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exercise 1.6

ZIP with MATLAB scripts and note:

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 example 1.1 notes:

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pozar_01_exercise_06_question.jpg
Reflection on ideal conductor


c0=299792458        % [m/s] light speed

mu0=4*pi*1e-7     % [H/m] permeability
mur=1  

etha0=120*pi         % [Ohm] air intrinsic impedance

E0=1                      % [V/m] incident electric field amplitude
f0=1e9                   % [Hz]
lambda0=c0/f0        % [m] wavelength
k0=2*pi/lambda0    % [m^-1] wave number

nk=[0 0 1]  % [x y z]  propagation direction incident wave

syms x y z

%% Incident fields

Ei=E0*[1 -1j 0]*exp(-1j*k0*z)  % incident electric field

Eix=Ei(1)
Eiy=Ei(2)
Eiz=Ei(3)

Hi=1j/etha0*cross(nk,Ei)           % incident magnetic field

Hix=Hi(1)
Hiy=Hi(2)
Hiz=Hi(3)

%% Conductor parameters

sigma_Cu=5.813e7                 % [S/m]  metal conductivity

% metal skin depth

ds=(2/(2*pi*f0*mu0*mur*sigma_Cu))^.5

% gamma : propagation constant

gamma=(1+j)*(2*pi*f0*mu0*sigma_Cu/2)^.5

% approximation of gamma for good conductor
gamma=(1+1j)/ds

% inside the metal
alpha=real(gamma)
beta=imag(gamma)

% wave intrinsic impedance

etha=(1+1j)/(sigma_Cu*ds)

% R : reflection coefficient metal surface 

R=(etha-etha0)/(etha+etha0)

% T : transmission coefficient metal surface

T=2*etha/(etha+etha0)

% check
1+R


%%  Reflected fields

Er=R*Ei

Erx=Er(1)
Ery=Er(2)
Erz=Er(3)

Hr=1j/etha0*cross(nk,Er)

Hrx=Hr(1)
Hry=Hr(2)
Hrz=Hr(3)
Lengthy rational expressions do not help see fields shape, but the field is there.
c0 =
   299792458
mu0 =
     1.256637061435917e-06
mur =
     1
etha0 =
     3.769911184307751e+02
E0 =
     1
f0 =
     1.000000000000000e+09
lambda0 =
   0.299792458000000
k0 =
  20.958450219516820
nk =
     0     0     1
Ei =
[exp(-(z*1474819821857485i)/70368744177664), -exp(-(z*1474819821857485i)/70368744177664)*1i, 0]
Eix =
exp(-(z*1474819821857485i)/70368744177664)
Eiy =
-exp(-(z*1474819821857485i)/70368744177664)*1i
Eiz =
0
Hi =
[-(3058219274252085*exp(-(z*1474819821857485i)/70368744177664))/1152921504606846976, (exp(-(z*1474819821857485i)/70368744177664)*3058219274252085i)/1152921504606846976, 0]
Hix =
-(3058219274252085*exp(-(z*1474819821857485i)/70368744177664))/1152921504606846976
Hiy =
(exp(-(z*1474819821857485i)/70368744177664)*3058219274252085i)/1152921504606846976
Hiz =
0
sigma_Cu =
    58130000
ds =
     2.087468689778981e-06
gamma =
      4.790491013811943e+05 + 4.790491013811943e+05i
gamma =
      4.790491013811943e+05 + 4.790491013811943e+05i
alpha =
     4.790491013811943e+05
beta =
     4.790491013811943e+05
etha =
  0.008240996067112 + 0.008240996067112i
R =
-0.999956280158039 + 0.000043717930620i
T =
      4.371984196094063e-05 + 4.371793061992124e-05i
ans =
      4.371984196083201e-05 + 4.371793061992124e-05i
Er =
[exp(-(z*1474819821857485i)/70368744177664)*(- 9006805461413065/9007199254740992 + 1612906955155755i/36893488147419103232), exp(-(z*1474819821857485i)/70368744177664)*(1612906955155755/36893488147419103232 + 9006805461413065i/9007199254740992), 0]
Erx =
exp(-(z*1474819821857485i)/70368744177664)*(- 9006805461413065/9007199254740992 + 1612906955155755i/36893488147419103232)
Ery =
exp(-(z*1474819821857485i)/70368744177664)*(1612906955155755/36893488147419103232 + 9006805461413065i/9007199254740992)
Erz =
0
Hr =
[exp(-(z*1474819821857485i)/70368744177664)*(27544786061532379213155122490525/10384593717069655257060992658440192 - 4932623137832573262812308499175i/42535295865117307932921825928971026432), exp(-(z*1474819821857485i)/70368744177664)*(- 4932623137832573262812308499175/42535295865117307932921825928971026432 - 27544786061532379213155122490525i/10384593717069655257060992658440192), 0]
Hrx =
exp(-(z*1474819821857485i)/70368744177664)*(27544786061532379213155122490525/10384593717069655257060992658440192 - 4932623137832573262812308499175i/42535295865117307932921825928971026432)
Hry =
exp(-(z*1474819821857485i)/70368744177664)*(- 4932623137832573262812308499175/42535295865117307932921825928971026432 - 27544786061532379213155122490525i/10384593717069655257060992658440192)
Hrz =
0
 1.- E and H in z>0

% Transmitted fields into metal

Et=T*Ei

Etx=Et(1)
Ety=Et(2)
Etz=Et(3)

Ht=1j/etha*cross(nk,Et)

Htx=Ht(1)
Hty=Ht(2)
Htz=Ht(3)



2.- S=E x H in z<0 and z>0

% module of the circular polarized wave
% E0 would actually be E0*absE2
E2=[1 1j 0]
absE2=(sum(E2(:).*conj(E2(:))))^.5

% S(z<0) directly with (1.115a)
Smin0=E0*conj(E0)*absE2^2*1/etha0*(1-R*conj(R)+R-conj(R))

% total S(z>0) directly with (1.115b)
Splus=E0*conj(E0)*absE2^2*T*conj(T)*1/conj(etha)*exp(-2*alpha*z)
Et =
[exp(-(z*1474819821857485i)/70368744177664)*(3225954942385999/73786976294838206464 + 1612906955155755i/36893488147419103232), exp(-(z*1474819821857485i)/70368744177664)*(1612906955155755/36893488147419103232 - 3225954942385999i/73786976294838206464), 0]
Etx =
exp(-(z*1474819821857485i)/70368744177664)*(3225954942385999/73786976294838206464 + 1612906955155755i/36893488147419103232)
Ety =
exp(-(z*1474819821857485i)/70368744177664)*(1612906955155755/36893488147419103232 - 3225954942385999i/73786976294838206464)
Etz =
0
Ht =
[exp(-(z*1474819821857485i)/70368744177664)*(- 55090776429410296037682570617123/10384593717069655257060992658440192 + 1204253695759877252659859183i/10384593717069655257060992658440192), exp(-(z*1474819821857485i)/70368744177664)*(1204253695759877252659859183/10384593717069655257060992658440192 + 55090776429410296037682570617123i/10384593717069655257060992658440192), 0]
Htx =
exp(-(z*1474819821857485i)/70368744177664)*(- 55090776429410296037682570617123/10384593717069655257060992658440192 + 1204253695759877252659859183i/10384593717069655257060992658440192)
Hty =
exp(-(z*1474819821857485i)/70368744177664)*(1204253695759877252659859183/10384593717069655257060992658440192 + 55090776429410296037682570617123i/10384593717069655257060992658440192)
Htz =
0
E2 =
  Column 1
  1.000000000000000 + 0.000000000000000i
  Column 2
  0.000000000000000 + 1.000000000000000i
  Column 3
  0.000000000000000 + 0.000000000000000i
absE2 =
   1.414213562373095
Smin0 =
      4.638616506593357e-07 + 4.638616506605997e-07i
Splus =
exp(-(1028750111805209*z)/1073741824)*(4381049423536409/9444732965739290427392 + 4381049423536409i/9444732965739290427392)
 
3.- show complex power is conserved at z=0

% power on air / metal interface

% from right : metal
Splus0=double(subs(Splus,z,0))

% Same power on interface

%% adding fields :  instead of 1.115a/b

Smin2=cross(Ei+Er,conj(Hi+Hr))


Smin20=double(subs(Smin2,z,0))


Splus2=cross(Et,conj(Ht))


Splus20=double(subs(Splus2,z,0))
Splus0 =
      4.638616506605998e-07 + 4.638616506605998e-07i

Smin2 =
[0, 0, -(exp(-(z*1474819821857485i)/70368744177664)*exp((conj(z)*1474819821857485i)/70368744177664)*15912420005586018318662059257509027593386095565i)/784637716923335095479473677900958302012794430558004314112]
Smin20 =
   1.0e-10 *
  Column 1
  0.000000000000000 + 0.000000000000000i
  Column 2
  0.000000000000000 + 0.000000000000000i
  Column 3
  0.000000000000000 - 0.202799580779530i
Splus2 =
[0, 0, exp(-(z*1474819821857485i)/70368744177664)*exp((conj(z)*1474819821857485i)/70368744177664)*(177716477804014917002290762852207227539240764547/383123885216472214589586756787577295904684780545900544 - 177716477804014917002290762852207227539240764547i/383123885216472214589586756787577295904684780545900544)]

Splus20 =
   1.0e-06 *
  Column 1
  0.000000000000000 + 0.000000000000000i
  Column 2
  0.000000000000000 + 0.000000000000000i
  Column 3
  0.463861650660600 - 0.463861650660600i
As expected Splus0 and Splus02 are the same.
4.- polarization of reflected wave

Erx=Er(1)
Ery=Er(2)
Erz=Er(3)

double(subs(Erx,z,0))
double(subs(Ery,z,0))
double(subs(Erz,z,0))

 
Erx =
exp(-(z*1474819821857485i)/70368744177664)*(- 9006805461413065/9007199254740992 + 1612906955155755i/36893488147419103232)

Ery =
exp(-(z*1474819821857485i)/70368744177664)*(1612906955155755/36893488147419103232 + 9006805461413065i/9007199254740992)

Erz =
0

ans =
-0.999956280158039 + 0.000043717930620i
ans =
  0.000043717930620 + 0.999956280158039i
ans =
     0
[-1 1j 0] is same as -[1-1j 0] that is same RHCP as incident.

The negative reflection coefficient changes amplitude sign of bounced back wave,

but does not change the polarization rotation direction.


Additional

%% electric current density on the surface generated by incident electric

fieldJt=sigma_Cu*T*E0*exp(-gamma*z)    % [A/m^2]

%% 2D current density

Js=double(int(Jt,z,0,Inf))

%% metal surface resistance

Rs=real(etha)1/(sigma_Cu*ds)
Js =
  0.005305048799247 - 0.000000115965413i

Rs =
   0.008240996067112

ans =
   0.008240996067112
 
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