# Equivalent circuit capacitance and inductance

When drawing the equivalent circuits of a real capacitor or inductance, usually the capacitor is modeled as ideal capacitance with a resistor in parallel and the inductance as an ideal inductance with a resistor in series.

Why is it done like this (i.e. in parallel/series)? What conditions need to be satisfied that I can do it like this?

I can sort of understand the reasoning in the inductance case. If I have a coil it would consist of a rather long wire which does have a resistance (in series). Not sure about the reasoning in the capacitor case.

The resistor parallel with the capacitance represents the dc leakage current through and the higher RF frequency dissipation in the dielectric. At RF frequencies the the copper losses in the leads and plates are also important and are modeled as series resistors.

You're right about the inductor's case.

For the capacitor, it's equally easy. Real capacitors suffer from leaks that discharge the capacitor. You simulate these leaks by creating an alternative channel for current to pass to the other pole, that is, an alternative parallel branch. A resitor in series would do nothing to create that alternative path.

Why is it done like this (i.e. in parallel/series)?

I'll try a different approach (and one of my favorites) from the answers already given.

Start with what you find intuitive: the physical inductor is (simply) modeled as an ideal inductor of inductance $L$ in series with a resistor of resistance $R$. This reduces to an ideal inductor in the limit as the resistance $R \rightarrow 0$

Now, exploit electric circuit duality where, for the present case, the relevant duals are

$$L \Leftrightarrow C$$

$$R \Leftrightarrow G$$

$$\mathrm{Series} \Leftrightarrow \mathrm{Parallel}$$

So, the electrical dual of the ideal inductor of inductance $L$ in series with a resistor of resistance $R$ is... (think about it first)... an ideal capacitor of capacitance $C$ in parallel with a conductor of conductance $G$. This reduces to an ideal capacitor in the limit as the conductance $G \rightarrow 0$