AND8112/D

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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.

d

s

a

k

2

5

3

1

C

R

Figure 3. A Flyback Power Supply where Switches

have been Isolated

0

Control

+

1:N

Figure 4.

0and Individual Signals Separately Plotted.

* V

N

V

N

) Vg

*

V

N

@ LP

Vg

LP

Vg

LP

*

V

N2

@ LP

d

@ TS

d

@ TS

N

@ Vg ) V

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+

IP

2

N

d

(eq. 5)

t V1(t) u+

V

) N

Vg

N

d

(eq. 6)

t V2(t) u+ [V ) N

Vg]

d

(eq. 7)

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.