The Z transform is typically used in digital filtering and signal processing problems. The paper illustrates the use of the Z transform in implementing the FDTD method where it is useful to avoid dealing with convolution integral in the time domain by going immediately to Z domain. This work study the pulse propagation in free space that comes upon a plasma. Plasma is very interesting medium and the results illustrated that at low frequencies, it looks like a metal, and higher frequency, it become transparent just like a dielectric medium

Albatniji, Amal Y.

Elwasife, Khitam Y.

English

Institutional Repository of the Islamic University of Gaza

 
American Journal of Electromagnetics and Applications 
2016; 4(1): 8-13 
http://www.sciencepublishinggroup.com/j/ajea 
doi: 10.11648/j.ajea.20160401.12 
ISSN: 2376-5968 (Print); ISSN: 2376-5984 (Online)  
 
Z Transform and FDTD Method in Unmagnetized Plasma 
Khitam Y. Elwasife, Amal Y. Albatniji 
Physics Department, Islamic University of Gaza, Gaza, Palestinian Authority 
Email address: 
abatniji@iugaza.edu.ps (A. Y. Albatniji), kelwasife@iugaza.edu.ps (K. Y. Elwasife) 
To cite this article: 
Khitam Y. Elwasife, Amal Y. Albatniji. Z Transform and FDTD Method in Unmagnetized Plasma. American Journal of Electromagnetics 
and Applications. Vol. 4, No. 1, 2016, pp. 8-13. doi: 10.11648/j.ajea.20160401.12 
Received: October 7, 2016; Accepted: October 28, 2016; Published: November 16, 2016 
 
Abstract: The Z transform is typically used in digital filtering and signal processing problems. The paper illustrates the use 
of the Z transform in implementing the FDTD method where it is useful to avoid dealing with convolution integral in the time 
domain by going immediately to Z domain. This work study the pulse propagation in free space that comes upon a plasma. 
Plasma is very interesting medium and the results illustrated that at low frequencies, it looks like a metal, and higher frequency, 
it become transparent just like a dielectric medium. 
Keywords: Z Transform, Finite Difference Time Domain, Unmagnified Plasma 
 
1. Introduction 
The finite-difference time-domain (FDTD) solution 
procedure developed by Yee [1], Recently has in been 
extended to dispersive and anisotropic media to handle 
materials such as magnetized ferrites and plasmas. The 
solution for dispersive media has been accomplished through 
a recursive update of a convolution integral in the 
constitutive relations for the fields of FDTD simulation [2], 
[3], [4], [5] by numerical solution of the differential equation 
form of the constitutive relations [6] and [7] and using Z 
transforms [8]. The basic techniques and important concepts 
of the one-dimensional electromagnetic particle are studied 
[9]. Maxwell’s equations and equations of motions for a large 
number of super particles are solved. Electromagnetic 
propagation through a magnetized plasma is studied with the 
finite-difference time-domain (FDTD) method by means of Z 
transforms. The propagation direction is perpendicular to the 
biasing magnetic filed and the electric field is parallel to the 
biasing magnetic filed. The dependence of the reflection and 
transmission coefficients on frequency is obtained by discrete 
Fourier transform. A comparison with analytical values of 
homogeneous magnetized plasma is made. The reflection and 
transmission coefficients in an inhomogeneous magnetized 
plasma layer are calculated [10]. A novel and normalized Z-
transform-based finite-difference time-domain (ZTFDTD) 
method is presented for simulating the interaction of the 
electromagnetic (EM) wave with unmagnetized plasma [11]. 
The 2-D ZTFDTD formulations for unmagnetized plasma are 
derived. Using a simplified 2-D model for a perfectly 
conducting cylinder covered with unmagnetized plasma, the 
stealth effect of unmagnetized plasma is studied in different 
thicknesses of plasma, electron densities of plasma, EM wave 
frequencies, and plasma collision frequencies. Numerical 
results indicate that plasma stealth is effective in theory and 
reasonable selection for the plasma parameters can greatly 
enhance its effectiveness. 
2. Theory 
The dielectric constant and conductivity of most media vary 
at different frequencies. The model consider is a plasma 
medium cover with free space in both side as shown in Fig. 1. 
 
Figure 1. Unmagnetized Plasma sandwich between two free space layer, it 
represent the region from 300-500 position in X-axis. 
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 American Journal of Electromagnetics and Applications 2016; 4(1): 8-13 9 
 
The pulse we have using is the Gaussian pulse which 
contain spectrum of frequencies. One of the most significant 
development in the FDTD method by means to simulate 
frequency dependent material. We simulate the of 
electromagnetic wave in unmagnified plasma, the 
permittivity of the unmagnetized plasma is given as [1] 
2
1
( )
p
civ
ω
ε
ω ω
∗ = +
−
                          (1) 
Where: 
ωp =2πfp 
fp is the plasma frequency 
vc is the electron collision frequency. 
Using partial fraction expansion Equation 1, becomes 
2 2/ /
1
p c p c
c
v v
i v i
ω ω
ε
ω ω
∗ = + −
+
                   (2) 
And lets start by taking Z transform of equation (2) to 
obtain 
2 2
1 . 1
/ /1
1
1 1 c
p c p c
v t
v v
t z e z
ω ω
ε ∗ − − −= + −− − △△
            (3) 
We use Maxwell's equation in the form of flux density as 
( ) ( ). ( )
r
D Eω ε ω ω∗=                       (4) 
Wher E is the electric field. By the convolution theorem 
the Z -transform of equation (4) is 
( ) ( ). ( ).D z z E z tε ∗= △                      (5) 
Inserting equation (3) in equation (4) we obtain 
2
1 . 1
1 1
( ) ( ) ( )
1 1 c
p
v t
c
t
D z E z E z
v z e z
ω
− − −
 = + − − − △
△
     (6) 
2 . 1
. .1 2
(1 )
( ) ( ) ( )
1 (1 )
c
c c
v t
p
v t v t
c
t e z
D z E z E z
v e z e z
ω − −
− −− −
 −= +  − − + 
△
△ △
△
 (7) 
E(z), the electric field as a function of Z can be solved for 
by 
1( ) ( ) ( )E z D z z S z−= −                       (8) 
Where S(z) can written as 
2 . 1
. .1 2
(1 )
( ) ( )
1 (1 )
c
c c
v t
p
v t v t
c
t e z
S z E z
v e z e z
ω − −
− −− −
 −=  − − + 
△
△ △
△
   (9) 
Therefore, the FDTD Simulation becomes: 
[ ] [ ] [ ]ex k dx k sx k= −
 
[ ] (1 exp( * )* 1[ ]
2)* / )*
exp( * )* 2[ ] ( * [ ]
(1 exp( * ))
sx k vc dt sxm k
omega dt vc
vc dt sxm k pow ez k
vc dt
= − −
− − +
− −
 (10) 
3. Results and Discussion 
The finite-different time-domain method has been widely 
used to simulate the transient solutions of electromagnetic 
wave propagation in many media according to its constant 
properties. Here we have assumed one dimensional 
unmagnetized plasma with permittivity dependent frequency. 
In time-domain, the electromagnetic propagation process of 
a Gaussian pulse through an unmagnetized plasma We have 
been simulated the Plasma medium by sinusoidal pulse with 
different frequencies such as 500π THz, 2000500π THz, 
4000500π THz, and 5000500π THz. Absorbing material 
boundary condition is used in these simulations. vc is the 
electron collision frequency. The electric field of different 
time steps has been plotted. The low or high frequencies 
which we use is equal 2000 THz and Vc=57 THz. It means 
below or above the Plasma frequency. Fig. 1 Simulation of 
electromagnetic waves propagation in free space and striking 
a plasma medium. The cell size is dx=75µm and vc is 0.2 
THz. The propagation wave has a center frequency of 
500πGHz. After 650 time steps, it is starts to strike plasma 
layer and start to reflect, but at 750 steps it is reflected as in 
Fig. 2. and in Fig. 3. it completely reflected when it reach 
850 time steps. Fig. 4 Illustrates simulation of wave 
propagation in free space and stricking a plasma medium. 
The cell size is dx=1nm and vc is 0.2 THz. The propagation 
wave has a center frequency of 4000πTHz. After 800 time 
steps, it is almost inter plasma medium. In Fig. 5. and Fig. 6. 
we illustrated the pulse with frequency of 4000πTHz. And 
5000πTHz respectively After 1300 time steps, it has 
completely passed plasma medium but each signal is 
different in their amplitude the transmission for pulse of 
sinusoidal electromagnetic waves shown in Fig. 6 and Fig. 
7 after 1300 and 1200 time steps respectively. In Fig. 8 The 
propagation wave has a center frequency of 2000πTHz. After 
565 time steps, it start to strike plasma medium. Also The 
propagation wave has a center frequency of 2000πTHz. After 
900 time steps, it has passed completely the plasma medium 
without any reflected as shown in Fig. 9. We show that in 
Fig. 10 and 11, the propagation wave has a center frequency 
of 2000πTHz. After 1100 and 1300 time steps respectively, it 
has passed completely the plasma medium without any 
reflected. 
 
 
 
 
 
10 Khitam Y. Elwasife and Amal Y. Albatniji:  Z Transform and FDTD Method in Unmagnetized Plasma  
 
 
Figure 2. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=75nm and vc is 0.2 THz. The propagation wave has 
a center frequency of 500πGHz. After 650 time steps, it is almost completely reflected. 
 
Figure 3. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=75nm and vc is 0.2 THz. a center frequency of 
500πGHz. After 750 time steps, it is almost completely reflected. 
 
Figure 4. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=1nm and vc is 0.2 THz. The propagation wave has a 
center frequency of 4000πTHz. After 800 time steps, it is completely inter plasma medium. 
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
 Plsama Material   Free Space    Free Space
Time = 650
Time steps(650)
E
le
c
tr
ic
 f
ie
ld
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
 Plsama Material   Free Space    Free Space
Time = 800
Time steps(800) inter plasma layer 
E
le
c
tr
ic
 f
ie
ld
 American Journal of Electromagnetics and Applications 2016; 4(1): 8-13 11 
 
 
Figure 5. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=1nm and vc is 0.2 THz. The propagation wave has a 
center frequency of 4000πTHz. After 1300 time steps, it has completely passed plasma medium. 
 
Figure 6. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=1nm and vc is 0.2 THz. The propagation wave has a 
center frequency of 5000πTHz. After 1300 time steps, it has passed completely plasma medium. 
 
Figure 7. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=1nm and vs. is 75 THz. The propagation wave has a 
center frequency of 5000πTHz. After 1200 time steps, it has passed completely plasma medium. 
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
 Plsama Material   Free Space    Free Space
Time = 1300
Time steps (1300) completely passed
E
le
c
tr
ic
 f
ie
ld
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
 Plsama Material   Free Space    Free Space
Time = 1300
Time steps(1300),vc=75THz
E
le
c
tr
ic
 f
ie
ld
12 Khitam Y. Elwasife and Amal Y. Albatniji:  Z Transform and FDTD Method in Unmagnetized Plasma  
 
 
Figure 8. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=1nm and vc is 75 THz. The propagation wave has a 
center frequency of 2000πTHz. After 565 time steps, it starts to strike plasma medium. 
 
Figure 9. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=1nm and vc is 75 THz. The propagation wave has a 
center frequency of 2000πTHz. After 900 time steps, it has passed completely plasma medium without any reflected. 
 
Figure 10. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=1nm and vc is 75 THz. The propagation wave has a 
center frequency of 2000πTHz. After 1100 time steps, it has passed completely plasma medium without any reflection. 
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
 Plsama Material   Free Space    Free Space
Time = 565
Time steps(565),w=2000THz
E
le
c
tr
ic
 f
ie
ld
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
 Plsama Material   Free Space    Free Space
Time = 900
Time steps(900),w=2000THz
E
le
c
tr
ic
 f
ie
ld
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
 Plsama Material   Free Space    Free Space
Time = 1100
Time steps(1100),w=2000THz
E
le
c
tr
ic
 f
ie
ld
 American Journal of Electromagnetics and Applications 2016; 4(1): 8-13 13 
 
 
Figurer 11. Simulation of wave propagation in free space and striking a plasma medium. The cell size is dx=1nm and vc is 75 THz. The propagation wave has 
a center frequency of 2000πTHz. After 1300 time steps, it has passed completely plasma medium without any reflection. 
4. Conclusion 
In this work we use sinusoidal pulse to simulate and study 
the behavior of electromagnetic waves sticking unmagnetized 
plasma medium for different frequencies and many time 
steps. We study the electric field through a plasma medium 
which a magnetic field is not strong enough to influence the 
motion of the charged particles. At low frequency the 
medium looks like a metal and the medium becomes 
transparent just like a dielectric medium at high frequency. 
The frequency which is high or low with respect to plasma 
frequency. 
 
References 
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problems involving Maxwell’s equations in isotropic media", 
IBEE. Antennas Propagate., Vol. AP-14, 1966, pp. 302-307. 
[2] R. J. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler 
and M. Schneider, "A fiequencydependent finite-difference 
time-domain formulation for dispersive materials", hms. 
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[3] R. J. Luebbers and F. P. Hupsberger, "FDTD for Nth-Order 
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[4] K. S. Kunz and R. J. Luebbers, "The Finite Diflemnce Time 
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[8] D. M. Sullivan, "A Frequency-Dependent FDTD Method for 
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[9] Yoshiharu Omura,"One-dimensional Electromagnetic Particle 
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[10] Huang Shou-jiang, "A Finite-Difference Time-Domain 
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[11] Ming Yan,"Z-Transform-Based FDTD Analysis of Perfectly 
Conducting Cylinder Covered With Unmagnetized Plasma", 
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2007, P. 2968 -2970. 
 
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
 Plsama Material   Free Space    Free Space
Time = 1300
Time steps(1300),w=2000THz
E
le
c
tr
ic
 f
ie
ld


Z Transform and FDTD Method in Unmagnetized Plasma

https://iugspace.iugaza.edu.ps/bitstream/20.500.12358/25949/1/Elwasife%2c%20Khitam%20Y._29.pdf

Z Transform and FDTD Method in Unmagnetized Plasma

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