example 3.1.1

ZIP with MATLAB scripts and note:

example 3.1.1 notes:

Parallel plates waveguide

Although it's 3rd chapter 1st example, [POZAR] doesn't give a numeral as with all other examples. Therefore I have numbered this example as 3.1.1.

Example 3.1.2 is the explanation of why 75 Ohm or 50 Ohm popular values of coaxial Z0. Example 3.1.3 corresponds to [POZAR] 3rd chapter 1st example.

Parallel metal plates:

TEM Transversal Electric and Magnetic

Ez=Hz=0

TM Transversal Magnetic

no magnetic field on direction of travelling wave

Hz=0

TE Transversal Magnetic

no electric field on direction of travelling wave

Ez=0

pozar_03_example_parallel_plates_table.m

mu0=4*pi*1e-7; e0= 8.8539e-12;er=1;mur=1;f0=10e9 % Hz

tand=.004;Rs=.01;etha0=377;etha=etha0/er^.5;

d=.01 % distance between plates [m]

N=5

n=[1:1:N] % 5 1st modes

k_TEM=2*pi*f0*(mu0*e0*mur*er)^.5*ones(1,N)

k_TE=2*pi*f0*(mu0*e0*mur*er)^.5*ones(1,N)

k_TM=2*pi*f0*(mu0*e0*mur*er)^.5*ones(1,N)

kc_TEM=zeros(1,N)

kc_TEn=n*pi/d

kc_TMn=n*pi/d

beta_TEM=2*pi*f0*(mu0*e0*mur*er)^.5*ones(1,N)

beta_TMn=imag((k_TM.^2-kc_TMn.^2).^.5)

beta_TEn=imag((k_TM.^2-kc_TMn.^2).^.5)

lambdac_TEM=Inf*ones(1,N)

lambdac_TMn=2*d./n

lambdac_TEn=2*d./n

lambdag_TEM=2*pi./k_TEM

lambdag_TMn=2*pi./beta_TMn

lambdag_TEn=2*pi./beta_TEn

vp_TEM=2*pi*f0./k_TEM

vp_TMn=2*pi*f0./beta_TMn

vp_TEn=2*pi*f0./beta_TEn

alphad_TEM=.5*k_TEM*tand

alphad_TMn=.5*k_TM.^2*tand./beta_TMn

alphad_TEn=.5*k_TE.^2*tand./beta_TEn

alphac_TEM=Rs/(etha*d)*ones(1,N)

alphac_TMn=2*k_TM*Rs./(beta_TMn)*etha*d

alphac_TMn=2*k_TE*Rs./(beta_TEn)*etha*d

% Fields for given mode only n(n1)

n1=1

W=10*d % plates width [m]

L=20*d % plates depth [m]

dx=d/10

x=[0:dx:W];y=[0:dx:d];z=[0:dx:L];

Ez_TEM=zeros(numel(x),numel(y),numel(z));

Hz_TEM=zeros(numel(x),numel(y),numel(z));

Ex_TEM=zeros(numel(x),numel(y),numel(z));

Ez_TM=ones(1,numel(x))'.*sin(pi/d*n(n1)*y).*reshape(exp(-1j*beta_TMn(n1)*z),1,1,numel(z));

Hz_TM=zeros(numel(x),numel(y),numel(z));

Ex_TM=zeros(numel(x),numel(y),numel(z));

Ez_TE=zeros(numel(x),numel(y),numel(z));

Hz_TE=ones(1,numel(x))'.*cos(n(n1)*pi*y/d).*reshape(exp(-1j*beta_TEn(n1)*z),1,1,numel(z));

Ex_TE=1j*2*pi*f0*mu0*mur/kc_TEn(n1)*ones(1,numel(x))'.*sin(n1*pi*y/d).*reshape(exp(-1j*beta_TEn(n1)*z),1,1,numel(z));

V0=1; % V0 Voltage applied between parallel plates, for instance V0=1Volt

Ey_TEM=-V0/d*ones(1,numel(x))'.*ones(1,numel(y)).*reshape(exp(-1j*beta_TEM(n1)*z),1,1,numel(z));

Hx_TEM=V0/(d*etha)*ones(1,numel(x))'.*ones(1,numel(y)).*reshape(exp(-1j*beta_TEM(n1)*z),1,1,numel(z));

Hy_TEM=zeros(numel(x),numel(y),numel(z));

Ey_TM=-1j*beta_TMn(n1)/kc_TMn(n1)*ones(1,numel(x))'.*cos(n(n1)*pi*y/d).*reshape(exp(-1j*beta_TMn(n1)*z),1,1,numel(z));

Hx_TM=-1j*2*pi*f0*e0*er./kc_TMn(n1).*ones(1,numel(x))'.*cos(n(n1)*pi*y/d).*reshape(exp(-1j*beta_TMn(n1)*z),1,1,numel(z));

Hy_TM=zeros(numel(x),numel(y),numel(z));

Ey_TE=zeros(numel(x),numel(y),numel(z));

Hx_TE=zeros(numel(x),numel(y),numel(z));

Hy_TE=1j*beta_TEn(n1)/kc_TEn(n1)*ones(1,numel(x))'.*sin(n(n1)*pi*y/d).*reshape(exp(-1j*beta_TEn(n1)*z),1,1,numel(z));

% intrinsic impedance of all modes

Z_TEM=etha*d/W

Z_TMn=etha*beta_TMn./k_TM

Z_TEn=etha*k_TE./beta_TEn

pozar_03_example_parallel_plates_attenuation_conductor.m

% k/kc=k*d/pi

% TE: beta=(k^2-kc^2)^.5=(k^2-(n*pi/d)^2)^.5

% alpha_c=2*kc^2*Rs/(k*beta*etha*d) , rearranging

% alpha_c*etha*d//Rs=2/(x*(x^2-1)^.5) with x=k/kc

c0=299792458

syms x

alpha_c_TE=2/(x*(x^2-1)^.5)

figure(1);h=fplot(alpha_c_TE);grid on

ax=h.Parent

ax.XLim=[0 10];ax.YLim=[0 3];

xlabel('k/kc=k*d/pi=2*d/lambda');ylabel('alpha\_conductor*etha*d/Rs')

% TM: alpha_c=2*kc*Rs/(beta*etha*d) , rearranging

% alpha_c*etha*d/Rs=2/(1-1/x^2)^.5 with x=k/kc

hold on

alpha_c_TM=2/(1-1/x^2)^.5

fplot(alpha_c_TM);grid on

legend('TE1','TM1');title('Attenuation [Np/m]')

alpha_c_TE_dB_over_f=alpha_c_TE*Np2dB

alpha_c_TM_dB_over_f=alpha_c_TM*Np2dB

syms x2

x2=1/(2*pi)*c0*x

Np2dB=10*log10(exp(1)^2);

alpha_c_TE_dB=alpha_c_TE*Np2dB

alpha_c_TM_dB=alpha_c_TM*Np2dB

figure(2);h2=fplot(alpha_c_TE_dB);grid on

hold on

ax2=h2.Parent

fplot(ax2,alpha_c_TM_dB);grid minor

legend(ax2,'TE1','TM1');title(ax2,'Attenuation [dB/m]')

ax2.XLim=[0 15];ax2.YLim=[0 30];

xlabel('k/kc=k*d/pi=2*d/lambda');ylabel('alpha\_conductor*etha*d/Rs')

Only 1m of ideal waveguide causes 5dB metal attenuation 1 octave abov cut-off

Single surface waveguides suffer more attenuation compared to combined (finite)

structures like pair wires, coax, etc..

TEM between parallel metal plates with W>>d |E| does not have [x y z] dependencies, that should be true between capacitor plates.

pozar_03_example_parallel_plates_fields.m

f0=2e10 % carrier frequency [Hz]

d=0.43e-2;W=10*d % waveguide cross-section

length_waveguide=5*d; % waveguide length [m]

er=2.08; % fill material: Teflon

dx=d/50;dy=dx;dz=dx; % set space resolutions

c0=299792458;

V0=1; % 1Volt between plates

etha0=377

etha=etha0/er^.5

lambda0=c0/f0;lambda=lambda0/er^.5;

beta=2*pi/lambda;

x=[0:dx:W];

y=[0:dx:d];

z=[0:dx:length_waveguide];

[Y,X,Z]=meshgrid(y,x,z);

ExTEM=zeros(numel(x),numel(y),numel(z));

EyTEM=zeros(numel(x),numel(y),numel(z));

EzTEM=zeros(numel(x),numel(y),numel(z));

HxTEM=zeros(numel(y),numel(y),numel(z));

HyTEM=zeros(numel(x),numel(y),numel(z));

HzTEM=zeros(numel(x),numel(y),numel(z));

% |E| n=0 TEM

one way to build a volume filled with field values is directly multiplying 1 dimension only vectors

EyTEM=ones(1,numel(x))'.*ones(1,numel(y))*…

(-V0/d).*reshape(exp(-1j*beta*z),1,1,numel(z));

HxTEM=(ones(1,numel(x))'.*…

ones(1,numel(y)).*…

reshape(exp(-1j*beta*z),1,1,numel(z)))*(-V0/(etha*d));

absE_TEM=((abs(EyTEM)).^2+(abs(EzTEM)).^2).^.5;

n=0;

xslice=[x(1) x(floor(numel(x)/2))];yslice=y(1);

zslice=z(floor(numel(z)/2));

colormap('jet');shading interp

daspect([1 1 1]);axis tight;camlight

figure(1);h1=slice(Y,X,Z,absE_TEM,yslice,xslice,zslice);

% figure(1);h1=slice(X,Y,Z,absE_TEm,xslice,yslice,zslice);

colorbar;ax1=gca;

ax1.DataAspectRatio=[1 1 1]

h1(1).EdgeColor='none';

h1(2).EdgeColor='none';

h1(3).EdgeColor='none';

colormap(ax1,'jet');shading interp

daspect(ax1,[1 1 1]);axis tight;

xlabel(ax1,'y:[0 d]');ylabel(ax1,'x:[0 W]');

zlabel(ax1,'z:[0 sheets depth');

title(ax1,['TE' n ' |E| parallel metal sheets waveguide']);

campos(ax1,[.1732 .0799 -.1491]);

Z_TEM=etha*d/W*ones(numel(x),numel(y),numel(z));

TE10 shape of |E| between metal parallel plates:

% |E| n=1, TE1 Transversal Electric field

n=1;

Another way to build a volume filled with a vector field is using meshgrid variables, simplifying notation but cubic increase of data to process

ExTE1=sin(n*pi*Y/d).*exp(-1j*beta*Z);

EyTE1=sin(n*pi*Y/d).*exp(-1j*beta*Z);

EzTE1=zeros(size(ExTE1));

absE_TE1=real(((ExTE1).^2+(EyTE1).^2+(EzTE1).^2).^.5);

colormap('jet');shading interp

daspect([1 1 1]);axis tight;camlight

figure(2);h2=slice(Y,X,Z,absE_TE1,yslice,xslice,zslice);

colorbar;ax2=gca;

ax1.DataAspectRatio=[1 1 1];

h2(1).EdgeColor='none';

h2(2).EdgeColor='none';

h2(3).EdgeColor='none';

colormap(ax2,'jet');shading interp

daspect(ax2,[1 1 1]);axis tight;

xlabel(ax2,'y:[0 d]');ylabel(ax2,'x:[0 W]');

zlabel(ax2,'z:[0 length waveguide');

title(ax2,['TE ' num2str(n) ' |E| parallel metal sheets waveguide']);

campos(ax2,[.1732 .0799 -.1491]);

n=2;

ExTE2=sin(n*pi*Y/d).*exp(-1j*beta*Z);

EyTE2=sin(n*pi*Y/d).*exp(-1j*beta*Z);

EzTE2=zeros(size(ExTE2));

absE_TE2=real(((ExTE2).^2+(EyTE2).^2+(EzTE2).^2).^.5);

colormap('jet');shading interp

daspect([1 1 1]);axis tight;camlight

figure(3);h3=slice(Y,X,Z,absE_TE2,yslice,xslice,zslice);

colorbar;ax3=gca;

ax1.DataAspectRatio=[1 1 1];

h3(1).EdgeColor='none';

h3(2).EdgeColor='none';

h3(3).EdgeColor='none';

colormap(ax3,'jet');shading interp

daspect(ax3,[1 1 1]);axis tight;

xlabel(ax3,'y:[0 d]');ylabel(ax3,'x:[0 W]');zlabel(ax3,'z:[0 length waveguide');

title(ax3,['TE ' num2str(n) ' |E| parallel metal sheets waveguide']);

campos(ax3,[.1732 .0799 -.1491]);