Capacitors and inductors begin to have influence on circuit behaviour as soon as there is non-zero frequency, which includes switched DC.
Using the Laplace transform and complex math (i.e. imaginary numbers) one can combine resistors, capacitors and inductors into a single quantity called impedance.
Impedance is made up of resistance (R) and reactance (X), where capacitors and inductors are called reactive components. Note that resistance is mathematically real while reactance is imaginary.
\(Z(f) = R(f) + jX(f)\)
Alternatively, this can be written in exponential form:
\(Z(s) = |Z|e^{j\theta}\)
The magnitude of Z is given as:
\(|Z| = \sqrt{R^2 + X^2}\)
And the angle as:
\(\theta = \arctan{\frac{X}{R}}\)
Reactance
For a capacitor the reactance is :
\(X_C(f) = -\frac{1}{2\pi fC}\)
And for an inductor:
\(X_L(f) = 2\pi fL\)
When in exponential form, capacitors have an angle of -90° and inductors +90°.
Series Combinations
Two element series circuits are very straightforward, just adding the two terms together.
RC circuit:
\(Z_{RC}(f) = R – j\frac{1}{2\pi fC}\)
RL circuit:
\(Z_{RL}(f) = R + j2\pi fL\)
An RLC circuit is a bit more complicated:
\(Z_{RLC} = R + j2\pi f – j\frac{1}{2\pi fC} = R -j\frac{1 + (2\pi f)^2LC}{2\pi fC}\)