AND8112 Datasheet, PDF (2/12 Pages) ON Semiconductor – A Quasi-Resonant SPICE Model Eases Feedback Loop Designs

AND8112/D
As in any sinusoidal signal, there are peaks and valleys. When you re−start the switch in one valley, where the voltage is
minimum, the MOSFET is no longer the seat of heavy turn−on losses engendered by capacitive effects: this is the so−called
Quasi−Resonance operation where the switching frequency depends on the peak current, the various slopes ON and OFF and
the number of valleys you choose after the core reset. In our study, we will first concentrate on a simplified SPICE version where
the power switch is actuated right after the reset detection point (parasitic ringings are neglected) and later on, a more
sophisticated declination will incorporate parasitic delays.
Modeling the Switch Network
Figure 3 depicts a Flyback topology where the switching
elements generating the above waveforms have been
highlighted: the power switch (usually a MOSFET
transistor) and a diode, performing a rectification job.
During the converter operation, the Pulse Width Modulator
controller (PWM) instructs the transistor to turn ON, in
order to store energy in the primary side. The primary
current builds up until the setpoint imposed by the feedback
loop is reached. At this time, the controller toggles the
transistor to the OFF state and energy transfers to the
secondary side. If the ON and OFF states can be described
by a set of linear equations, there exists a discontinuity
linking these two events. Despite the presence of linear
elements in the converter (capacitors, inductors and
resistors), the presence of the commuting switch clearly
introduces the non−linearity that prevents us from directly
writing the small−signal equations…
When learning electronic circuits at school, there were
some exercises in which we were asked to reveal the transfer
function of bipolar amplifiers. At that time, we learned to
replace the transistor symbol by its equivalent small−signal
model: the schematic turned into the simple association of
current and voltage sources that greatly simplified the
analysis. In the average circuit modeling technique, we also
follow the same philosophy: the exercise lies in isolating and
replacing the switch network with a set of current and
voltage sources whose electrical architecture do not vary
with time. Therefore, plugging the equivalent model back
into the converter of interest allows us to resolve its transfer
characteristics.
I_L(t) 1:N
1
L_P
+
V_g
I_1(t)
d
a
2
3
V_1(t)
s
k⁵
C
Control
I_2(t)
V_2(t)
R V_(t)
Figure 3. A Flyback Power Supply where Switches
have been Isolated0
I_L(t)
I_P
Vg
LP
*
N
V
@ LP
I_1(t)
I_P
Vg
LP
I_2(t)
I_P/N
*
N2
V
@
LP
V_1(t)
V
N
)
Vg
V_2(t)
N @ Vg ) V
(V(L_P)
V_g
*
V
N
d @ TS TS
d @ TS
T_S
Figure 4. 0and Individual Signals Separately Plotted.
Deriving Equations
The object of deriving a model consists in writing the
equations that describe the switch network averaged input
and output quantities that a) depend from each other b) obey
to the control input. Let us draw the various waveforms
before starting any line of algebra (Figure 4). From this
picture, we can develop equations that will finally describe
the averaged evolution of the values of interest, the input /
output voltage and current of our switch network:
t
I1(t)
u+
IP
2
d
(eq. 4)
t
I2(t)
u+
2
IP
N
dȀ
(eq. 5)
V)N
t V1(t) u+
N
Vg
dȀ
t V2(t) u+ [V ) N Vg] d
(eq. 6)
(eq. 7)
where: V is the output voltage, V_g the input voltage, I_P the
primary peak current, N the N_S / N_P turn ratio and d the
duty−cycle (d′ = 1 –d).
Please note that in this first approach, we do not consider
any delay occurring at the switch opening or induced by
equation 3. These events will considered later on, in a more
complex model.
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