Buck Converter: Crossover Frequency vs. LC Resonance and ESR Effects

[kraw]

Hi everyone! I'm new here — I’ve been looking for a good forum where I could post some technical questions and hopefully also help out where I can, and EEVblog seems like the perfect place. Thanks for letting me be part of the community! I hope I can become an active member here. 

I’ve designed quite a few DC-DC converters before, mostly buck types, and honestly, I’ve always just focused on getting the output voltage and current right. But now I’m seeing some weird behavior in a design based on an eval board, and I’ve realized I need to understand the dynamic side of things much better — especially how the control loop behaves.

This converter switches at 450 kHz, and at around 2 A input, I’m seeing a big ripple (about 1.2 A peak-to-peak) at around 5 kHz, which is pretty much the LC resonance of the output filter. I’m using all ceramic caps, and I’ve taken into account that they drop about 20–30% in value due to bias.

So I’ve got a few conceptual questions I’m hoping someone can help with:

Does the crossover frequency (fc) always need to be higher than the LC resonance? If yes, how much higher is safe or recommended to avoid issues? Is there a common rule of thumb?

I know the crossover frequency is where the loop gain hits 0 dB, but does that mean that after that point, the system basically can’t correct disturbances anymore because it has no authority?

Since I’m seeing strong ripple at the LC resonance frequency (5 kHz), does that mean my crossover frequency is too close to it — or even below it — and that’s why the system isn’t damping it properly?

The compensation setup comes from the eval board and includes slope compensation, but I didn’t calculate or tweak it myself. I started reading TI’s AN-1162 app note, which explains quite a lot, but I still have some doubts on how to apply it correctly to my case.

Also, I’ve noticed that the ESR of the output caps have a pretty big impact on how the loop behaves. I get that ESR introduces a zero, but I don’t fully get why that has such a strong effect on stability, especially when using low-ESR ceramics.

If anyone can share some advice or point me to a resource that explains:

How far above the LC resonance the crossover should be.
Why ESR plays such a key role in stability.
General tips for compensating buck converters (voltage-mode).

I am using the LM5146 from Texas Instruments (Voltage mode regulator)!

[MariuszD]

This article comprehensively describes various approaches to buck converter compensation.
https://web.cecs.pdx.edu/~tymerski/chap1.pdf

[petemate]

What you are interested in, is the dynamic performance of switchmode supplies and this is basically a subject of control theory. The book "Fundamentals of power electronics" covers this well in section two. You can find the instructors slides here: https://imserv.org/MFKForum/Datasheets/Fundamentals%20of%20power%20electronics/ecee.colorado.edu/copec/book/slides/slidedir.html

What you want is basically chapter 8 and 9.

To answer your specific questions:

Does the crossover frequency (fc) always need to be higher than the LC resonance? If yes, how much higher is safe or recommended to avoid issues? Is there a common rule of thumb?

No, I don't believe it does. The reason for the LC filter is to filter out the switching wavefrom frequency components to a degree that makes you think the ripple is small enough, while the crossover is defined as the point where your compensation loop stops being able to influence the converter performance. So separate items. But you can't have your resonance frequency higher than the converter switching frequency, for obvious reasons. The only rule-of-thumb that I am aware of in this regard is that your crossover frequency is supposed to be no higher than about 1/5 of your switching frequency.

Its perfectly fine to have a very low crossover frequency: When doing initial design of a compensation network, its not uncommon to just put a really large capacitor in the compensation network. This basically result in a low-frequency pole(essentially an integrator), which then makes your closed loop system very stable, but also very slow.

I know the crossover frequency is where the loop gain hits 0 dB, but does that mean that after that point, the system basically can’t correct disturbances anymore because it has no authority?

For an "input-to-output closed-loop transfer function", the crossover frequency is the point where disturbances can no longer be attenuated by the compensation.

Since I’m seeing strong ripple at the LC resonance frequency (5 kHz), does that mean my crossover frequency is too close to it — or even below it — and that’s why the system isn’t damping it properly?

That may very well be, but its impossible to say for sure without seeing the design. The design of compensation networks is tricky business and the math is a bit heavy. Its probably easier for you to simulate your converter using e.g. LTspice and get a bode plot of the loop gain to check for stability.

Also, I’ve noticed that the ESR of the output caps have a pretty big impact on how the loop behaves. I get that ESR introduces a zero, but I don’t fully get why that has such a strong effect on stability, especially when using low-ESR ceramics.

Thats because the ESR zero changes the transfer function: It changes the magnitude response and the phase. The zero "lifts up" the magnitude and phase of the loop(meaning that it is no longer dropping with frequency at the same rate - see more about this in the book), so you'll need something to bring it back down. You typically place some poles around your ESR zero. Its not magic, its just another aspect that you'll need to include in your compensation loop.

General tips for compensating buck converters (voltage-mode).

Don't do voltage mode. Always do current mode control. In voltage mode, you have only an outer control loop that sets the PWM based on output voltage, but in current mode you actually have two control loops. The inner control loop sets the inductor current based on the error from the outer loop, which sets the voltage. This is beneficial for several reasons: You can't control inductor current with voltage-mode control, which means that you can risk saturation of your inductor. Also, current mode effectively eliminates the LC filter double pole because the inductor current is no longer inpendent, but turned into a current source. This simplifies compensator design significantly.

[kraw]

What you are interested in, is the dynamic performance of switchmode supplies and this is basically a subject of control theory. The book "Fundamentals of power electronics" covers this well in section two. You can find the instructors slides here: https://imserv.org/MFKForum/Datasheets/Fundamentals%20of%20power%20electronics/ecee.colorado.edu/copec/book/slides/slidedir.html

What you want is basically chapter 8 and 9.

To answer your specific questions:

Does the crossover frequency (fc) always need to be higher than the LC resonance? If yes, how much higher is safe or recommended to avoid issues? Is there a common rule of thumb?

No, I don't believe it does. The reason for the LC filter is to filter out the switching wavefrom frequency components to a degree that makes you think the ripple is small enough, while the crossover is defined as the point where your compensation loop stops being able to influence the converter performance. So separate items. But you can't have your resonance frequency higher than the converter switching frequency, for obvious reasons. The only rule-of-thumb that I am aware of in this regard is that your crossover frequency is supposed to be no higher than about 1/5 of your switching frequency.

Its perfectly fine to have a very low crossover frequency: When doing initial design of a compensation network, its not uncommon to just put a really large capacitor in the compensation network. This basically result in a low-frequency pole(essentially an integrator), which then makes your closed loop system very stable, but also very slow.

I know the crossover frequency is where the loop gain hits 0 dB, but does that mean that after that point, the system basically can’t correct disturbances anymore because it has no authority?

For an "input-to-output closed-loop transfer function", the crossover frequency is the point where disturbances can no longer be attenuated by the compensation.

Since I’m seeing strong ripple at the LC resonance frequency (5 kHz), does that mean my crossover frequency is too close to it — or even below it — and that’s why the system isn’t damping it properly?

That may very well be, but its impossible to say for sure without seeing the design. The design of compensation networks is tricky business and the math is a bit heavy. Its probably easier for you to simulate your converter using e.g. LTspice and get a bode plot of the loop gain to check for stability.

Also, I’ve noticed that the ESR of the output caps have a pretty big impact on how the loop behaves. I get that ESR introduces a zero, but I don’t fully get why that has such a strong effect on stability, especially when using low-ESR ceramics.

Thats because the ESR zero changes the transfer function: It changes the magnitude response and the phase. The zero "lifts up" the magnitude and phase of the loop(meaning that it is no longer dropping with frequency at the same rate - see more about this in the book), so you'll need something to bring it back down. You typically place some poles around your ESR zero. Its not magic, its just another aspect that you'll need to include in your compensation loop.

General tips for compensating buck converters (voltage-mode).

Don't do voltage mode. Always do current mode control. In voltage mode, you have only an outer control loop that sets the PWM based on output voltage, but in current mode you actually have two control loops. The inner control loop sets the inductor current based on the error from the outer loop, which sets the voltage. This is beneficial for several reasons: You can't control inductor current with voltage-mode control, which means that you can risk saturation of your inductor. Also, current mode effectively eliminates the LC filter double pole because the inductor current is no longer inpendent, but turned into a current source. This simplifies compensator design significantly.

First of all, thank you very much for your detailed answer.
Regarding the article, it seems really interesting — I’ll definitely read it to better understand the dynamic behavior of the converter.

As for the crossover frequency, I thought it should be greater than the LC resonance in order to control and damp it (and, as you said, still quite lower than the switching frequency to avoid being disturbed by the nature of the switching noise).
If I set, for instance, a very low crossover, my system will be very stable — but the current at the resonance frequency will go entirely through the input, right?
That’s what I’m seeing: with 2 A at the input, there’s a current ripple going from about 1.4 to 2.6 A, with a "perfect" sinusoidal shape at the resonance frequency.
If I had a crossover frequency higher than that (where the gain is still sufficient and with proper phase margin), it should damp that behavior, or am I completely off?

Regarding the ESR topic, I now understand it much better thanks to your explanation (and as you suggested, I’ll also take a look at the book).

And about voltage mode — I didn’t know it was considered the "worst" of both approaches. I just assumed both had the same loop tools but used them differently. Now I get the difference between them.

Thank you very much!

[mtwieg]

Does the crossover frequency (fc) always need to be higher than the LC resonance? If yes, how much higher is safe or recommended to avoid issues? Is there a common rule of thumb?

Crossover frequency is a target that the designer should choose and aim to meet. If 10Hz is enough, then it's enough. If you need 100kHz, then you need 100kHz (but will likely have to work to get it).

I know the crossover frequency is where the loop gain hits 0 dB, but does that mean that after that point, the system basically can’t correct disturbances anymore because it has no authority?

Basically yes. But the behavior below, at, and above fc can be described quantitatively with control theory.

Since I’m seeing strong ripple at the LC resonance frequency (5 kHz), does that mean my crossover frequency is too close to it — or even below it — and that’s why the system isn’t damping it properly?

In general, when the voltage control loop is unstable it will tend to oscillate at the phase crossover frequency (where the open loop gain has a total phase of 360). If the LC filter is undamped, then this will tend (but not always) to happen near the LC resonant frequency. This is because the phase response vs frequency will be changing fastest there.

The compensation setup comes from the eval board and includes slope compensation, but I didn’t calculate or tweak it myself. I started reading TI’s AN-1162 app note, which explains quite a lot, but I still have some doubts on how to apply it correctly to my case.

Not sure what you're referring to here... LM5146 is a voltage mode controller, slope compensation does not apply.

Also, I’ve noticed that the ESR of the output caps have a pretty big impact on how the loop behaves. I get that ESR introduces a zero, but I don’t fully get why that has such a strong effect on stability, especially when using low-ESR ceramics.

ESR in the capacitor can definitely have a large effect on stability. Like you said, it creates a zero in the open loop transfer function. The higher the ESR, the lower the frequency of the zero. How this influences stability is best seen by graphically observing its effect on the open loop transfer function (especially its phase).

For further reading I highly recommend the white papers from Venable. They require you provide a name and email address to download them. I've never gotten spam from them, and they don't actually verify the email at all.