Solved Examples on Telegrapher’s Equation

Here, we will understand Telegrapher’s Equation by using some examples with explanation.

Example 1

In designing a high-speed data communication network, Engineers are compelled to select the appropriate coaxial cable for transmitting signals between network devices. Amongst the instructions is that they need to ensure that the selected cable would have the correct characteristic impedance to minimize signal distortion, loss and most appropriately ensure efficient data transmission. Assuming that the standard impedance used in the communication systems is 65Ω, investigate the best and more appropriate cable.

Cable A: Per-unit-length inductance (L) = 0.15 mH/m, Per-unit-length capacitance (C) = 60 nF/m

Cable B: Per-unit-length inductance (L) = 0.2 mH/m, Per-unit-length capacitance (C) = 50 nF/m

To determine the best coaxial cable between Cable A and B, based on Characteristic impedance, we use the Telegrapher’s equation:

Analysis for cable A

For the conditions to be met – minimizing signal distortion, loss and most appropriately ensuring efficient data transmission – we shall approach it as a lossless transmission line.

Hence, we shall apply equ (18):

Z_ο = (L / C)^1/2

Z_οA = ((0.15 mH/m) /(60 nF/m) )^1/2 = ((0.15 x 10 ^-3/m) /(60 x 10 ^-9/m) )^1/2
Z_οA = (2500)^1/2= 50Ω

Analysis for cable B

Similarly,

Z_ο = (L / C)^1/2

Z_ο = ((0.2 mH/m) /(50 nF/m) )^1/2 = ((0.2 x 10 ^-3/m) /(50 x 10 ^-9/m) )^1/2

Z_οB = (4000)^1/2= 63.25Ω

From the analysis of cable A and B,

Since the characteristic impedance of Cable B, Z _0B = 63.25Ω is closer to the standard impedance used in the communication systems typically for the above question 65Ω, the Engineers should choose Cable B in order to better meet the requirements – minimizing signal distortion, loss and most appropriately ensuring efficient data transmission, which simply means better impedance matching.

Example 2

As a system Engineer, upon designing a system for a satellite network, you are told to select an appropriately better transmission line to ensure efficient and flawless signal transmission. In doing so, you must calculate these parameters: the propagation constant, attenuation constant, and phase constant for the selected transmission line. Given parameters are:

Resistance (R) = 0.5 ohms per meter

Inductance (L) = 0.2 microhenries per meter

Capacitance (C) = 100 picofarads per meter

Conductance (G) = 0.1 ohms per meter

Operating frequency = 1 GHz

Recalling the propagation constant from equ (7):

Ɣ ² = (R+jωL)(G+jωC)

where, ω = 2πf and from given parameter, f = 1GHz = 1 × 10⁹Hz

ω = 2π × 1 x 10⁹ rad/s = 2π × 10⁹rad/s

Ɣ ² = (0.5 + ȷ400π)(0.1 + ȷ0.2π)

Ɣ ² = 0.5(0.1) + 0.5(ȷ0.2π) + ȷ400π(0.1) + ȷ400π(ȷ0.2π)

where, ȷ × ȷ = – 1

Ɣ ² = −789.52 + ȷ125.98

Employing “DeMoivre’s theorem” we can find the value of the propagation constant, Ɣ :

Z^1/n = |Z|^1/n< θ ̸n

Ɣ ² = −789.52 + ȷ125.98 = 799.51 < 170.93^ο

Ɣ = (799.51 )^1/2< (170.93)/2

Ɣ = 28.28 < 85.47 ^ο = 2.23 + ȷ28.19

Recall, Ɣ = α + ȷβ, and by comparing, α = 2.23 Nepers/metre, and β = 28.19 rad/m.

Thus:

Propagation constant, Ɣ = 2.23 + ȷ28.19

Attenuation constant, α = 2.23 Nepers/metre,

Phase constant, β = 28.19 rad/m.

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.