i APENDIX A PERSAMAAN MAXWELL

APENDIX A
PERSAMAAN MAXWELL
Persamaan Maxwell dalam bentuk differensial.
⃗
Hukum Gauss
⃗ ⃗
Hukum Gauss untuk magnetisme
⃗ ⃗
Hukum Induksi Faraday
⃗
⃗
Hukum Ampere
⃗
⃗
(A.1)
(A.2)
⃗
(A.3)
⃗
(A.4)
Persamaan Maxwell dalam bentuk integral
Hukum Gauss
∮ ⃗
Hukum Gauss untuk magnetisme
∮ ⃗
Hukum Induksi Faraday
∮⃗
Hukum Ampere
∮⃗
⃗
∫
(A.5)
(A.6)
∫⃗
∫⃗
(A.7)
∫
(A.8)
Karena (teorema Gauss dan Stokes)
∮ ⃗
∮
∮ ⃗
∮
Untuk medan listrik pada persamaan (A.5) dan (A.7) dalam keadaan konservatif
sehingga dapat ditulis :
⃗
⃗ ⃗
(A.9)
⃗
(A.10)
Selanjutnya hubungan vektornya
⃗
⃗
(A.11)
⃗
⃗ ⃗
(A.12)
dan
i
⃗
∫⃗
(A.13)
⃗ = kerapatan fluks listrik (C/m2)
= rapat volume muatan (C/m3)
⃗ = Medan Listrik (V/m)
⃗ = tegangan listrik (V)
 = permitivitas dielektrik (F/m)
Hubungkan persamaan (A.9), (A.11) dan (A.12) akan didapat persamaan Poisson :
⃗
⃗⃗
(A.14a)
Jika  konstan,
⃗
Jika
(A.14b)
maka menjadi persamaan Laplace :
⃗
⃗⃗
(A.15a)
⃗
(A.15b)
Karena (hukum Biot-Savart)
∮ ⃗
∮
∮⃗
⃗ = fluks magnet (A/m)
⃗ = Medan magnet (T)
Maka persamaan (A.6) dan(A.8) menjadi
⃗
⃗
(A.16)
Dan
⃗ ⃗
(A.17)
Dengan hubungan vector
⃗
 = permeabilitas magnetic (H/m)
=konduktivitas (mhos/m)
⃗
(A.18)
⃗
(A.19)
Medan magnet pada potensial A
⃗
⃗
(A.20)
Dengan menggunakan vektor identitas
⃗
(⃗
⃗ (⃗
)
)
Jika (A.17) dan (A.18) dihubungkan dalam kondisi tanpa arus (ruang hampa) maka
⃗
sehingga didapat persamaan Poisson :
(A.21)
Ketika
maka
(A.22)
i
APENDIX B
PERSAMAAN GELOMBANG
Gelombang Elektromagnetik merambat secara transversal. Dengan medan listrik dan
medan magnet yang merambat saling tegak lurus. Misalkan gelombang elektromagnet
merambat ke sumbu-x dan arah gerak medan listrik ke sumbu-y sementara medan
magnetik ke sumbu-z sehingga diperoleh ⃗
⃗
⃗
⃗ ⃗
dan ⃗
⃗
⃗ . Maka persamaan Maxwell akan berubah menjadi :
⃗ ⃗
⃗
(B.1)
⃗ ⃗
⃗
(B.2)
⃗
⃗
⃗
⃗
(B.3)
⃗
⃗
⃗
(B.4)
⃗
⃗
⃗
(B.5)
⃗⃗
⃗
(B.6)
Persamaan (B.4) dan persamaan (B.6) bisa disimpulkan medan listrik dan medan
magnet hanya berpengaruh pada sumbu x dan gayut waktu.
⃗
⃗
⃗
⃗
Jika digabungkan maka menjadi :
⃗
⃗
Bandingkan dengan persamaan gelombang umum :
Atau
,
=0
Maka diperoleh nilai
√
Dengan v = c = kecepatan cahaya = 3 x 108 m
Dengan mengubah daerah waktu menjadi frekuensi maka
Dengan
Maka persawaan Maxwell dapat dituliskan
⃗( )
( )⃗ ( )
⃗( )
( )⃗ ( )
Dengan vektor perambatan elektromagnetik sebagai berikut :
( )
⃗( )
⃗( )
( )
⃗( )
⃗( )
⃗( )
⃗( )
( )
( )⃗ ( )
(
( )
( )
) ( ) ⃗ ( )⃗ ( )
( )
(
) ( )⃗ ( )
( ) ( ) ⃗ ( )⃗ ( )
Maka dapat disimpulkan bahwa
√
Dengan n adalah indeks bias.
√
i
APENDIX C
METODE FDTD (FINITE DIFFERENCE TIME DOMAIN)
Metode ini menggunakan turunan fungsi yang dihubungkan dengan beda hingga
seperti berikut
(
( )
)
( )
(C.1)
Dengan dihubungkan ke dalam medan listrik dan medan magnet, akan dihasilkan
persamaan FDTDnya misalkan untuk waktu penjalaran medan E dan medan H sebagai
berikut
untuk ⃗
(
untuk ⃗
)
dan letak pada sumbu z misalkan
untuk ⃗
(
untuk ⃗
)
Telah diketahui persamaan Maxwell
Dengan notasi ⃗ ( )
)
⃗(
⃗
⃗
⃗
⃗
(C.2)
(C.3)
) dan ⃗
(
)
⃗ ((
)
Maka didapat :
⃗
⃗ (
⃗
⃗
⃗ ( )
)
( )
⃗ ( )
⃗
⃗
(
)
⃗
(
)
⃗
⃗
(
)
⃗
(
)
)
(
Dengan menghubungkan ke dalam persamaan (C.2) dan (C.3) persamaan diatas
menjadi
⃗
(
⃗
⃗
)
(
⃗ ( )
( )
(⃗ (
)
(⃗
(
⃗
Dengan
dan
⃗
)
⃗ ( )
)
(
)
(C.4)
⃗
(C.5)
dalam model Drude diberikan dalam persamaan
( )
Substitusi model Drude dalam persamaan (C.5) akan didapat
⃗
⃗
⃗
⃗
)⃗
(
(
(
⃗
⃗
(
)
(
)
(
(
)⃗
(
)⃗
)
)⃗
)
⃗
⃗
)⃗
(
(C.6)
nilai fungsi frekuensi dikonversikan menjadi fungsi waktu dengan hubungan
dan
. Dengan perkalian
⃗
merupakan waktu rata-rata. Maka :
⃗
⃗
⃗
⃗
Dengan cara yang sama sebelumnya akan di dapat :
⃗
(⃗
( ⃗
⃗
⃗
⃗
)
(⃗
)
(⃗
⃗
( ⃗
)
)
⃗
(C.7)
Nilai variabel tambahan dalam persamaan yang baru merupakan skalar yaitu
(
)
(
)
(
)
i
(
(
)
)
)
dan
⃗
(⃗
⃗
⃗
)
( ⃗
⃗
)
(⃗
(⃗
⃗
( ⃗
)
)
⃗
(
)
)
Maka persamaan FDTD gelombang yang merambat dalam arah z sebagai berikut
⃗
⃗ ( )
( )
(⃗ ( )
⃗ (
))
(
)
dan
⃗
⃗ ( )
( )
(⃗ ( )
⃗ (
))
(
)
Persamaan diatas memberikan algoritma FDTD untuk ⃗ dan ⃗ .
Untuk indeks bias FDTD dihubungkan dengan transformasi Fourier sebagai pengubah
ke FDTD :
√
Untuk medium yang homogen nilai  = 1 sehingga indeks bias hanya bergantung 
dan  bergantung kepada medan listrik.
̃(
̃(
∫ ⃗(
)
)
∫ ⃗( )
)
(
(
)
)
(
)
Sehingga didapat bahwa hasil transformasi Fouriernya adalah
̃(
)
⃗( )
(
)
Indeks bias merupakan perbandingan kecepatan antara dua medium maka persamaan
diatas dapat diubah menjadi :
̃(
)
̃(
)
|
(
(
⃗( )
⃗( )
)
|
)
(
(
)
(
)
)
(
|
)
⃗ (
⃗ (
)
)
|
Untuk simulasi dalam jubah silinder maka diperlukan persamaan dalam koordinat
silinder sebagai berikut:
( )
( )
( )
(
)
(
)
(
)
(
)
Dari transformasi kartesius :
(C.13)
(
)
(C.14)
(C.15)
Sehingga
⃗
( ⃗ )
(
⃗
)
(
)
(
)
(
)
⃗
( )
⃗
⃗
( )
⃗
i
APENDIX D
PROGRAM MATLAB
clc
clear all
% Adimas Agung
% 110801001
% Simulasi Parameter.
SIZE = 4*1024; % Angka beda langkah
SlabLeft = round(SIZE/3); % Letak sebelah kiri lempengan
SlabRight = round(2*SIZE/3); % Letak sebelah kanan lempengan
MaxTime = 4*SIZE; % Jumlah langkah waktu
PulseWidth = round(SIZE/8); % Lebar Pulsa Gauss
td = PulseWidth; % tempo pulsa.
source = 10; % Letak Sumber
SaveFields = 1; % 0. No, 1. Ya menyimpan.
SnapshotInterval = 32; % jumlah waktu untuk memfoto (menangkap
layar).
% Memilih Sumber Gelombang.
% 1. Gaussian 2. Sine wave 3. Ricker wavelet
SourceChoice = 2;
% Konstanta yang digunakan.
c = 3e8;
pi = 3.141592654;
e0 = (1e-9)/(36*pi);
u0 = (1e-7)*4*pi;
dt = 0.5e-11;
dz = 3e-3;
Sc = c * dt/dz
l = PulseWidth*dz;
f = c/l
fmax = 1/(2*dt)
w = 2*pi*f;
k0 = w/c; % bilangan gelombang.
% Parameter dalam Ricker wavelet.
if SourceChoice == 3
fp = f; % puncak ke puncak frekuensi
dr = PulseWidth*dt*2; % Waktu tunggu
end
% Inisialisasi.
Ex = zeros(SIZE, 3); % komponen x untuk medan E
Dx = zeros(SIZE, 3); % komponen x untuk D
Hy = zeros(SIZE, 3); % komponen y untuk medan H
By = zeros(SIZE, 3); % komponen y medan B
% Medan datang dan transmisi.
Exi = zeros(MaxTime, 1);
Ext = zeros(MaxTime, 1);
Extt = zeros(MaxTime, 1);
x1 = SlabLeft+1; % letak observasi (posisi pengamatan).
% Penghitungan index Bias.
Z1 = SlabLeft + 50;
z1 = Z1*dz;
Z2 = SlabLeft + 60;
z2 = Z2*dz;
Exz1 = zeros(MaxTime, 1);
Exz2 = zeros(MaxTime, 1);
einf = ones(SIZE,1);
einf(SlabLeft:SlabRight) = 1; % einf(Drude) atau er di lempengan.
uinf = ones(SIZE,1);
uinf(SlabLeft:SlabRight) = 1; % uinf(Drude) atau ur di lempengan.
wpesq = zeros(SIZE,1);
wpesq(SlabLeft:SlabRight) = 2*w^2; % nilai DNG(Drude) untuk wpe
kuadrat pada lempengan.
wpmsq = zeros(SIZE,1);
wpmsq(SlabLeft:SlabRight) = 2*w^2; % nilai DNG(Drude) untuk wpm
kuadrat pada lempengan.
ge = zeros(SIZE,1);
ge(SlabLeft:SlabRight) = w/32; % frekuensi listrik pada lempengan.
gm = zeros(SIZE,1);
i
gm(SlabLeft:SlabRight) = w/32; % frekuensi magnet pada lempengan.
a0 = (4*dt^2)./(e0*(4*einf+dt^2*wpesq+2*dt*einf.*ge));
a = (1/dt^2)*a0;
b = (1/(2*dt))*ge.*a0;
c = (e0/dt^2)*einf.*a0;
d = (-1*e0/4).*wpesq.*a0;
e = (1/(2*dt))*e0*einf.*ge.*a0;
am0 = (4*dt^2)./(u0*(4*uinf+dt^2*wpmsq+2*dt*uinf.*gm));
am = (1/dt^2)*am0;
bm = (1/(2*dt))*gm.*am0;
cm = (u0/dt^2)*uinf.*am0;
dm = (-1*u0/4).*wpmsq.*am0;
em = (1/(2*dt))*u0*uinf.*gm.*am0;
if SaveFields == 1
ExSnapshots = zeros(SIZE, MaxTime/SnapshotInterval); % Data yang
akan diplot (bentuk kurva).
frame = 1;
end
n1 = 1;
n2 = 2;
linecount = 0;
% waktu keluar dari pengulangan.
tic
% uji pengulangan untuk medan datang pada ruang bebas.
for q = 0:MaxTime
% Penghitungan Hy menggunakan turunan persamaan Hy. Beda waktu q.
Hy(1:SIZE-1,n2) = Hy(1:SIZE-1,n1) + ( ( Ex(1:SIZE-1,n1) Ex(2:SIZE,n1) ) * dt/(u0*dz) );
% ABC untuk ukuran H.
Hy(SIZE,n2) = Hy(SIZE-1,n1) + (Sc-1)/(Sc+1)*(Hy(SIZE-1,n2) Hy(SIZE,n1) );
% Penghitungan Ex menggunakan turunan persamaan Ex. Beda waktu
q+1/2.
Ex(2:SIZE,n2) = Ex(2:SIZE, n1) + ( dt/(e0*dz)*(Hy(1:SIZE-1, n2) Hy(2:SIZE, n2)) );
% ABC untuk E di 1.
Ex(1,n2) = Ex(2,n1) + (Sc-1)/(Sc+1)*(Ex(2,n2) - Ex(2,n1));
% Sumber.
if SourceChoice == 1
Ex(source,n2) = Ex(source,n2) + exp( -1*((q-td)/(PulseWidth/4))^2
) * Sc;
elseif SourceChoice == 2
Ex(source,n2) = Ex(source,n2) + sin(2*pi*f*(q)*dt) * Sc;
elseif SourceChoice == 3
Ex(source,n2) = Ex(source,n2) + (1-2*(pi*fp*(q*dt-dr))^2)*exp(1*(pi*fp*(q*dt-dr))^2) * Sc;
end
Exi(q+1) = Ex(x1,n2); % Medan datang di kiri lempengan.
temp = n1;
n1 = n2;
n2 = temp;
end
% Pengulangan inisialisasi medan untuk simulasi.
Ex = zeros(SIZE, 3); % komponen x untuk medan E
Hy = zeros(SIZE, 3); % komponen y untuk medan H
% Simulasi dimulai.
fprintf ( 1, 'Simulation started... \n');
for q = 0:MaxTime
% Indikator.
fprintf(1, repmat('\b',1,linecount));
linecount = fprintf(1, '%g %%', (q*100)/MaxTime );
Ex(:,3) = Ex(:,n2);
Dx(:,3) = Dx(:,n2);
Hy(:,3) = Hy(:,n2);
By(:,3) = By(:,n2);
i
% Penghitungan Hy menggunakan turunan persamaan Hy. Beda waktu q.
By(1:SIZE-1,n2) = By(1:SIZE-1,n1) + ( ( Ex(1:SIZE-1,n1) Ex(2:SIZE,n1) ) * dt/(dz) );
Hy(:,n2) = am.*(By(:,n2)-2*By(:,n1)+By(:,3))+bm.*(By(:,n2)By(:,3))+cm.*(2*Hy(:,n1)Hy(:,3))+dm.*(2*Hy(:,n1)+Hy(:,3))+em.*(Hy(:,3));
% ABC untuk ukuran.
Hy(SIZE,n2) = Hy(SIZE-1,n1) + (Sc-1)/(Sc+1)*(Hy(SIZE-1,n2) Hy(SIZE,n1) );
By(SIZE,n2) = u0*Hy(SIZE,n2);
% Penghitungan Ex menggunakan persamaan diferensial Ex. Beda
waktu q+1/2.
Dx(2:SIZE,n2) = Dx(2:SIZE, n1) + ( dt/(dz)*(Hy(1:SIZE-1, n2) Hy(2:SIZE, n2)) );
Ex(:,n2) = a.*(Dx(:,n2)-2*Dx(:,n1)+Dx(:,3))+b.*(Dx(:,n2)Dx(:,3))+c.*(2*Ex(:,n1)Ex(:,3))+d.*(2*Ex(:,n1)+Ex(:,3))+e.*(Ex(:,3));
% ABC untuk E di 1.
Ex(1,n2) = Ex(2,n1) + (Sc-1)/(Sc+1)*(Ex(2,n2) - Ex(2,n1));
Dx(1,n2) = e0*Ex(1,n2);
% Sumber.
if SourceChoice == 1
Ex(source,n2) = Ex(source,n2) + exp( -1*((q-td)/(PulseWidth/4))^2
) * Sc;
elseif SourceChoice == 2
Ex(source,n2) = Ex(source,n2) + sin(2*pi*f*(q)*dt) * Sc;
elseif SourceChoice == 3
Ex(source,n2) = Ex(source,n2) + (1-2*(pi*fp*(q*dt-dr))^2)*exp(1*(pi*fp*(q*dt-dr))^2) * Sc;
end
Dx(source,n2) = e0*Ex(source,n2);
if (SaveFields == 1 && mod(q,SnapshotInterval) == 0)
ExSnapshots(:,frame) = Ex(:,n2);
frame=frame+1;
end
Ext(q+1) = Ex(x1,n2);
Extt(q+1) = Ex(SlabRight+10,n2);
% Medan untuk Penghitungan index bias.
Exz1(q+1) = Ex(Z1, n2);
Exz2(q+1) = Ex(Z2, n2);
temp = n1;
n1 = n2;
n2 = temp;
end
fprintf ( 1, '\nSimulation complete! \n');
toc
% Pengolahan.
Fs = 1/dt;
% pengambilan sampel frekuensi
T = dt;
% waktu sampel
L = length(Exi);
% Panjang sinyal
t = (0:L-1)*T;
% Vektor waktu
fspan = 100;
% Titik alur dalam daerah frekuensi
figure(1)
subplot(211)
plot(Fs*t, Exi, 'LineWidth', 2.0, 'Color', 'b')
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Medan Listrik Datang', 'FontSize', 12, 'FontWeight', 'b')
xlabel('waktu', 'FontSize', 11, 'FontWeight', 'b')
grid on
figure(2)
subplot(211)
plot(Fs*t, Ext, 'LineWidth', 2.0, 'Color', 'b')
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Transmisi Medan Listrik', 'FontSize', 12, 'FontWeight', 'b')
xlabel('waktu', 'FontSize', 11, 'FontWeight', 'b')
grid on
figure(3)
subplot(211)
i
plot(Fs*t, Extt, 'LineWidth', 2.0, 'Color', 'b')
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Transmisi Medan Listrik Melalui Lempeng', 'FontSize', 12,
'FontWeight', 'b')
xlabel('waktu', 'FontSize', 11, 'FontWeight', 'b')
grid on
NFFT = 2^nextpow2(L); % untuk tenaga kedua dari Exi
% Medan datang dan transmisi.
EXI = fft(Exi,NFFT)/L;
EXT = fft(Ext,NFFT)/L;
EXTT = fft(Extt,NFFT)/L;
% Penghitungan Index Bias.
EXZ1 = fft(Exz1,NFFT)/L;
EXZ2 = fft(Exz2,NFFT)/L;
f = Fs/2*linspace(0,1,NFFT/2+1);
% Membuat alur spektrum amplitudo satu bagian.
figure(1)
subplot(212)
EXIp = 2*abs(EXI(1:NFFT/2+1));
plot(f(1:fspan), EXIp(1:fspan), 'LineWidth', 2.0, 'Color', 'r')
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Spektrum Amplitudo Exi(t)', 'FontSize', 12, 'FontWeight', 'b')
xlabel('Frekuensi (Hz)', 'FontSize', 11, 'FontWeight', 'b')
ylabel('|EXI(f)|', 'FontSize', 11, 'FontWeight', 'b')
grid on
figure(2)
subplot(212)
EXTp = 2*abs(EXT(1:NFFT/2+1));
plot(f(1:fspan), EXTp(1:fspan), 'LineWidth', 2.0, 'Color', 'r')
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Spektrum Amplitudo Ext(t)', 'FontSize', 12, 'FontWeight', 'b')
xlabel('Frekuensi (Hz)', 'FontSize', 11, 'FontWeight', 'b')
ylabel('|EXT(f)|', 'FontSize', 11, 'FontWeight', 'b')
grid on
figure(3)
subplot(212)
EXTTp = 2*abs(EXTT(1:NFFT/2+1));
plot(f(1:fspan), EXTTp(1:fspan), 'LineWidth', 2.0, 'Color', 'r')
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Spektrum Amplitudo Extt(t)', 'FontSize', 12, 'FontWeight',
'b')
xlabel('Frekuensi(Hz)', 'FontSize', 11, 'FontWeight', 'b')
ylabel('|EXT(f)|', 'FontSize', 11, 'FontWeight', 'b')
grid on
% Koefisien Transmisi.
figure(4)
subplot(211)
TAU = abs(EXT(1:NFFT/2+1)./EXI(1:NFFT/2+1));
plot(f(1:fspan), TAU(1:fspan), 'LineWidth', 2.0, 'Color', 'b')
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Koefisien Transmisi', 'FontSize', 12, 'FontWeight', 'b')
xlabel('Frekuensi (Hz)', 'FontSize', 11, 'FontWeight', 'b')
ylabel('|EXT(f)/EXI(f)|', 'FontSize', 11, 'FontWeight', 'b')
axis([-1 1 -2 2])
axis 'auto x'
grid on
subplot(212)
plot(f(1:fspan), 1-TAU(1:fspan), 'LineWidth', 2.0, 'Color', 'b')
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Koefisien Refleksi', 'FontSize', 12, 'FontWeight', 'b')
xlabel('Frekuensi (Hz)', 'FontSize', 11, 'FontWeight', 'b')
ylabel('1-|EXT(f)/EXI(f)|', 'FontSize', 11, 'FontWeight', 'b')
axis([-1 1 -2 2])
axis 'auto x'
grid on
% Penghitungan Indeks Bias.
nFDTD = (1/(1i*k0*(z1-z2))).*log(EXZ2(1:NFFT/2+1)./EXZ1(1:NFFT/2+1));
figure(5)
subplot(211)
plot(f(1:fspan), real(nFDTD(1:fspan)), 'LineWidth', 2.0, 'Color',
'b');
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Indeks Bias re(n)', 'FontSize', 12, 'FontWeight', 'b')
xlabel('Frekuensi (Hz)', 'FontSize', 11)
ylabel('re(n)', 'FontSize', 11)
grid on
subplot(212)
i
plot(f(1:fspan), imag(nFDTD(1:fspan)), 'LineWidth', 2.0, 'Color',
'r');
set(gca, 'FontSize', 10, 'FontWeight', 'b')
title('Indeks Bias im(n)', 'FontSize', 12, 'FontWeight', 'b')
xlabel('Frekuensi (Hz)', 'FontSize', 11, 'FontWeight', 'b')
ylabel('im(n)', 'FontSize', 11, 'FontWeight', 'b')
grid on
if SaveFields == 1
% Simulasi Animasi.
for i=1:frame-1
figure (6)
% Batas penyebaran.
hold off
plot([SlabLeft SlabLeft], [-1 1], 'Color', 'r');
hold on
plot([SlabRight SlabRight], [-1 1], 'Color', 'r');
plot(ExSnapshots(:,i), 'LineWidth', 2.0, 'Color', 'b');
set(gca, 'FontSize', 10, 'FontWeight', 'b')
axis([0 SIZE -1 1])
title('Simulasi Fungsi Waktu', 'FontSize', 12, 'FontWeight',
'b')
xlabel('Beda Langkah(k)', 'FontSize', 11, 'FontWeight', 'b')
ylabel('Medan Listrik (Ex)', 'FontSize', 11, 'FontWeight',
'b')
grid on
end
end
APENDIX E
PROGRAM INVISIBLE CLOAK
clear all; close all;
% Nilai Konstanta
%*******************************************************************
c_0
= 3.0e8;
% Kecepatan cahaya
mu_0 = 4.0*pi*1.0e-7;
% Permeabilitas
eps_0 = 8.8542e-12;
% Permitivitas
losstangent = 0.0;
ie = 401;
je = 400;
% # Besar sel dalam arah x
% # Besar sel dalam arah y
ib = ie + 1;
jb = je + 1;
npmls = 20;
ip = ie - npmls;
jp = je - npmls;
xc = round(ie/2);
yc = round(je/2);
is = 25;
js = 25;
% Letak Sumber Gelombang
it = 30;
jt = 30;
% Daerah Medan atas atau bawah
% Daerah medan kiri atau kanan
freq = 6.0e14;
omega = 2*pi*freq;
k_0 = omega/c_0;
dx = c_0/freq/400;
dt = dx/c_0/sqrt(2);
% Beda waktu
R1 = c_0/freq/8/dx
R2 = 1.43*R1;
% dimensi Jubah
nmax = 400000;
% Jumlah beda waktu
aimp = sqrt(mu_0/eps_0);
% impedansi gelombang
threshold = 0.1;
step = ceil(nmax/10);
% FDTD error
% freq = 2.0e9;
% omega = 2*pi*freq;
tau = 1/freq;
delay = 3*tau/dt;
i
N = round(tau/dt); M = round(c_0/freq/dx);
source = zeros(1,nmax);
clear j;
ST = 300;
st = 20e3;
for n=1:nmax
if n < ST*N
x = 1.0 - (ST*N-n)/(ST*N);
g = 10.0*x^3 - 15.0*x^4 + 6.0*x^5;
%
sumber(n) = g * sin(2*pi*freq*n*dt);
source(n) = g * exp(j*2*pi*freq*n*dt);
else
%
sumber(n) = sin(2*pi*freq*n*dt);
source(n) = exp(j*2*pi*freq*n*dt);
end
%
sumber (n) = exp(-(((n-delay)*dt)/tau)^2) .*
exp(j*2*pi*freq*n*dt);
end
%*******************************************************************
% Inisialisasi matriks untuk komponen medan
%*******************************************************************
Dx = zeros(ie,jb); Dx_h1 = zeros(ie,jb); Dx_h2 = zeros(ie,jb);
Dxsynch = zeros(ie,jb); Dx_h1synch = zeros(ie,jb); Dx_h2synch =
zeros(ie,jb);
caDx = ones(ie,jb); cbDx = ones(ie,jb).*(dt/eps_0/dx);
Ex = zeros(ie,jb); Ex_h1 = zeros(ie,jb); Ex_h2 = zeros(ie,jb);
Ex_h1synch = zeros(ie,jb); Ex_h2synch = zeros(ie,jb);
Dy = zeros(ib,je); Dy_h1 = zeros(ib,je); Dy_h2 = zeros(ib,je);
Dysynch = zeros(ib,je); Dy_h1synch = zeros(ib,je); Dy_h2synch =
zeros(ib,je);
caDy = ones(ib,je); cbDy = ones(ib,je).*(dt/eps_0/dx);
Ey = zeros(ib,je); Ey_h1 = zeros(ib,je); Ey_h2 = zeros(ib,je);
Ey_h1synch = zeros(ib,je); Ey_h2synch = zeros(ib,je);
Bz = zeros(ie,je); Bz_h1 = zeros(ie,je); Bz_h2 = zeros(ie,je);
Bzx = zeros(ie,je); Bzy = zeros(ie,je); Hz_p = zeros(ie,je);
daBzx = ones(ie,je); dbBzx = ones(ie,je).*(dt/mu_0/dx);
daBzy = ones(ie,je); dbBzy = ones(ie,je).*(dt/mu_0/dx);
daBz = ones(ie,je); dbBz = ones(ie,je).*(dt/mu_0/dx);
Hz = zeros(ie,je); Hz_h1 = zeros(ie,je); Hz_h2 = zeros(ie,je);
%
%
%
%
HzsourceTF = zeros(length(it:ie-it),nmax);
HztransTF = zeros(length(it:ie-it),nmax);
HzsourceSF = zeros(length(it:ie-it),nmax);
HztransSF = zeros(length(it:ie-it),nmax);
% Coefficients for cloak
a0x = ones(ie,jb); b0x = ones(ie,jb); b0xy = zeros(ie,jb);
a1x = zeros(ie,jb); b1x = zeros(ie,jb); a1xy = zeros(ie,jb); b1xy =
zeros(ie,jb);
a2x = zeros(ie,jb); b2x = zeros(ie,jb); a2xy = zeros(ie,jb); b2xy =
zeros(ie,jb);
a0y = ones(ib,je); b0y = ones(ib,je); b0yx = zeros(ib,je);
a1y = zeros(ib,je); b1y = zeros(ib,je); a1yx = zeros(ib,je); b1yx =
zeros(ib,je);
a2y = zeros(ib,je); b2y = zeros(ib,je); a2yx = zeros(ib,je); b2yx =
zeros(ib,je);
a0z = ones(ie,je); b0z = ones(ie,je);
a1z = zeros(ie,je); b1z = zeros(ie,je);
a2z = zeros(ie,je); b2z = zeros(ie,je);
% Medan Datang
Ex_inc
= zeros(1,jb);
Hzx_inc = zeros(1,jb);
Ey_inc
= zeros(1,ib);
Hzy_inc = zeros(1,ib);
caEx_inc = ones(1,jb); cbEx_inc = ones(1,jb).*(dt/eps_0/dx);
caEy_inc = ones(1,ib); cbEy_inc = ones(1,ib).*(dt/eps_0/dx);
daHzx_inc = ones(1,je); dbHzx_inc = ones(1,je).*(dt/mu_0/dx);
daHzy_inc = ones(1,ie); dbHzy_inc = ones(1,ie).*(dt/mu_0/dx);
%*******************************************************************
% Jubah
%*******************************************************************
for i = 1 : ie
for j = 1 : je
r = sqrt((i - xc + 0.5).^2 + (j - yc).^2);
if (r<=R2) && (r>=R1);
erx = ((r-R1) / (r));
ethetax = ((r) / (r-R1));
sigma_px =
ethetax*losstangent*((tan((omega*dt)/2))/(dt/2));
gamma_px = (2*losstangent*erx*sin((omega*dt)/2))/((1erx)*dt*cos((omega*dt)/2));
omega_px = sqrt((2*sin((omega*dt)/2)*(-2*(erx1)*sin((omega*dt)/2)+...
losstangent*erx*gamma_px*dt*cos((omega*dt)/2)))/((dt^2)*((cos((omega*
dt)/2))^2)));
sinx = (j - yc)/r;
cosx = (i - xc + 0.5)/r;
ax = ((cosx.^2) + (ethetax.*(sinx.^2))) ./ (dt^2);
bx = ((gamma_px.*(cosx.^2)) + ((sigma_px +
(ethetax*gamma_px)).*(sinx.^2))) ./ (2*dt);
cx = (((cosx.^2).*((omega_px)^2)) +
(sigma_px*gamma_px.*(sinx.^2))) ./ (4);
wx = ((1-ethetax).*sinx.*cosx) ./ (dt^2);
fx = ((gamma_px-sigma_px-(ethetax*gamma_px)).*sinx.*cosx)
./ (2*dt);
vx = ((((omega_px)^2)-(sigma_px*gamma_px)).*sinx.*cosx)
./ (4);
kx = 1 / (dt^2);
lx = gamma_px / (2*dt);
i
tx = ((sinx.^2) + (ethetax.*(cosx.^2))) ./ (dt^2);
qx = ((gamma_px.*(sinx.^2)) + ((sigma_px +
(ethetax*gamma_px)).*(cosx.^2))) ./ (2*dt);
px = (((sinx.^2).*((omega_px)^2)) +
(sigma_px*gamma_px.*(cosx.^2))) ./ (4);
Ax1
Bx1
Cx1
Dx1
=
=
=
=
ax+bx+cx; Ax2 = (2.*ax)-(2.*cx); Ax3 = ax-bx+cx;
wx+fx+vx; Bx2 = (2.*wx)-(2.*vx); Bx3 = wx-fx+vx;
kx+lx; Cx2 = (2.*kx); Cx3 = kx-lx;
tx+qx+px; Dx2 = (2.*tx)-(2.*px); Dx3 = tx-qx+px;
a0x(i, j) = Ax1 - (((Bx1).^2) ./ Dx1);
a1x(i, j) = Ax2 - ((Bx1.*Bx2) ./ Dx1);
a2x(i, j) = ((Bx1.*Bx3) ./ Dx1) - (Ax3);
a1xy(i, j) = Bx2 - ((Bx1.*Dx2) ./ Dx1);
a2xy(i, j) = ((Bx1.*Dx3) ./ Dx1) - (Bx3);
b0x(i, j) = Cx1;
b1x(i, j) = -Cx2;
b2x(i, j) = Cx3;
b0xy(i, j) = -(Bx1.*Cx1) ./ Dx1;
b1xy(i, j) = (Bx1.*Cx2) ./ Dx1;
b2xy(i, j) = -(Bx1.*Cx3) ./ Dx1;
elseif (r<R1)
cbDx(i, j) = (dt/(3*eps_0)/dx);
end
r = sqrt((i - xc).^2 + (j - yc + 0.5).^2);
if (r<=R2) && (r>=R1)
ery = ((r-R1) / (r));
ethetay = ((r) / (r-R1));
sigma_py =
ethetay*losstangent*((tan((omega*dt)/2))/(dt/2));
gamma_py = (2*losstangent*ery*sin((omega*dt)/2))/((1ery)*dt*cos((omega*dt)/2));
omega_py = sqrt((2*sin((omega*dt)/2)*(-2*(ery1)*sin((omega*dt)/2)+...
losstangent*ery*gamma_py*dt*cos((omega*dt)/2)))/((dt^2)*((cos((omega*
dt)/2))^2)));
siny = (j - yc + 0.5)/r;
cosy = (i - xc)/r;
ay = ((cosy.^2) + (ethetay*(siny.^2))) ./ (dt^2);
by = ((gamma_py*(cosy.^2)) + ((sigma_py +
(ethetay*gamma_py)).*(siny.^2))) ./ (2*dt);
cy = (((cosy.^2).*((omega_py)^2)) +
(sigma_py*gamma_py.*(siny.^2))) ./ (4);
wy = ((1-ethetay).*siny.*cosy) ./ (dt^2);
fy = ((gamma_py-sigma_py-(ethetay*gamma_py)).*siny.*cosy)
./ (2*dt);
vy = ((((omega_py)^2)-(sigma_py*gamma_py)).*siny.*cosy)
./ (4);
ky = 1 / (dt^2);
ly = gamma_py / (2*dt);
ty = ((siny.^2) + (ethetay*(cosy.^2))) ./ (dt^2);
qy = ((gamma_py*(siny.^2)) + ((sigma_py +
(ethetay*gamma_py)).*(cosy.^2))) ./ (2*dt);
py = (((siny.^2).*((omega_py)^2)) +
(sigma_py*gamma_py.*(cosy.^2))) ./ (4);
Ay1
By1
Cy1
Dy1
=
=
=
=
ay+by+cy; Ay2 = (2.*ay)-(2.*cy); Ay3 = ay-by+cy;
wy+fy+vy; By2 = (2.*wy)-(2.*vy); By3 = wy-fy+vy;
ky+ly; Cy2 = (2*ky); Cy3 = ky-ly;
ty+qy+py; Dy2 = (2.*ty)-(2.*py); Dy3 = ty-qy+py;
a0y(i, j) = Dy1 - (((By1).^2) ./ Ay1);
a1y(i, j) = Dy2 - ((By1.*By2) ./ Ay1);
a2y(i, j) = ((By1.*By3) ./ Ay1) - (Dy3);
a1yx(i, j) = By2 - ((By1.*Ay2) ./ Ay1);
a2yx(i, j) = ((By1.*Ay3) ./ Ay1) - (By3);
b0y(i, j) = Cy1;
b1y(i, j) = -Cy2;
b2y(i, j) = Cy3;
b0yx(i, j) = -(By1.*Cy1) ./ Ay1;
b1yx(i, j) = (By1.*Cy2) ./ Ay1;
b2yx(i, j) = -(By1.*Cy3) ./ Ay1;
elseif (r<R1)
cbDy(i, j) = (dt/(3*eps_0)/dx);
end
i
r = sqrt((i - xc).^2 + (j - yc).^2);
if (r<=R2) && (r>=R1)
muz =
(((r-R1) / (r)) * (((R2) / (R2-R1))^2));
if (muz < 1)
gamma_pz = (2*losstangent*muz*sin((omega*dt)/2))/((1muz)*dt*cos((omega*dt)/2));
omega_pz = sqrt((2*sin((omega*dt)/2)*(-2*(muz1)*sin((omega*dt)/2)+...
losstangent*muz*gamma_pz*dt*cos((omega*dt)/2)))/((dt^2)*((cos((omega*
dt)/2))^2)));
a0z(i,
a1z(i,
a2z(i,
b0z(i,
b1z(i,
b2z(i,
j)
j)
j)
j)
j)
j)
=
=
=
=
=
=
1/dt^2 + gamma_pz/(2*dt) + ((omega_pz^2))/4;
-2/dt^2 + ((omega_pz^2))/2;
1/dt^2 - gamma_pz/(2*dt) + ((omega_pz^2))/4;
1/dt^2 + gamma_pz/(2*dt);
-2/dt^2;
1/dt^2 - gamma_pz/(2*dt);
elseif (muz >= 1)
sigma_pz =
mu_0*muz*losstangent*((tan((omega*dt)/2))/(dt/2));
daBzx(i, j) = (1-((sigma_pz*dt)/(2*mu_0*muz))) /
(1+((sigma_pz*dt)/(2*mu_0*muz)));
dbBzx(i, j) = (dt/(mu_0*muz)/dx) /
(1+((sigma_pz*dt)/(2*mu_0*muz)));
daBzy(i, j) = (1-((sigma_pz*dt)/(2*mu_0*muz))) /
(1+((sigma_pz*dt)/(2*mu_0*muz)));
dbBzy(i, j) = (dt/(mu_0*muz)/dx) /
(1+((sigma_pz*dt)/(2*mu_0*muz)));
daBz(i, j) = (1-((sigma_pz*dt)/(2*mu_0*muz))) /
(1+((sigma_pz*dt)/(2*mu_0*muz)));
dbBz(i, j) = (dt/(mu_0*muz)/dx) /
(1+((sigma_pz*dt)/(2*mu_0*muz)));
end
end
end
end
%*******************************************************************
% Konstanta Berenger
%********************************************************************
***
sigmax = -3.0*eps_0*c_0*log(1.0e-5)/(2.0*dx*npmls);
rhomax = sigmax*(aimp^2);
for m=1:npmls
sig(m) = sigmax*((m-0.5)/(npmls+0.5))^2;
rho(m) = rhomax*(m/(npmls+0.5))^2;
end
%*******************************************************************
% Koefisien Berenger
%*******************************************************************
for m=1:npmls
re = sig(m)*dt/eps_0;
rm = rho(m)*dt/mu_0;
ca(m) = exp(-re);
cb(m) = -(exp(-re)-1.0)/sig(m)/dx;
da(m) = exp(-rm);
db(m) = -(exp(-rm)-1.0)/rho(m)/dx;
end
%*******************************************************************
% Inisialisasi matriks Berenger
%*******************************************************************
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<< Medan Hz >>>>>>>>>>>>>>>>>>>>>>>>>>>
for j=1:je
% Depan & belakang
for i=1:npmls
m = npmls+1-i;
daBzx(i,j) = da(m);
dbBzx(i,j) = db(m);
end
for i=ip+1:ie
m = i-ip;
daBzx(i,j) = da(m);
dbBzx(i,j) = db(m);
end
end
for i=1:ie
% kiri dan kanan
for j=1:npmls
m = npmls+1-j;
daBzy(i,j) = da(m);
dbBzy(i,j) = db(m);
end
for j=jp+1:je
m = j-jp;
daBzy(i,j) = da(m);
dbBzy(i,j) = db(m);
end
end
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<< Medan Ex>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
for i=1:ie
% Kiri dan kanan
for j=2:npmls+1
m = npmls+2-j;
caDx(i,j) = ca(m);
cbDx(i,j) = cb(m);
end
for j=jp+1:je
m = j-jp;
caDx(i,j) = ca(m);
cbDx(i,j) = cb(m);
end
end
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<< Medan Ey>>>>>>>>>>>>>>>>>>>>>>>>>>>>
i
for j=1:je
for i=2:npmls+1
m = npmls+2-i;
caDy(i,j) = ca(m);
cbDy(i,j) = cb(m);
end
for i=ip+1:ie
m = i-ip;
caDy(i,j) = ca(m);
cbDy(i,j) = cb(m);
end
end
% Depan dan belakang
% Algoritma 1D FDTD
%*******************************************************************
for j=1:npmls
m = npmls+1-j;
daHzx_inc(j)
dbHzx_inc(j)
end
for j=jp+1:je
m = j-jp;
daHzx_inc(j)
dbHzx_inc(j)
end
= da(m);
= db(m);
= da(m);
= db(m);
for j=2:npmls+1
m = npmls+2-j;
caEx_inc(j) = ca(m);
cbEx_inc(j) = cb(m);
end
for j=jp+1:je
m = j-jp;
caEx_inc(j) = ca(m);
cbEx_inc(j) = cb(m);
end
%----------------------------------------------for i=1:npmls
m = npmls+1-i;
daHzy_inc(i) = da(m);
dbHzy_inc(i) = db(m);
end
for i=ip+1:ie
m = i-ip;
daHzy_inc(i) = da(m);
dbHzy_inc(i) = db(m);
end
for i=2:npmls+1
m = npmls+2-i;
caEy_inc(i) = ca(m);
cbEy_inc(i) = cb(m);
end
for i=ip+1:ie
m = i-ip;
caEy_inc(i) = ca(m);
cbEy_inc(i) = cb(m);
end
%*******************************************************************
% inisialisasi Movie
%*******************************************************************
figure('position',[200 200 700 520]); set(gcf,'color','white');
rect = get(gcf,'position'); rect(1:2) = [0 0];
Dx_h1 = Dx; Dy_h1 = Dy; Bz_h1 = Bz; Ex_h1 = Ex; Ey_h1 = Ey; Hz_h1 =
Hz;
clear j;
%*******************************************************************
% Pengulangan FDTD
%*******************************************************************
% sy = 500;
% Perambatan persamaan Gauss arah y
% sx = 500;
% Perambatan persamaan Gauss arah x
% n = 1; ne = 1; err = 0;
n = 1; ne = 1; err = 100;
% Dengan (n<nmax) && (err(ne)>threshold)
while (n<ST*N) || (n<nmax) && (err(ne)>threshold)
%*******************************************************************
% Distribusi Medan
%*******************************************************************
Dx_h2 = Dx_h1; Dx_h1 = Dx; Dy_h2 = Dy_h1; Dy_h1 = Dy; Bz_h2 =
Bz_h1; Bz_h1 = Bz;
Ex_h2 = Ex_h1; Ex_h1 = Ex; Ey_h2 = Ey_h1; Ey_h1 = Ey; Hz_h2 =
Hz_h1; Hz_h1 = Hz;
%sumber(n) = exp(-(((n-delay)*dt)/(0.8*tau))^2)
.*sin(2*pi*freq*n*dt);
%*******************************************************************
% Ex_inc datang (1-D FDTD)
%*******************************************************************
Ex_inc(2:jb) = caEx_inc(2:jb).*Ex_inc(2:jb) +
cbEx_inc(2:jb).*(Hzx_inc(1:je)-Hzx_inc(2:jb));
%*******************************************************************
% Medan Dx dan Dy field
%*******************************************************************
Dx( : ,2:je) = caDx( : ,2:je).*Dx( : ,2:je) + cbDx( :
,2:je).*(Hz( :
,2:je) - Hz( : ,1:je-1));
Dy(2:ie, : ) = caDy(2:ie, : ).*Dy(2:ie, : ) + cbDy(2:ie, :
).*(Hz(1:ie-1,: ) - Hz(2:ie, :
));
i
%*******************************************************************
% Ruang penghubung D dan E
%*******************************************************************
Dysynch(1:ie, 2:je) = 0.25*(Dy(1:ie, 2:je)+Dy(2:ib,
2:je)+Dy(1:ie, 1:je-1)+Dy(2:ib, 1:je-1));
Dy_h1synch(1:ie, 2:je) = 0.25*(Dy_h1(1:ie, 2:je)+Dy_h1(2:ib,
2:je)+Dy_h1(1:ie, 1:je-1)+Dy_h1(2:ib, 1:je-1));
Dy_h2synch(1:ie, 2:je) = 0.25*(Dy_h2(1:ie, 2:je)+Dy_h2(2:ib,
2:je)+Dy_h2(1:ie, 1:je-1)+Dy_h2(2:ib, 1:je-1));
Ey_h1synch(1:ie, 2:je) = 0.25*(Ey_h1(1:ie, 2:je)+Ey_h1(2:ib,
2:je)+Ey_h1(1:ie, 1:je-1)+Ey_h1(2:ib, 1:je-1));
Ey_h2synch(1:ie, 2:je) = 0.25*(Ey_h2(1:ie, 2:je)+Ey_h2(2:ib,
2:je)+Ey_h2(1:ie, 1:je-1)+Ey_h2(2:ib, 1:je-1));
Dxsynch(2:ie, 2:je) = 0.25*(Dx(2:ie, 2:je)+Dx(2:ie,
3:jb)+Dx(1:ie-1, 2:je)+Dx(1:ie-1, 3:jb));
Dx_h1synch(2:ie, 2:je) = 0.25*(Dx_h1(2:ie, 2:je)+Dx_h1(2:ie,
3:jb)+Dx_h1(1:ie-1, 2:je)+Dx_h1(1:ie-1, 3:jb));
Dx_h2synch(2:ie, 2:je) = 0.25*(Dx_h2(2:ie, 2:je)+Dx_h2(2:ie,
3:jb)+Dx_h2(1:ie-1, 2:je)+Dx_h2(1:ie-1, 3:jb));
Ex_h1synch(2:ie, 2:je) = 0.25*(Ex_h1(2:ie, 2:je)+Ex_h1(2:ie,
3:jb)+Ex_h1(1:ie-1, 2:je)+Ex_h1(1:ie-1, 3:jb));
Ex_h2synch(2:ie, 2:je) = 0.25*(Ex_h2(2:ie, 2:je)+Ex_h2(2:ie,
3:jb)+Ex_h2(1:ie-1, 2:je)+Ex_h2(1:ie-1, 3:jb));
%*******************************************************************
% Medan Ex dan Ey
%*******************************************************************
Ex(1:ie, 2:je) = ((b0x(1:ie,2:je).*Dx(1:ie,2:je) +
b0xy(1:ie,2:je).*Dysynch(1:ie,2:je) + ...
b1x(1:ie,2:je).*Dx_h1(1:ie,2:je) +
b1xy(1:ie,2:je).*Dy_h1synch(1:ie,2:je) + ...
a1x(1:ie,2:je).*Ex_h1(1:ie,2:je) +
a1xy(1:ie,2:je).*Ey_h1synch(1:ie,2:je) + ...
b2x(1:ie,2:je).*Dx_h2(1:ie,2:je) +
b2xy(1:ie,2:je).*Dy_h2synch(1:ie,2:je) + ...
a2x(1:ie,2:je).*Ex_h2(1:ie,2:je) +
a2xy(1:ie,2:je).*Ey_h2synch(1:ie,2:je)) ...
./ (a0x(1:ie,2:je)));
Ey(2:ie, 1:je) = ((b0y(2:ie, 1:je).*Dy(2:ie, 1:je) +
1:je).*Dxsynch(2:ie, 1:je) + ...
b1y(2:ie, 1:je).*Dy_h1(2:ie, 1:je)
1:je).*Dx_h1synch(2:ie, 1:je) + ...
a1y(2:ie, 1:je).*Ey_h1(2:ie, 1:je)
1:je).*Ex_h1synch(2:ie, 1:je) + ...
b2y(2:ie, 1:je).*Dy_h2(2:ie, 1:je)
1:je).*Dx_h2synch(2:ie, 1:je) + ...
a2y(2:ie, 1:je).*Ey_h2(2:ie, 1:je)
1:je).*Ex_h2synch(2:ie, 1:je)) ...
./ (a0y(2:ie, 1:je)));
b0yx(2:ie,
+ b1yx(2:ie,
+ a1yx(2:ie,
+ b2yx(2:ie,
+ a2yx(2:ie,
%*******************************************************************
% Nilai Ex datang (1-D & 2-D)
%*******************************************************************
Ex(it+1:ie-it,
jt+1) = Ex(it+1:ie-it,
jt+1) +
(dt/eps_0/dx).*Hzx_inc(
jt+1);
%.*exp(-((it+1:ie-it)-ic).^2./(2*sy^2))';
% Left
Ex(it+1:ie-it,je-jt+1) = Ex(it+1:ie-it,je-jt+1) (dt/eps_0/dx).*Hzx_inc(je-jt+1);
%.*exp(-((it+1:ie-it)-ic).^2./(2*sy^2))';
% Right
%*******************************************************************
% Nilai Ey datang(1-D & 2-D)
%*******************************************************************
Ey(
it+1,jt+1:je-jt) = Ey(
it+1,jt+1:je-jt) (dt/eps_0/dx).*Hzx_inc(jt+1:je-jt);
%.*exp(-((
it+1)-ic).^2./(2*sy^2))'; % Bottom
Ey(ie-it+1,jt+1:je-jt) = Ey(ie-it+1,jt+1:je-jt) +
(dt/eps_0/dx).*Hzx_inc(jt+1:je-jt);
%.*exp(-((ie-it+1)-ic).^2./(2*sy^2))'; % Top
%*******************************************************************
% Penghitungan Hzx_inc (1-D FDTD)
%*******************************************************************
Hzx_inc(1:je) = daHzx_inc(1:je).*Hzx_inc(1:je) +
dbHzx_inc(1:je).*(Ex_inc(1:je)-Ex_inc(2:jb));
%*******************************************************************
%
..... Eksitasi Sumber Datang(1-D FDTD).....
%*******************************************************************
Hzx_inc(js) = Hzx_inc(js) + 1*source(n);
%*******************************************************************
% medan Bzx dan Bzy (Bz)
%*******************************************************************
Bzx(1:ie,: ) = daBzx(1:ie,: ).*Bzx(1:ie,: ) + dbBzx(1:ie,:
).*(Ey(1:ie,: ) - Ey(2:ib, :));
Bzy( :, 1:je) = daBzy( :, 1:je).*Bzy( :, 1:je) + dbBzy( :,
1:je).*(Ex( :, 2:jb) - Ex( :, 1:je));
Bz = Bzx + Bzy;
%*******************************************************************
% Medan Hz
%*******************************************************************
Hz = (b0z.*Bz + b1z.*Bz_h1 + b2z.*Bz_h2 - (a1z.*Hz_h1 +
a2z.*Hz_h2)) ./ a0z;
%*******************************************************************
% Nilai Hz datang (1-D & 2-D)
%*******************************************************************
Hz(it+1:ie-it,
jt+1) = Hz(it+1:ie-it,
jt+1) (dt/mu_0/dx).*Ex_inc(
jt+1);
% Left
Hz(it+1:ie-it,je-jt+1) = Hz(it+1:ie-it,je-jt+1) +
(dt/mu_0/dx).*Ex_inc(je-jt+1);
% Right
i
%HzsourceTF(1:length(it:ie-it),n)=Hz(it:ie-it,jt+1);
%HztransTF(1:length(it:ie-it),n)=Hz(it:ie-it,je-jt-1);
%HzsourceSF(1:length(it:ie-it),n)=Hz(it:ie-it,js);
%HztransSF(1:length(it:ie-it),n)=Hz(it:ie-it,je-js);
%*******************************************************************
% Visualisasi Medan
%*******************************************************************
if mod(n,10)==0
timestep = num2str(n);
pcolor((1:je)./M,(1:ie)./M,real(Hz)); shading interp; axis
image; colorbar; axis([1/M je/M 1/M ie/M]);
%pcolor((1:je)./M,(1:ie)./M,real(Hz)/max(max(real(Hz)))); shading
interp; axis image; colorbar; axis([1/M je/M 1/M ie/M]);
line((R1*sin((0:360)*pi/180)+yc)./M,(R1*cos((0:360)*pi/180)+xc)./M,'c
olor','k');
line((R2*sin((0:360)*pi/180)+yc)./M,(R2*cos((0:360)*pi/180)+xc)./M,'c
olor','k');
%xlabel('y/\lambda'); ylabel('x/\lambda'); title(['amplitude of
H_z']);
xlabel('y/\lambda'); ylabel('x/\lambda'); title(['H_z at
timestep ', timestep]);
getframe(gcf,rect);
end
%*******************************************************************
% Penghitungan fungsi eror untuk konvergensi pulsa
%******************************************************************
if mod(n,N)==0
ne = n/N;
if (20*log10(sum(sum(abs(Hz(it:ie-it,jt:je-jt)))))) >
(20*log10(sum(sum(abs(Hz_p(it:ie-it,jt:je-jt))))))
err(ne) = 0;
else
err(ne) = (20*log10(sum(sum(abs(Hz(it:ie-it,jt:je-jt)))))) (20*log10(sum(sum(abs(Hz_p(it:ie-it,jt:je-jt))))));
end
Hz_p = Hz;
end
%*******************************************************************
%Penghitungan fungsi eror untuk konvergensi pulsa
%*******************************************************************
if mod(n,N)==0
ne = n/N;
if max(max(abs(Hz_p(npmls+10:ip-10,npmls+10:jp-10))))==0
err(ne) = 100;
else
err(ne) = max(max(abs(abs(Hz(npmls+10:ip-10,npmls+10:jp10))-abs(Hz_p(npmls+10:ip-10,npmls+10:jp-10))))) /
max(max(abs(Hz_p(npmls+10:ip-10,npmls+10:jp-10)))) * 100;
end
Hz_p = Hz;
end
%*******************************************************************
% Menampilkan
%*******************************************************************
if mod(n,step)==0
%save results_2D.mat dx dt freq nmax ie je is js err N M Ex Ey Hz;
c = round(clock);
disp(['Current time step: ',int2str(n-mod(n,step)),',
',int2str(n/nmax*100),'% completed at
',num2str(c(4)),':',num2str(c(5)),':',num2str(c(6))]);
end
n = n + 1;
end
%*******************************************************************
% Menyimpan
%******************************************************************
save results_2DTEidealcloakdielcore3.mat dx dt freq n nmax ie je xc
yc R1 R2 is js err N M Ex Ey Hz;
i
APENDIX F
MENGUBAH PERSAMAAN-PERSAMAAN KE PROGRAM
Persamaan (4.40) :
(
(
)
)
(
(
)
)
(
)
Diubah dalam program menjadi :
(
)
(
)
a = 4/4*e0*einf + e0*wpesq*dt^2 + e0*einf*ge*2*dt*
a = 4/e0*4*einf+e0*dt^2*wpesq+e0*2*dt*einf*ge
a = (1/dt^2)*(4*dt^2)/(e0*(4*einf+dt^2*wpesq+2*dt*einf*ge)
a = (1/dt^2)*a0
a0 = (4*dt^2)/(e0*(4*einf+dt^2*wpesq+2*dt*einf*ge))
(
)
(
(
)
)
(
)
b = (ge*2*dt)/(4*e0*einf+e0*wpesq*dt^2+e0*einf*ge*2*dt)
b = (ge*2*dt)/(e0*4*einf+e0*dt^2*wpesq+e0*2*dt*einf*ge)
b = (1/(2*dt))*ge*(4*dt^2)/(e0*(4*einf+dt^2*wpesq+2*dt*einf*ge))
b = (1/(2*dt))*ge.*a0
(
)
(
)
c = 4*e0*einf/(4*e0*einf+e0*wpesq*dt^2+e0*einf*ge*2*dt)
c = (e0/dt^2)*einf*(4*dt^2)/e0*(4*einf+dt^2*wpesq+2*dt*einf*ge)
c = (e0/dt^2)*einf.*a0;
( )
( )
(
)
d = (-e0*wpesq*dt^2)/(4*e0*einf+e0*wpesq*dt^2+e0*einf*ge*2*dt)
d = (-1*e0/4)*wpesq*(4*dt^2)/e0*(4*einf+dt^2*wpesq+2*dt*einf*ge)
d = (-1*e0/4)*wpesq*a0
(
)
(
)
(
)
e = e0*einf*ge2*dt/(4*e0*einf+e0*wpesq*dt^2+e0*einf*ge*2*dt)
e = (1/(2*dt))*e0*einf*ge
*(4*dt^2)/e0*(4*einf+dt^2*wpesq+2*dt*einf*ge)
e = (1/(2*dt))*e0*einf*ge*a0
Selanjutnya untuk medan magnet persamaanya
(
(
)
)
(
(
)
Diubah dalam program menjadi :
i
(
)
)
(
)
(
)
am = 4/4*u0*uinf + u0*wpesq*dt^2 + u0*uinf*gm*2*dt*
am = 4/u0*4*uinf+u0*dt^2*wpesq+u0*2*dt*uinf*gm
am = (1/dt^2)*(4*dt^2)/(u0*(4*uinf+dt^2*wpesq+2*dt*uinf*gm)
am = (1/dt^2)*am0
am0 = (4*dt^2)/(u0*(4*uinf+dt^2*wpesq+2*dt*uinf*gm))
(
)
(
(
)
)
(
)
bm = (gm*2*dt)/(4*u0*uinf+u0*wpesq*dt^2+u0*uinf*gm*2*dt)
bm = (gm*2*dt)/(u0*4*uinf+u0*dt^2*wpesq+u0*2*dt*uinf*gm)
bm = (1/(2*dt))*gm*(4*dt^2)/(u0*(4*uinf+dt^2*wpesq+2*dt*uinf*gm))
bm = (1/(2*dt))*gm.*am0
(
)
(
)
cm = 4*u0*uinf/(4*u0*uinf+u0*wpesq*dt^2+u0*uinf*gm*2*dt)
cm = (u0/dt^2)*uinf*(4*dt^2)/u0*(4*uinf+dt^2*wpesq+2*dt*uinf*gm)
cm = (u0/dt^2)*uinf.*am0
(
)
(
)
(
)
dm = (-u0*wpesq*dt^2)/(4*u0*uinf+u0*wpesq*dt^2+u0*uinf*gm*2*dt)
dm = (-1*u0/4)*wpesq*(4*dt^2)/u0*(4*uinf+dt^2*wpesq+2*dt*uinf*gm)
dm = (-1*u0/4)*wpesq*am0
(
(
)
)
(
)
em = u0*uinf*gm2*dt/(4*u0*uinf+u0*wpesq*dt^2+u0*uinf*gm*2*dt)
em = (1/(2*dt))*u0*uinf*gm
*(4*dt^2)/u0*(4*uinf+dt^2*wpesq+2*dt*uinf*gm)
em = (1/(2*dt))*u0*uinf*gm*am0
persamaan (4.35) :
⃗
[
⃗
]
[
(⃗
]
⃗
)
Hy(1:SIZE-1,n2) = Hy(1:SIZE-1,n1) + ( ( Ex(1:SIZE-1,n1)
- Ex(2:SIZE,n1) ) * dt/(u0*dz) )
persamaan (4.36) :
⃗
⃗
(⃗
[
]
⃗
[
])
Ex(2:SIZE,n2) = Ex(2:SIZE, n1) + ( dt/(e0*dz)*(Hy(1:SIZE-1, n2)
- Hy(2:SIZE, n2)) )
persamaan (4.39), (4.41),(4.42) (4.43) dan (4.44) :
⃗
(⃗
⃗
( ⃗
⃗
⃗
(⃗
)
(⃗
)
⃗
( ⃗
)
⃗
)
)
Ex(:,n2) = a.*(Dx(:,n2)-2*Dx(:,n1)+Dx(:,3))+b.*(Dx(:,n2)Dx(:,3))+c.*(2*Ex(:,n1)-Ex(:,3))+d.*(2*Ex(:,n1) +Ex(:,3))
+e.*(Ex(:,3))
⃗
(⃗
⃗
⃗
)
(⃗
( ⃗
⃗
)
(⃗
⃗
( ⃗
)
)
Hy(:,n2) = am.*(By(:,n2)-2*By(:,n1)+By(:,3))+bm.*(By(:,n2)By(:,3))+cm.*(2*Hy(:,n1)-Hy(:,3))+dm.*(2*Hy(:,n1)
+Hy(:,3))+em.*(Hy(:,3))
⃗
( )
⃗ ( )
(⃗ ( )
⃗ (
))
By(1:SIZE-1,n2) = By(1:SIZE-1,n1) + ( ( Ex(1:SIZE-1,n1) –
Ex(2:SIZE,n1) ) * dt/(dz) )
⃗
( )
⃗ ( )
(⃗ ( )
⃗ (
))
Dx(2:SIZE,n2) = Dx(2:SIZE, n1) + ( dt/(dz)*(Hy(1:SIZE-1, n2) –
i
⃗
)
Hy(2:SIZE, n2)) )
(
|
)
̃ (
̃ (
)
|
)
nFDTD = (1/(1i*k0*(z1-z2)))
.*log(EXZ2(1:NFFT/2+1)./EXZ1(1:NFFT/2+1))
( )
( )
erx = ((r-R1) / (r));
( )
ethetax =
( )
((r) / (r-R1));
⃗
[
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
]
⃗
⃗
[
⃗
⃗
⃗
[
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
(
)
]
⃗
⃗
⃗
⃗
⃗
⃗
]
⃗
⃗
[
⃗
⃗
(
Ax1 = ax+bx+cx
⃗
)
⃗
⃗
⃗
]
(
)
Ax2 = (2.*ax)-(2.*cx)
(
)
(
)
Ax3 = ax-bx+cx
ax = ((cosx.^2) + (ethetax.*(sinx.^2))) ./ (dt^2);
bx = ((gamma_px.*(cosx.^2)) + ((sigma_px +(ethetax*gamma_px))
.*(sinx.^2))) ./ (2*dt);
cx = (((cosx.^2).*((omega_px)^2)) + (sigma_px*gamma_px
.*(sinx.^2))) ./ (4);
(
)
(
)
(
)
(
)
Bx1 = wx+fx+vx
(
)
(
)
Bx2 = (2.*wx)-(2.*vx)
(
)
(
)
Bx3 = wx-fx+vx;
wx = ((1-ethetax).*sinx.*cosx) ./ (dt^2)
fx = ((gamma_px-sigma_px-(ethetax*gamma_px)).*sinx.*cosx)./(2*dt)
vx = (((omega_px)^2)-(sigma_px*gamma_px)).*sinx.*cosx) ./ (4)
Cx1 = kx+lx
Cx2 = (2.*kx)
Cx3 = kx-lx
i
kx = 1 / etheta0*(dt^2);
lx = gamma_px / etheta0* (2*dt);
(
)
(
)
(
)
Dx1 = tx+qx+px
(
)
Dx2 = (2.*tx)-(2.*px)
(
)
Dx3 = tx-qx+px
tx = ((sinx.^2) + (ethetax.*(cosx.^2))) ./ (dt^2)
qx = ((gamma_px.*(sinx.^2)) + ((sigma_px +(ethetax*gamma_px))
.*(cosx.^2)))./(2*dt)
px = (((sinx.^2).*((omega_px)^2)) + (sigma_px*gamma_px
.*(cosx.^2))) ./(4)
a0x(i, j) = Ax1 - (((Bx1).^2) ./ Dx1);
a1x(i, j) = Ax2 - ((Bx1.*Bx2) ./ Dx1);
a2x(i, j) = ((Bx1.*Bx3) ./ Dx1) - (Ax3);
a1xy(i, j) = Bx2 - ((Bx1.*Dx2) ./ Dx1);
a2xy(i, j) = ((Bx1.*Dx3) ./ Dx1) - (Bx3);
b0x(i, j) = Cx1;
b1x(i, j) = -Cx2;
b2x(i, j) = Cx3;
b0xy(i, j) = -(Bx1.*Cx1) ./ Dx1;
b1xy(i, j) = (Bx1.*Cx2) ./ Dx1;
b2xy(i, j) = -(Bx1.*Cx3) ./ Dx1;
untuk bagian y :
(
)
(
)
Ay1 = ay+by+cy
(
)
Ay2 = (2.*ay)-(2.*cy)
(
)
(
)
Ay3 = ay-by+cy
ay = ((cosy.^2) + (ethetay.*(siny.^2))) ./ (dt^2);
by = ((gamma_py.*(cosy.^2)) + ((sigma_py +(ethetay*gamma_py))
.*(siny.^2))) ./ (2*dt);
cy = (((cosy.^2).*((omega_py)^2)) + (sigma_py*gamma_py
.*(siny.^2))) ./ (4);
(
)
(
)
(
)
(
)
By1 = wy+fy+vy
(
)
(
)
By2 = (2.*wy)-(2.*vy)
(
)
(
)
i
By3 = wy-fy+vy;
wy = ((1-ethetay).*siny.*cosy) ./ (dt^2)
fy = ((gamma_py-sigma_py-(ethetay*gamma_py)).*siny.*cosy)./(2*dt)
vy = (((omega_py)^2)-(sigma_py*gamma_py)).*siny.*cosy) ./ (4)
Cy1 = ky+ly
Cy2 = (2.*ky)
Cy3 = ky-ly
ky = 1 / etheta0*(dt^2);
ly = gamma_py / etheta0* (2*dt);
(
)
(
)
(
)
Dy1 = ty+qy+py
(
)
Dy2 = (2.*ty)-(2.*py)
(
)
Dy3 = ty-qy+py
ty = ((siny.^2) + (ethetay.*(cosy.^2))) ./ (dt^2)
qy = ((gamma_py.*(siny.^2)) + ((sigma_py +(ethetay*gamma_py))
.*(cosy.^2)))./(2*dt)
py = (((siny.^2).*((omega_py)^2)) + (sigma_py*gamma_py
.*(cosy.^2))) ./(4)
a0y(i, j) = Ay1 - (((By1).^2) ./ Dy1);
a1y(i, j) = Ay2 - ((By1.*By2) ./ Dy1);
a2y(i, j) = ((By1.*By3) ./ Dy1) - (Ay3);
a1yx(i, j) = By2 - ((By1.*Dy2) ./ Dy1);
a2yx(i, j) = ((By1.*Dy3) ./ Dy1) - (By3);
b0y(i, j) = Cy1;
b1y(i, j) = -Cy2;
b2y(i, j) = Cy3;
b0yx(i, j) = -(By1.*Cy1) ./ Dy1;
b1yx(i, j) = (By1.*Cy2) ./ Dy1;
b2yx(i, j) = -(By1.*Cy3) ./ Dy1;
i