

C. Larouci, J.P.
Ferrieux, L. Gerbaud, and J. Roudet are a group of professors and
researchers at the laboratoire d'electrotechnique de Grenoble.
Their principal research orientations include power electronics
structures, electromagnetic compatibility (EMC), and the
development of the tools dedicated to sizing in electrical
engineering.



Simulation programs such as Saber, Pspice,
and Simplorer are effective tools for timedomain analysis of power
circuits. However, if these structures have an AC input and a high
switching frequency (various time scales), timedomain simulation
becomes a long, memoryintensive process before you get a
steadystate solution. The use of analytical models is a good
compromise between the tool speed and the model accuracy.
This article presents the timedomain study of the Flyback
structure. We will demonstrate the optimization of the total volume
of this structure under EMC and loss constraints using analytical
models.
TimeDomain Study of a Flyback
Circuit
You use the Flyback circuit (Figure 1) as a singlestage
converter. To ensure a sinusoidal input current, you use a
mixedcontrol circuit that combines two conduction modes:
discontinuous and continuous. This circuit controls the
instantaneous average value of the input current (I1 in Figure
1) to a sinusoidal reference. On the other hand, you obtain the regulation of
the output voltage by using a bulky capacitor C.
Figure 1: Schematic of the flyback circuit
Figures 2 and 3 present the filtered input current
and the output voltage simulated by Pspice and Gentiane software with the following parameters:
 Switching frequency: Fs = 50kHz
 Primary inductance: L1 = 2mH
 Transformation ratio: m = 0.5
 Output capacitance: C = 7mF (sized to have 1% ripple at
48V)
 The inductance and the capacitance of the input filter are Lf =
2mH and Cf = 0.1µF.
Figure 2: Input current for the flyback circuit
shown in Figure 1
Figure 3: Output voltage for the flyback circuit
shown in Figure 1
In this case, you obtain the steady state after three to four
hours of simulation with Pspice, which confirms the difficulty of
the timedomain study for such an application. The analytical
modeling approach provides results in less time than
simulation.
Analytical Method for Optimizing the
Flyback Circuit
The optimization technique consists of varying the parameters of
interest in an analytical model.
Analytical Model of the MixedControl Circuit
During discontinuous conduction of the flyback circuit, the duty
cycle a is constant and is a function of
the output power Po, the switching frequency Fs, the primary
inductance L1, and the input voltage amplitude Vmax:
However, this duty cycle is variable in the continuous
conduction mode according to Equation 2:
The EMC Spectrum Analytical Model
The analytical model of the mixedcontrol circuit lets you estimate
the primary current spectrum (in other words, the current spectrum
in the switch). You use this spectrum like a differential mode
generator to perform frequency modeling.
Figure 4: This schematic shows an equivalent
diagram of the Flyback circuit in the differential mode. In this
circuit p is the Laplace operator; Z1(p) and Z2(p) are the Line
Impedance Stabilization Networks (LISNs); Z3(p) and Z4(p) are input
filters; and Ih(p) is the differential mode generator.
From Figure 4, the EMC spectrum V_{LISN}(p) in
the Laplace space is expressed as:
Figures 5 and 6 present the superposition of the
analytically estimated and simulated primary current spectrum and
the difference between these two spectrums.
Figure 5: Simulated and analytically estimated
results for the primary current spectrum
Figure 6: The difference between the simulated and
estimated spectrums shown in Figure 5
The results of Figure 6 validate the analytical
estimation of the primary current spectrum for the mixedcontrol
circuit and differential mode generator.
Volume Analytical Models for Optimizing the Objective
Function
The volumes of the transformer and inputfilter inductor are
related by the component winding areas and other parameters. These volumes are expressed with an analytical
formula (Equation 4) by introducing:
 Electrical quantities (the primary and secondary RMS and
maximum current values)
 Technological quantities (the current density and the peak flux
density)
 Geometrical quantities (physical coefficients that depend on
the magnetic shaping circuit and conductor insulation).
From the constructor abacus data, the volume of the input filter
capacitor for the nominal voltage is analytically evaluated by
(Equation 5):
Having an analytical model of volumes, the sum is an objective
function. The aim of the optimization algorithm is to minimize this
objective function under EMC and loss constraints.
Optimizing the Flyback Circuit
The developed analytical models are integrated in an
optimization process. The optimization parameters are:
 Primary inductance (L1)
 Transformer ratio (m)
 Inductance and the capacitance of the input filter (Lf and
Cf)
 Switching frequency (Fs).
The objective is to seek the best combination of these
parameters to minimize the total volume of the structure, meet EMC
requirements, and operate with good efficiency (minimize total
circuit losses). These models have been introduced into two
environments that offer different optimization algorithms (EDEN, Mathcad).
We use the parameters of the timedomain study as an initial set
of values to start optimization (Equation 6). After the
optimization procedure, the algorithm converges towards a new set
of values (Equation 7) where the EMC constraint is met
(Max_EMC_spectrum < 79 dbµv, imposed by the ISM 55011
standard) and circuit volume is 1.66 times smaller with
almost the same efficiency.