ZIP with MATLAB scripts and note:

example 6.7

example 6.7 notes:

It would be accurate to calculate frequency deviations with the exact expression: (310pg):

same as

But when fields E H and band parameters not known, but for

then :

the numerator integral of the initial expression can be constrained to the volume where the material is introduced in the cavity, therefore

Wm We: stored magnetic and electric energy,

c0=299792458

f0=5e9

er=3;e0=8.854e-12;mu0=4*pi*10^-7

A=1

etha0=120*pi;lambda0=c0/f0;k0=2*pi/lambda0

a=.02;b=.01;d=.1

t=.0025

dx=a/200;

x=[0:a/200:a];y=[0:b/200:b];z=[0:d/200:d];

since relative magnetic permeability is null, the function to integrate in the numerator is |Ey| only.

% or with anonymous function: df0(t)/f0

df_rel=[]

for k=1:1:numel(y)

fun1=@(xsym,ysym,zsym)...

(er-1)*e0*(A*sin(pi*xsym/a)*sin(pi*zsym/d)).^2;

N1=triplequad(fun1,0,a,0,y(k),0,d);

fun2=@(xsym,ysym,zsym)...

(e0*(A*sin(pi*xsym/a)*sin(pi*zsym/d)).^2.+...

mu0*((1j*A/(k0*etha0*a)*sin(pi*xsym/a).*cos(pi*zsym/d)).^2.+...

(1j*pi*A/(k0*etha0*a)*cos(pi*xsym/a).*sin(pi*zsym/d)).^2));

D1=triplequad(fun2,0,a,0,b,0,d);

df_rel=[df_rel N1/D1];

end

df_rel(1)=[];df_rel(end)=[];

figure(1);plot(y([2:end-1]),df_rel);grid on;

xlabel('t');ylabel('df0')

title('frequency relative deviation with integral')

% df0(t)/f0 from direct expression

df_rel2=[]

for k=1:1:numel(y)

E1=(er-1)*e0*A^2*a*y(k)*d/4;

E2=a*b*d*e0/2*A^2;

df_rel2=[df_rel2 -E1/E2];

end

figure(2);plot(y,df_rel2);grid on;

xlabel('t');ylabel('df0')

title('frequency relative deviation with direct expression')

note the graph shows |df0|>1 but given the long list of approximations this calculation should only be applied for frequencies close to f0.