By beginning with the inductor current waveform, we have derived the closed-loop transfer function for peak-current-loop switching converters. Now the goal is to find the transfer functions of the individual blocks in the loop.
The incremental valley-current waveform equation for d (k ) was derived in Part 3 as
Solving for the steady-state il equation, it converts to the z -domain as
As a transfer-function for the power stage,
This can be transformed directly to the s -domain in that z /(z – 1) transforms as 1/s . It can alternatively be transformed to the sampled s -domain by substitution of Z = esTs to result in
Then the sampled il *(s ) can be recovered in stepped form by the inseparable zero-order hold, H 0 (s ) which follows the sampler;
The resulting il /d transfer function is independent of converter configuration in that it is derived from the closed-loop waveforms. The interpretation of GidV is that it is simply an inductor, L , integrating Voff to result in il .
The next historical development from that of Ridley’s sampled-loop model was the “unified” model of Middlebrook and Tan. It put the sampling function where circuit sampling occurs, in or adjacent to the PWM function of the forward path. (Ridley’s He is in the feedback path.) Then a “simplified unified model” derivation proceeds as follows in finding the PWM transfer function, FmV .
Given the feedback loop transfer function, we can equate it to the well-known closed-loop feedback equation,
The feedback path, HC xRS , of the converter block diagram in Part 3 converts sensed voltage to inductor current, and in the equations, we are working directly in current. Thus, the feedback path has a gain of 1 and the forward path is FmV (s )xGidV . With the incremental valley-current sampled-loop expression as derived in Part 4 for the sampled-loop model,
equate corresponding expressions,
GidV (s ) is the converter transfer function il /d which was derived previously in this Part from the valley current;
Having GidV , make the substitution for GidV into FmV (s ) and solve;
The quasistatic FmV (0 ) is
FmV has a single pole at
which is the same pole as in the unified model of Tan and Middlebrook. Note that the D ≥ 1/2criterion for loop instability (which causes a non-LHP pole pair) appears in the Fm block of the unified model. This is not the final refinement of current-loop modeling, however. The sampled-loop and unified models both are based on the valley current value of il at the end of the cycle. This is not really the quantity that is of interest to us in designing power supplies; we want to know how the cycle-average current behaves. We are interested in its response because it is what we see at the converter output.
In the next part of this article, the author’s refined model is summarized, based on beginning with average il in deriving the transfer function. The resulting Fm agrees with other models but not the unified model. If you have come this far and basically have followed the presentation of concepts, you know more about converter dynamics than many converter designers. The sampled-loop and unified models include the subharmonic instability effect and can guide design of slope compensation , a method of altering D to keep the poles in the LHP. There is more to the story, in the next part.