**Table of Contents**

We already know how to calculate the impedance of a capacitor:

At 10kHz a 1nF capacitor has an impedance of 15.9k.

When we connect a 10k resistor in series with this capacitor, we may expect that the total impedance will be 25.9k is. But that's not the case, because a capacitor causes a -90 degrees phase shift in the current flow. The equation

doesn't show this. But what if we plot the impedance as a vector? The length will be the (absolute) impedance and the angle the phase shift:

This component has an absolute impedance of 1 ohm and causes a 45 degrees phase shift between voltage and current.

A resistor doesn't cause a phase shift; this vector will be on the
x-axis. a capacitor causes a -90 degrees phase shift and will be on the
(negative) y-axis. The total impedance of R and C is R +
X_{C}. However, we have to add vectors instead of
plain numbers. Since we have 90 degrees angles, it's easy to calculate the
(absolute) impedance: we can just use Pythagoras' theorem.
Z_{t} =
√(R^{2}+X_{C}^{2}).
The phase shift is equal to arctan(-X_{C}/R)

Wouldn't it be nice if we had a more simple way for saying: the
impedance is x ohms and causes a y degrees fase shift? A 180 degrees phase
shift is easy; in that case we could say: the impedance equals -x ohms. A
180 degrees phase shift equal to multiplying by -1. Now suppose that a
phase shift of 90 degrees is equal to multiplying by j. A 180 degrees
phase shift will be the same as multiplying by
j^{2}. This means that
j^{2} = -1. A negative square is only possible in
so called 'complex math'. (Mathematicians among us may be accustomed to
use i instead of j. But we already use i as a symbol for current, so
that's confusing.)

Every impedance can be written as: a + bj. Number a is called the real part and is plotted on the x-axis. Number b is called the imaginary part and is plotted on the y-axis.

A resistor doesn't cause a phase shift and is therefore purely real.

We know that a capacitor causes a -90 degrees phase shift; its impedance is therefore purely imaginary, and can be written as:

As an example, we'll look again at our 10k resistor in series with a
15.9k capacitor. The total (complex) impedance is 10k - 15.9kj. The
absolute value (|Z|) equals √(10k^{2} +
15.9k^{2}) = 18.78k. The phase shift it causes in
the current (arg(Z)) is arctan(-15.9k/10k) = -57.8 degrees.

So, when we connect a resistor and capacitor in series:
Z_{R+C} = R - X_{C}j.

The absolutie value is: |Z_{R+C}| =
√(R^{2}+X_{C}^{2}).

The phase shift is: arg(Z_{R+C}) =
arctan(-X_{C}/R).