Question 8.22: The circuit in Figure P8-22 is operating in the sinusoidal s......
The circuit in Figure P8-22 is operating in the sinusoidal steady state with v_{\mathrm{S}}(t)=V_{\mathrm{A}} cos (ωt). Derive a general expression for the phasor response \rm{I}_{\mathrm{L}} and the voltage \rm{V}_{\mathrm{O}}.
There are several ways to solve this problem. One approach is to combine the impedances of the parallel inductor and resistor and then use voltage division to find the output voltage in phasor form. We can then calculate the inductor current in phasor form. The following MATLAB code completes the calculations
Script File
clear all
format short eng
syms R L positive
syms w VA t IL Vo Vs ZL Zeq
% Create a phasor for the source voltage
Vs = VA*exp(0*j*pi/180);
% Calculate the inductor impedance and the equivalent
% impedance of the parallel inductor-resistor combination
ZL = j*w*L;
Zeq = 1/(1/ZL + 1/R);
% Use voltage division to find the output voltage phasor
Vo = simplify(Zeq*Vs/(Zeq+R))
% Find the phasor inductor current
IL = simplify(Vo/ZL)
Vo =
-VA*w*L/(-2*w*L+i*R)
IL =
i*VA/(-2*w*L+i*R)
\begin{aligned}& \rm{I}_{\mathrm{L}}=\frac{V_{\mathrm{A}}}{R+2 j \omega L}\\ \\& \rm{V}_{\mathrm{O}}=\frac{j \omega L V_{\mathrm{A}}}{R+2 j \omega L}\end{aligned}