The power stage of switching converters can be modeled essentially as a PWM-switch. A switch seems to be far from the kind of circuit element that can be linearized, but it can be. This Part 2 develops a linear model of the PWM-switch.
In 1986, Richard Tymerski developed an incremental model of the PWM-switch, shown below, where
Then the on-time of the switch is ton = D•Ts and the off-time is toff = D’•Ts.
The PWM-switch has three terminals, labeled P for passive, A for active, and, C, for the common or inductor terminal. The switch is in the A position for the duty ratio, D, fraction of the cycle period and in the P position for D' = (1 – D).
This three-terminal circuit fragment can be regarded as an active device like a transistor, and has three two-port configurations for which input and output ports share a common terminal. The common-emitter (CE), common-base (CB), and common-collector or "emitter-follower" (CC) configurations of transistors have analogs with PWM-switches. Each of the three PWM-switch configurations, based on a common terminal, results in one of the three basic PWM-switch converter circuits.
The PWM-switch terminal voltages and currents are averaged over a switching cycle. This limits the applicable frequency range of the switch model to the Nyquist frequency, fs /2. The switching frequency is the basic parameter affecting allowable system bandwidth, and a model that is valid up to the Nyquist frequency covers the frequency range of interest for linear circuit analysis.
From Part 1, the total duty ratio is
where D is the steady-state operating-point value and d is the incremental variation around D.
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 PWM-switch model. When the switch is in the δ (or D for the operating-point) position, the cycle-average voltage from C to P is δ times that from A to P, because vA = vC during δ. That is,
Similarly, during the off-time (δ'xTs), vC = vP and
For currents, during the on-time (δxTs), the average current into A must equal the average current out of C, or
while during the off-time, while C is connected to terminal P,
The incremental model of the switch can be constructed from these PWM-switch equations. Keeping in mind that the symbol, d, is used as d = dδ where δ is the total duty ratio and d is the incremental duty ratio (and dX is the differential of X, not d •X.) Taking the differential of iA results in
Standard electronics notation is used here whereby the total iC = IC + ic . IC is the static and ic the incremental value.
Applying differential calculus similarly to the other three equations results in the incremental-model equations:
These equations can be expressed graphically by an equivalent circuit, the PWM-switch linear model.
The operating point of the incremental model is set by the static (dc) model parameters, IC, VAP, and D, not unlike biasing a transistor. 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 PWM-switch in a circuit and the dynamic behavior derived using the usual linear methods of analog circuit analysis, as shown below for the CP configuration.
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. One reason is that it forms a low-pass filter with the output capacitor, and this output circuit is in the voltage feedback loop. Because the PWM-switch model is a structural and not a black-box (behavioral) model, it is independent of (and thus unaffected by) the circuit it is placed in.
The assumptions of the model should be remembered. It is valid below the Nyquist frequency (fs/2), it assumes constant values for its three parameters, and assumes inductor flux balance, which is the case in steady-state operation. During startup and other transient conditions, flux balance is still a reasonable approximation whenever the change in switch parameters is small over a switching period.
With the incremental PWM-switch model, it is possible to determine the loop dynamic behavior for voltage-mode control, whereby D is varied to control Vo. It can be substituted into circuit simulations. What remains is to model the inner current loop from its waveform equations.