Purpose
Capacitors and inductors store energy, and their behavior depends on frequency. This note records their defining relations, then the complex number machinery (impedance and phasors) that lets you analyze AC circuits with the same rules you use for resistors.
Capacitors
A capacitor stores energy in the electric field between two plates. It passes AC and blocks DC, which is why a series capacitor acts as a high pass filter.
Capacitance is the ratio of charge to voltage, measured in Farads:
The energy stored in a capacitor is
Differentiating with respect to time gives the current through the capacitor:
Inductors
An inductor stores energy in the magnetic field around a coil of wire, built up by the current flowing through it. Units are Henrys (H). It passes DC and blocks AC, acting as a low pass filter in series.
The energy stored in an inductor is
and the voltage across it is
Complex numbers review
With defined by , Euler’s formula connects exponentials to sinusoids:
Complex numbers for AC signals
Pretend signals are complex during analysis, then take the real part at the end. Multiplying by a real number scales the amplitude. Multiplying by a unit complex number shifts the phase. Multiplying by a general complex number does both.
For example, a shifted cosine is the real part of a complex exponential:
Impedance
Impedance is the AC generalization of resistance:
The capacitor’s impedance shrinks as frequency grows, which matches “passes AC, blocks DC” above. The inductor’s impedance grows with frequency, matching “passes DC, blocks AC”.
Sanity check at the frequency extremes
As , (open circuit) and (short circuit). As the roles flip. Evaluating a derived transfer function at these two limits catches most algebra mistakes before you plot anything.
Phasors
A phasor is a complex number that represents the amplitude and phase of a sinusoidal signal. Since impedances obey Ohm’s law with phasors, the differential equations from and turn into algebra, and series and parallel combination rules carry over from resistors.