5] E field shape of the first 3 modes on the excitation port:
5
First-mode electric field shape:
Second-mode electric field shape:
Third-mode electric field shape:
6
Comment:
As we can when a dielectric material is put inside the waveguide the cut off frequency of modes is decreased so more modes may appear in wave guide & this is due to the following equation:
Fcnm=(c/ (εr)^(0.5))((n/a)^2+(m/b)^2)^(0.5).
So more modes like TE11 & TM11 appear in waveguide at same operating frequency in vacuum these 2 modes doesn’t exist.
7
Using MATLAB: With perfect conductor and vacuum inside:
1] The propagation constant (β) Vs frequency:Code:vmin=7*10^9;vmax=2*10^10;samples=10^5;vs=(vmax-vmin)/samples;v=[vmin:vs:vmax];mu=1.2566*10^-6;e=8.854*10^-12;n1=1;m1=0;n2=0;m2=1;n3=1;m3=1;n4=2;m4=0;a=18*10^-3;b=15.5*10^-3;beta1=sqrt((((2*pi*v).^2)*mu*e)-((((n1*pi)/a)^2)+(((m1*pi)/b)^2)));beta2=sqrt((((2*pi*v).^2)*mu*e)-((((n2*pi)/a)^2)+(((m2*pi)/b)^2)));beta3=sqrt((((2*pi*v).^2)*mu*e)-((((n3*pi)/a)^2)+(((m3*pi)/b)^2)));beta4=sqrt((((2*pi*v).^2)*mu*e)-((((n4*pi)/a)^2)+(((m4*pi)/b)^2)));figure(1)plot(v,beta1,v,beta2,v,beta3,v,beta4);xlabel('f(Hz)');ylabel('beta');title('Propagation constant vs freq.');
Simulation:
0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 1010
0
50
100
150
200
250
300
350
400
f(Hz)
beta
Propagation constant vs freq.
TE10TE11&TM11
TE20
TE01
8
2] Characteristic impedance Vs frequency:
Code:vmin=1*10^9;vmax=5*10^10;samples=10^5;vs=(vmax-vmin)/samples;v=[vmin:vs:vmax];c=3*10^8;mu=1.2566*10^-6;e=8.854*10^-12;ztem=sqrt(mu/e);n1=1;m1=0;n2=0;m2=1;n3=1;m3=1;n4=2;m4=0;a=18*10^-3;b=15.5*10^-3;fc10=((c/2)*sqrt(((n1/a)^2)+((m1/b)^2)));fc01=((c/2)*sqrt(((n2/a)^2)+((m2/b)^2)));fc11=((c/2)*sqrt(((n3/a)^2)+((m3/b)^2)));fc20=((c/2)*sqrt(((n4/a)^2)+((m4/b)^2)));zte10=ztem.*(1./sqrt(1-((fc10./v).^2)));zte01=ztem.*(1./sqrt(1-((fc01./v).^2)));zte11=ztem.*(1./sqrt(1-((fc11./v).^2)));zte20=ztem.*(1./sqrt(1-((fc20./v).^2)));figure(1)plot(v,zte10,v,zte01,v,zte11,v,zte20);axis([2*10^9 4*10^10 0 0.2*10^4]);xlabel('f(Hz)');ylabel('zte(ohm)');title('ZTE vs freq.');
Simulation:
0.5 1 1.5 2 2.5 3 3.5 4
x 1010
0
200
400
600
800
1000
1200
1400
1600
1800
2000
f(Hz)
zte(
ohm
)
ZTE vs freq.
TE01
TE11&TM11
TE20
TE10
9
With dielectric constant: 1] The propagation constant (β) Vs frequency:Code:vmin=3*10^9;vmax=8*10^9;samples=10^5;vs=(vmax-vmin)/samples;v=[vmin:vs:vmax];mu=1.2566*10^-6;e=8.854*10^-12;n1=1;m1=0;n2=0;m2=1;n3=1;m3=1;n4=2;m4=0;a=18*10^-3;b=15.5*10^-3;beta1=sqrt((((2*pi*v).^2)*mu*e*6.15)-((((n1*pi)/a)^2)+(((m1*pi)/b)^2)));beta2=sqrt((((2*pi*v).^2)*mu*e*6.15)-((((n2*pi)/a)^2)+(((m2*pi)/b)^2)));beta3=sqrt((((2*pi*v).^2)*mu*e*6.15)-((((n3*pi)/a)^2)+(((m3*pi)/b)^2)));beta4=sqrt((((2*pi*v).^2)*mu*e*6.15)-((((n4*pi)/a)^2)+(((m4*pi)/b)^2)));figure(1)plot(v,beta1,v,beta2,v,beta3,v,beta4);xlabel('f(Hz)');ylabel('beta');title('Propagation constant vs freq.');
Simulation:
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
x 109
0
50
100
150
200
250
300
350
400
f(Hz)
beta
Propagation constant vs freq.
TE01
TE10 TE11&TM11
TE20
10
2] Characteristic impedance Vs frequency:
Code:vmin=1*10^9;vmax=5*10^10;samples=10^5;vs=(vmax-vmin)/samples;v=[vmin:vs:vmax];c=3*10^8;mu=1.2566*10^-6;e=8.854*10^-12;ztem=sqrt(mu/(e*6.15));n1=1;m1=0;n2=0;m2=1;n3=1;m3=1;n4=2;m4=0;a=18*10^-3;b=15.5*10^-3;fc10=((c/2)*sqrt(((n1/a)^2)+((m1/b)^2)))/sqrt(6.15);fc01=((c/2)*sqrt(((n2/a)^2)+((m2/b)^2)))/sqrt(6.15);fc11=((c/2)*sqrt(((n3/a)^2)+((m3/b)^2)))/sqrt(6.15);fc20=((c/2)*sqrt(((n4/a)^2)+((m4/b)^2)))/sqrt(6.15);zte10=ztem.*(1./sqrt(1-((fc10./v).^2)));zte01=ztem.*(1./sqrt(1-((fc01./v).^2)));zte11=ztem.*(1./sqrt(1-((fc11./v).^2)));zte20=ztem.*(1./sqrt(1-((fc20./v).^2)));figure(1)plot(v,zte10,v,zte01,v,zte11,v,zte20);axis([0.5*10^9 1*10^10 0 0.2*10^4]);xlabel('f(Hz)');ylabel('zte(ohm)');title('ZTE vs freq.');
Simulation:
1 2 3 4 5 6 7 8 9 10
x 109
0
200
400
600
800
1000
1200
1400
1600
1800
2000
f(Hz)
zte(
ohm
)
ZTE vs freq.
TE20
TE11&TM11
TE10
TE01
11
Comparing MATLAB & HFSS results:
As we can see the HFSS is a simulation program for microwaves so it takes in consideration another factors than mat lab as we neglect some terms (make approximation) in mat lab to get familiar equation that can be used easily in hand analysis.
