INTRODUCTION
FDTD method uses FDM (Finite Difference Method) by discretizing time and space derivatives form of Maxwell’s equations. This method was first formulated by Yee (1966) and later developed by Taflove (1980). More recently, it has been successfully applied to various microwave circuit problems (Choi et al., 1986), but there is no any explanation of one dimensional FDTD implementation in the literature and the present work is the new one. By the way , understanding the present research helps how to implement the three dimensional FDTD method.
In this research an exact numerical solution of the transmission line equations
via the FDTD method is provided when the space discretization, Δz and the
time discretization, Δt, are chosen as Δt = p [Δz/v],
where v is the velocity of the wave propagation on the line. For stability conditions,
it is necessary that (Kneppo et al., 1980). Then a Gaussian voltage pulse
feeds the line as initiation. Therefore, by using the two discretized equations,
voltage and currents at any discretized points of the line and time will be
calculated. To have no reflection of voltage and current pulses, a matched load
is implemented at the end of the line.
DISCRETIZATION OF VOLTAGECURRENT EQUATIONS OF THE TRANSMISSION LINE
The relation for voltage and current of a differential length, dz, of the line (Fig. 1) are:
and
Where R, L, C. G are resistance, inductance, capacitance and conductance per unit length of the line respectively.
For an ideal line (R = G = 0):
and
Now an exact numerical solution of Eq. 3 and 4
via FDTD method is provided, when the space discretization and the time discretization,
Δz, are chosen such that, so:
and
By obtaining from
Eq. 5 and 6 we have

Fig. 1: 
Equivalent circuit of a transmission line for a differential
length dz 
and
CHOICE OF EXCITATION PULSE
For feeding the line a Gaussian voltage pulse is chosen (Fig. 2). This pulse has a smooth waveform in distance (or time) and its Fourier transform (spectrum) is also a Gaussian pulse centered at zero frequency. Initially, at voltage nodes which starts at k = 0, this pulse is defined as:
and
IMPLEMENTATION OF DISCRETIZED VOLTAGE AND CURRENT EQUATIONS OF THE LINE
To calculate voltage at thevoltage nodes (k, k + 1, k + 2, …….) and at
the time steps (n  1, n, n + 1, ……..) and currents at current nodes
at time steps
the codes has been written and Eq. 7 and 8
are solved by iterating k for a fixed time and then iterating time. For example
consider Fig. 3, if we have current
at
point, 
(meaning at current node on distance k + 1/2 and time n  1/2) and voltage
at point (k + 1, n) and voltage at point
we can calculate (by Eq. 7)
at point
. 
After that by Eq. 8, using this new current ,
current at
point and
voltage at
point (k, n), we can calculate voltage at
point (k, n + 1).

Fig. 2: 
Initial Gaussian voltage pulse for feeding the line 

Fig. 3: 
Mesh for VI transmission line 
Initially, at t = 0 (n = 0), the Gaussian voltage pulse defines voltages at
voltage node (k = 0, 1, 2, 3, …..) and currents at at
current node
The parameters in code are:
m 
= 
12 
b 
= 
10 
Δz 
= 
10^{3} 
Δt 
= 
10^{12} sec 
le 
= 
80x10^{3} m (lenght of the line) 
N_{t} 
= 
200 (number of time steps) 
N_{z} 
= 
100 (number of space steps) 
Nu 
= 
le/Δz = 80 (number of position setps for whole length) 
IMPLEMENTATION MATCHED LOAD AT END LOAD
When z_{L }= z_{0} , matched load is provided and there
is no reflection at the load. Such a load absorbs all received energy from the
source. The characteristic of this load is similar to the line with infinite
length.

Fig. 4: 
Voltage pulses at a) k = Nu/2 b) k = Nu  1 c) k = Nu, when
z_{L} = z_{0} (matched load) 

Fig. 5: 
Current pulses at a) k = Nu/2 b) k = Nu  1 c) k = Nu, when
z_{L} = z_{0} (matched load) 
Therefore, to implement this case we put the voltage at time and position
step k = Nu to voltage at last time and position step, meaning .
In this case voltage and currents at any points of the line and relevant time
are equal to the initial pulse. Figure 4 and 5
show voltage and current pulses, respectively:
a) at center of the line (k = Nu/2) b) at one position step before c) at the end of the line (k = Nu) when z_{L} = z_{0}
CONCLUSIONS
FDTD method can be successfully used to calculate voltages and currents numerically at discretized spaces and time for transmission lines. To have no reflection of voltage and current, meaning no reflection of electrical energy, putting is sufficient where Nu is the number of position steps for whole length.
ACKNOWLEDGMENT
The author wishes to thank Isfahan University of Technology for financial support of this project.