Harmonic Balance Method and Methods of Scaling for Electronics and Communication Engineering

This article discusses the Harmonic Balance method and Methods of scaling for Electronics and Communication Engineering. It includes simplified equations and Matlab code for implementation. References are also provided.

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