3-6
Topic 3
Traditional Modeling Methods Do Not Work Well
To develop a transfer function, all variables
should be defined by equations governed by the
LLC converter topology shown in Fig. 5a. These
equations are then solved to get the transfer
function. Conventional methods such as state-
space averaging have been successfully used in
modeling pulse-width-modulated switching
converters, but from a practical viewpoint they
have proved unsuccessful with resonant converters,
forcing designers to seek different approaches.
Modeling with Approximations
As already mentioned, the LLC converter is
operated in the vicinity of series resonance. This
means that the main composite of circulating
current in the resonant network is at or close to the
series resonant frequency. This provides a hint that
the circulating current consists mainly of a single
frequency and is a pure sinusoidal current.
Although this assumption is not completely
accurate, it is close—especially when the square
wave’s switching cycle corresponds to the series
resonant frequency. But what about the errors?
If the square wave is different from the series
resonance, then in reality more frequency
components are included; but an approximation
using the single fundamental harmonic of the
square wave can be made while ignoring all higher-
order harmonics and setting possible accuracy
issues aside for the moment. This is the so-called
first harmonic approximation (FHA) method, now
widely used for resonant-converter design. This
method produces acceptable design results as long
as the converter operates at or close to the series
resonance.
The FHA method can be used to develop the
gain, or the input-to-output voltage-transfer
function. The first steps in this process are as
follows:
Represent the primary-input unipolar square- •
wave voltage and current with their funda men tal
components, ignoring all higher-order
harmonics.
Ignore the effect from the output capacitor and •
the transformer’s secondary-side leakage
inductance.
Refer the obtained secondary-side variables to •
the primary side.
Represent the referred secondary voltage, which •
is the bipolar square-wave voltage (V
so
), and the
referred secondary current with only their
fundamental components, again ignoring all
higher-order harmonics.
With these steps accomplished, a circuit model
of the LLC resonant half-bridge converter in Fig.
5a can be obtained (Fig. 5b). In Fig. 5b, V
ge
is the
fun damental component of V
sq
, and V
oe
is the fun-
damental component of V
so
. Thus, the non linear
and nonsinusoidal circuit in Fig. 5a is approx i-
mately transformed into the linear circuit of Fig.
5b, where the AC resonant circuit is excited by an
effective sinusoidal input source and drives an
equivalent resistive load. In this circuit model,
both input voltage V
ge
and output voltage V
oe
are
in sinusoidal form with the same single frequency—
i.e., the fundamental component of the square-
wave voltage (V
sq
), generated by the switching
operation of Q1 and Q2.
This model is called the resonant converter’s
FHA circuit model. It forms the basis for the
Fig. 5. Model of LLC resonant half-bridge converter.
I
r
L
r
+
–
+
+
–
+
C
r
V
oe
L
m
I
m
I
oe
R
e
I
r
I
os
L
r
C
r
V
sq
V
ge
V
sq
V
so
V
so
L
m
I
m
R´
L
a. Nonlinear nonsinusoidal circuit.
b. Linear sinusoidal circuit.