Research Article

# Implementation of Matched Load in a Transmission Line by Using FDTD Method Javad Pourabadeh

ABSTRACT

In a transmission line, when the characteristic impedance of the line (Z0) is equal to the end load impedance (ZL), the voltage and current reflection coefficients are zero, such a load absorbs all electrical energy from the source. This case, which is called Matched Load is very important in electrical transmission lines. This study describes how to implement the matched load in a transmission line when calculating voltages and currents numerically at different discretized time and space of the line via FDTD (Finite Difference Time Domain) method.

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 How to cite this article: Javad Pourabadeh , 2006. Implementation of Matched Load in a Transmission Line by Using FDTD Method. Journal of Applied Sciences, 6: 1314-1316. DOI: 10.3923/jas.2006.1314.1316 URL: https://scialert.net/abstract/?doi=jas.2006.1314.1316

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 VOLTAGE-CURRENT EQUATIONS OF THE TRANSMISSION LINE

The relation for voltage and current of a differential length, dz, of the line (Fig. 1) are: (1)

and (2)

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): (3)

and (4)

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: (5)

and (6)

By obtaining from Eq. 5 and 6 we have Fig. 1: Equivalent circuit of a transmission line for a differential length dz (7)

and (8)

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: (9)

and (10)

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 V-I 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) Nt = 200 (number of time steps) Nz = 100 (number of space steps) Nu = le/Δz = 80 (number of position setps for whole length)

IMPLEMENTATION MATCHED LOAD AT END LOAD

When zL = z0 , 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 zL = z0 (matched load) Fig. 5: Current pulses at a) k = Nu/2 b) k = Nu - 1 c) k = Nu, when zL = z0 (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 zL = z0

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.

REFERENCES
1:  Choi, D.H. and W.J.R. Hoefer, 1986. The finite difference time domain method and its application to eigen value problems. IEEE Trans. Microwave Theor. Techniques, 34: 1464-1470.
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2:  Kneppo, I., 1994. Microwave Integrated Circuits. Chapman and Hall, London.

3:  Taflove, A., 1980. Application of the finite difference time domain method to sinusoidal steady state electromagnetic penetration problems. IEEE Trans. Electromagnetic Compatibility, 22: 191-202.
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4:  Yee, K., 1966. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans. Antennas Propag., 14: 302-307.
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