Power converters and motor current-control loops contain switching elements – nonlinear functions that require linearization for incremental (small-signal) modeling of loop dynamic response. As with transistors, an operating point is chosen and the incremental model is developed using the static (dc) values of currents and voltages at this point. This article develops the incremental switch model.

In 1986, Richard Tymerski, while a doctoral student at VPI, developed a small-signal model of a power switch that assumes average currents and voltages during the switching cycle. Vatche Vorperian, also of VPI, had discovered that the three basic switching converter topologies were the three possible configurations of a three-terminal device, the inductor in series with a current switch, as shown below.

The switch cell has three terminals labeled P for passive, A for active, and, C, for the common or inductor terminal. The duty ratio is D for the active switch position and D' = (1 – D) for the passive position.

This three-terminal "device" is reminiscent of the transistor, with its three configurations. Its input and output ports share a common terminal. The common-emitter (CE), common-base (CB), and common-collector or "emitter-follower" (CE) configurations of transistors have analogs with converter power switches. Each of the three switch-cell configurations, based on a common terminal, results in one of the three basic first-order converter topologies.

The switch-cell terminal voltages and currents are averaged over a switching cycle. This limits the applicable frequency range of the switch model to half the switching frequency, which is the Nyquist frequency. At and above it, aliasing occurs. The switching frequency is the basic parameter affecting allowable system bandwidth, and a model that is valid to the Nyquist frequency covers the frequency range of interest for linear circuit analysis.

The average inductor voltage over a switching cycle must be zero to maintain flux balance. That is,

where, in general,

With zero average voltage across the inductor, it can be removed from the switch-cell model, which simplifies to the Tymerski switch model, shown below.

When the switch is in the D position, the average voltage from C to P is D times that from C to P, because v_A = v_C during D. That is,

Similarly, during the off-time (D'), v_C = v_P and

For currents, during the on-time (D), the average current into A must equal the average current out of C, or

while during the off-time, when C is connected to terminal P,

The incremental (small-signal) model of the switch-cell can be constructed from these switch-cell equations. To avoid confusion with the symbol, d, for incremental duty-ratio, the alternative symbol, d , is used here instead to denote the differential of (which is approximately a "small change in") the following quantity. Taking the differential if i_A results in

Applying differential calculus to the other three equations similarly results in the incremental-model equations:

These equations can be expressed in the form of an equivalent circuit, as shown below.

As an incremental model, the operating point is established by the static (dc) model parameters, I_C, V_AP, and D. The duty-ratio is a large-signal parameter; the model is valid for the nominal value of D of the circuit. Changes around D are d D (or d), the incremental model variable for D. The transformer, with turns ratio of 1:D, converts voltages and currents by the ratio of D.

This circuit model can now be substituted for the switch in a converter topology, and the dynamic behavior of the circuit derived using the usual linear methods of circuit analysis. The series inductor must be included in the circuit model, though it was unnecessary in deriving the switch model under the per-cycle flux-balance assumption. The parameter values are chosen based on the operating point of the circuit. These parameters are readily available from the static design of the converter. Because the switch model is a physical and not a black-box model, it is independent (and thus unaffected by) the circuit it is placed in.

The assumptions of the model should be remembered. It is valid to the Nyquist frequency, it assumes constant values for its three parameters, and assumes inductor flux balance, which must be the case in steady-state operation or the current would change without bound. During start-up and other transient conditions, flux balance is still a reasonable approximation whenever the change in switch parameters is small over a switching period.

Just as transistors can be linearized around an operating point, the Tymerski switch – the core of power converters – can be linearized around a fixed value of the duty ratio, D. This results in a linear circuit (assuming the passive components have linear models) that can be analyzed using frequency-response methods and linear circuits and control theory.

Ó Dennis L. Feucht, 2001