# How to simulate a continuous signal passing through a capacitor (simple coupling capacitor)

the differential equation for the current flowing through a serial capacitor (see for example https://www.allaboutcircuits.com/textbook/direct-current/chpt-13/capacitors-and-calculus/) indicates that one must take a derivative of the signal but in the literature, a serial capacitor (for instance a coupling capacitor) is often considered as a unit that just eliminates a continuous component of the signal. Well. Is it necessary to take a derivative or just get the first moment of the stochastic signal and eliminate it in order to simulate the passing of that signal through the coupling capacitor?

• I recommend Papoulis' "Circuits and systems: a modern approach". It deals extensively with this kind of problem.
– MBaz
Mar 18 at 15:26
• Are you looking to make a digital DC blocking filter? I suggest a search on that phrase -- "DC blocking filter". Mar 18 at 22:42
• No. Sooner I make a coupling capacitor but do not know how to do it. Clearly, if I need to take a derivative then I change the signal dramatically. If I need to apply just a high-pass filter with a cut-off freq standing below "any signal frequency", then I need just eliminate a mean value. Mar 20 at 11:09

Is it necessary to take a derivative

Yes

or just get the first moment of the stochastic signal and eliminate

That's just a very crude approximation of taking the derivative. In some cases that's sufficient, but in most it's not.

For DC blocking the capacitor is typically used together with a resistor to form a high pass filter. The approximation is often "good enough" if the corner frequency of the high pass is well below any signal frequency of interest.

• I get the impression OP is talking about the full impulse response, i.e. transient plus steady-state, not just the cap as a DC blocker. So what OP refers to as the "first moment of the stochastic signal" might be the transient response leading into the steady-state (0) (therefore no frequency involved). In which case considering the cap, alone, is not enough. But only if I'm not mistaken. Mar 18 at 18:38
• Let's focus on a coupling capacitor. Can it be seen as a DC blocker only? And of course, there is a resistor (in the form of the input impedance of the stage, for example). Well. Is it advisable, in this case, to simulate a coupling capacitor as a high-pass filter (Butterworth, for example) of order one, if it influences a signal as a high-pass filter (together with the input impedance)? Mar 18 at 19:52
• Actually it's not necessary to take the derivative; that's just an artifact of how we usually solve such problems. You can integrate the average, and subtract. You can certainly view that as the action of a capacitor in such a circuit. Mar 18 at 22:41
• Does integration need to be made on the scale t, where t is the time constant of the high-pass filter (which is formed by the capacitor in question and the input impedance of the stage)? If yes then it is advisable to use a readily available function of Butterworth's high-pass filter. Mar 20 at 11:14