1 power electronics by dr. carsten nesgaard small-signal converter modeling and frequency dependant...

1

Power ElectronicsPower Electronics

by

Dr. Carsten Nesgaard

Small-signal converter modeling and frequency dependant behavior in controller synthesis

2

AgendaAgenda• Small-signal approximation

• Voltage-mode controlled BUCK

• Converter transfer functions dynamics of switching networks

• Controller design (voltage-mode control)

• Discrete time systems

• Measurements

3

Small-signal approximationSmall-signal approximation

• An analytical evaluation of equipment performance

Advantages of small-signal approximation of complex networks:

• An analysis of equipment dynamics

• Stability

• Bandwidth

• A design oriented equipment synthesis

The linearization of basic AC equivalent circuit modeling corresponds to the mathematical concept of series expansion.

4

Small-signal approximationSmall-signal approximation

• Limited to rather low frequencies (roughly fS/10)

Drawbacks of small-signal approximation of complex networks:

• Inability to predict large-signal behavior

• Transients

• High frequency load steps

• Calculation complexity increases quite rapidly

5

Voltage-mode controlled BUCKVoltage-mode controlled BUCK

a

DriveDrive

Cmp V p

t

EA

V r+

-

+

-

V e

u 1

S 1

S 2

L

+

- Cx2

R ESR

R Load y2

+

-

y1 x1

iC ILoad

Basic BUCK topology with closed feedback loop

Not included in ‘Fundamentals of Power Electronics’ SE.

6

Voltage-mode controlled BUCKVoltage-mode controlled BUCKConverter waveforms:

DriveDriveu 1

S 1

S 2

L

+

- Cx2

R ESR

R Load y2

+

-

y1 x1

iC ILoadtd .T T

Drive S 1

td .T T

Drive S 2

t

I in = y 1

d .T T

t

vL

d .T T

-v OUT = -y 2

v in - v OUT = u 1 - y 2

iL = x 1

td .T T

ILoad IL

td .T T

iC

IL

td .T T

vC = x 2

vOUT = y 2

7

ESRCLin1 R)(

C (t) v (t) v (t) v (t)u dt

tdvC

(t)y (t)v 2OUT

AC modelingAC modelingConverter states:

u 1

L

+

- Cx2

R ESR

R Load y2

+

-

y1 x1

iC ILoad

u 1

L

+

- Cx2

R ESR

R Load y2

+

-

y1 x1

iC ILoad

0 < t < dT: dT < t < (1 – d)T

KCL: KVL:

(t)I (t)i (t)i (t) x (t)y LOADCL11

dt

tdvC )(C iC

(t)x 2(t)x 2 dt

tdiL )(L (t)vL

LOAD

2LOAD R

(t)y (t)I

8

AC modelingAC modelingAveraging and linearization (in terms of input and output variables):

Inductor equation:

Capacitor equation (same for both intervals):

Input current equation:

LLOAD

OUT i - R

v C

dt

dvC

d-1 v- d v- v L OUTOUTin dt

diL

di i Lin LLin iD Id (t)i

OUTinin v Vd vD ˆ

L dt

id L

LLOAD

OUT i - R

v

ˆC dt

vd C

x X x

x X x

x X x

DC and 2nd order terms are removed from the equations to the right.

9

AC modelingAC modelingResulting AC equivalent circuit:

DC transformer relating input voltage and inductor current, thus behaving ‘almost’ like a real transformer.

R ESR

dt

vd CˆC

R LOAD

+

-

C

OUTv

+-+ -

dt

id LL

inVd

L

LIdinv+

-

in i 1 : D

?D - 1 ?d ?D ˆ

L dt

id L LOAD

OUT

R

v - ?d ?D - 1-

ˆC dt

vd C

?D ?d iin

10

R ESR

R LOAD

+

-

C

OUTv

L

inv+

-

in i 1 : D

LOAD

OUT

R

Vd

+-

2OUT

D

Vd

Canonical AC modelCanonical AC modelRearranging the AC equivalent circuit found on the previous slide by the use of traditional circuit theory a universal model can be established:

A similar model applies to a wide variety of other converter topologies. In Fundamentals of Power Electronics SE a table containing coefficients for the different sources can be found.

11

Converter transfer functionsConverter transfer functions

Basic control system

State equation

Control equation

Output equation

P1

Source (u)

Feed forward

State (x) Output (y)

Feedback

Con

trol (

d)

d u - s EBxAI

State equation:

u d Q xF

Control equation:

u y NxM

Output equation:

A : State matrix B : Source matrixE : Control matrix F : Feedback matrix I : Identity matrix Q : Feed forward matrixM : Output-state-matrix N : Output-source-matrix d : Control variable u : Source variable x : State variable y : Output variable

12

State equation

Control equation

Output equationSource (u) = 0

Feed forward

State (x) Output (y)

Feedback

Con

trol (

d)

q

x

Converter transfer functionsConverter transfer functions

Opening the loop and rewriting the system equations the following trans-fer functions can be obtained:

FEAIq

x 1

xq - s T

Open loop transfer function:

NQEBFEAIM - - s u

y T 1

yu

Closed loop transfer function:

13

State-space averagingState-space averaging

a

DriveDrive

Comp V p

t

EA

V r+

-

+

-

V e

u 1

S 1

S 2

L

+

- Cx2

R ESR

R Load y2

+

-

y1 x1

iC ILoad State variables:

Inductor current x1

Capacitor voltage x2

Output variable y (dependent)

In order to contain past information all variables are functions of time

By definition the following apply:

Source variable u (independent)

14

State-space averagingState-space averagingAveraging the equations previously found results in the following non-linearized matrices:

C)R (R

1 -

C

1L)R (R

R -

L

R -

'

LOADESR

LOADESR

LOADESR

A d

0

L

1

'

B

The use of linearization requires:

d D d u Uu x X x

d Du U' x X' x BA

Insertion into the state equation results in:UD' X' 0 :DC BA

dU' uD' x' x :AC BBA

15

State-space averagingState-space averagingThe averaged and linearized matrices can now be identified:

UD' X' 0 :DC BA

dU' uD' x' x :AC BBA d u x x EBA

U X 0 BA

Comparing the above A matrix with the non-linearized A’ matrix found on the previous slide, it can be seen that no changes have occurred.

C)R (R

1 -

C

1L)R (R

R -

L

R -

LOADESR

LOADESR

LOADESR

A

0

L

D

B

0

L

U

E

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State-space averagingState-space averagingAveraging and linearizing the control variable d (PWM controller) in terms of state variables gives the following relation:

2ESR

2

PP

e x RCx

V

K-

V

v d

dt

dK

VP is the sawtooth peak voltage

is the EA gain, ‘a’ factor and comp.

Collecting terms in accordance with the control equation, and realizing that multiplication by ‘s’ in the frequency domain is the equivalent to differentiation in the time domain, the matrices F and Q can be identified:

0 sCR 1V

K-0 u d ESR

P

QFQx F and

Since feedback is the only means of converter control applied, Q is (as expected) zero.

17

sCR 10R

DD

R

Ls 1

ESR

LOADLOADM

sCR 1

V

K-0 ESR

P

F

C)R (R

1 -

C

1L)R (R

R -

L

R -

LOADESR

LOADESR

LOADESR

A

0

L

D

B

0

L

U

E 0 Q

State-space averagingState-space averagingSummarizing the voltage-mode controlled BUCK matrices:

LOAD

2

R

D- N

0

1

1/L

1/RLOADC -K/VP

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Nyquist stability requirement for closed loop systems:

Prh(GCL(s)) = Prh(GOL(s)) + = 0

Where: Prh = number of right half-plane poles

= number of times the Nyquist contour of the open- loop transfer function circles the point (-1,0)

GCL = Closed-loop transfer function

GOL = Open-loop transfer function

StabilityStability

Minimum open-loop transfer function gain margin: 6 - 8 dB

Minimum open-loop transfer function phase margin: 30 - 60

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Circuit data:

L = 300 H

C = 69 F

RESR = 0,2

ILoad,m = 1 A

U1 = 12 V

y2 = 5 V

IL = 0,2 A

f = 50 kHz

Additional data:

Vp = 2,45 V

a = 0,5

a

DriveDrive

Comp V p

t

EA

V r+

-

+

-

V e

u 1

S 1

S 2

L

+

- Cx2

R ESR

R Load y2

+

-

y1 x1

iC ILoad

20

10 100 1 103

1 104

1 105

60

40

20

0

20

40

60

80

20 log GH m n( )

f m n( )10 100 1 10

31 10

41 10

540

30

20

10

0

10

20

30

20 log GH m n( )

f m n( )

10 100 1 103

1 104

1 105

180

135

90

45

0

45

90

135

180

180

arg GH m n( )( )

f m n( )10 100 1 10

31 10

41 10

5180

135

90

45

0

45

90

135

180

180

arg GH m n( )( )

f m n( )


A plot of the open-loop transfer function is shown below (K = 1):

21

3

3.5

4

4.5

-150

-100

-50

0

3

3.5

4

4.532 kHz

10 kHz

3,2 kHz

1 kHz

0 dB

-50 dB

-100 dB

-150 dB

f

A

R ESR

0,1

0,3

0,5

0,2

0,4

0°

-45°

-90°

-135°


A 3D plot of the converter filter transfer function is shown to the right.

Note: The zero caused by RESR increases the phase (green curve) as a function of frequency and RESR. Unfortunately due to the same zero the filter attenuation drops (red curve).

22

PI-comp.

Lag-comp.

PD comp.

Lead-comp.

A PI-Lead-comp. (PID) will be used in this presentation

CompensationCompensation-9020log(K p)

1/

1/

20log(K p)

1/i1/i

-90

1/i1/i

20log(K p/)

9020log(K p)

1/d 1/d

1/d

20log(K p)

20log(K p/)90

1/d 1/d 1/d

s

1 sK G(s)

i

ip

1 1 s

1 sK G(s)

i

ip

1 sK G(s) dp

1 1 s

1 sK G(s)

d

dp

23

CompensationCompensation

Widely accepted error amplifier configuration:

V ref

V OUT

V error

C 3

C 2 C 1R 2

R 1EA

-

+

sRCC C CRs

RCs 1RCs 1 K(s)

232321

2211

Pole at f = 0 for increased DC gain

Pole at fESR for compensation

Double zero at resonance peak for increased phase margin

222

111 RC2

1 z

RC2

1 z

232

3221 RCC2

C C p 0 p

1

2

R

R gain DC

24

H m n( )Vp fc

2

V1 fn2

fn

10

1 jf m n( ) 10

fn

1 jf m n( )

fc

j f m n( )( ) 1 j 2 f m n( ) R2 C

10 100 1 103

1 104

1 105

1 106

80

60

40

20

0

20

40

6060

80

20 log GH m n( )

20 log H m n( )

10610 f m n( )

Compensator and converter transfer functions:

CompensationCompensation

10 100 1 103

1 104

1 105

1 106

180

135

90

45

0

45

90

135

180

180

arg GH m n( )( )

180

arg H m n( )( )

f m n( )

Amplitude: Phase:

0 dB/dec-40 dB/dec -20 dB/dec

-20 dB/dec +20 dB/dec 0 dB/dec

Red : Converter transfer function

Blue : Compensation transfer function

H m n( )Vp fc

2

V1 fn2

fn

10

1 jf m n( ) 10

fn

1 jf m n( )

fc

j f m n( )( ) 1 j 2 f m n( ) R2 C

10 100 1 103

1 104

1 105

1 106

80

60

40

20

0

20

40

60

20 log GH m n( )

20 log H m n( )

f m n( )10 100 1 10

31 10

41 10

51 10

6180

135

90

45

0

45

90

135

180

180

arg GH m n( )( )

180

arg H m n( )( )

f m n( )

10 100 1 103

1 104

1 105

180

135

90

45

0

45

90

135

180

180

arg GH m n( )( )

f m n( )10 100 1 10

31 10

41 10

512

0

12

24

36

48

60

72

20 log GH m n( )

f m n( )

fC 4.0 kHz

56,1

25


Using the previously derived matrices an expression for the input impedance Zin can be established:

10 100 1 103

1 104

1 105

1

10

100

Z1 m n( )

f m n( )

11

-1

1

1yu,11 - - s

(s)u

(s)y T

NQEBFEAIM

10 100 1 103

1 104

1 105

180

135

90

45

0

45

90

135

180

180

arg Z1 m n( )

f m n( )

26

10 100 1 103

1 104

1 105

80

60

40

20

0

20

40

60

20 log GH m n( )

20 log H m n( )

f m n( )

10 100 1 103

1 104

1 105

180

135

90

45

0

45

90

135

180

180

arg GH m n( )( )

180

arg H m n( )( )

f m n( )

DCM reduces the converter transfer function to a first order system, since the time derivative of the small-signal inductor current is zero and thus disqualifies the inductor current as a state variable.


A plot of the open-loop transfer function during Discontinuous Conduction Mode (red curve) and EA compensation (blue curve)

10 100 1 103

1 104

1 105

80

60

40

20

0

20

40

60

20 log GH m n( )

20 log H m n( )

f m n( )

10 100 1 103

1 104

1 105

180

135

90

45

0

45

90

135

180

180

arg GH m n( )( )

180

arg H m n( )( )

f m n( )

10 100 1 103

1 104

1 105

80

60

40

20

0

20

40

60

20 log U m n( )

f m n( )

10 100 1 103

1 104

1 105

180

135

90

45

0

45

90

135

180

180

arg U m n( )( )

f m n( )10 100 1 10

31 10

41 10

580

60

40

20

0

20

40

60

20 log U m n( )

f m n( )

27

Discrete time systemsDiscrete time systems

Transient response and the relationship between the s-plane and the z-plane:

Discrete time: Continuous time:ti eC y(t) C y(i)

At the sampling instants:Tii e

Inserting into the expression to the left, it can be seen that the continuous time stability requirement maps onto the z-plane in form of the unit circle.

Thus, the dynamics of the two systems are identical at the sampling instants:Tse z j s

jer z

T e r T

28


Arithmetic and operations:

• Integration and differentiation

• Plotting the frequency response

• Tustin’s rule

• Sampling rate

1 zT

1 - z2 s

T)j(Ts e )j s ( e zz

1 - z

zT (z)

X

Y x(i)T 1) - y(i y(i)

BandwidthsBandwidth f times20 to5 f

2

f

29

10 100 1 103

1 104

1 105

90

70

50

30

10

10

30

5050

90

Cont m n( )

Disc_1 m n( )

Disc_2 m n( )

10510 f m n( )

10 100 1 103

1 104

1 105

0

5

10

15

20

25

30

3535

0

Cont m n( )

Disc_1 m n( )

Disc_2 m n( )

10510 f m n( )


Plot of the discrete compensation transfer function:

Cont Continuous time

Disc_1 Discrete time with sample frequency = 50 kHz (no prewarping)

Disc_2 Discrete time with sample frequency = 100 kHz (no prewarping)

30

10 100 1 103

1 104

1 105

20

10

0

10

20

30

40

50

20 log GH m n( )

20 log GD_2 m n( )

Meas_Amplitude

f m n( ) f m n( ) Range

GH Continuous time

GD_2 Discrete time with sample frequency = 50 kHz (no prewarping)

Meas Actual measurement

MeasurementsMeasurements

Below is a comparison of the predicted continuous time loop gain, predicted discrete time loop gain and an actual measurement of the loop gain:

10 100 1 103

1 104

1 105

20

10

0

10

20

30

40

50

20 log GH m n( )

20 log GD_2 m n( )

Meas_Amplitude


10 100 1 103

1 104

1 105

180

135

90

45

0

45

90

135

180

180

arg GH m n( )( )

180

arg GD_2 m n( )( )

Meas_Phase


31

10 100 1 103

1 104

1 105

180

135

90

45

0

45

90

135

180

180

arg U m n( )( )

Meas_Phase

f m n( ) Range

GH Continuous time

Meas Actual measurement

MeasurementsMeasurements

The same transfer function as before, but during Discontinuous Conduction Mode:

10 100 1 103

1 104

1 105

80

60

40

20

0

20

40

60

20 log U m n( )

Meas_Amplitude

f m n( ) Range

1 power electronics by dr. carsten nesgaard small-signal converter modeling and frequency dependant...

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