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In many datasheets and application guides there is always an equation for ripple current, because it is fundamental to the operation of a buck or boost converter, but I never see a direct equation for ripple voltage.

I know the ripple comes from the finite capacitance that needs to discharge the load current during the off-part of the duty cycle, so I suspect the equation is a second order (since the derivative of the capacitors voltage depends on the derivative of the inductor current). I also know the capacitor’s ESR plays a role.

I ask out of experience. Once I built a boost converter and the output voltage ripple grew to over 100mV at max load, whereas another time I build a boost converter and the ripple was constant at 4mV across all loads. These were clearly satisfying very different power requirements, but what is the interplay between capacitance ESR and the pure inductive/capacitance relationship in determining output ripple? And when does it start to vary significantly across loads?

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I suspect the equation is a second order (since the derivative of the capacitors voltage depends on the derivative of the inductor current)

This is fundamental to understanding the ripple voltage. The inductor and capacitor form a 2nd order low-pass filter with an input that can be assumed to be a square wave of variable duty cycle to accomodate load and incoming supply variations.

Given that the inductor and capacitor have a natural resonance significantly below the switching frequency (maybe at one tenth) you can expect that the 1st fundamental of the square wave to be attenuated by 40 dB because a 2nd order low-pass filter attenuates at 40 dB per decade.

So if the square wave (switching waveform) was 12 volts p-p then the fundamental would be reduced by 40 dB to 120 mV p-p and the harmonics reduced even more.

If the switching frequency is 100 times the LC natural frequency then the 12 volts p-p is attenuated by 80 dB to 1.2 mV p-p. However, there is a trade-off - to obtain maximum dynamic control when dealing with load current variations, it is desirable to have the LC natural frequency as high as possible so it's a compromise.

With capacitive ESR the attenuation of the LC filter will not continue at 40 dB/decade but will eventually turn into 20 dB/decade as the LC becomes an LR single order low-pass filter.

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  • \$\begingroup\$ Thank you for this answer! Specifically the last bit with the ESR damping the effect of the filter at high frequency was the missing piece for me. I suppose SMPS are just LC filters, when one looks past the ripple current and all of the other factors attributed to them. A last question: it seems that you're saying at all frequencies the SMPS is a filter. Doesn't that mean that the ripple voltage is always dependent on the load? \$\endgroup\$ – Michael E Jul 26 '18 at 6:59
  • \$\begingroup\$ The ripple voltage (across a broad range of loads from very light to quite heavy) is largely determined by the cut-off frequency and the 40 dB/decade roll-off of the second order filter. Adding more load reduces the Q of the filter and this can alter the waveform shape of the ripple but not markedly so. Try using a simulator to prove this to your own satisfaction. It's a simple sim to setup and I always encourage folk to gain sim experience. \$\endgroup\$ – Andy aka Jul 26 '18 at 7:46
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Quite likely all the discrete and the parasitic Cs and Ls interact, and cause ripple/ringing. Thus you should expect "ripple" at 150MHz because of 10nH (0.010uH) interacting with 100pF Cout of onchip power FETs.

And all other possible LC resonators may be a problem, depending upon the amount of resistive dampening in the energy-circulation loop.

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