Derivation of Telegrapher’s Equation

To derive the Telegrapher’s equation with lucid understanding and step, let’s begin with the fundamental assumptions of transmission line:

Basic Assumptions

• Considering only a small segment of transmission line of length labeled Δ(x).
• Applying Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) at the chosen segment.

Analyzing Voltage

Applying KVL to the small segment, accounting for the voltage drop across resistive and inductive elements:

ΔV= R Δ(x) I + jωLΔ(x)I ……. (equ (1)

Here, R is resistance per unit length, L is inductance per unit length, ω is the angular frequency; ΔV is the change in voltage; I is the current flowing through the segment.

Analyzing the Current

Applying KCL to the small segment, accounting for the variation in current due to conductive and capacitive effects, we have:

ΔI= GΔ(x)V + (jωCΔ(x))⋅V ……. (equ (2)

Here, G is conductance per unit length; C is capacitance per unit length; V is the voltage applied, and ΔI is the change in current.

Differentiation

Differentiating with respect to X, and taking the limit as Δ(x) approaches zero(0), equ (1) and equ (2) become:

dV/dx = − RI − jωLI = – (RI + jwLI) = – (R+ jwL) I ……. equ (3)

dI/dx = − GV − jωCV = – (GV + jwCV) = – (G + jwC) V……. equ (4)

Since, the propagation of these parameter is defined in the form of wave, hence, and it is generally known that the wave equation is typically expressed, analyzed and explained using second order differential equation. As such, we shall take another derivative of equ (3).

Finding the second derivative of the voltage expression and substituting the current expression:

Taking the second derivative of both sides – voltage and current – of equ(3) we have: d² v/dx²= -( R + jωL) dI/dx …….. equ (5)

but recall that, dI/dx = − (GV + jωCV );

thus, substituting it into equ (5), we have:

d² v/dx²= – ( R + jωLI) . – (GV + jωCV ) = (R+jωL)(G+jωC)V ……. equ (6)

Introduction of Propagation Constant

Defining the propagation constant as Ɣ

Ɣ ² =(R+jωL)(G+jωC) ……. equ (7)

(Where, Ɣ = α + ȷβ)

where α is Attenuation constant, measured in Nepers per meter; and, β is Phase constant, measured in radian per meter.

Finally, rewriting the equation (6) expression using the propagation constant:

d² v/dx² = Ɣ²V ……. equ (8)

similarly, for current also, we have;

d² I/dx² = Ɣ²I ……. equ (9)

Characteristics Impedance(Z_ο)

In Transmission line, Characteristic impedance (Z_ο) is a fundamental property to denote the impedance the line appears to have when the transmission line is infinitely long. It is mathematically defined as the ratio of voltage to current in a travelling wave along the line.

To derive the characteristic impedance (Z_ο) from the Telegrapher’s equation, let’s start with the equation itself:

Recall from equ (8) and (9), we had:

d² v/dx² = Ɣ²V , and similarly, for current, d² I/dx² = Ɣ²I

where Ɣ² = (R+jωL)(G+jωC) is the propagation constant.

Having solved equ (8) and (9), that is the solution to the second order differentiate equation, we have:

For voltage: V(x)= V ⁺e^−γx + V ^–e^γx …… equ (10)

And current: I(x)= I⁺e^−γx + I ^–e^γx …… equ (11)

where V ⁺e^−γx and I⁺e^−γx describe the forward travelling wave, and V ^–e^γx and I ^–e^γx describe the backward travelling wave: (V⁺, I⁺) and (V ⁻, I ⁻ ) are the differential constants.

Working with equ(3),

dV/dx = – (R+ jωL) I ,

but recall that in equ (10) and (11) , we have the equation for voltage and current respectively, thus, substituting into above equ (3), we have:

Only considering the forward traveling wave:

d/dx (V ⁺e^−γx ) = – (R+ jωL) (I⁺e^−γx ) …… equ (12)

Differentiating the left hand side with respect to x, we have:

− γV ⁺ e ^−γx = – (R+ jωL) (I ⁺e ^−γx ) …… equ (13)

simplifying equ (13), we have:

V ⁺ / I ⁺= (R+ jωL)/ γ …… equ (14)

V ⁺ / I ⁺= (R+ jωL)/ ((R+jωL)(G+jωC))^1/2 (where Ɣ= ((R+jωL)(G+jωC))^1/2)

V ⁺ / I ⁺= (R+ jωL)^1/2/ (G+jωC)^1/2 …… equ (15)

The above equation is called the Characteristic Impedance equation, also mathematically called the ratio of voltage to current – either forward or backward traveling defines it. However, for forward traveling wave, the Characteristic Impedance (Z_ο) is positive, and backward, it’s negative value.

Thus, for forward, Z_ο = (R+ jωL)^1/2/ (G+jωC)^1/2 …… equ (16)

And, for backward, following similar approach, from equ(12) to equ(15), we will arrive at:

V ^– / I ^– = Z_ο = – (R+ jwL)^1/2/ (G+jωC)^1/2 …… equ (17)

Both equation (16) and (17) are called Characteristic Equation (Z_ο).

Note:

For lossless transmission line, R = G = 0, and the Characteristics Equation becomes :

Z_ο = (L / C)^1/2 …… equ (18), and for backward, Z_ο = – (L / C)^1/2 …… equ (19)

In the world of wireless Communication, the Telegraph Equation is a vastly talked about concept in the study of transmission lines, particularly in Electrical engineering and telecommunications. Much specifically, for high-frequency transmission lines, It precisely describes the propagation of electrical signals along transmission lines, such as wires, cables, or waveguides, to mention a few.

Many equations are important in the world of Telecommunication, but, the one developed in August 1876 by Oliver Heaviside, the Telegrapher equation provides the mathematical framework for understanding how voltage and current propagate along transmission lines. It entirely analyzes the characteristics of parameters such as resistance, inductance, conductance, and capacitance, while it hides the complexity between these electrical properties. It also helps to predict signal transmission characteristics and to address challenges in signal integrity and attenuation.

In this article, we are delving into the concepts of the Telegraph Equation. We will start with an explanation of its fundamental principles, extending to its derivation and practical applications. We will also examine the characteristics of impedance associated with transmission lines, while we explore solved examples to illustrate the application of the equation even in practical scenes. In addition, we shall discuss both the advantages and disadvantages of using the Telegrapher equation in engineering practice.