Software Makes Transfer Functions More Manageable

Jun 1, 2008 12:00 PM
By Ekrem Cengelci, Senior System Engineer, Microsemi, Garden Grove, Calif.

Small-signal analysis of switching dc-dc converters determines their specific transfer functions, for example, for stability analysis or design of a proper input filter. Transfer functions of interest for switching dc-dc converters are input impedance, output impedance, duty cycle (or control voltage) to output voltage, and input voltage to output voltage. Two pieces of literature about accomplishing small-signal analysis of switching dc-dc converters suggests two fundamental methods:

1. The state-space averaging method, by Dr. Middlebrook et al^[1], employs algebraic manipulation of a set of state-space equations to derive average model equations of a switching converter topology. Linearizing the averaged model at a dc operating point derives the desired transfer function. With this method, the complete circuit must be averaged and linearized every time the topology is changed.

2. The method of PWM switch model, by Dr. Vorperian^{[2-4, 8]}, averages and linearizes the dc-dc converter's switch pair, which is its only nonlinear portion, and invariantly uses this model in different topologies. The main advantages of a PWM switch approach are:

   • The PWM switch model is already averaged and linearized, so no averaging and linearization is necessary, which differs from the state-space averaging method.

   • As a circuit-oriented approach, it is very convenient for use with numerical or symbolic circuit-analysis software.

   • The PWM switch model's properties are the same for a given conduction and control mode, so they can be used invariantly from one topology to another.

Because of these advantages, particularly that it is circuit oriented, the PWM switch model is best suited for obtaining symbolic and numeric small-signal equations of switching dc-dc converters. The small-signal model of a switching converter with a PWM switch model yields a linear circuit, and finding the transfer function of interest requires analyzing this linear circuit.

PSPICE has built-in PWM switch models for a numerical analysis of switching dc-dc converters. The models are simple enough to remodel on any other numerical electrical simulation tools, even if PSPICE is not available. However, obtaining circuit equations of such a linear circuit (symbolic analysis) is done by manually analyzing the circuit.

Depending on the level of complexity with the equivalent linear circuit, obtaining the analytical transfer function of interest may turn out to be a very difficult task. This can occur if there are a large number of terms in the resultant transfer function when using standard linear circuit-analysis techniques (nodal or loop).

Alternatively, Middlebrook suggested the extra element theorem (EET) to analyze linear circuits in a more efficient manner.^[5-7] This method reduces the number of mathematical manipulations and simplifies obtaining the transfer function of interest.

Vorperian's book, Fast Analytical Techniques for Electrical and Electronic Circuits, is a good reference for the EET and provides various application examples.^[8] Although the EET method is much more practical and useful compared to nodal or loop analysis methods, it still requires manual manipulation of circuit equations and becomes impractical when the number of circuit elements exceeds a certain number.

For example, let's say a designer is interested in the small-signal PWM switch-equivalent circuit model of a buck converter to obtain duty-cycle to output-voltage transfer function with voltage-mode control in continuous conduction mode (CCM). By including the inductor's dc resistance and the capacitor's ESR, there are eight circuit elements in the equivalent circuit, and the resultant transfer function contains only about 25 terms of circuit parameters. EET easily analyzes this circuit to find the transfer function of interest.

On the other hand, the complexity increases when determining the small-signal PWM switch-equivalent circuit model of, for example, the nonisolated, single-ended primary inductor converter (SEPIC) to obtain its duty-cycle to output-voltage transfer function with voltage-mode control in discontinuous conduction mode (DCM). Now, by including the inductors' dc resistances and the capacitors' ESR, there are 14 circuit elements, and the resultant transfer function contains more than 700 terms of circuit parameters when expanded and collected with “s.”

Obviously, it is a major task to obtain the transfer function of the SEPIC converter topology manually even with the EET method. Besides, it is a good possibility to inadvertently introduce errors during this manual analysis process.

Needless to say, it also requires learning and digesting the EET method. Besides, the complexity level of the equivalent circuit analysis is much higher with the current-mode control if all the parasitic elements of the reactive components and subharmonic oscillation model parameters are included (the final transfer function contains thousands of parameters with the SEPIC converter topology). Analyzing such a circuit even with the EET method is an extremely long (or impossible) task unless you omit some parasitic elements to simplify the circuit.

There is a computer-based method to analyze such linear circuits and obtain analytical expressions for the transfer functions of interest. The method works even for transfer functions containing thousands of parameters in it. It is fully computer based, so a designer can obtain and analyze the resultant transfer functions much faster and more reliably than manual analysis methods. Tools needed for such analysis are:

• Linear circuit-analysis software that can perform symbolic circuit analysis

• Math software to post-process the transfer function equation generated by the circuit-analysis program

• An executable program to format the transfer function equation generated by the circuit-analysis program in the form required by the math software.

The details of this method are explained and demonstrated next.

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