Q:

Impedance in a multi-frequency AC signal?

Consider a full-wave rectified voltage \$V=V|\sin(\omega t)|\$ being supplied to a capacitance \$C\$ and a load \$R\$ in parallel. The voltage across the capacitor is the voltage across the resistance. To find that voltage we need the current. To represent the supplied (rectified voltage) I used a Fourier series (only the real part represents the actual physical solution) and got an output which is a superposition of 3 different frequency AC (\$\omega=2,4,6\$) and a DC \$2/\pi\$. To solve for the current I need \$Z\$ so that \$I=V/Z\$. In normal cases, \$Z_c=\dfrac{1}{i\omega C}\$, but with multiple frequencies, I am unsure what to use?
How can I get a function for the variation of voltage across the
 capacitor?

A:

The impedance of a circuit element (or parallel combination of circuit elements) is the ratio of the phasor voltage to the phasor current.
But, this presumes sinusoidal excitation, i.e., a single frequency.
When there is a sum of sinusoids (as in a Fourier series decomposition), there is no notion of a single valued impedance.
Instead, what one must do is apply the notion of impedance to each sinusoidal component separately.
For example, if you know the sinusoidal voltage components, you can find the phasor current components for each frequency separately, using the impedance for each respective frequency, and then superpose the associated sinusoidal current components for the total current.

