Question 8.52: The dependent source circuit in Figure P8-52 is operating in......
The dependent source circuit in Figure P8-52 is operating in the sinusoidal steady state with ω = 1 krad/s and µ = 100. Find the phasor gain K=\mathrm{V}_{\mathrm{O}} / \mathrm{V}_{\mathrm{S}} and the input impedance Z_{\mathrm{IN}} seen by {V}_{\mathrm{S}}. Validate your answer using OrCAD.
Use node-voltage analysis to determine expressions for the voltages in the circuit and then find the input-output ratio. Once we know the voltages, we can also compute the input impedance. The solution using MATLAB is presented below.
Script File
clear all
syms Vs Vo Vx V1
mu = 100;
Eqn1 = (V1-Vs)/10e3 + (V1-Vx)/10e3 + (V1-mu*Vx)/(-10e3j);
Eqn2 = (Vx-V1)/10e3 + Vx/(-10e3j);
Soln = solve(Eqn1,Eqn2,V1,Vx);
Vx = Soln.Vx;
Vo = mu*Vx;
K = double(Vo/Vs)
MagK = abs(K)
PhaseK = angle(K)*180/pi
% Compute the input impedance
V1 = Soln.V1;
Is = (Vs-V1)/10e3;
Zin = double(simplify(Vs/Is))
K =
0.0000e+000 + 1.0309e+000i
MagK =
1.0309e+000
PhaseK =
90.0000e+000
Zin =
9.8969e+003 +100.9891e+000i
Verify the MATLAB results using OrCAD. First compute the capacitance at ω = 1 krad/s.
% Compute the capacitance
w = 1e3;
wC = 1/10e3;
C = wC/w
C =
100.0000e-009
The required circuit is shown below.
The results are shown below.
FREQ VM(N05696,0) VP(N05696,0) VR(N05696,0) VI(N05696,0)
1.592E+02 1.031E+00 9.000E+01 1.227E-11 1.031E+00
FREQ IM(V_PRINT1) IP(V_PRINT1) IR(V_PRINT1) II(V_PRINT1)
1.592E+02 1.010E-04 -5.846E-01 1.010E-04 -1.031E-06
We can compute the input impedance by taking the ratio of the input voltage to the input current. The input voltage is 1 ∠0° V.
% Compute input impedance
Iin2 = 1.01e-4 - 1.031e-6j;
Zin2 = 1/Iin2
Zin2 =
9.9000e+003 +101.0580e+000i
The OrCAD results agree with the MATLAB calculations.
\begin{aligned}& K=\rm{V}_{\mathrm{O}} / \mathrm{V}_{\mathrm{S}}=j 1.0309=1.0309 ~\angle ~90^{\circ}. \\\\& Z_{\mathrm{IN}}=9.8969+j 0.101 ~\mathrm{k} \Omega.\end{aligned}
