Section 3.1 Sinusoidal Signals
In this chapter, we will focus on circuit behavior that results from sinusoidal signals
\begin{align*}
V(t) =\amp V_0 \cos\left(\omega t + \delta_V\right) \\
I(t) =\amp I_0 \cos\left(\omega t + \delta_I\right)
\end{align*}
where \(\omega=2\pi f=2\pi/T\) is the angular frequency, \(f\) is the frequency, \(T\) is the period, and \(\delta_V, \delta_I\) are the phase of the signals at time \(t=0\text{.}\) (We could have used the sine function instead of cossine, but these are the same function but shifted from each other by \(\pm\pi/2\) radians in their arguments.)
While it may seem like focusing on a sinusoidal signal with a single frequency will limit the applicability of the analysis we develop in this chapter, this is not actually the case. Fourier’s Theorem states that any well-behaved function can be represented by an integral (sum) of sinusoidal signals.
\begin{equation*}
I(t)=\int\limits_{-\infty}^{\infty} a(\omega) \cos\left(\omega t + \delta_I(\omega)\right) \text{d}\omega
\end{equation*}
where \(a(\omega)\) and \(\delta_I(\omega)\) are weights and phase shifts for each \(\omega\) value. We can use the trigonometric identity
\begin{equation*}
\cos\left(\omega t + \delta_I\right) = \cos(\omega t)\cos\left(\delta_I(\omega)\right) + \sin(\omega t)\sin\left(\delta_I(\omega)\right)
\end{equation*}
to express this in a more common form
\begin{equation*}
I(t)=\int\limits_{-\infty}^{\infty} \left[\alpha(\omega) \cos\left(\omega t\right) +
\beta(\omega) \sin\left(\omega t\right)\right]\text{d}\omega
\end{equation*}
where \(\alpha(\omega)=a(\omega)\cos\left(\delta_I(\omega)\right)\) and \(\beta(\omega)=a(\omega)\sin\left(\delta_I(\omega)\right)\text{.}\)
Since any general (well-behaved) function of time can be represented as a sum of sinusoidal signals, we can use the superposition theorem to find the response of a linear circuit to a general time signal. If we examine the behavior of a linear circuit in response to a single sinusoidal signal with frequency \(\omega\text{,}\) we can then use superposition to add up (integrate) responses for many sinusoidal signals (each with its own \(\omega\)), weighting the results in the sum using the same weighting factors that appear above.
Thus, it is sufficient to examine the response of linear circuits to a single sinusoidal signal. INCLUDE FURTHER DISCUSSION OF SINE/COSINE SERIES
