Question 8.55: Given the circuit in Figure P8-55: (a) Use node-voltage or m......
Given the circuit in Figure P8-55:
(a) Use node-voltage or mesh-current analysis to develop a set of matrix equations for the circuit.
(b) Use MATLAB to solve the matrix equations and then find the phasor gain K=\mathrm{V}_{\mathrm{O}} / \mathrm{V}_{\mathrm{S}} and input impedance Z_{\mathrm{IN}} of the circuit.
(c) Without using the matrix equations, use the MATLAB command solve on the original node-voltage or mesh-current equations to solve the equations and then find the phasor gain and input impedance.
(a) The node-voltage equations are presented below.
\begin{aligned}& \frac{\mathrm{V}_{\mathrm{A}}-\mathrm{V}_{\mathrm{S}}}{15}+\frac{\mathrm{V}_{\mathrm{A}}-\mathrm{V}_{\mathrm{B}}}{60}+\frac{\mathrm{V}_{\mathrm{A}}-\mathrm{V}_{\mathrm{O}}}{j 50}=0\\ \\& \frac{\mathrm{V}_{\mathrm{B}}-\mathrm{V}_{\mathrm{A}}}{60}+\frac{\mathrm{V}_{\mathrm{B}}}{-j 90}+\frac{\mathrm{V}_{\mathrm{B}}-\mathbf{V}_{\mathrm{O}}}{50}=0 \\\\& \frac{\mathrm{V}_{\mathrm{O}}}{100}+\frac{\mathrm{V}_{\mathrm{O}}-\mathrm{V}_{\mathrm{B}}}{50}+\frac{\mathrm{V}_{\mathrm{O}}-\mathrm{V}_{\mathrm{A}}}{j 50}=0\end{aligned}These equations can be re-written as follows:
These equations can be written in matrix form, as will be shown in the MATLAB code that follows.
(b) Use MATLAB to create the matrix equations and solve for the required values
Script File
clear all
syms Vs Va Vb Vo
A = [1/15+1/60+1/50j, -1/60, -1/50j; ...
-1/60, 1/60+1/(-90j)+1/50, -1/50; ...
-1/50j, -1/50, 1/100+1/50+1/50j];
B = [Vs/15; 0; 0];
Vans = A\B;
Va = Vans(1);
Vb = Vans(2);
Vo = Vans(3);
K = double(simplify(Vo/Vs))
MagK = abs(K)
PhaseK = angle(K)*180/pi
% Compute the input current to find the input impedance
Iin = (Vs-Va)/15;
Zin = double(simplify(Vs/Iin))
K =
646.1342e-003 -405.7975e-003i
MagK =
762.9948e-003
PhaseK =
-32.1304e+000
Zin =
84.9075e+000 - 20.1201e+000i
(c) Use the MATLAB command solve to verify the results
clear all
syms Vs Va Vb Vo
Eqn1 = (Va-Vs)/15+(Va-Vb)/60+(Va-Vo)/50j;
Eqn2 = (Vb-Va)/60+Vb/(-90j)+(Vb-Vo)/50;
Eqn3 = Vo/100+(Vo-Vb)/50+(Vo-Va)/50j;
Soln2 = solve(Eqn1,Eqn2,Eqn3,Va,Vb,Vo);
Va2 = Soln2.Va;
Vo2 = Soln2.Vo;
K2 = double(simplify(Vo2/Vs))
Iin2 = simplify((Vs-Va2)/15);
Zin2 = double(simplify(Vs/Iin2))
K2 =
646.1342e-003 -405.7975e-003i
Zin2 =
84.9075e+000 - 20.1201e+000i
The results agree with those calculated above.
(a) The equations are presented above.
(b) K = 0.6461 − j0.4058 = 0.76299 ∠−32.13°.
Z_{\mathrm{IN}}=84.907-j 20.12~ \Omega .
(c) The answers using the second MATLAB approach match those found previously.