The PWM switch model introduced by Vatché Vorpérian in ... · The PWM switch model introduced by Vatché Vorpérian in 1986 describes a way to model a voltage-mode switching converter

The PWM switch model introduced by Vatché Vorpérian in 1986 describes a way to model a voltage-mode switching converter with the VM-PWM switch model. The large-signal model is equivalent to a dc transformer whose turns ratio is D, the controlled duty ratio. Later, in 1990, Vatché published another model, the CM-PWM switch model, targetting peak current mode controlled converters. The model nicely predicts subharmonic oscillations and works in large-signal conditions: it can compute its bias point and it works in transient analysis. This current mode model was preceded by that developed by Ray Ridley using sampled data analysis. Ridley’s model predicts the same dynamic response than what the CM-PWM switch does, but it only works in ac and cannot compute its bias point. The CM-PWM switch is sometimes prone to convergence problems and Vatché published a revised netlist in his 2004 book Fast Analytical Techniques for Electrical and Electronic Circuits. The fix is to add a coefficient in front of the two sources setting the peak and average values. It appears below:

Christophe Basso – June 2017

For some designers, the CC-PWM switch was troublesome given its difficulty – in some cases – to converge. The VM-PWM switch model is more robust and converges well. Ridley used the small-signal version of this VM-PWM switch to which he added a pair of RHP zeros to represent the sampling action in the current loop.

This extra circuit uses a negative resistance value to produce a negative quality factor and thus RHP zeroes.

Rather than implementing a complex circuitry, the idea here is to reuse the classical VM-PWM switch and by adding a simple extra dummy source plus a capacitor, simulate the response of a current mode converter subject to subharmonic oscillations.

Ca=1/Fsw/3.14159La=1/Fsw/3.14159


The implementation is very simple. Build a clamped source driving the duty ratio input of the PWM switch model. It is a voltage source clamped between 10 mV and 999 mV (1 to 99% duty ratio).

((V(Vc)-I(VIC)*{Ri})*{Fsw})/({Se}+{Ri}*(V(a,c)+1u)/(2*{L})) > 0.99 ? 0.99 :+((V(Vc)-I(VIC)*{Ri})*{Fsw})/({Se}+{Ri}*(V(a,c)+1u)/(2*{L})) < 10m ? 10m :+((V(Vc)-I(VIC)*{Ri})*{Fsw})/({Se}+{Ri}*(V(a,c)+1u)/(2*{L}))

It is actually the equation described in the book Switch-Mode Power Supplies: Spice Simulations and Practical Designs sec. ed. on page 199. Add a dummy voltage source in series with the c terminal that you call VIC plus a resonating capacitor between terminals c and p. The capacitor value is classically

Cs=1/(L*(Fsw*3.14)^2)

The control voltage Vc is transformed into a duty ratio via the in-line equation to generate the correct duty ratio D. The following slide shows the final implementation around the VM-PWM switch model. This is a CCM simulation only here and I am not sure it would be easy to have the D generator toggle into a DCM expression easily.


Vstim{Vc}AC = 1

DVc

Cres{Cs}

4

L2{L}

11

C3100uF

R45

R5100m

Vout

parameters

Fsw=100k

L=100u

Cs=1/(L*(Fsw*3.14)^2)

Ri=-250m

Se=0

Vc=1.6

R61u

c

a

p

B1Voltage

((V(Vc)-I(VIC)*{Ri})*{Fsw})/({Se}+{Ri}*(V(a,c)+1u)/(2*{L})) > 0.99 ? 0.99 :+((V(Vc)-I(VIC)*{Ri})*{Fsw})/({Se}+{Ri}*(V(a,c)+1u)/(2*{L})) < 10m ? 10m :+((V(Vc)-I(VIC)*{Ri})*{Fsw})/({Se}+{Ri}*(V(a,c)+1u)/(2*{L}))

5

rL10m

p

c

Vin10

dac

PW

M s

witch V

Mp

7X1PWMVM

c

a D

p

VIC

Clamped expression for the duty ratio expression in CCM current-mode:

((V(Vc)-I(VIC)*{Ri})*{Fsw})/({Se}+{Ri}*(V(a,c)+1u)/(2*{L})) > 0.99 ? 0.99 :

+((V(Vc)-I(VIC)*{Ri})*{Fsw})/({Se}+{Ri}*(V(a,c)+1u)/(2*{L}))

1.60V

9.94V

17.5V

9.94V

0V

2.68uV

433mV

10.0V

9.94V

This is a CCM current mode boost converter built around the voltage mode PWM switch model to which a duty ratio generator has been added, together with a resonating capacitor.

Dummy 0-V source VIC

Resonating cap.

Duty ratio generator


This is the CCM current mode boost converter built around the original CM-PWM switch purposely thought to model current mode control power supplies. It only uses current sources. As you can see from the schematic, all operating points are similar to those from the previous slide.

5

4

1

vc

p

L1{L}

R15

C1100uF

R2100m

Vout

parameters

Fsw=100k

L=100u

Cs=1/(L*(Fsw*3.14)^2)

Ri=-250m

Se=0

Vc=1.6

a

R31u

B1Voltage

D

B2Current

V(D)*I(VIC)

B3Current

V(Vc)/{Ri}

VIC

B4Current

{Se}*V(D)/({Ri}*{Fsw}) + v(c,p)*(1-V(D))*({1/Fsw}/(2*{L}))

v(c,p)/v(a,p)

c

c

Vstim{Vc}AC = 1

C2{Cs}

a

p

9

rL10m

Vin10V

9.94V

9.94V

17.5V

0V

9.94V

2.68uV

433mV1.60V

10.0V


3.00

9.00

15.0

21.0

27.0

vdb

ou

t#1

, vd

bo

ut

in d

b(v

olts

)P

lot1

13

10 100 1k 10k 100kfrequency in hertz

-360

-270

-180

-90.0

0

ph

_vo

ut#

1,

ph

_vo

ut

in d

eg

ree

sP

lot2

24

The graphs are well coincident and a small deviation appears as we approach Fsw but it is reasonably low.

out

c

V f

V f

out

c

V f

V f


Here in LTSpice, I compared a buck converter in CM driven by theVM-PWM switch model developped by Didier to which I added theresonating cap.


Vin210V

L

{L}

C

220µF

R

5

AC 1

VInj{Vc_set}

PWM-VM

D

U1

Lsw={L} Fsw={Fsw}DCMin=0.05 DCMax=0.95

VIC

0

V=if( V(D_raw) > 0.99, 0.99, if( V(D_raw) < {D_lo}, {D_lo}, V(D_raw) ) )B1

B2V= ( (V(Vc) - I(VIC)*{Ri} ) * {Fsw}) / ( {Se}+{Ri}*(V(a,c)+1u) / (2*{L}) )

Cres{Cs}

PWM-CM

D

VCtrl

DCMin=0.05 DCMax=0.95

Lsw={L} Fsw={Fsw}RISense={Ri} Slope={Se}

X1

Vin110V

L1

{L}

C1

220µF

R1

5

a OutVMc

Du

ty

D_rawVc

dc

Vc

OutCM

.Param L=75uH

.ac dec 1000 10 1000kHz

.param Vc_set = 1

.param Fsw=100k

.param Ri=0.8

.param Se=0

.param D_lo = 7u

.param D_hi = 0.99

.param Cs = {1/ ( L * (Fsw * 3.14)**2 ) }

5.4224815V

542.46503mV

5.4226832V

542.48518mV

Dynamic responses from the VM-PWM switch with the addedcapacitor and the CM-PWM switch model. Fsw = 100 kHz.


10Hz 100Hz 1KHz 10KHz 100KHz 1MHz-81dB

-72dB

-63dB

-54dB

-45dB

-36dB

-27dB

-18dB

-9dB

0dB

9dB

18dB

-90°

-60°

-30°

0°

30°

60°

90°

120°

150°

180° V(outcm) V(outvm)

Vin210V

L

{L} C

220µF

R

5

AC 1

VInj{Vc_set}

PW

M-V

M D


U1

VIC

0



Cres

{Cs}

PW

M-C

M

D

VC

trl

DCMin=0.05 DCMax=0.95Lsw={L} Fsw={Fsw}RISense={Ri} Slope={Se}

X1

Vin110V

L1

{L}

C1

220µF

R1

5

a

OutVM

c

Du

ty

D_rawVc

Vc

OutCM

dc

.Param L=75uH

.ac dec 1000 10 1000kHz

.param Vc_set = 2

.param Fsw=100k

.param Ri=0.3

.param Se=0

.param D_lo = 7u

.param D_hi = 0.99

.param Cs = {1/ ( L * (Fsw * 3.14)**2 ) }

-13.407595V

573.32522mV

573.33124mV

-13.407927V

A buck-boost converter with the two differentmodels. Dc points are identical.



-20dB

-16dB

-12dB

-8dB

-4dB

0dB

4dB

8dB

12dB

16dB

20dB

60°

90°

120°

150°

180°

210°

240°

270°

300°



Fsw


Vin210V

L

{L}

C

220µF

R

5

AC 1

VInj{Vc_set}

PW

M-V

M

D


U1

VIC

0



Cres

{Cs}

PW

M-C

M D

VC

trl

DCMin=0.05 DCMax=0.95Lsw={L} Fsw={Fsw}

RISense={Ri} Slope={Se}

X1

R2

1µ

Vin110V

L1

{L}

C1

220µF

R1

5

OutVM

c

Du

ty

D_rawVc

a

Duty

Vc

dc

OutCM

.Param L=50uH

.ac dec 1000 10 1000kHz

.param Vc_set = 2.2

.param Fsw=300k

.param Ri=-0.3

.param Se=0

.param D_lo = 7u

.param D_hi = 0.99

.param Cs = {1/ ( L * (Fsw * 3.14)**2 ) }

18.92837V

472.45076mV

0V

18.928526V

0V

A boost converter with the two differentmodels. Dc points are identical.



-4dB

0dB

4dB

8dB

12dB

16dB

20dB

24dB

28dB

32dB

36dB

-270°

-240°

-210°

-180°

-150°

-120°

-90°

-60°

-30°



Fsw


Special thanks to Marc Dimattina for helping me on this approach and of

course to Didier Balocco for porting my models to LTSpice.

Documents

The PWM switch model introduced by Vatché Vorpérian in ... · The PWM switch model introduced by Vatché Vorpérian in 1986 describes a way to model a voltage-mode switching converter