Static Converter Optimization

	ABOUT THE AUTHORS 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.

This article presents the time-domain 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.

You use the Flyback circuit (Figure 1) as a single-stage converter. To ensure a sinusoidal input current, you use a mixed-control 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 time-domain study for such an application. The analytical modeling approach provides results in less time than simulation.

The optimization technique consists of varying the parameters of interest in an analytical model.

Analytical Model of the Mixed-Control 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 mixed-control 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 mixed-control circuit and differential mode generator.

Volume Analytical Models for Optimizing the Objective Function
The volumes of the transformer and input-filter 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.

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 time-domain 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.