Determining Phase Boost needed to Compensate SMPS

Question

I'm attempting to find the Control-To-Output Bode Plot (open-loop) of a SMPS designed around the TPS54160 so that I can design the type II compensator. The first step in designing a compensator is finding the attenuation and phase boost needed at the crossover frequency (4Khz) using the open-loop gain. Thus, I need the open-loop gain (or the gain of the system without the compensation added). I used the circuit below to find open-loop bode plot, and according to it I need 7.36dB of attenuation at the crossover frequency and some amount of phase boost. I'm not sure about how much phase boost I need (given a phase margin requirement of 60 degrees) because it appears there is 100 degrees of phase margin already since the phase has only shifted 80 degrees at the crossover frequency, since this is a single pole system when open-loop. Perhaps, I'm just confused. Are my model and bode-plot correct? And if so, how do I calculate how much phase boost I need?

In the circuit above G1 is the idealized OTA that the TPS54160 uses for its error amplifier, where R1 and C1 model the bandwidth and DC Open-loop gain of the amplifier. G2 and the output filter form the power stage. I think this model is correct based on the datasheet, which gave relatively simple instructions.

In the bode plot above, I measured V(b) (the output) with an AC source at the input of the power stage. According to this bode plot the phase has shifted 80 degrees at the crossover frequency of 4Khz. The gain is 7.36dB at the desired crossover frequency so it needs to be attenuated.

Answer 1

The phase boost in a type-2 compensator depends on the distance between the zero and the pole. If they are spread apart, the maximum boost goes up to 90° in theory while it goes back to zero when they are coincident. To determine the amount of boost, have a look at the below picture:

Boosting the phase actually means that you will tailor the compensator G so that its argument at the selected crossover frequency \$f_c\$ is less than -270°. These -270° are due to the pole at the origin (integrator) and the inverting op-amp which brings an additional 180° phase shift. In the example, the plant H argument at \$f_c\$ is -175°. Should you try to close the loop without boost at all with a simple integrator, you would end up with a total shift of -175° -270° = -360°-85°, showing a negative margin.

What you want is a positive phase margin implying that the loop gain stays away from the -360° limit at the 0-dB crossover frequency. Selecting this margin depends on the transient response you want but assuming a 70° PM is your goal, then the sum of the plant argument and the newly-designed compensator must be -115°: you must boost the phase by -270° + boost = -115° implying a boost in phase of 155°. You will find more details in the APEC seminar but also in here with a comprehensive coverage of the subject.

Now coming back on your model, it is oversimplified to me as it does not include the sub-harmonic poles contribution. I would advantageously switch to the CM PWM switch model described extensively in Vorpérian's papers or in this APEC seminar.