Switching Power Supply Source Impedance: Calculating Input Capacitance for Stability

I have been enjoying the recent Circuit Surgery columns on Power Supplies and Potential Dividers in EPE Magazine.  I love the way that Ian Bell tackles each subject, explaining the theory clearly from first principles using worked examples to aid understanding.  I wondered whether Circuit Surgery would take a look at a subject that has always been a little fuzzy for me: Switching Power Supply Source Impedance and how to mitigate for it.  We all know the rule of thumb that the source impedance of the input to a voltage regulator must be lower than the negative input impedance of the converter (negative because the line voltage drops when more current is taken by the regulator) by a factor of at least 10 times.  We also know that if the source has a high impedance (e.g. long wires from the source to the regulator) that this can be mitigated for with some input capacitance, usually a large value electolytic.  What's not clear to me is how big that capacitor should be for a given source impedance.  Explanations range from the highly mathematical to the generic "use a 470uF for everything".  Is there a way to work out a suitable minimum capacitance value, based on the estimated or calculated impedance of the source?

Hi Martin,

Thanks for the suggestion which I'm forwarding on to Ian.  I'm really glad to hear you enjoy the Circuit Surgery column which Ian has been writing for more than 20 years now. So I'm sure Ian will take a look at your question for a future column.

Best wishes  Alan W

I separate this into two comments, here the high-frequency effect of the switching power supply, and later the low-frequency effect of the primary power source.Most switching power supplies draw current in pulses at a rate equal to the switching frequency. One job of the input capacitor is to convert these pulses into a relatively steady current from the primary power source.I suggest that the capacitor be made at least large enough so that the voltage across the capacitor will drop at most 5% over one cycle of the switching frequency, assuming all the current is coming from the capacitor.The capacitor must also have low enough effective series resistance (ESR) to deliver this current with negligible resistive drop. The capacitor must be rated to handle this much AC current.

Now for effects of the primary power source. When the load on the switching power supply suddenly increases, the power supply immediately tries to draw more power from the primary power source. Some primary sources, such as nearby lead-acid batteries, handle sudden demand easily. Others may handle sudden demand very badly, by dropping their output voltage, which causes the switching power supply to draw even more current. Rotating electromagnetomechanical generators and other sources with high series inductance, are notorious for bad behavior, especially if they are barely large enough for the steady load.Enlarging the capacitor across the input to the switching power supply, lowers the resonant frequency of the primary tuned circuit and thus lowers the reactance at the resonant frequency. The series resistance is relatively independent of frequency, so lowering the resonant frequency lowers the Q. Lowering the Q enough, will prevent oscillation. In the case of rotating electromagnetomechanical generators, the resonance can be in part mechanical (including the effect of whatever is providing the rotation).In the case of long cables, at frequencies high enough so the cable length is comparable to the wavelength of the frequency in the cable, radio transmission-line effects become important.

A few more comments of a very general nature related to real world SMPS'sThe capacitance is only one of a number of parameters that needs to be considered.
There is a topological dependency, for example  I have designed both Sepic and Zeta SMPS  with the same output requirements, which have low and high input ripple respective. For the same input ripple voltage, that immediately throws out the idea of one capacitor fits all, since you normally want to keep the capacitor value, volume, and ESR as low as possible while meeting your design requirements. The trend towards using ceramic capacitors on lower current SMPS's introduces another problem, that of the large sensitivity to voltage of the capacitance value.
Poor PCB layout is also an issue, additional inductance  may become a big  problem.Many SMPS's are not battery fed, their power is derived from a high voltage DC bus created by rectifying the mains voltage. This presents an impedance nothing like that of a battery.EMC is a big issue, this often dictates that the input side has many additional frequency sensitive components to minimize both radiated and conducted noise.Cheers,Richard