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In the context of control loop stability of DC-DC Converters,

A switch mode power supply is essentially a sampled-data system, therefore the theoretical maximum bandwidth is one half the switching frequency. Practically the phase and transport lag there make it impossible to close the loop there, so 1/5 to 1/10th the switching frequency is a good rule of thumb.

Could some one tell me what does the words in blockquote mean when it says, "there make it impossible to close the loop there"? What does "close the loop" mean in electrical terms?

winny
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    where does that quote come from? Context **always** helps (and I know it's not the first time I mention that to you), and this –standing alone– is a gross simplification that's not generally true. So, without you adding the source of the cited statement, I'll vote to close this as unclear, since from an overly generalizing statement, anything can be inferred. – Marcus Müller Nov 21 '19 at 08:28
  • but, regarding "close the loop": That's not meant in an electrical sense. That is meant as a figure of speech meaning "we can't reach that". – Marcus Müller Nov 21 '19 at 08:30
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    Extremely sorry for this mistake. "https://electronics.stackexchange.com/questions/126568/relationship-between-switching-frequency-and-controllers-crossover-frequency-in" –  Nov 21 '19 at 08:45

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Crossover frequency selection is a bit more complex than to simply say "1/5th or 1/10th of the switching frequency" for the loop 0-dB crossover point. For a buck converter operated in voltage-mode control, there is a resonance in the small-signal output impedance at \$f_0=\frac{1}{2\pi\sqrt{LC}}\$ and the system must have gain to fight an oscillatory step response. Simply said, no gain, no feedback. Beyond crossover, when the gain is less than 1, the converter runs in ac open loop. If output oscillations occur at \$f_0\$ during a load step, then the system must have enough gain to reject them. Therefore, a good recommendation is to have \$f_c>3f_0\$ and be less than \$\frac{F_{sw}}{2}\$.

In current-mode control, the resonance is gone and the theoretical upper limit is \$\frac{F_{sw}}{2}\$ provided the sub-harmonic poles of fixed-switching-frequency operation are gone as in constant on-time (COT) for instance. However, if widening the bandwidth surely brings a better response time, it reduces the power supply robustness to noise: think of it as funnel, the wider it is, the more noise susceptible the power supply becomes. So tailor crossover to meet the acceptable undershoot for a given output capacitor and not more than that.

For converters like boost or buck-boost (flyback for the isolated version), there is delay in the conversion process which is modeled as a right-half-plane zero: you must store energy in the inductor during \$t_{on}\$ before you feed the output capacitor during \$t_{off}\$. Therefore, if a sudden power demand occurs, the current in the inductor must grow cycle by cycle at a sufficient pace to keep up with the demand. If the demand is too fast or if the inductance is too large, the output voltage momentarily drops and oscillations take place. To avoid this problem, you have to purposely slow down the converter to always give enough time for the inductor energy to build up. You do that by limiting the crossover value for these two converters. A good recommendation is to have \$f_c<0.3f_{RHPZ}\$ meaning that you calculate the lowest position of the RHP zero (lowest input voltage and highest output current) and you limit \$f_c\$ below 20-30% of this value. Trying to go beyond will reduce phase margin. Finally, as with a buck in voltage-mode-control, the minimum crossover is \$f_c>3f_0\$ but this time, \$f_0\$ moves in relationship with \$D\$, the duty ratio.

In current mode, the resonance at \$f_0\$ disappears but the RHPZ occupies a similar position so you can't have a faster converter unlike what people believe. As you can read, crossover selection is not a magic number pulled out of a hat, it has to be determined based on the converter you want to stabilize and the performance you want to achieve. The below picture offers a summary for continuous conduction mode (CCM) converters.

enter image description here

Verbal Kint
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  • Excellent insights! Thx! – joribama Nov 22 '19 at 00:53
  • I was trying to understand the relationship between the crossover frequency and the switching frequency of the converter since you answered. I'm not able to get the relationship clear. Could you please explain that too in clear, simple and easy understandable terms –  Nov 24 '19 at 07:14
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    You could, in theory, choose a crossover frequency \$f_c\$ up to half the switching frequency. However, there are some physical limits that I detailed in the answer. If you have a boost or a buck-boost converter (or a flyback), 30% of the right-half-plane zero is the upper limit. For a buck or a forward, you can choose a high \$f_c\$ as long as you stay well above \$f_0\$ (in VM). However, the more gain you have at high frequency, and the more your converter becomes sensitive to external noise. High \$f_c\$ is also demanding on the compensator op-amp GBW. – Verbal Kint Nov 24 '19 at 10:19
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What does "close the loop" mean in electrical terms ?

This is a control theory concept and simply means that you measure some physical quantity you want to keep at a certain value and then use this measurement to tweak some sort of control to cause this quantity to get closer to the desired value.

Let's be a little less abstract... think about a thermostat in a house air conditioner. This is a closed loop because the thermostat will measure the room temperature and if the room is hotter than desired it will turn on the air conditioner so that the temperature drops and get closer to the desired value. Once the temperature is low enough the air conditioner turns off to prevent the room from becoming too cold. If you simply turn on the air conditioner with the fan at a certain speed and leave it on all the time regardless of the actual room temperature, then you have an open loop.

Now back to your DC-DC converter... closed loop means that you'll measure the output voltage and will increase or decrease the duty cycle of the switcher if the output voltage is too low or too high in an attempt to keep the output voltage at the desired value.

"there make it impossible to close the loop there"

My understanding is that "there" means "at frequencies close to the switching frequency".

The idea is that in a switched circuit you only have the opportunity to "fix things" once per cycle. If you want the output voltage to go up, you increase the duty cycle (I'm thinking about a Buck converter) for that particular period; if you want the output voltage to go down, you decrease the duty cycle for that particular period. This means that the switching frequency imposes a limitation on how fast you can change the output voltage - you can never react faster than the switching period. That's why the text suggests using 1/5 or 1/10 of the switching frequency as an upper limit in terms of how fast you should design your control circuit to react in order to keep the output voltage constant.

joribama
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  • Thank you very much. Before I accept this answer, can you just provide the numericals as example for your explanation at the last paragraph. –  Nov 21 '19 at 09:22
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    Are you asking for actual frequency values? Let's say your switching frequency is 100kHz. This means your loop should not try to correct the output voltage faster than 20,000 times per second (1/5 ->20kHz) or 10,000 times per second (1/10 -> 10kHz). – joribama Nov 21 '19 at 09:29
  • Thank you for the wonderful and clear answer. Really helped me with the analogy –  Nov 21 '19 at 09:32
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    I'm glad I could help :) – joribama Nov 21 '19 at 09:34
  • just curious to know. What would happen if I try to correct the output voltage faster than 20kHz and 10kHz (as you mentioned) with respect to your example? –  Nov 21 '19 at 09:37
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    That's a very good question. As you go above these recommended frequencies for loop bandwidth, you may start to see an increase in the output voltage switching noise, or even worse, the system may become unstable, i. e., the output voltage may start to oscillate on its own. – joribama Nov 21 '19 at 09:58
  • Thank you for the clarification –  Nov 21 '19 at 10:37
  • In your answer above, in the last line, you mentioned that we should design "our control circuit to react in order to keep the output voltage constant". Could you please tell me what components constitute the control circuitry in a typical buck converter circuit (consider a TPS54260 IC)? is the output inductor and capacitor constitute the control circuit or the internal Op-amps present inside the buck converter IC constitute the control circuitry? –  Nov 03 '20 at 16:56
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    Hi Newbie - wow, it's been almost a year! "Control circuit" is a somewhat vague term, and someone can make the case that everything (IC, inductor, capacitors, etc.) is a part of the control circuitry. Others may prefer to define control circuitry as containing only the components that don't carry the bulk of the power, in the case of the TPS54260 it would exclude the internal FET, the external diode, output inductor and input and output capacitors. – joribama Nov 07 '20 at 02:37
  • Thank you joribama –  Nov 07 '20 at 04:27
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The observing part should have as little delay / filtering possible (Maximize bandwidth & linearity). If there is a significant delay in the observation , you are very likely creating an oscillator.

OT: Commonly, bandwidth of the control loop must be less than the switching frequency. (Not all topologies need that though) Depending on topology, Buck , Boost, Flyback, Forward, LLC, things may need very specific adjustments to their control loop stable & perform as intended.

In general, control loops are stabilized by reducing their output Slew-rate, slowing down response from integrator and amplifiers. This provides time for the whole system (DC/DC) to react to a 'small' change from the controller.

That all said, Id dare to say it is possible to create a controller with a bandwidth equal to the switching frequency. Hysteric PWM controller does this. It has a delay in the observer to cause intentional oscillations. :-) Iirc, this was an invention by Philips, they created a high quality audio amplifier using this method.

Steven Boks
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