# Understanding how bypass caps work, mathematically

Assume there is a band of noises that you you want to filter with center frequency Fn. Then The complex voltage representing noise at frequency F is V_(F, T) = V(F) e^(i W(F) T), where V(F) is the peak value of the sinusoidal voltage and W(F) is the angular frequency = 2 pi F, and T is time. I went ahead and LaTeXified this to be more readable:

If this voltage were applied to a bypass capacitor C, then

is the current through C that associated with the noise voltage at that frequency. This is valid for complex voltage and current since we already know that i = C dv/dt in real quantities, but

After some math we see that there's a corresponding complex equation for every real equation involving linear combinations of derivatives of voltage and current. I'm not so much interested in making this rigorous as figuring out now how to calculate the best value for C if it even matters.

Let me know if I can clarify anything.

• Everything you say is true, but a capacitor doesn't "filter" anything unless it is working in conjunction with other impedances (such as resistors or coils) in the circuit. When talking about bypass capacitors in particular, you need to consider the parasitic impedances associated with the power distribution network, both inside and outside a chip -- not to mention, the non-ideal characteristics of the capacitor itself. – Dave Tweed Feb 2 '14 at 17:07
• without all the maths, can you succinctly describe what you want to know. – Andy aka Feb 2 '14 at 17:07
• Please explain more guys. You're leaving me hangin... – BananaCats Author Feb 2 '14 at 17:09
• @DaveTweed That is probably true, but please work within this simpler model. Or optionally write a long article on what you describe. But that would be hard I think. – BananaCats Author Feb 2 '14 at 17:10
• There's really nothing to say about your simple model, except that the impedance ($V_n/I_n$) of an ideal capacitor drops with increasing frequency. Ed Nisley wrote a couple of installments of his Circuit Cellar "Above the Ground Plane" column on the topic of bypass capacitors in the real world (issues 259 and 260) that would probably be a good place for you to start. – Dave Tweed Feb 2 '14 at 17:21