Section 3.4 Complex Impedance
When analyzing AC circuits with complex signals \(\tilde{V}(t)\text{,}\) we will need to expand beyond the concept of resistance. We define impedance \(Z\) as
\begin{align*}
Z \amp\equiv \frac{\tilde{V}}{\tilde{I}} \\
\amp \equiv R + i X
\end{align*}
where \(R\) is our normal resistance and \(X\) is reactance. The impedance is the factor that relates \(\tilde{I}\) to \(\tilde{V}\) as seen in Figure 3.4.1. Generally, \(Z\) is frequency-dependent.
Let’s look at a few examples. Ohm’s law continues to hold for AC signals, so it is trivial to demonstrate that \(Z_R=R\text{.}\)
Now, let’s work to determine the impedance for a capacitor. In introductory physics, you likely learned that capacitance \(C\) is a quantity that links the voltage difference \(V\) between two objects and the charge \(Q\) on each object through the expression
\begin{equation*}
Q=CV\text{.}
\end{equation*}
Pairing this expression with the definition of current
\begin{equation*}
I=\frac{\text{d}Q}{\text{d}t}
\end{equation*}
we find that
\begin{equation*}
I=C\frac{\text{d}V}{\text{d}t}\text{.}
\end{equation*}
For a voltage \(\tilde{V}(t)=\tilde{V}_0 e^{i\omega t}\text{,}\) we thus see that the resulting current through the capacitor is \(\tilde{I}(t)=i\omega C \tilde{V}_0 e^{i\omega t}\text{.}\) Using these expressions, we find that
\begin{equation*}
Z_C = \frac{\tilde{V}}{\tilde{I}}=\frac{\tilde{V}_0 e^{i\omega t}}{i\omega C\tilde{V}_0 e^{i\omega t}}
\end{equation*}
so that
\begin{equation}
Z_C = \frac{1}{i\omega C} = -\frac{i}{\omega C}\text{.}\tag{3.4.1}
\end{equation}
Note that the impedance \(Z_C\) is frequency-dependent and complex. Examining this expression in limiting cases, \(Z_C\rightarrow \infty\ (\text{as }\omega\rightarrow 0)\) and \(Z_C\rightarrow 0\ (\text{as }\omega\rightarrow \infty)\text{.}\)
Now, let’s work to determine the impedance for an inductor. Inductors, fundamentally, are coils and act in a way as an anti-capacitor. The fundamental equation governing inductor behavior is
\begin{equation*}
V=L\frac{\text{d}I}{\text{d}t}
\end{equation*}
which means that the voltage drops across an inductor when \(\text{d}I/\text{d}t\ne 0\text{.}\) If we assume a current
\begin{equation*}
\tilde{I}(t)=\tilde{I}_0 e^{i\omega t}
\end{equation*}
then
\begin{equation*}
\tilde{V}(t)=i\omega L \tilde{I}_0 e^{i\omega t}
\end{equation*}
so that
\begin{equation*}
Z_L = \frac{\tilde{V}}{\tilde{I}} = \frac{i\omega L \tilde{I}_0 e^{i\omega t}}{\tilde{I}_0 e^{i\omega t}}\text{.}
\end{equation*}
Simplifying, this leaves us with
\begin{equation*}
Z_L = i\omega L\text{.}
\end{equation*}
Note that the impedance \(Z_L\) is frequency-dependent and complex. Examining this expression in limiting cases, \(Z_L\rightarrow 0\ (\text{as }\omega\rightarrow 0)\) and \(Z_L\rightarrow \infty\ (\text{as }\omega\rightarrow \infty)\text{.}\)
The results above can be used with a generalized Ohm’s law
\begin{equation*}
I=\frac{V}{Z}
\end{equation*}
and also with equations for series and parallel impedances
\begin{align*}
Z_\text{series} \amp = Z_1 + Z_2 + \cdots \\
\frac{1}{Z_\text{parallel}} \amp = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots
\end{align*}
