An average model for the phase-shifted converter
Christophe Basso
May 2009
Revised April 2010
Introduction
A small-signal model has been described by Fu-Sheng Tsai in 1993
Despite all efforts, I could not make it work (sorry about that).
Based on the PWM switch, I modified the auto-toggling model described
in my last book to cope with phase-shifted requirements.
The model is now fully auto-toggling between CCM and DCM.
Fu-Sheng Tsai, «Small-signal and transient analysis of a zero-voltage switched, Phase-Controlled PWM Converter
Using Averaged Switch Model», IEEE Transactions, Vol. 29, n°3, May/June 1993
Cycle-by-cycle simulation to understand the signals
The switching model uses the simplied version of a phase shift controler
parameters
fc=1k
pm=60
Gfc=-7
pfc=-81
Vout=12
G=10^(-Gfc/20)
boost=pm-(pfc)-90
pi=3.14159
K=tan((boost/2+45)*pi/180)
Vin
240V
D1
MUR460
parameters
10
DT=250n
CDT=DT/(0.69*1k)
Fsw=500k
D3
MUR460
X2
PSW1
C7
33p
X4
PSW1
Vxfmr C6
33p
∆
8
9
D4
MUR460
VHBL
Vramp
2
Fzero=fc/k
Fpole=fc*k
outA
ramp
outA
FB
outB
outB
CS
outC
outC
outD
B2
Voltage
V(outC)
Ileak
L1
15.7u
VHBR
12
D2
MUR460
X1
Config_1
1
lleak
15
7
B1
Voltage
V(outA)
outD
outB
L2
10mH
outD
X3
PSW1
C4
33p
C5
33p
X5
PSW1
X6
XFMR-TAP
RATIO = 1/6
11
13
RLED=CTRmin*Rpullup/G
Czero=1/(2*pi*Fzero*Rupper)
CpoleX=1/(2*pi*Fpole*Rpullup)
Cpole=CpoleX-Copto
VTL431min=2.5
Vf=1
Vdd=5V
Vcesat=300m
Ibias=1m
IL3
Vout
D6
out
17
Rupper
{Rupper}
Rled
{Rled}
23
Vint
L3
3.47u
IC = 20
out
R2
1.88m
18
Verr
CTRmin=0.3
Pole=4k
Copto=1/(2*pi*Pole*Rpullup)
Rpullup=4.7k
Rupper=(Vout-2.5)/250u
RLED1=Vout-Vf-VTL431min
RLED2=Vdd-Vcesat+Ibias*CTRmin*Rpullup
RLEDmax=(RLED1/RLED2)*Rpullup*CTRmin
14
D5
R1
0.01
1: n L
R6
1k
C3
{Cpole}
X8
Optocoupler
Fp = Pole
CTR = CTRmin
Czero
{Czero}
IC = 12
20
21
X10
TL431_G
Rlower
10k
C1
11.8m
IC = 11
Rload
500m
Resulting signals from the switching simulation
3.9 A
0
-100
-200
2.00
I L , peak
Doff
I1
I L (t ) n
I L , peak
I L , valley
4.35 A
3.55 A
0
2
-2.00
− I1
∆I = I1 + I L ,valley
I lleak ( t )
V prim ( t )
-4.00
1
t
491 ns
40.0
∆D
20.0
Plot2
p, vint in volts
vxfmr in volts
100
6
4.00
p, ileak in amperes
200
Tsw
Deff
2
0
4
5
498 ns
-20.0
-40.0
9.9959m
9.9968m
9.9977m
time in seconds
9.9986m
10.000m
Equations to derive the leakage inductance reset time
Voltage across the transformer primary
∆ DTsw Deff Tsw
Doff = 1 − D
Vin
D = ∆D + Deff
Vout
Deff =
nVin
0
DTsw
Tsw
Doff Tsw
Doff + D = 1
−Vin
Output inductor equations
∆I L =
I L , peak

Vout 1 − Deff
∆I L 

= n  I out +
= n  I out +


2 
2 Fsw L


 



Vout 1 − Deff  

−

2 Fsw L

I L , valley

∆I L 

= n  I out −
= n  I out


2 


Vout (1 − Deff
)
LFsw
Fsw is the output
ripple period which is
twice the primary
leakage period
Equations to derive the leakage inductance reset time
Vout
I1 = I L , peak − Doff Tsw S f = I L , peak − (1 − D ) nTsw
L
Vout
∆I L 

I1 = n  I out +
− (1 − D ) nTsw

2 
L


Vout (1 − Deff )
Vout

I1 = n I out +
− (1 − D ) Tsw

2 LFsw
L





The current excursion across the leakage inductor is
∆I = I L , valley + I1
The voltage excursion across the leakage inductor is
Vin
The current slope across the leakage inductor is
The leakage inductor reset time is then:

Vout 1 − Deff
∆I = n  I out −

2 Fsw L

Vin lleak
t = ∆I lleak Vin

 
Vout (1 − Deff )
Vout
 + n  I out +
− (1 − D ) Tsw


2 LFsw
L






Equations to derive the leakage inductance reset time
∆I
∆D =
Vin
Tsw
lleak

Vout 1 − Deff
n  I out −

2 Fsw Lout

∆D =

 
Vout (1 − Deff )
Vout
 + n  I out +
− (1 − D ) Tsw


2 Lout Fsw
L


Vin
Tsw
lleak




Vout

n  2 I out − Tsw
(1 − D ) 
L


∆D =
Vin
Tsw
lleak
The two formulas exactly match, Ts is the controller switching
period whereas Tsw is the output ripple period: Ts = 2 Tsw
V. Vlatkcovic and al., «Small-Signal Analysis of the Phase Shifted PWM Converter», IEEE transactions, Vol. 7, n°1, January 1992
Equations to derive the leakage inductance reset time
Numerical application:
1
n = , I out = 24 A, Fsw = 500 kHz , lleak = 15.7 µH
6
L = 3.43 µH , Vin = 240 V , Vout = 12 V
V


n  2 I out − Tsw out (1 − D ) 
L
 = 0.245
∆D = 
Vin
Tsw
lleak
∆DTsw = 490 ns
D = ∆D + Deff
Simulation gives 491 ns
Vout
12
= ∆D +
= 0.245 +
= 0.54
nVin
166m × 240
I L , peak = 4.408 A
I L ,valley = 3.591 A
I1 = 3.9 A
In the PWM switch model, the ripple is neglected
In the previous expression, the leakage inductor current swinged between the
peak value and I2.
On average, we consider the inductor current ripple free
Therefore, the leakage current swings between –nIout and +nIout
In the buck average model, Iout = Ic.
The average equation for ∆D is therefore:
∆Davg
2lleak I out Fsw
=
nVin
Gives 0.262
In the PWM switch, Ic is the output current terminal and Vin is applied across
Vap:
∆Davg
2lleak I c Fsw
=
nVap
Conduction Mode Transition Point Calculations
Discontinuous Conduction Mode, classical buck:
The critical point is the same as for a classical
buck except that Deff must be used:
Rcrit
nVin
0.166 × 240
= 2 LFsw
= 2 × 3.47u × 1Meg
= 9.9 Ω
nVin − Vout
0.166 × 240 − 12
This is the output ripple
frequency or twice the
oscillator value.
Model transition point:
Doff = 1 − Deff
∆D
Deff
Doff
Tsw
Doff = 1 − ( D − ∆D )
Doff = 1 − D + ∆D
Conduction Mode Transition Point Calculations
The mode transition can also be derived by cancelling the equation
describing ∆D:
V
V


n  2 I out − Tsw out (1 − D ) 
L
 =0
∆D = 
Vin
Tsw
lleak
∆D =
∆D +
out
nVin
lleak (TswVout 2 − nTswVoutVin + 2nI out LVin )
TswVin ( LVin − nlleakVout )
=0
lleak (TswVout 2 − nTswVoutVin + 2nI out LVin ) = 0
I out , crit =
Rcrit =
Vout
I out ,crit
TswVout ( nVin − Vout )
2nLVin
2nLVin
2 × 0.166 × 3.47u × 240
=
==
= 9.9 Ω
Tsw ( nVin − Vout )
1u × ( 0.166 × 240 − 12 )
The phase-shifted converter has been loaded by a 9.9-Ω resistor
The switching ripple on the output inductor is 1 MHz
2.00
1.00
0
1
I Lout ( t )
Rload = 9.9 Ω
-1.00
Boundary
condition
-2.00
50.0544m
50.0550m
50.0556m
time in seconds
50.0561m
50.0567m
A formula including the effect of the leakage inductance has been
derived by Monsieur Schutten from General Electric:
Rcrit = 2 Fsw
(l
2
n
+ L)
leak
Vout
1−
nVin
It is similar to those derived before except that the total leakage
(reflected to the secondary side) is accounted in series with the output
inductor.
For large turns ratios, the contribution of the reflected leakage term
is weak. However, for smaller ratios, it can make a difference between
the results delivered by the first set of equations and Mr Schutten’s
equation.
Conduction Mode Transition Point Calculations
Deff
c
a
D + Doff = 1 CCM
D − ∆D
D + Doff
1
Vap
∆D = 0
DCM
1: D − ∆D
D
1:
D + Doff
Vcp
To match
d − ∆D
the book notation: 1: 1
d1 + d 2
p
In the DCM PWM switch model described page 166 of my book:
d2 =
2 LFsw I c
− d1
D Vac
d2 is the off duty-cycle in DCM, d1 is the on duty-cycle
When d2 hits 1 − d1 + ∆D
The model leaves DCM and enters CCM
By clamping d2 between 0 and 1 − d1 + ∆D the models auto-toggles.
Average model implementation
.SUBCKT PWMswitchPS a c p d dL params: L=10u Ll=1u Fs=100k
*
* This subckt is DCM-CCM phase shift average model
*
* 1 - active ; 2 - passive ; 3 - common ; 4 - duty-cycle ; 5 - dL
*
.subckt limit d dc params: clampH=0.99 clampL=16m
Gd 0 dcx d 0 100u
Rdc dcx 0 10k
V1 clpn 0 {clampL}
V2 clpp 0 {clampH}
D1 clpn dcx dclamp
D2 dcx clpp dclamp
Bdc dc 0 V=V(dcx)
.model dclamp d n=0.01 rs=100m
.ENDS
*
.subckt limit2 d2nc d d2c dL
Gd 0 d2cx d2nc 0 100u
Rdc d2cx 0 10k
V1 clpn 0 7m
BV2 clpp 0 V=1-V(d)+V(dL)-6.687m
D1 clpn d2cx dclamp
D2 d2cx clpp dclamp
B2c d2c 0 V=V(d2cx)
.model dclamp d n=0.01 rs=100m
.ENDS
*
Xd d dc limit params: clampH=0.99 clampL=16m
BVcp 6 p V=((V(dc)-V(dL))/(V(dc)-V(dL)+V(d2)+1u))*V(a,p)
BIap a p I=((V(dc)-V(dL))/(V(dc)-V(dL)+V(d2)+1u))*I(VM)
Bd2 d2X 0 V=(2*I(VM)*{L}-v(a,c)*V(dc)^2*{1/Fs}) / ( v(a,c)*V(dc)*{1/Fs}+1u )
Xd2 d2X dc d2 dL limit2
BdL dL 0 V=2*{Ll}*I(VM)*{Fs}/(V(a,p)+1u) < 0 ? 0 : 2*{Ll}*I(VM)*{Fs}/(V(a,p)+1u)
VM 6 c
*
2lleak I c Fsw
.ENDS
2
Leakage inductance
Accounts for ∆D
∆Davg =
Clamp of d
nVap
2 I c L − Vac d12Tsw
d2 =
Vac d1Tsw
Average model implementation, IsSpice
X4
PWMswitchPS
L = L1/N^2
Ll = Ll
Fs = Fs
Calculations give 0.245
dL
236mV
Vout
a dL
c
12.0V
72.3V
3
8
240V
538mV
1
Vin
{Vin}
5
p
out
12.0V
7
12.0V
X2
XFMR
RATIO = N
d
PWM switch VM
R3
1m
L1
{L1}
R1
1.88m
Rload
0.544
12.0V
2
C1
11.8m
.544
full load
Vdd
5
out
GAIN
5.00V
13
X1
GAIN
K = 333m
R7
{Rpullup}
1.61V
Rupper
{Rupper}
Rled
{Rled}
9.70V
9
10
Verr
LoL
1kH
1.61V
R6
1k
R9
1
X8
Optocoupler
Fp = Pole
CTR = CTRmin
1.61V
19
CoL
1kF
0V
C3
{Cpole}
8.48V
Czero
{Czero}
14
2.50V
20
Vstim
AC = 1
15
X10
TL431_G
Rlower
10k
Ac simulations versus prototype values
M. Schutten
model
IsSpice
IsSpice
Control to output
M. Schutten and al., “Improved Small-Signal Analysis for the Phase-Shifted PWM Power Converter”, IEEE Transactions, Vol. 18, n°2, 2003
Ac simulations versus prototype values
IsSpice
Audio susceptibility
M. Schutten and al., “Improved Small-Signal Analysis for the Phase-Shifted PWM Power Converter”, IEEE Transactions, Vol. 18, n°2, 2003
Mode transition point
The load resistor is increased until the ac response changes
1
3
6
7
CCM mode
Rload = 8 Ω
40.0
20.0
0
-20.0
-40.0
H (s)
DCM
-35.6
DCM mode
Rload = 10 Ω
-46.5
CCM
-57.5
-68.5
∠H ( s )
-79.5
10
100
1k
frequency in hertz
10k
100k
Transient simulations versus cycle-by-cycle results
12.07
Cycle by
cycle
Plot1
vout, vout#a in volts
12.03
11.99
Average model
11.95
11.91
9.70m
11.1m
12.5m
time in seconds
13.9m
15.3m
Conclusion
A simple and efficient average model has been derived
It automatically toggles between CCM and DCM
It predicts the mode transition point
It matches, in CCM, ac prototype measurements
It can easily be ported to different simulator languages