



Lesson 3: Complete characterization of the input impedance of a DC-input-Port Converter

Outline

Input impedance of the feedback-regulated DC/DC converter

Input impedance of the feedback-regulated DC/AC converter

The effect of the feedforward




Outline






Basic concepts of the input impedance of regulated DC input port converters

)(

)(

si

svZ

i

ii =

Vi V0

+

-

L

CR0

Vref+

- Vfb=β·V0

+

-Modulator

Compensator

d

Sensor

iv

ii

iI

iV

+

-iv

ii

cplR−

ii

The simple small-signal model is not valid in the whole frequency rangeThe feedback loop shoul be studied

CPL behavior at low frequency

negative resistor

The constant powerload behavior is onlyvalid until cross-overfrequency of thefeedback-loop



Basics of calculation of the close-loop input impedance: Single loop Voltage Mode Control

Compensator

Sensor

+

-Modulator

OvREFv d

FBv

ERRv ++

i

Ovv v

vG =

iv

load

Ovi

i

vZG == O

loadi

G(s)

H

+PlantGVd

0

To calculate close-loop input impedance Zi,

it should be taken into account:

• The open-loop duty cycle to output voltage transfer

function (plant)

• The open-loop Audiosusceptibility

• The effect of the output voltage feedback-loop that

affects the duty cycle and hence the input current.

0

Vi V0

+

-

L

CR0

Vref+

- Vfb=β·V0

+

-Modulator

Compensator

d

Sensor

ii

Rv ModModRHX v ⋅−⋅= )(

i

ii i

vZ =



Step 1: Derive the small-signal model of the converter

+-

CiAi Bi

Co

Ao BoZs(s)

Rc

C

d(s)

)(svo

)(svi

Zi

ZL(s)

i i(s) i0(s)

The same small-signalmodel used to derive theanalytical output impedancecan be used



Step 2: Calculate the required transfer functions (I)

d

+-

CiAi Bi

Zi

i i(s)

Oviv

oiiii vBdACivi ⋅−⋅+⋅=Input current expression1



83

Step 2: Calculate the required transfer-functions (II)

d

+-

CiAi Bi

Gvd

Zi

i i(s)

+-

+-

Gvv +

-

Oviv

)()(1

)()(ˆ

ˆ)(

sZsB

sZsA

d

vsG

LO

LOOvd ⋅+

⋅==

)()(1

)()(ˆ

ˆ)(

sZsB

sZsC

v

vsG

LO

LO

i

Ovv ⋅+

⋅==

Open-loop Audiosusceptibility

Input current expression

dGvdvGvvv io ⋅+⋅=2

Output voltage expression

Plant

oiiii vBdACivi ⋅−⋅+⋅=1



oiiii vBdACivi ⋅−⋅+⋅=

84

Step 2: Calculate the required transfer-functions (III)

H

Mod Rvd

+-

CiAi Bi

Gvd

Zi

i i(s)

+-

+-

Gvv +

-

Oviv

-1

Input current expression1

)()(1

)()(ˆ

ˆ)(

sZsB

sZsA

d

vsG

LO

LOOvd ⋅+

⋅==

)()(1

)()(ˆ

ˆ)(

sZsB

sZsC

v

vsG

LO

LO

i

Ovv ⋅+

⋅==

Open-loop Audiosusceptibility

dGvdvGvvv io ⋅+⋅=2

Output voltage expression

Plant

ModHRvd vo ⋅⋅−⋅= )(3 Duty cycle expression



Voltage-Mode Control (VMC), closed-loop

)()( XvModHRvd

dGvdvGvvv

vBdACivi

ovo

io

oiiii

−⋅=⋅⋅−⋅=⋅+⋅=

⋅−⋅+⋅=(1)

(2)

(3)

(3)(2)

io

oio

vT

Gvvv

TvvGvvv

⋅+

=

−⋅+⋅=

1

)(

(4)

(4)(3)

i

i

vT

T

Gvd

Gvvd

XvT

Gvvd

⋅+

⋅−=

−⋅⋅+

=

1

)(1

(5)

iiiiii vT

GvvBv

T

T

Gvd

GvvAvCii ⋅

+⋅−⋅

+⋅−⋅+⋅=

11)(

(4)&(5)(1):

H

Mod Rvd

+-

CiAi Bi

Gvd

Zi

i i(s)

+-

+-

Gvv +

-

Oviv

H(s)(s)GMod(s)(s)R)()·(GT(s) vdv ⋅⋅⋅== sHs

Analytical derivation of the close-loop input impedance is quite complex and tedious

Analytical calculation of the close-loop input impedance in Voltage Mode Control (VMC)

-1

Input current expression



T

GvvBi

T

T

Gvd

GvvAiCii

vZ

i

ii

+⋅−

+⋅−⋅+

==

11)(

1

-180º

H

Mod Rvd

+-

CiAi Bi

Gvd

Zi

i i(s)

+-

+-

Gvv +

-

Oviv

-1

fc

Constant

Negative resistor

A regulated converter is not a negative resistor load (CPL) in the whole frequency rangeThe system can be stable with a proper design of the output impedance of the source converter

Voltage-Mode Control (VMC), closed-loop

Frequency response of the close-loop input impedance in VMC



Open-loop versus Closed-Loop Input Impedance in VMCBuck converter (CCM)

fcT

Zi_CL

0º

-180º

Zi_OL

Constant

“Low frequency”

+

-

inv

ii

cplR−

Open loop (T=0)

close loop

Positive resistorup to fres

Negative resistor(CPL) up to fc

+

-

ivii fres

“High frequency”

Open loop (T=0) close loopOutput

converterfilter at f higher

than fres

Output converter

filter at f higherthan fc

+

-

iv

1 : d

1 : dii

CLf res ⋅⋅⋅

=π2

1




Outline






Basic concepts of the feedback-regulated three-phase DC/AC converter

L

Carga

+

-

vO_A

iO

iC

iL_A

C

Ph A

L

Carga

+

-

vO_A

iO

iC

iL_A

C

Ph B

L

+

-

vO_C

iL_C

Ph CABC

Ph A iL_A

Ph B iL_B

Ph C iL_C

IL ref

iL measured

+

-

Sensingand

filteringPWM Modulator

compensator

It is not possible to force the currents iA, iB, iC, independently since they are tied by the expression : iA+ iB+ iC = 0

Vi

Neutral point

Neutral point

Voa

Vob

Voc

iL_A

iL_B

iL_C



Rotating Space Vector

The 3-ph instant values of voltage and currents can be represented as projections of a rotating

“space vector” over the axes of a 3D space (Park space)

a

b

X

abc coordinates

ω

a b c

xa

xc

xb

c

xa

xb

xc

• For 3-ph direct sequence magnitudes, X is a vector that rotates counterclockwise at a rotating speed ω, that is the frequency grid



a

b

c

X

α

X

β

Three stationary axes Two stationary axes Two rotating axes

abc αβ

Rotating Space Phasor

ω ω

Stationary Space Phasor

Stationary reference frame

α

X

β d-q

ω

Rotatingaxes

XdXq

Synchronous reference frame (SRF)

But the main concept deals with joint the reference system to the rotating phasor

(dq coordinates system)

Reference frame transformation (I)

A three-phase wire system of signals “a, b, c” fulfills a+b+c = 0 Map the system into a bi-dimensional plane

defined by orthogonal coordinates αβ



Three-phase instantaneous voltages and currents are seen as two DC quantities in the Synchronous

reference frame

Reference frame transformation (II)



PI

PI a

b

c

d

q

PWM

pref

qref

Id_ref

Iq_ref

+

+

-

-

id

iq

a

b

c

d

q

Vi

223

2

qd

qdd vv

vqvpi

+⋅+⋅

⋅=

223

2

qd

dqq vv

vqvpi

+⋅−⋅

⋅=

Modulating signals in dqcoordinates

Modulating signals in abccoordinates

Simple dq Control of Three-phase Grid-Tied DC/AC converter

( )qqdd ivivp ⋅+⋅⋅=2

3

( )qddq ivivq ⋅−⋅⋅=2

3

The control is based on “tracking” the output active and reactive power injected by the DC/AC converter to the grid

For a given 3-ph voltage generator, the instant active and reactive powers can be expressed on dq currents and voltages

The current references are obtained from the active and reactive power references

d



The average model of the output port of the

converter are coupleddq components coupling)

dd di ⋅2

3qq di ⋅

2

3

+

vi

-

ii

Vi

Neutral point

+

vi

-

ii

Ai

Bi

Ci

AdBd Cd

Average Model of Three-phase Grid-Tied DC/AC converter

di dV ⋅

qi dV ⋅



Small-Signal Model of the output port of Three-phase Grid-Tied DC/AC converter

+-

L r

Component d

+-

L r

Component q

+ -

+-

rsLiLvG

Vi qd

id +⋅

⋅

⋅⋅+⋅⋅= 1ˆˆ2

ˆmod_mod ω

rsLiLvG

Vi dq

iq +⋅

⋅

⋅⋅−⋅⋅= 1ˆˆ2

ˆmod_mod ω

di qiL ˆ⋅ωqi diL ˆ⋅ω

di vG

Vmod_mod ˆ

2⋅⋅ q

i vGV

mod_mod ˆ2

⋅⋅

VG rsL +⋅1

++

ωL

di

rsL +⋅1

+

-

qi

ωL

Power

stage

+-

REFdi _ˆ

+-

REFqi _ˆ

Compensator

(R)

dvmod_ˆmod2

GVi ⋅

VGmod2

GVi ⋅

qvmod_ˆ

dε

qε

Multiple-input multiple-output (MIMO) system is obtained making difficult the regulator design



dq Components Decoupling Concept

dεdRi VG rsL +⋅

1+

-

ωL

di

qi

PLANT

mod2G

Vi ⋅dvmod_ˆ

A single-input single-output (SISO) system is obtained in order to easy design of the compensator

rsLvG

Vi d

id +⋅

⋅

⋅⋅= 1ˆ

2ˆ

mod_mod

dεdRi VG rsL +⋅

1di

PLANT

mod2G

Vi ⋅dvmod_ˆ

+-

REFdi _ˆ

+-

REFdi _ˆ

rsLiLvG

Vi qd

id +⋅

⋅

⋅⋅+⋅⋅= 1ˆˆ2

ˆmod_mod ω



dq Components Decoupling by means of Feed-Forward Compensation

dεdRi VG rsL +⋅

1+

-

ωL

di

qi

PLANT

mod2G

Vi ⋅dvmod_ˆ

rsLiLvG

Vi qd

id +⋅

⋅

⋅⋅+⋅⋅= 1ˆˆ2

ˆmod_mod ω

Compensate the “undesired” behavior feeding-forward the plant with its opposite effect

rsLvG

Vi d

id +⋅

⋅

⋅⋅= 1ˆ

2ˆ

mod_mod

dεdRi VG rsL +⋅

1di

PLANT

mod2G

Vi ⋅dvmod_ˆ

+-

REFdi _ˆ

+-

REFdi _ˆ

+

+

ωLqi

FeedForward

Compensation



Feed-Forward Compensation of the input voltage

Use feed-forward compensation to remove the plant variations under input voltage perturbations

dεdRi

rsL +⋅1

di

PLANT

dvmod_ˆ+-

REFdi _ˆ

dεdRi VG rsL +⋅

1+

-

ωL

di

qi

PLANT

mod2G

Vi ⋅dvmod_ˆ+

-REFdi _

ˆ

+

+

ωLqi

FeedForward

Compensation

iV

mod

12

GVi

⋅

rsLiLvG

Vi qd

id +⋅

⋅

⋅⋅+⋅⋅= 1ˆˆ2

ˆmod_mod ω

( )rsL

vi dd +⋅⋅= 1

ˆˆmod_



Small-Signal Model of the output port of Three-phase Grid-Tied DC/AC converter with feedback and feedforward regulation

VG rsL +⋅1+

-

ωL

di

rsL +⋅1

+

-

qi

ωL

Power

stage

+REFdi _

ˆ

+

-

REFqi _ˆ

dε

di

-

dRi

ωL

+

+

qεqRi

qi

++

ωL

mod

12

GVi

⋅mod2

GVi ⋅

mod

12

GVi

⋅VGmod2G

Vi ⋅

dvmod_ˆ

qvmod_ˆ

Feedforward regulationFeedback regulation



Small-signal model of the input port of Three-phase Grid-Tied DC/AC converter

qqqqddddi iDdIiDdIi ˆ2

3ˆ2

3ˆ2

3ˆ2

3ˆ ⋅⋅+⋅⋅+⋅⋅+⋅⋅=Input current expression

+

vi

-

ii

dd dI ˆ2

3 ⋅⋅dd iD ˆ

2

3 ⋅⋅ qq dI ˆ2

3 ⋅⋅ qq iD ˆ2

3 ⋅⋅

Input impedance expression

i

ii i

vZ =



Simulation Example of Three-phase Grid-Tied DC/AC converter

Reference Frame

Transformation

Modulator

dq Current

components

Reference

generationCompensators

Active and reactive powercalculation

Input voltage

perturbation

i

ii i

vZ =

iiPower

Stage



Simulated Close-loop input impedance of three-phase inverter

0

5

10

15

20

25

30

35amp(Zi_promediado) amp(Zi_modelo)

0

-100

-200

100

200phase(Zi_promediado) phase(Zi_modelo)

-180º

Constant

The input impedance is a negative resistor (CPL) even beyond the

crossover frequency of the feedback-loop

As expected, the input impedance is a negative resistor, since active power is tracked, henceconstant-power load (CPL) behavior of the converter is obtained

Frequency (Hz)1

Magnitude

Phasei

ii i

vZ =

Crossover frequency



Review of the control regulation effects on the input impedance

dεdRi VG rsL +⋅

1+

-

ωL

di

qi

PLANT

mod2G

Vi ⋅dvmod_ˆ+

-REFdi _ˆ

+

+

ωLqi

FeedForward

Compensation

iV

mod

12

GVi

⋅

Vi

Neutral point

+

vi

-

ii

Ai

Bi

Ci

AdBd Cd

i

ii i

vZ =

pref

qref

223

2

qd

qdd vv

vqvpi

+⋅+⋅

⋅=

223

2

qd

dqq vv

vqvpi

+⋅−⋅

⋅=

We have analyze the effect of the feedback but not the

feedforward regulation effect over input impedance




Outline






viThe input voltage can change


iO

Input Perturbation Output Perturbation+-

LOAD

Power converter

Measured magnitude

Reference

control magnitude -

+Control

Feedforward objective

FeedForward technique can be used to decouple the plant from the external perturbations



Example Feedforward of the input voltage

Feedforward is a fast correction (very high bandwidth) of transient perturbations in the input voltage

The perturbation “immediately” changes the control quantity (most of cases the modulating signal), without waiting for the response of the compensator

VI s1

L

C R

+

vO(t)

-

F(vi)

Feedforward

Compensation

of input voltageGMOD

PWMModulator

R(s)+-

+

+

REFv

KV Sensor

modv

Power converter

KV



Feedback and Feedforward

Vin Feedforward

Compensator

Sensor

+

-Modulator

OvREFv d

FBv

ERRv MODv Power converterPlant

Converter with feedback regulation

Converter with feedback and

feedforward regulation

IvOi

Compensator

Sensor

+

-Modulator

OvREFv d

FBv

ERRv MODv Power converterPlant

IvOi

Feedback

Feedback

+-

iOFeedforward

-



Without FF

Effect of the input voltage feedforward in a DC/DC converter

With FF



Input voltage

Without FF

With FF

Output voltage

Output voltage

Effect of the feedforward of the input voltage

VI s1

L

C R

+

vO(t)

-

F(vi)

Feedforward

Compensation

of input voltageGMOD

PWMModulator

R(s)+-

+

+

REFv

KV Sensor

modv

Power converter

KV

Due to the action of the

feedback regulation the

converter behaves as

CPL at frequency lower

than the crossover

frequency


feedforward regulation

the converter behaves

as constant power load

in “almost” whole

frequency range

The output voltage is “almost” constant independently of the input voltage Variations Po=ctePi=cte



dεdRi VG rsL +⋅

1+

-

ωL

di

qi

PLANT

mod2G

Vi ⋅dvmod_ˆ+

-REFdi _ˆ

+

+

ωLqi

Ideal

FeedForward

Compensation

iV

mod

12

GVi

⋅

Vi

Neutral point

+

vi

-

ii

Ai

Bi

Ci

AdBd Cd

i

ii i

vZ =

pref

qref

223

2

qd

qdd vv

vqvpi

+⋅+⋅

⋅=

223

2

qd

dqq vv

vqvpi

+⋅−⋅

⋅=

The plant to be controlled does not

depend on the input voltage, Vi

( )rsLv

i

C

d

+⋅= 1

ˆ

ˆ

Cv

Effect of the feedforward of the input voltage on Inverters



Three-phase inverter has a “ideal feedforward” of the input voltage

Vi

IA(Line Current)

IA(Line Current)With FF compensation of Vi

Without FF compensation of Vi

Vi

Neutral point

+

vi

-

ii

Ai

Bi

Ci

AdBd Cd


feedforward regulation the

converter behaves at

constant power load in the

whole frequency range




Summary

We need to know, many different transfer-functions.

The analytical derivation of the close-loop input impedance is more complex and tedious than output impedance calculations

H

Mod Rvd

+-

CiAi Bi

Gvd

Zi_cli i

+-

+-

Gvv +

-

Oviv

T

GvvBi

T

T

Gvd

GvvAiCii

vZ

i

icli

+⋅−

+⋅−⋅+

==

11)(

1_

Close-loop input impedance is a negativeresistor only up to crossover frequency of the feedback-loop

The feedforward of the input voltage makesthe converter behaves as negative resistor in the whole frequency range

feedback

feedforward

F(Vin)



Lesson 4: Input impedance estimation of commercial DC-input-Port of regulated Converters

Outline

Effects at the DC-input-Port of regulated converters

Close-loop Input Impedance Estimator Experimental validation of the estimator with

commercial converters




Outline






DC - DC

Auxiliary DC Bus

Auxiliary services

DC - DCCascade converters

DC - ACSOURCE CONVERTER

LOAD CONVERTERS

Unknown loads (input impedance)

Problem of input impedance analytical calculation in case of commercial converters• Consider as context, the problem of designing the voltage loop of the “Source Converter” in a

typical distributed supply system for auxiliary services:

• Analytical techniques are not suitable since internal parameters are unknown.

We need an Input Impedance Estimator



Pi=Po=Vo·Io+

-busv

• A regulated converter behaves at low frequency as a constant-power load (CPL) in large signal. If the input voltage increases, the input current decreases and vice versa

• In small-signal, the CPL behavior at low frequency can be modeled as a negative resistor (negative slope)

• The negative resistor value depends on the steady-state operating conditions

+

-busv

ii

cplR−

ii

bus

i

v

p+

-busv

ii

iv

ii

iI

iV

Effect of Constant Power Load Behavior



Small-signal model of constant power load

Pi

bus

i

v

p+

-busv

ii

Large signal model:

bus

ii v

pi =

Small-signal model:

ibus

busbus

ii

i

Pp

Vvi

ibusbus

Pp

Vvbus

ibusi

pV

vV

Pi

pp

pvFv

v

pvFi

ii

busbus

ii

busbus

ˆ1

ˆˆ

ˆ),(

ˆ),(ˆ

2⋅+⋅−=

=⋅∂

∂+⋅∂

∂===

==

busbus

ii

busbus

bus

ii v

V

Pp

Vv

V

Pi ˆˆ

1ˆˆ

22⋅−=⋅+⋅−=If output power does not change,

input power does not change

busbus

ii v

V

Pi ˆˆ

2⋅−=

0ˆ =ip

Small signal model

+

-busv

ii+

-busv

i

buscpl P

VR

2

−=

Large-signal Model Small-signal Model



Effect of feedback-loop

ZiNegative R @ low frequencyLoop effect @ high frequency

fc

fc

T

T

Zi

Zi

The behaviour as constant power load (negative

resistance) is only valid until crossover frequency



Constant power load & feedback-loop effects (different control)

Mag

nitu

de(d

BΩ

)P

hase

(deg

rees

)

Average Current Mode Control

Peak Current Mode Control

Voltage Mode Control

Average Current Mode Control

Peak Current Mode Control

Voltage Mode Control

fc

Below crossover frequency, the behavior of the converter is the same negative resistor (CPL)

Above crossover frequency, the behaviour is not a negative resistor and depends on the feedback-loop



CoCi

ii

Sourceconverter

Closed-Loop input impedance, Zi(s)

RVbus

Most commercial Dc-input-port converters has a capacitor connected at the input to reduce input

current ripple

Load Converter

Input capacitor of Dc input port converters



Effect of the of the input capacitor

fc

The effect of the input capacitor (fcpl) is predominant over the feedback loop effect

Zi (with Cin)

Zi(without Cin)

Negative R @ low frequencyInput capacitor @ high frequency

Negative R @ low frequencyLoop effect @ high frequency

Zi (without Cin)

fcpl

-180º -90º

Zi (with Cin)

i

icpl P

VR

2

−=CinR

fcpl

cpl ⋅⋅⋅=

π2

1

fc



Behavioral model of the close-loop input impedance

-

oV

refv

d

+

+

-

L

C

iVR

vK

vRmodG

CinR

inC+

-

Cross frequency

Cin=100µµµµF

Constant magnitude

-180º

Decreasing magnitude

-90º

iZ

Y i

ii

vi

+

-

Zi

Modeling of the input port of the converter

Required information should be obtained from converter measurements or

datasheet, no internal converter parameters are needed

Behavioral model of the close-loop input impedance

iZ




Outline






Behavioral models

G-parameters based input-output model

Input Norton Network Output Thevenin Network

+-

+

vo

-

io

io·HiY i

ii

vi

+

-

vi·Go

Zo

Input admittance Yi: relationship between ii and vi

Back-current gain Hi: relationship between ii and io

Audiosusceptibility Go: relationship between vo and vi

Output impedance Zo: relationship between vo and io



Analytical Identification techniques

Frequency

Behavioral modelsG-parameters based input-output model

...( )

...

nn o

o nn o

a s aZ s

b s b

⋅ + +=⋅ + +

DC-DC+vo-

io

Phase Zo

Magnitude Zo


EXPERIMENTAL SET-UPPower amplifier

Complex measurements plus powerful but complicated identification techniques can be used to obtain

the complete information of the load converters input port

Difficult, expensive and time consuming to develop a behavioral model



Estimator of the close-loop input impedance

The estimator proposed, is the equivalent circuit composed by the input capacitor (Cin) in parallel with the negative resistor (Rcpl)

+

-

inVCinR

inC

CPLR

)(estimatorZ iFilter-

oV

refv

d

+

+

-

L

C

iVR

vK

vRmodG

CinR

inC+

-

iFilterZ

Crossover frequency

iFilterZ

Cin=100µµµµF

Constant magnitude

-180º

Decreasing magnitude

-90º

sCRR

sCRRestimatedZ

inCincpl

inCincpliFilter ⋅⋅++

⋅⋅+⋅=

)(1

)1()(

The close-loop input impedance can be easily obtained

Based on the main effects observed at the input of the regulated DC-input-port converters

ONLY THE CONSTANT POWER LOAD RESISTOR VALUE

AND THE CAPACITOR VALUE SHOULD BE INDENTIFIED



Close-loop impedance estimator versus analytical input impedance

Zi_analytical(green line)

Zi_estimator(black line)

Mag

nitu

de (

dB)

Pha

se (º

)

Zi_analytical(green line)

Zi_estimator(black line)

CinP

Vf

i

icpl

⋅⋅⋅= 2

2

1

πT

GvvBi

T

T

Gvd

GvvAiCii

vZ

i

ii

+⋅−

+⋅−⋅+

==

11)(

1

Exact calculation for Buck CCM in Voltage Mode control

Estimated calculation for Buck CCM in Voltage Mode control

sCRR

sCRRestimatedZ

inCincpl

inCincpliFilter ⋅⋅++

⋅⋅+⋅=

)(1

)1()(

Validation with Buck CCM converter in VMC



Validation with different input capacitor values

The simple estimator is valid even when the cutt-off estimator frequency fcpl is close to crossover frequency fc

(less than one decade of the fc)

fc fsw fres Cin RCPL fCPL

20KHz 250KHz 2.2KHz 47uF (A) -14.4 250Hz

20KHz 250KHz 2.2KHz 4.7uF (B) -14.4 2.5KHz

-

oV

refv

d

+

+

-

L

C

iVR

vK

vRmodG

CinR

inC+

-

iFilterZ

Buck converter CCM Voltage Mode control

Different input capacitor values

iZCase A

Case B



Same input capacitor design but different converter and control strategy

Cin designed to obtain in any power converter a 5 % input current ripple

Different converter topologies

(Buck and Boost)

Different operating modes

(CCM and DCM)

Different control techniques

(VMC and ACMC)

oVd

+

-

L

C

iVR

CinR

inC+

-

Buck converter Boost converter

VI s1

L

C R

+

vO

-

To evaluate if thedifferent design of input capacitor dominates the high frequency effects over feedback-loop

CinR

inC

To evaluate if the different control dominates the high frequency effects over input capacitor



Analysis of the worst-case for input impedance estimator validation

The input impedance estimator

has similar results in VMC and

in ACMC

CCM

Cin=18µµµµF

Cin=18µµµµF

ACMC

VMC

CCM The input impedance estimator

has been validated with a boost

converter in CCM that is the

worst-case regarding the effect

of input capacitor over the

feedback-loop effects




Outline






Impedance estimator parameters from datasheet

+

-iv

i

icpl P

VR

2

−=

Vi :Input Voltage

Pi :Input powerηη

oooi

IVPP

⋅==

Cin :Input capacitor

The datasheet efficiency and the operating conditions Vo and Po can be used to calculate the negative resistor value

Only the type of input filter is specified



Additional simple measurements for estimate input capacitor

Very simple setup is required sincethe measurement can be done whilethe converter is not working

Only a multimeter is required

zo

Vdc

DC power supply

R=2Ω

C=100mF

DC/DC

converter

Rload

Zi

+

vi

-

i i


EXPERIMENTAL SET-UP

Power amplifier

Input impedance measurement



Accuracy of the estimator in commercial DC/DC converters

The input capacitor has been measured, and the input impedance estimator has been compared with

the input impedance measurement at two different output power levels showing good accuracy



Lesson 4: Input impedance estimation of commercial DC-input-Port Converters

Summary

• Three effects can be observed at DC-input-port converter:

• Negative resistance due to Constant-Power Load (CPL) behavior

• Effect of the feedback loop• Effect of the input

capacitor used to filter input current ripple

-

Sensor

+

+

-

L

C R

Compensator Reference

Zi magnitude

Decreasing magnitude(capacitor)

Constant magnitude(negative resistance, CPL)

The effect of the input capacitor dominates the input impedance at high frequency.A simple input impedance estimator can be considered, which parameters can be obtained from the datasheet and simple aditional measurements

+

-

Zi

ViRcin

Cind

Mod

Vo

+

-

Zi

ViRcin

CinRcpl

sCRR

sCRRestimatedZ

inCincpl

inCincpli ⋅⋅++

⋅⋅+⋅=

)(1

)1()(



Lesson 1: Stabilitiy of Dc Power

Distribution Systems

Lesson 2: Feedback-loop design considerations

for the BUS converter

Lesson 3: Complete characterization of the

input impedance of a DC-input-Port

Converter

Lesson 4: Input impedance estimation of

commercial DC-input-Port Converters

Outline



Final Remarks (I)

Closed-Loop “unterminated” output impedance, Zo(s)

Closed-Loop input impedance, Zi(s)

R

The main stability problem is due to the interaction of cascade converters. The easy way to design the feedback-loop of the source converter is to apply the Middlebrook´simpedance criterion

1)(

)()( <=

sZi

sZosT

Stable if

Vi+Vo-

+Vbus

-

LOAD converter

SourceconverterSourceconverter

Zin

Zo



Final Remarks (II) Analytical calculations of the input impedance are quite tedious and difficult in some

cases

Moreover, analytical techniques are not valid in the problem of designing the voltage loop of the “Source Converter” in a typical distributed supply system for auxiliary services where unknown converter loads and not valid for commercial converters.

The alternative is to develop behavioral models of the converter. But they are difficult and time consuming since complex measurements plus complicated identification techniques should be used to obtain the complete information of the load converters input port.

DC -DC

Auxiliary DC Bus

Auxiliary services

DC -DC

DC -ACSOURCE CONVERTER

LOAD CONVERTERS

Unknown loads (input impedance )



Final Remarks (III)• A simple input impedance estimator has been proposed based on the similar

effects observed in a converter that has a DC input port, • Negative resistor at low frequencies due to Constant-Power Load (CPL)

behavior • Effect of the compensation of the feedback-loop at high frequencies• Usually, the effect at high frequencies is dominated by the input capacitor

Zi magnitude

Decreasing magnitude(capacitor)

Constant magnitude(negative resistor, CPL)

Input Impedance Estimator is a very effective first-approach tool to predict cascade converters instability. The estimator Is based on:

• Datasheet information (Rcpl)

• Simple measurements using a multimeter (Cin)

+

-

Zi

ViRcin

CinRcpl

sCRR

sCRRestimatedZ

inCincpl

inCincpli ⋅⋅++

⋅⋅+⋅=

)(1

)1()(



Main References and Readings

A. Riccobono and Enrico Santi “Comprenhesive Review of Stability Criteria for DC Power Distribution System”, IEEE Trans. On Industry Applications, vol.50, no.5, págs.3525-3535, November 2014

J.Sun, “Small-Signal Methods for AC Distributed Power Systems: A review”, IEEE Trans. Power Electron., vol.24, no.11, págs.2547, November 2009

M. Sanz, V. Valdivia, P. Zumel, D. López del Moral, C. Fernández, A. Lázaro, A. Barrado, “Analysis of the Stability of Power Electronics Systems: a Practical Approach”, 29 th IEEE Proc. Applied Power Electronics Conference (APEC), pp. 2682 – 2689, 2014

A. Kislovski, R. Redl, and N. Sokal, “Dynamic Analysis of Switching-Mode DC/DC Converters”, New York: Van Nostrand Reinhold, 1991

L. Arnedo, D. Boroyevich, R. Burgos, F. Wang, “Black-Box Terminal Characterization Models for the Analysis and Simulation of Distributed Power Electronic Systems”, IEEE Proc. Power Electronics Specialists Conference (PESC), pp. 1968-1973, 2007

V. Valdivia “Behavioral Modeling and Identification of Power Electronics Converters and Subsystems Based on Transient Response” PhDissertation, Carlos III University of Madrid, January, 2013.

Y. Panov, M. Jovanovic, “Practical issues of input/output impedance measurements in switching power supplies and application of measured data to stability analysis”, Proc. IEEE Applied Power Electronics Conference and Exposition (APEC), pp: 1339 - 1345, 2005

M. Sanz, A. Lázaro, C. Fernández, P. Zumel, D. López del Moral, I. Quesada, A. Barrado, “Practicing Design Method of Regulators for Cascaded Converters”, IEEE Proc. 14th Workshop on Control and Modeling for Power Electronics (COMPEL), pp. 1-5, 2014

M. Sanz; A. Lazaro; M. Bermejo; D. Lopez del Moral; P. Zumel;C. Fernandez; A. Barrado, Low-cost input impedance estimator of Dc-to-Dc converters for designing the control loop in cascaded converters, 2016 IEEE Applied Power Electronics Conference and Exposition (APEC), pp. 3090-3096, March 2016

M. Sanz, M. Bermejo, A. Lázaro, D. López del Moral, C. Fernández, P. Zumel, A. Barrado, Simple Input Impedance Converter Model to Design Regulators for Dc-Distributed System, 2016 Seventeenth IEEE Workshop on Control and Modeling for Power Electronics (COMPEL), pp. 1-6, June 2016

R. Miftakhutdinov, "Power distribution architecture for tele- and data communication system based on new generation intermediate bus converter," INTELEC 2008

M. Blanco, G. Navarro, M. Lafoz, ”Control of power electronics driving a switched reluctance linear generator in wave energy applications”. Proc. European Conference on Power Electronics and Applications, EPE '09, pp. 1 – 9, 2009

A. Emadi et al, "Power electronics intensive solutions for advanced electric, hybrid electric, and fuel cell vehicular power systems," IEEE Trans. Power Electron., 2006.

Y. Xie , et. al "A PC-cluster based real-time simulator for all electric ship integrated power systems analysis and optimization", IEEE ESTS 2007



Contact: E-mail: [email protected]

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