# Seemingly Simple Circuits, Part 5: Current-Loop Converter Transfer Function

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.

## 5 comments on “Seemingly Simple Circuits, Part 5: Current-Loop Converter Transfer Function”

1. etnapowers
February 23, 2015

“incremental valley-current sampled-loop expression as derived in Part 4 for the sampled-loop model”

@Dennis: interesting blog, I wonder if the approximated expression of the sampled-loop model introduces second order effects appearing outside the normal range of working of the switching converter.

2. D Feucht
February 23, 2015

I am not sure what the “normal range” of current-loop converters is supposed to be, though the model is valid up to the Nyquist frequency. The approximation of significance is for the delay within a switching cycle, approximated by a quadratic zero and pole pair which introduces a little error but not enough to detract from capturing the major dynamic behavior of the current loop.

3. etnapowers
February 24, 2015

@Dennis: I'm wondering what happens if the zero-pole couple is splitted  for the presence of a drift of a parameter, as an example?

4. D Feucht
February 24, 2015

Let me recount a little. The sample point within a switching cycle where the PWM comparator senses the peak current and changes output state does so at a time that is in advance of the valley point of the current at the end of the cycle. This time difference, like a delay, is accounted for in the average current expression by a rational polynomial approximation of a delay, which is an infinite number of poles and zeros. I reduce it to 2 with a Pade' approximation of one pole-pair and one zero-pair. They do not drift relative to each other because they are an approximation of the delay. It doesn't “drift” either because it is part of the loop dynamics and changes with it. The only drift is in the circuit elements and voltage reference.

5. 