Particular Solution

In general, the frequency response function (FRF) of the system is defined as the ratio of the output to the input of the system, both represented as complex exponentials. In this specific case, we have

$\displaystyle H=\frac{{\bf I}}{{\bf V}}=\frac{1}{Z}$ (102)


$\displaystyle \vert H\vert=\frac{1}{\sqrt{R^2+\omega^2L^2}},\;\;\;\;\;\angle H=-\phi$ (103)

Therefore the steady state output can be found to be:

$\displaystyle i(t)=\vert H\vert \cos(\omega t+\angle H)
=\frac{1}{\sqrt{R^2+\omega^2 L^2}}\cos(\omega t-\phi)$ (104)

The second method, much easier than the first one, is actually a short cut representation of the first DE method. This is the justification of the complex variable or phasor method for analyzing AC circuits. However, note that the phasor method can only find the steady state solution. The homogeneous differential equation will have to be solved to obtain the transient solution.