Section 3.3 Complex Signals
Signal analysis with sinusoidal functions can be challenging due to the need for trigonometric identities. The mathematical manipulations in circuit analysis can more easily be done if signals are instead represented as complex exponentials.
Using Euler’s formula
\begin{equation*}
e^{\pm i\theta}=\cos(\theta)\pm i\sin(\theta)
\end{equation*}
we can see that
\begin{equation*}
V_0 \cos\left(\omega t + \delta'\right)
\end{equation*}
is the real part of
\begin{equation*}
V_0 e^{i\left(\omega t + \delta'\right)}\text{.}
\end{equation*}
Using this notation, it is easy to see that the phase information can be absorbed into the amplitude, resulting in a complex amplitude:
\begin{equation*}
\tilde{V}(t)=V_0 e^{i\omega t + i\delta'} = V_0 e^{i\delta'}e^{i\omega t} = \tilde{V}_0 e^{i\omega t}\text{.}
\end{equation*}
With this notation, the measured signal \(V(t)\) is the real part of the complex signal \(\tilde{V}(t)\text{.}\) So, we can use \(\tilde{V}(t)\) in equations describing circuit behavior, and the real part of the result represents what we would measure for the parameter of interest.
