ECE6124 Assignment: Harmonic Balance and Multiple Scales Analysis

This assignment solution covers the application of the Harmonic Balance method to determine the current through a resistor in a non-linear circuit, along with the implementation of scaling techniques. The Harmonic Balance method involves reducing the non-linear circuit to a linear one and solving for the current using Matlab code, including parameter definitions and frequency determination. Additionally, the solution addresses the simplification and analysis of an equation using the method of multiple scales, again implemented in Matlab to generate a graphical representation of the solution. References to relevant literature on numerical methods in electromagnetics and multiple scale methods are also provided. Desklib provides solved assignments and past papers for students.

Electronics and Communication Engineering

Contents
1. Harmonic Balance method for given non- linear circuit................................................................2
2. Methods of scaling.............................................................................................................................4
3. References..........................................................................................................................................5
1

Matlab Code:
1. Harmonic Balance method for given non- linear circuit
The below equations are found from the given circuit. The non-linear circuit is reduced to linear
circuit [1]. Then the current flow through the resistor R2 is find.
% Analysis parameters
ha = 8; %% harmonic order
nu = 2^7; %% Number of samples
ws = .5; %% Starting frequency
we = 1.6; %% Ending frequency
Q= 0.1;
%% component values
Vin= sinwt;
Rd= 1;
Xd= 3;
R2= 6;
Xc= 8;
%% From the given circuit
a = (Xc*R2)/ (Xc+R2);
2

b = a + (Rd + Xd); %% ((Xc*R2) + (Rd + Xd)*(Xc +R2))/ (Xc + R2)
Id = Vin / b; %% (Vin (Xc+R2)) / ((Xc*R2) + ((Rd+Xd)*(Xc+R2)))
IR2 = (Id * Xc) / (R2 + Xc); %% current through the resistance R2 from the given circuit
y0 = [0;real(IR2);-imag(IR2);zeros(2*(ha-1),1)];
%% Solving with respect to ‘w‘
ps = .01; % Path step size
S_opt = struct('jacob','none'); % Jacobian is not used in this program
Y = solve_and_continue(X0,... @(X) HB_DOF_Spring(R2,Id,Xc,Xd,H,N),... ws,we,ps,S_opt);
%%Determination of the frequency and amplitude
w = Y(end,:);
c = sqrt(Y(3,:).^2 + Y(4,:).^2);
function R = HB_single_DOF_Spring(R2,Id,Xc,Xd,H,N) % Real is converted to the complex
%% form in the harmonics.
IR2 = [Y(1);Y(2:2:end-1)-1i*Y(3:2:end-1)]; % frequencies which are excited
w = Y(end); % harmonic occurs fundamentally due to the force from external
Fex = [0;Q;zeros(ha-1,1)]; % IFFT
t = (0:2*pi/nu:2*pi-2*pi/nu)';
q = real(exp(1i*t*(0:ha))*IR2); % Evaluation of the nonlinear force
f = gamma*q.^3; % DFT
Fc = fft(f)/n;
F = [real(Fc(1));2*Fc(2:ha+1)]; % force occurs in the state of equilibrium
3

Rc = ( -((0:ha)'*w).^2 * mu + 1i*(0:ha)'*w * delta + kappa ).*IR2+f-Fex;
R = [real(Rc(1));real(Rc(2:end));-imag(Rc(2:end))];
P = harmonic (IR2);
figure(1);
title ('Harmonic balance for non-linear circuit');
fplot (P, [-5,5]), grid on;
2. Methods of scaling
The above equation is simplified as,
The above equation is the first order equation which is derived from the given equation of
methods of multiple scales [2].
Matlab Code
%% Assume the values of x, l, m
x = 6;
l = 8;
m = 1;
%% First derivative of the given equation
syms x(t)
4

eqn = diff(x,2) == ((k*x)/(m*sqrt(x^2+l^2)))*((l/2) – (1/sqrt(x^2+l^2)))
V = odeToVectorField(eqn)
%% Matlab function
M = matlabFunction (V,'vars', {'t','X'})
interval = [0 20];
x0 = [2 0];
xSol = ode45(M,interval,x0)
%% Generate graph
tValues = linspace(0,20,100); %% Generate t values in the interval in linespace function
xValues = deval(xSol,tValues,1);
plot(tValues,xValues)
3. References
[1]"Introduction to Numerical Methods in Electromagnetics", Advanced Modeling in
Computational Electromagnetic Compatibility, pp. 80-122.
[2]Math.ucdavis.edu, 2018. [Online]. Available:
https://www.math.ucdavis.edu/~hunter/m204/ch6.pdf. [Accessed: 27- Nov- 2018].
5