In this last part of a long article, we look at the final refinement for waveform-based modeling which is based on average switching-cycle current, not on peak or valley current values within the cycle. This is significant because the current of interest to us in power supply design is the average current. Power-supply ratings are based on it.

The sampled-loop model falls short in several ways. One of them is the lack of unification in derivation of the PWM transfer function with the current closed-loop function. In its construction, a separate developmental argument was given for why *F*_{m} should be what it is, and it was then incorporated into the closed-loop function. In the unified model of Middlebrook and Tan, they derived *F*_{m} as we did in Part 5.

Yet the “unified” model is not really unified but is a piecemeal adaptation of the quasistatic average current of the low-frequency average (lf-avg) model and the dynamics of the sampled-loop model. The two are fitted together into a single model but are not derived from a single set of fundamental equations. *F*_{m} appears in a “unified” way, but the unification of the dynamics with the lf-avg model - the first generation model which is accurate at low frequencies - was not accomplished. The reason is simple: the lf-avg model is based on average current, and the dynamics of the per-cycle average is not the same as that of the valley current. In the unified model, the phase introduced by sampling does not directly shift average incremental inductor current ̅*i*_{l} in the cycle. The constant factor in the PWM transfer function, *F*_{m0}, was derived from the lf-avg inductor current while the dynamics came from the sampled-loop model.

For a truly unified model, the full frequency response of the blocks in the block diagram of the model - including both quasistatic and dynamic factors of transfer functions - should be derived from a single set of general equations describing converter circuits. Tymerski achieved the unification of dynamic per-cycle-average inductor current with sampled-loop dynamics in a single set of state-variable equations. Elsewhere, the static *F*_{m0} was extracted from the discrete-time duty-ratio equations. In the unified model, *F*_{m0} is extracted out of ̅*i*_{l} from lf-avg equations but is not used in the dynamics derivations.

The direction taken here is to express ̅*i*_{l} in the discrete time-domain early in the analysis so that subsequent development results in dynamics based on it, thereby modeling ̅*i*_{l} dynamically and allowing *F*_{m0} to fall out of the derivation. This is what I refer to as the *refined* model.

The substitution of ̅*i*_{l} for *i*_{l} in the waveform-derived current-loop transfer function not only changes the constant gain of *G*_{idV} by ˝ but it also introduces additional *i*_{l} terms in the difference equation of ̅*i*_{l}(*k*). We now derive *G*_{id} for average current. Because *i*_{l} is the incremental valley current and not ̅*i*_{l}, for the refined model the average and not the valley current is used. Substitute from the equation that relates average and valley currents for a triangle-wave,

into the incremental current equation,

Setting *i*_{i} to zero, transforming to the *z*-domain, and simplifying,

In the *s*-domain,

This is the converter power-stage function, *G*_{id}. It differs from the valley-current *G*_{idV} by the ˝ factor expected of an average triangle-wave current.

The average-current transfer-function in *z* is then

*T*_{C} (

*z*) transforms to the sampled

*s*-domain as

Sampling is inseparably followed by the hold function. To produce a stepped version of this sampled function, it is multiplied by *H*_{0}(*s*):

Applying the two-point fit of *H*_{e}(*s*),

the resulting approximation is

*T*_{C} has the pole-pair of the sampled-loop *T*_{CV} with stability for *D* < ˝. The numerator accounts for the phase shift of ̅*i*_{l}(*k*)
from the valley value of *i*_{l}(*k*). By applying the two-point “modified Padé” approximation for the exponential,

then

The complete transfer function is

In addition to the “half-*D*” pole-pair found in the sampled-loop and unified models, this function has a pole-pair at a fixed damping of ζ= π/4 ≈ 0.785 and pole angle of about 38.24^{o}. It also has a LHP complex zero-pair with damping

ζ_{z} varies with *D* = [0, ˝, 1] by ζ_{z}= [π/4, π/8, 0] corresponding to zero angles of *φ*_{z} ≈ [38.24^{o}, 66.88^{o}, 90^{o}].

Following the construction of the simple unified model, equate the expression of *T*_{C}(*s*) with that of the closed feedback loop:

while substituting

Then solve for the new expression for *F*_{m}(*s*), which is

,

Substituting,

In normalized form,

from which

Both magnitude and phase of *F*_{m}(*s*) are flat, increasing significantly in the last decade before the Nyquist frequency.

Simplifying *F*_{m}, the cubic denominator of *F*_{m}(*s*) factors into

where for [*s*/(*ω*_{s}/2)]^{2} << 1, the rightmost term is approximately zero and

This is not unlike the single-pole *F*_{m}(*s*) of the unified model. Compared to the sampled-loop static forward-path gain for which G_{id} = ˝ xG_{idV}, *F*_{m0} is effectively 2/
Δ
*I*_{L0}x*D*’. Unlike the simple unified model, at *D* = ˝, *F*_{m0} does not go to infinity and is

The simple unified model *F*_{m0} goes to infinity at *D* = ˝ whereas alternative *F*_{m} expressions in other models, including the refined model, remain finite.

This last part of this article is not at rock-bottom depth yet. What has not been considered is slope compensation of subharmonic instability nor the inclusion of incremental input and output voltages in the model. The peak-current controller is a structurally simple circuit that is anything but simple to analyze and understand in depth. I have written more on this elsewhere (web-search on my name) as has Ray Ridley and more recently, Robert Sheehan of NSC and now TI. His modeling effort begins with circuits, not waveforms. Consequently, he can circuit-simulate his modeling and it is more accurate but not as general. He has been trying to generalize it. Perhaps the final model will be a convergence and harmonization of circuit- and waveform-based models.