Basso SMPS Book#2 Comment 2 - Chris Basso Generic Averaged Controller models

About the writer: Harvey Morehouse is a contractor/consultant with many years of experience using circuit analysis programs. His primary activities are in Reliability, Safety, Testability and Circuit Analysis. He may be reached at harvey.annie@verizon.net. Simple questions for which I know the answer are free. Complex questions, especially where I am ignorant of the answers, are costly!!!

Summary: As a result of my previous book comments, some correspondence with Christophe Basso ensued, and most of the comments were covered in the first comment article. This article deals with another of his comments, which dealt with compensation of the error amplifier. Here it is shown how confusion can easily arise, that could easily have been avoided had a modulator gain block been used.

Chapter 1-2, FLYBACKVM.CPM CIRCUIT:

Chris provided the following four figures to me in a correspondence.

Figure 1
FLYBACK circuit

A phase-gain plot of the circuit is shown in the following Figure 2:

Figure 2
Phase-Gain plot

Figure 3
Step Load circuit

10 kHz BW, 70° phase margin in CCM, Rload = 1 Ohm.

Figure 4
Step Load Circuit graph

The purpose of the Figures was to show (as IF I thought otherwise) that the circuit could indeed be stabilized and the step load response found.

It is instructive, however, to examine these results a bit closer. Consider Figure 1. In that circuit, X1 is shown as an op-amp with a maximum voltage output of 4V. Now, in Figure 3, the same op-amp is shown with a maximum voltage excursion of 0.99V! Are the circuits identical in performance? If so why, and if not why not?

The first thing to consider is Figure 2. Now in Figure 2, this is an AC sweep. SPICE performs this by first performing a DC analysis of the circuit. All capacitors are opened, all inductors are shorted, and a DC operating point determined.

EXCEPT for a case where this would result in a solution with the op-amp output stuck at the low or high rail, it will result in the op-amp operating in the linear region. Actually, this region will be somewhere between zero (0%) and one (100%) for commonly used modulator and switch modules. And, in the linear region, the gain is 30,000 in both circuits.

Now in a realistic circuit, as modeled in the general case, with a 4V error amplifier maximum output, and the switch circuit expecting that 1V control input would represent 100% output, there is a missing 'modulator gain' block in series with the modulation control input. This block has a nominal gain equal to the inverse of the error amplifier output high voltage. For an error amplifier swing between zero to 4V, the gain would be ¼ or 0.25. This enables a 4V error amplifier output to command the modulator to 100%.

Now the question arises, is using a modulator gain block with a gain of 0.25 and an error amplifier output high level of 4V identical to using an error amplifier output of 1V and a unity gain modulator block? Clearly not, but let us use two instances of a circuit similar to that of Figure 1, one having an error amplifier max voltage of 4V and a 0.25 modulator gain, and the second with an error amplifier max output voltage of 1V and a modulator gain of unity. This is shown in Figure 5 following:

Figure 5
Comparison circuits

In Figure 5 we have replaced the source batteries with voltage sources, as we may have need of their expanded capabilities later. Arbitrary sources B1 and B2 provide modulator gain functions, with gains of 0.25 and 1.0 respectively.

Here we will be plotting the AC gain from the output of the error amplifier(s) to the voltage output(s). This is shown in Figure 6 following:

Figure 6
Comparison circuit AC plot

In Figure 6 it may be seen that the curves have identical shapes (graphing differences offset the gain curves slightly). However, the curve for the left circuit of Figure 5 has 12 dB less gain than that of the right circuit. Remember, the AC plots are linearized plots about the DC operating point, which is where the op-amp is in its linear region with a gain of 30,000 in both cases. But neglecting the modulator gain overstates the 'plant' gain by 12 dB, as the modulator gain is always present. In this case, -12dB represents a gain of ¼.

Now one CAN overstate the gain, compensate the op-amp, perform an analysis on the circuit (again without the modulator gain) and get results that indicate that the circuit is stable. But when the modulator gain element is then used, the results are wrong!!

The next question is, regarding the frequency compensation for the error amplifier. Were it determined with an erroneously large plant gain, the compensation would almost certainly be sufficient to stabilize the circuit with or without a modulator gain block, however, the results would differ. The differences would show up in transient situations, and also in the closed loop curves.

For maximum versatility in graphing, the circuit was modified somewhat to allow the close loop response to be determined, as well as to allow a meaningful transient analysis of the two circuits to be performed. This is shown in the circuit of Figure 7 following:

Figure 7
Closed loop comparison circuits

Sources vstim1 and vstim2 allow the closed loop gains to be determined in an AC analysis, but allow a transient analysis to be performed easily as well. The closed loop gain is the ratio of the error amplifier output to the modulator input, the voltage across the stimulator source. Two separate graphs, one for each circuit, will be generated. The AC analysis graph for the left side circuit with the modulator gain equal to 0.25 is shown in Figure 8 following:

Figure 8
Closed loop with modulator gain graph

The red curve in Figure 8 is the phase, whereas the black plot is the gain. There are two zero dB crossings near 4 kHz. At this region the phase margin is 150 and 110 degrees. At the zero phase point shown, the gain margin is about 27 dB. This is a stable plot.

Figure 9
Closed loop with unity modulator gain graph

In Figure 9 two gain crossings are again seen, at about 2.0 and 8.0 kHz. The phase margin is about 170 and 130 degrees. The phase crossing is seen at about 60 kHz, and the gain margin is about 12 dB. This is also stable.

Examining the transient performance now, Figure 3 shows a step loading from 1 to 10 ohms. Removing the load resistor, adding a switch with one ohm on-resistance and 10 ohms off-resistance and performing a transient analysis, the graph shown in Figure 10 following occurs

Figure 10
Transient analysis graph1

In Figure 10 can be seen significant differences in performance between the two circuits. Now the step loading results can be seen by delaying the start of the transient output waveforms, as shown in Figure 11 following:

Figure 11
Transient analysis graph2

Figure 11 shows that the compensation of the first circuit is inferior to that of the second circuit, as the peak voltage excursions and recovery times are longer than that of the second circuit, however, the first circuit with a modulator gain term is more representative of an actual circuit. It is the compensation that is in error.

Again, the technique is not wrong, however it could give the wrong impression to readers. Now the question arises, who am I to question Chris Basso? He is an author, both of several books and many articles!

I refer to an article entitled "Designing Stable Control Loops" by Dan Mitchell and Bob Mammano, a Unitrode document found on the web as slup173 Designing Stable Control loops.pdf. Specifically, a model of a typical converter is shown as:

Figure 12
Flow graph of linearized buck converter with voltge-mode control

Now, Vp is the max positive output of the error amplifier, which is part of the forward path between VE(s) to Vo(s). The 1/Vp term is part of the modulator. The article describes how to tailor the error amp compensation based on the open loop gain of the remainder of the circuit. Numerous other articles make this same point, including Chris Basso's excellent book. I cannot understand how he allowed this to happen.

Summary:

Notwithstanding that some persons might not catch the omission in the latest Chris Basso book and get confused, I still think it is one of the best books available. His technique as shown in the book is NOT in question! What is in question is the overall circuit he used it with. I still like his book. It is GREAT!!

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