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Compliance Engineering Journal ISSN NO: 0898-3577
Volume 10, Issue 12, 2019 Page No: 301
Double Laplace Transform Approach to the Electric Transmission Line with trivial
Leakages through electrical insulation to the Ground
1*Rohit Gupta, 2Dinesh Verma, 3Amit Pal Singh
Lecturer, Associate Professor, Assistant Professor
1, 2 Yogananda College of Engineering and Technology, Jammu
3Jagdish Saran Hindu (PG) College, Amroha U.P.
Abstract
Generally, the general equations of electric transmission line are analyzed by the Fourier
transform, Laplace Transform and Method of Variation of Parameter. These methods are very
useful for analyzing ordinary and partial differential equations. Among these transform
approaches, the double Laplace transform approach has been applied to analyze boundary value
problems arising in different areas of engineering and science like save equation, one
dimensional heat flow equation, Laplace equation, Harmonic Vibration of a beam supported at
its two ends. This paper deals with the analysis of the general equations of electric transmission
line with trivial leakages through electrical insulation to ground by double Laplace Transform
approach. This approach will come out to be very effective mathematical tool applied to analyze
general equations of electric transmission line.
Index terms: Double Laplace transform approach, Electric Transmission Line.
1. Introduction
The term electric transmission line signifies a set of wires made of good electrical conductors
like copper or aluminum provided with excellent electrical insulation, and used for transmission
of electrical energy. In general, an electric transmission line has a resistance R contributed by the
two wires taken together, an inductance L, a capacitance C and shunt conductance G. These four
quantities form the primary parameters of the electric transmission line and their values depend
on the type and construction of electric transmission line [1], [2]. In this paper, we will analyze
the general equations of electric transmission line with trivial leakages through electrical
insulation to ground by double Laplace Transform approach. For trivial leakages to ground on an
electric transmission line, the parameters like the conductance and the inductance are set to zero
[3], [4] in the general equations of electric transmission line to get the mathematical model which
will be analyzed, in this paper, by using the double Laplace transform approach.
2. Material and Method
Considering a semi-infinite electric transmission line with a constant voltage applied at its
sending end (y = 0) at t = 0. If are the voltage and the current at any point y
and at any instant t, then the equations describing the evolution of current and voltage on a lossy
electric transmission line [1], [2] are given by
…………….. (1)
……………… (2)
Differentiating equation (1) w.r.t. y and equation (2) w.r.t. t and simplifying the result, we have
……….. (3)
* Corresponding author: guptarohit565@gmail.com,
Compliance Engineering Journal ISSN NO: 0898-3577
Volume 10, Issue 12, 2019 Page No: 302
Differentiating equation (1) w.r.t. t and equation (2) w.r.t. y and simplifying the result, we have
………….. (4)
These equations represent the general wave equations for a lossy electrical transmission line [1],
[2].
For trivial leakages to ground on an electric transmission line, we put the conductance, G =0 and
the inductance, L = 0, because these parameters are responsible for leakages on the electric
transmission line [3], [4]. Therefore, we can rewrite equations (3) and (4) as follows:
……… (5)
………. (6)
The boundary conditions for the equations (5) and (6) are as follows:
We will solve equations (5) and (6) by Double Laplace transform approach.
3. Basic definitions:
The double Laplace Transform of g(y, t), a function two variables y > 0 and t > 0 is defined [5],
[6] as follows
The double Laplace transform of the first order partial derivatives is defined as follows:
The double Laplace transform of the second order partial derivatives is defined as follows:
4. Solution of electric transmission line equations:
Partially differentiate equation (5) w.r.t. y and using equation (6). We get
……… (7)
Taking double Laplace transforms of equation (7), we get
Compliance Engineering Journal ISSN NO: 0898-3577
Volume 10, Issue 12, 2019 Page No: 303
)] …….. (8)
As
therefore, equation
(8) gives
]
Rearranging the equation, we get
…………. (9)
Taking double inverse Laplace transform [7] of equation (9) w.r.t t and s, we get
On applying inverse Laplace transform [8], [9] w. r. t. r, we get
Or
Or
………. (10)
Since is finite as y , therefore, on putting
, in the equation (10) and simplifying, we get
Again on applying inverse Laplace transform [8], [9] w. r. t. s, we get
Or
Compliance Engineering Journal ISSN NO: 0898-3577
Volume 10, Issue 12, 2019 Page No: 304
……… (11)
Or
……… (12)
From equation (6), we have
… (13)
Using equation (12) in equation (13), we can write
Or
On simplifying the above equation, we get
……….. (14)
The equations (11) or (12) and (14) provide the solutions of general equations of electric
transmission line with trivial leakages through electrical insulation to the ground.
5. Conclusion
In this paper, we have successfully applied Double Laplace Transform approach for analyzing
the general equations of electrical transmission line with trivial leakages through electrical
insulation to the ground. The approach is an effective tool to analyze the general equations of
electrical transmission line and with its ease of application in different areas of engineering and
science, the other boundary value problems can also be analyzed easily.
References
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Journal of Engineering Science and Researches, Issue February, 2019, pp. 96-101. DOI- 10.5281/zenodo.2565939.
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