Finite Difference Method...dr. Hatem Mokhtari

Finite difference method and Pspice simulation applied to the coaxial cable in a linear temperature gradient H. Mokhtari A. Nyeck C. Tosser-Roussey A. Tosser-Roussey

Indexing terms. Simulation, Coaxial components

Abstract: An application of the finite difference method and a Pspice simulation for the calculation of the attenuation and phase in a coaxial cable submerged in a linear temperature gradient is proposed. The two methods thus used are then compared for various cable lengths. Also, both accuracy and run time for various cable lengths using the two described methods are compared to optimise the calculations and even choose which of them would be handled easier for further similar applications.

1

Introduction

The well known propagation equation of electromagnetic waves [l] in a coaxial cable for uniform temperature distribution is quite easy to solve, but the depth-dependent temperature is an open problem because the differential equations are more complicated. A typical application medium is the natural earth linear temperature gradient where the cable can be submerged for seismic activity studies or oil detection. The solution proposed in this paper concerns the finite difference method (FDM) applied to a coaxial cable in a linear temperature gradient. We have implemented a Pspice simulation [ 2 ] to compare both precision and efficiency of the two calculation methods. The value of the characteristic impedance is quite difficult to define because the analytical model requires numerical solution techniques taking into account the input and the output boundary conditions [3]. Since Pspice handles complicated matrix equations, we have studied a simple and versatile method, namely FDM, based on the subdivision of the cable into elementary quadripoles where, locally, the temperature is assumed to be constant but increases proportionally with the distance. 2

Theory

Starting from the usual propagation equation applied to a coaxial cable [l], we try to generalise the equations by superimposing the fact that the whole of the studied cable Paper 8439A (S8), first received 25th March and in revised form 8th July 1991 The authors are with Laboratoire de Mkcatronique Industrielle ENIM, Ile de Saulcy, 57045 Metz, France IEE PROCEEDINGS-A, Vol. 139, N o . I ,J A N U A R Y 1992

lies in a medium whose temperature increases linearly against the depth variable, say X . Thus, the leading equations can be written as follows:

dx

=

-(C + j C o ) V ( x )

It is assumed that the resistance can be written as R ( x ) = RO + Ax. However, the remaining distributed parameters are assumed to be constant only because their physical properties do not markedly vary with temperature. Let us now replace eqns. 1 and 2 by their corresponding finite difference equations [4]. Hence, the new equations are given by

V,,, - V, = - Ax(R, + j L o ) I , I,,, - I , = - Ax(G + jCw)V, Where,

R, = RO + AnAx V , = V(nAx) and I,

n

=

E

I(nAx)

[0, N - I]

According to Fig. 1, the boundary condition at the output (i.e. at x = L , L : cable length) is required for computing the voltage attenuation at a given frequency. Thus, we write the inverse of the attenuation in the form given beow (4)

The load admittance is given by the boundary condition GL =

1, v,

By substituting eqn. 5 in eqns. la and 2a, with n = N - 1, yields VN -= VN-,

1

+ LCw2Ax2 - j R N - l C o A x 2 + R N - , G,Ax + j L o C L A x

1

39

bearing in mind that G, the transverse conductance of the cable dielectric medium, is assumed to be negligible in most cases. o

V,"

studied case). A PC has been used for numerical evaluation of FDM equations; one megabyte is required for the computation. The FDM and Pspice computed variations in the attenuations and phase with the frequency are given in Fig. 2 for a series of cable lengths. From a

6or

-

P

50

'load

% 40

b

630 0

20 c

"10 I

I

I

I

I

l

l

I

I

Ax

2Ax

I

nAx (n+l)Ax (N-lIAx NAx

5

Fig. 2

Equivalent circuit for coaxial cable subdivision, definition 01 Fig. 1 elementary subcircuits for Pspice simulations andfinite-difference method

-@-

-0-A-A-

According to eqn. 4 the phase delay between the input and the output of the voltage is given by the following relation:

a)= k1% =l

(7)

where the general term of this finite summation, ch, is derived from eqn. 6

+ R k - , G , A x ) R k - , CoAx' +(1 + L C W ~ A ~ ~ ) L C W G , A X 1 + L C O ~ A X+~R,- G , Ax

]

(1

[

-tg-l

tg-l(u)

-

tg-'(u) = t g - 1

{lx}

(8)

(9)

where U and U are real variables. Results

Obviously, it is quite difficult to create a linear temperature gradient having the same temperature coefficient as the studied case (earth natural linear temperature gradient) and the use of measured data is thus impossible. The values of the coaxial cable parameters at room temperature, usually defined as R , L, C , G, are RO = 468e - 4 Ohm/m, C = 12e - 11F/m, L = 5.6e - 6H/m, G = 0 (Neglected in general but tan be added, for highloss transmission lines). These values are taken as an example and can be changed for any other special cable types. The distance-dependent temperature coefficient is taken to be 0.03"C/m. The value for this distancedependant temperature coefficient represents the variation coefficient for the earth natural temperature gradient introduced as an example of typical application. Both internal and external conductors are built from copper and the resistance-dependant temperature coefficient then takes the usual value of 1/240"C. For the sake of comparison the load impedance is taken to be equal to 50 Ohms (purely resistive in this 40

-=-0-

Attenuation against frequency for L

= 500,1000 and

1500 m

500mFDM 500 m Pspice

IOOOmFDM loo0 m Pspice 1500mFDM I500 m Pspice

rough physical point of view, the extended modelising cell must be much smaller than the wavelength in the range of frequencies of the experiment. The elementary cell length is 1 m and the shortest wavelength (i.e. for f= 100 kHz) is lambda = 3 km, which effectively satisfies the above condition. The FDM obtained results are in good agreement with the Pspice simulation. No marked deviation is observed except for lower values of cable lengths (i.e. for L = 500 m) where a more refined subdivision would be required especially for the highest frequencies because the wavelength decreases and consequently the above mentioned approximation for the discretised model slightly fails. In Fig. 2, the agreement betwene the Pspice simulation and the FDM results for L = lo00 m seem better than for L = 1500 m. It is an unexpected result but the differences between the two models are quite negligible. The differences, for the attenuations, between Pspice and the FDM results are for L = 1000 m, AAtt (Average) = 0.5 dB and 0.6 dB for L = 1500 m which are approximately equal. Even for higher frequencies, these differences do not markedly vary except for the shortest cables. As has been previously pointed out, the subdivision must be established according to both freqency of the propagated signal and the cable length. Where L = 500 m, taking Ax = 0.5 m for example, should provide a more representative attenuation and phase and consequently less discrepancies, especially for higher frequencies, with the Pspice simulation model. Furthermore, it is important to note that attenuation derives from a finite product eqn. 4 and the phase derives from a finite summation (eqn. 7) which would affect the precision of the computations especially for the longest cables for the attenuation, and for the shortest cables for the phase as is clearly seen in Figs. 2 and 3. The well known formulation of the following characteristic impedance: ~

As attempted, a negative value of the phase delay between the input and the output signals is obtained. The following mathematical identity [ 5 ] has been used for a convenient calculation of eqn. 8:

3

1

frequency, kHz

,

0

ch, =

1

100

IO

z,=

+

R(t"C) jLw G jCo

/[

+

1

(10)

cannot be conveniently applied for variations in temperature. However, for a uniformly distributed temIEE PROCEEDINGS-A, Vol. 139, N o . I , J A N U A R Y 1992

perature the variations in the characteristic impedance for a series of temperature can be obtained (Fig. 4). The run time computation ratio is given in Table 1. The results are in good agreement when the cable is suffi-

the phase does not vary when the cx2 term is added to the resistance function. The run time computation ratio, in this quadratic function case, is given in Table 2. This clearly shows that the FDM procedure is faster than the general Pspice for solving this propagation problem. 50r

100

10

frequency, k H z

Phase againstfrequency for L Fig. 3 -0SMmFDM -0- 5M m Pspice -A1000mkDM --AIMO m Pspice 1500mFDM -01500 m Pspiu

=

201 10

500, 1000 and 1500 m

-.-

I

I

1

I

I

1

1

1

1

100

frequency, kHz

Fig. 5 Aftenuation against frequency for a bx + cx’ -0- loo0 m Pspice -01OOOmFDM

+

L

=

IO00 m, R(x) =

110,

-500

100

10

60 10

frequency, k H z

100 frequency, k H z

Fig. 4 Characteristic impedance against frequency for uniformly varying temperature

Phase againstfrequency for L = 1000 m, R(x) = a Fig. 6 -0- l m m Pspice -HIOOOmFDM

+ bx + ex2

Table 2: Variations in the reduced calculation time with cable lenath Table 1 : Variations in the reduced calculation time with cable lenath -

~

Cable length f(Pspice)/f(FDM) L=500m L = 1000 m L=1500m

1322 1.541 1.659

ciently long to assume the discretising elementary cell representative. Fortunately, the linear temperature gradient increases the resistance slightly; the situation would be quite difficult for strong variation cases because the FDM model or the Pspice simulation program would require more subdivisions and consequently more run times. As a second example, a quadratic variation for the resistance with the distance (i.e. the resistance is assumed to vary as R ( x ) = a + bx + cx’, where c = 2e - 7 Q/m2) has been studied. The results are given in Figs. 5 and 6. The coefficients a, b are retained from the previous linear temperature variation case, however c is taken to be a weak increasing coefficient. As shown in Figs. 5 and 6, the new function R ( x ) = a bx + cx2 affects the attenuation as expected. However, the phase is not varying markedly from the linear temperature gradient case. The likely reason for this result is that the general expression term in eqn. 8 for

+

I E E PROCEEDINGS-A, Vol. 139, N o . I , J A N U A R Y 1992

Cable length f(Pspice)/f(FDM) L=1000m

4

1.755

Conclusion

The FDM procedure applied to the modelisation of the electromagnetic wave propagation is a fast alternative procedure of the Pspice simulation for the calculation of the attenuation and phase along the coaxial cable in either the studied case of a linearly temperature dependent resistance or quadratic function. This method could be extended to other variations in the resistance against the depth variable but the convergence must be reexamined in this case. The FDM procedure can be easily implemented with the aid of a PC and is faster than the Pspice simulation. 5

References

1 2 3 4

KRAUS, J.D.:‘Electromagnetics’ (McGraw-Hill, 1973) Pspice technical reference. MicroSim corporation, 1989 ANGOT, A.: ‘Compliments de mathematiques’ (Masson, 1982) HAMMING, R.W.: ‘Numerical methods for scientists and engineers’ (McGraw-Hill, 1962) 5 GRADSHTEYN, I.S., and RYZHIK, I.M.: ’Table of integrals, series, and products’ (Academic Press, 1980) 41