Numerical Analysis of Transmission Line Telegraph Equation ... · transmission line, the forward voltage increasing rate in x-direction is ux/ , the current increasing rate is ix

Numerical Analysis of Transmission Line Telegraph Equation Based on FDTD Method

1Jianhui Song, 2Yanju Liu, 3Yang Yu

*1School of Information Science and Engineering, Shenyang Ligong University, Shenyang, 110159, P. R. China, [email protected]

2School of Information Science and Engineering, Shenyang Ligong University, Shenyang, 110159, P. R. China, [email protected]

3School of Information Science and Engineering, Shenyang Ligong University, Shenyang, 110159, P. R. China, [email protected]

Abstract

In order to investigate the theory of the time domain reflectometry (TDR) cable length measurement, the transmission line transient analysis iteration formula based on the finite difference time domain (FDTD) is derived based on the transmission line theory. The electromagnetic wave propagation theory and the regular pattern of the TDR signal propagating in cables are studied. The theoretical calculations are consistent with the experimental results, which verify the correctness of the transmission line transient analysis iteration formula studied in this paper. To further illustrate the theory of the time domain reflectometry cable length measurement, the iteration formula is applied to the TDR cable length measurement. The impacts of the shape, width and amplitude of the excitation signal on the cable length measurement are analyzed.

Keywords: FDTD, Transmission Line, Telegraph Equation, TDR

1. Introduction

A number of different numerical techniques are used in computational electromagnetic interference problems, each of which has its own strong suit, as well as limitations. Among these, the finite difference method offers an important advantage over some of the other techniques because it is not only flexible as well as general-purpose in its scope of applications, but it is highly parallelizable as well[1]. The finite difference time domain (FDTD) is a very useful numerical simulation technique proposed by K. S. Yee in 1966[2-4], which is a common way to obtain transient responses of transmission lines[5-8]. The derivatives in the telegraph equations are discretized and approximated with various finite differences. In this method the position variable x is discretized as Δx and the time variable t is discretized as Δt[9,10]. FDTD has an advantage of simplicity, high precision and less computation and it is widely used in solving various kinds of electromagnetic problems[11-13]. Based on the transmission line theory, the transmission line transient analysis iteration formula based on the finite difference time domain (FDTD) is derived. The iteration formula is applied to the time domain reflectometry (TDR) cable length measurement.

2. TDR cable length measurement systems

TDR is a very useful measuring technology based on high-speed pulse technology. The cable length measuring principle is very simple. The test voltage pulse is injected into one end of the cable, and the pulse will be reflected at the end of the cable. By measuring the time interval between the injection pulse and the reflection pulse, the cable length can be obtained by assuming the velocity as constant. The formula for length measurement is

2

tvl

(1)

where l denotes the cable length, v denotes the signal propagating velocity, t denotes the time interval between the injection pulse and the reflection pulse.

Numerical Analysis of Transmission Line Telegraph Equation Based on FDTD Method Jianhui Song, Yanju Liu, Yang Yu

Journal of Convergence Information Technology(JCIT) Volume 7, Number 20, Nov 2012 doi : 10.4156/jcit.vol7.issue20.31

258

The TDR cable length measuring system mainly constitutes of signal generator, oscilloscope and computer components. Signal generator emits different pulse amplitudes and widths according to the actual measurement situation. The injection and reflection signal waveform data are collected by the oscilloscope and processed by computer. 3. Iteration formula based on FDTD

Cable is one of the transmission lines. Ideally, cable line (here in after referred to as cable) can be regarded as a uniform transmission line.

The transmission line equation is the starting point to analyze the transient process of the transmission line[14]. Because of the limitations of the existing mathematical tools, it is rather difficult to find the analytical solution of the transmission line equation. This paper uses finite difference time domain method to analyze the propagation characteristics of the TDR signal and the transmission line.

If the signal is injected into the transmission line, the electric field and magnetic field will be around the transmission line. After the time of x/v, the voltage and current at any point Z of the transmission line will change into 0, where the v is the propagation speed of the signal propagating along the transmission line, and the x is the distance from the point Z to the signal transmitter distance. It is obviously that the signal is not a simple spread along the transmission line, but the form of electromagnetic field propagating in the path between the transmission line and the reference surface. Therefore, when analyzing the long-term model and the equivalent circuit of the cable, the transmission line model must be considered as the distributed parameter model, which should not only consider the resistance and inductance of the line, but also consider the conductance and capacitance between the conductors.

Even at the same time, the voltage of the transmission line at different location is different, thus the voltage and current of the transmission line is not only a function of time t, but also a function of spatial coordinates x, the expression should be the binary function of time t and distance x, which can be expressed as

( , )u u x t (2)

( , )i i x t (3)

It is known as the uniform transmission line if the resistance, inductance, capacitance and conductance along the transmission line at any point are equal. The structure and the diameter of the uniform transmission line must be consistent. Based on the distributed parameter circuit theory, the original parameters of the uniform transmission line are represented as per-unit-length line parameters. The equivalent circuit diagram of per-unit-length transmission line is shown as in Figure 1.

Figure 1. Equivalent circuit diagram of per-unit-length transmission line

When the modes have been established a period of time after the excitation in the

transmission line, the forward voltage increasing rate in x-direction is /u x , the current increasing rate is /i x , the beginning of the voltage and the beginning of current are u and i


259

respectively. Therefore, the voltage at point A B is u

u dxx

, and the current at point A is

ii dx

x

. When the transmission line distribution parameters are independent of frequency, the

time domain telegraph equations are

0 0

( , ) ( , )( , ) ( , )F

u x t i x tL R i x t V x t

x t

(4)

0 0

( , ) ( , )( , ) ( , )F

i x t u x tC G u x t I x t

x t

(5)

where 0R , 0L , 0C and 0G are the per-unit-length parameters of transmission lines.

The transmission line is divided into N segments of length x . Similarly, the total solution time is divided into N segments of length t . To provide the accuracy of the discretization, the voltage points and current points are interlaced at alternating direction as 1u , 2u , 3u ,…, 1Nu ,

and 1i , 2i , …, 1Ni . Each voltage and adjacent current solution point is separated by / 2x . In

addition, the time points are also interlaced, and each voltage time point and adjacent current time point are separated by / 2t [15-17]. The spatial-time grid of transmission line is shown in Figure 2.

n

n+1

n-1

i

voltagecurrent

x

t

t

x

i+1i-1

Figure 2. Spatial-time grid of transmission line The Formula (4) and (5) can be expressed as

1/ 2 1/2 1/2 1/2 1/ 2 1/ 21 1/ 2 1/2 1/2 1/2

0 0 2 2

n n n n n n n ni i i i i i Fk Fku u i i i i V V

L Rx t

(6)

1/ 2 1/ 2 1 1 1

1/2 1/ 20 0 2 2

n n n n n n n ni i i i i i Fk Fki i u u u u I I

C Gx t

(7)

The transmission line transient analysis iteration formulas based on FDTD are simplified as

1/ 2 1/2 1/ 2 1/20 00 1/2 0 1/2 1( ) ( ) ( ) ( )

2 2 2n n n n n ni i i i Fk Fk

R Rx x xL x i L x i u u V V

t t

(8)

1 1/2 1/2 10 00 0 1/ 2 1/2( ) ( ) ( ) ( )

2 2 2n n n n n ni i i i Fk Fk

G Gx x xC x u C x u i i I I

t t

(9)

The initial conditions are


260

0

0

( )

( )x

x

i x

xu

(10)

4. The FDTD convergence and stability analysis

As the FDTD method is based on a set of finite difference equations instead of the solution of the original electromagnetic field partial differential equations, the solution of differential equations must be convergence and stability.

By considering the harmonic field situation, the Maxwell curl equations can be written as

0( , , , ) exp( )f x y z t f j t (11)

The steady state solution is

fj f

t

(12)

As the first derivative of the left end of the above equation is replaced by the difference

approximation, the above equation becomes

1 1

2 2n n

nf fj f

t

(13)

( , , , )nf f x y z n t (14)

where, t is the time interval. As the t is small enough, the value growth factor is defined as

1

2

1

2

n n

nn

f f

ff

(15)

So,

2 1 0j t (16)

The solution of the above formula is

21 ( )2 2

j t t (17)

For cable systems, the propagation of the electromagnetic waves is considered as one-dimensional.

Therefore, 2 2

2 20

f f

x v

(18)

Considering the solutions of the plane wave, that is

0( , ) exp[ ( )]xf x t f j k x t (19)


261

Using the finite difference approximation, the second derivative approximation is

2

2 2

( ) 2 ( ) ( )

( )

f f x x f x f x x

x x

(20)

By Taking Formula (18) into Formula (17), the follow expression can be obtained

22

2 22

sin ( )exp( ) 2 exp( ) 2( ) ( )

2

x

x x

k xjk x jk xf

f fxx x

(21)

Therefore, the discrete solution of Formula (17) is

22

22

sin ( )2 0

( )2

xk x

x v

(22)

After transformation, Formula (21) can be expressed as

2

2 2

2

sin ( )2( ) ( ) 1

2 2( )2

xk xv t t

x

(23)

The sufficient conditions for any xk is

1v t

x

(24)

Therefore, the time step must be small enough so that it satisfies the following courant stability

condition

/t x v (25)

5. Experimental results and analysis

The measurements were performed on RVV 300/300V PVC insulated cable. The parameters of the cable under test are as follow: cable length 86.751l m, 0 1.0154L μH/s, 0 24.982C pF/m. The

excitation source is pulse signal and ideal Gaussian signal. The actual measured waveform and the FDTD simulation waveform of different incident wave are shown in Figure 3. It can be seen from Figure 3 that the FDTD simulation waveforms are consistent with the actual measured waveform, which verifies the correctness of the transmission line transient analysis iteration formula studied in this paper.


262

Sig

nal a

mpl

itud

e(V

)

Figure 3. Actual measured waveform and the FDTD simulation waveform of different incident wave

Compared with the traditional measurement methods, TDR technology has an advantage of non-

destruction, portability and high-precision, which is an ideal cable length measurement method. TDR is a very useful measuring technology based on high-speed pulse technology. The cable length measuring principle is very simple. The test voltage pulse is injected into one end of the cable, and the pulse will be reflected at the end of the cable. By measuring the time interval between the injection pulse and the reflection pulse, the cable length can be obtained by assuming the velocity as constant.

To further illustrate the theory of the time domain reflectometry cable length measurement, the iteration formula is applied to the TDR cable length measurement. From Figure 2 it can be seen that ideally, the time interval between the injection wave and the reflected wave is the same no mater what the shape of the excitation signal is launched into the cable under test. That is the excitation signal shape does not affect the TDR cable length measurement results. When the excitation source is the pulse signal, the FDTD simulation waveforms are slightly different with the actual measured waveform. The rising edge of the actual pulse signal is close to the Gaussian waveform and the waveform becomes asymmetric with the fast rising and slow falling edge because of the attenuation and dispersion.

Figure 4 is the actual measured waveform and the FDTD simulation waveform of different pulse width. It can be seen from the picture that ideally, the width of the pulse signal does not affect the cable length measurement results. As the pulse width is wider, it carries the greater energy. Therefore, it can measure longer cable length. On the other hand, the pulse width determines the minimum cable length that can be measured. The solution to this problem is to choose a different pulse width for different measurement cable length.

-2 0 2 4 6 8 10 12x 10-7

-1

0

1

2

3

4

5

Time ( s )

Sig

nal a

mpl

itude

(V)

Actual measured waveformPulse width : 125nsPulse width : 250ns

Figure 4. Actual measured waveform and the FDTD simulation waveform of different pulse width


263

Figure 5 is the actual measured waveform and the FDTD simulation waveform of different pulse amplitude. It can be seen from the figure that ideally, the amplitude of the pulse signal does not affect the cable length measurement results. However, too small amplitude can not work in a noisy environment. On the other hand, too large amplitude may cause electromagnetic interference (EMI).

Figure 5. Actual measured waveform and the FDTD simulation waveform of different pulse amplitude 6. Conclusions

In order to study the principle of the TDR cable length measurement, the transmission line transient analysis iteration formula based on the finite difference time domain FDTD is derived. The iteration formula is applied to the TDR cable length measurement. The impacts of the shape, width and amplitude of the excitation signal on the cable length measurement are analyzed. The actual measured and simulation results show that ideally, the shape, width and amplitude of the excitation signal do not affect the cable length measurement results. However the actual measured waveforms are slightly different due to the combined effect of the cable attenuation and dispersion.

7. Acknowledgement

The study is funded by the 2011 Education Department of Liaoning Province (Project No.:L2011036)

8. References [1] Wenhua Yu, Raj Mittra, Xiaoling Yang, et al., “High-Performance Conformal FDTD Techniques”,

IEEE Microwave Magazine, vol. 85, no. 2, pp. 42-45, 2010. [2] T. Sekine, K. Kobayashi, S. Yokokawa, “Transient Analysis of Lossy Nonuniform Transmission

Line Using the Finite Difference Time Domain Method”, Electronics and Communications in Japan, vol. 85, no. 2, pp. 1062-1070, 2002.

[3] Lei Qi, Tiebing Lu, Xiang Cui, “Transient Analysis for Nonuniform Multiconductor Transmission Lines Based on the FDTD Method”, Proceedings of the CSEE, vol. 23, no. 7, pp. 102-106, 2003.

[4] Hiromichi Kobashi, Shigeo Kawata, Yasuhiko Manabe, et al., “PSE Park: Framework for Problem Solving Environments”, JCIT: Journal of Convergence Information Technology, vol. 5, no. 4, pp. 225~239, 2010.

[5] Wei Shao, Jialin Li, “An Efficient Laguerre-FDTD Algorithm for Exact Parameter Extraction of Lossy Transmission Lines”, Applied Computational Electromagnetics Society Journal, vol. 27, no. 3, pp. 223-228, 2012.

[6] M. Benkraouda, H. Ghamlouche, M. I. Hussein, et al., “Numerical Modeling of High-Tc


264

Superconducting Microstrip Line Using FDTD Technique”, International Journal of Modelling and Simulation, vol. 3, no, 1, pp. 67-72, 2010.

[7] Watanabe Yuta, Igarashi, Hajime, “Accelerated FDTD Analysis of Antennas Loaded by Electric Circuits”, IEEE Transactions on Antennas and Propagation, vol. 60, no. 2, pp. 958-963, 2012

[8] Cavka Damir, Harrat Basma, Poljak Dragan, et al., “Wire Antenna Versus Modified Transmission Line Approach to the Transient Analysis of Grounding Grid”, Engineering Analysis with Boundary Elements, vol. 35, no. 10, pp. 1101-1108, 2011.

[9] M. Tang, J. F. Mao, “A Precise Time-Step Integration Method for Transient Analysis of Lossy Nonuniform Transmission Lines”, IEEE Transactions on Electromagnetic Compatibility, vol. 50, no. 1, pp. 166-174, 2008.

[10] Lingfeng Shi, Kai Lin, Bing Yuan, et al., “Transient Analysis of Unequal Length Nonuniform Lossy Multiconductor Transmission Lines Based on FDTD”, Dianbo Kexue Xuebao, vol. 26, no. 4, pp. 666-671, 2011.

[11] Wei Tang, Peinan Jiao, Qingliang Li, et al., “Application ADI-FDTD in one Dimension PBG Structure”, Chinese Journal of Radio Science, vol. 18, no. 3, pp. 281-285, 2003.

[12] Yushun Guo, “An Efficient Transient Analysis Method for General Transmission Lines Based on the Time-Space Discretization of Telegrapher Equations”, Acta Electronica Sinica, vol. 29, no. 3, pp. 373-377, 2001.

[13] Bo Yang, Bihua Zhou, Cheng Gao, et al., “Using A Two-Step Finite-Difference Time-Domain Method to Analyze Lightning-Induced Voltages on Transmission Lines”, IEEE Transactions on Electromagnetic Compatibility, vol. 53, no. 1, pp. 256-260, 2011

[14] ZhenChong Wang, Yayu Chen. “Hazardous Waste Landfill Leakage Detection Based on Transmission lines Model”, AISS: Advances in Information Sciences and Service Sciences, Vol. 3, no. 9, pp. 17~24, 2011.

[15] Antonio Orlandi, R. Clayton, “FDTD Analysis of Lossy, Multiconductor Transmission Lines Terminated in Arbitrary Loads”, IEEE Transactions on electromagnetic compatibility, vol. 38, no. 3, pp. 388-399, 1996.

[16] Afrooz Kambiz, Abdipour Abdolali, “Efficient Method for Time-Domain Analysis of Lossy Nonuniform Multiconductor Transmission Line Driven by A Modulated Signal Using FDTD Technique”, IEEE Transactions on Electromagnetic Compatibility, vol. 54, no. 2, pp. 482-494, 2012.

[17] Jianhui Song, Feng Yuan, Zhenling Ding, et al., “Transient Analysis of Transmission Lines Based Unconditionally Stable FDTD Method”, Jilin Daxue Xuebao (Gongxueban), vol. 40, no. 5, pp. 1438-1441, 2010.


265

Documents

Numerical Analysis of Transmission Line Telegraph Equation ... · transmission line, the forward voltage increasing rate in x-direction is ux/ , the current increasing rate is ix