# What are Transmission Lines : Types, Equation and Applications

Transmission lines grew out of the work of James Clerk Maxwell (13 June 1831 – 5 Nov 1879) was a Scottish scientist, Lord Kelvin (26 June 1824 – 17 Dec 1907) and Oliver Heaviside was born on 18 May 1850 and died on 3 Feb 1925. In North America first transmission line is operated at 4000V in 1889 June-3. Some of the power transmission and distribution companies in India are NTPC in New Delhi, Tata Power in Mumbai, NLC India in China, Orient Green in Chennai, Neuron Towers or Sujana Towers Ltd in Hyderabad, Aster Transmission line construction, L.J.Technologies in cherlapalli, Mpower Infratech private limited in Hyderabad.

## What are Transmission Lines?

The transmission lines are part of the system that gets electricity from the power stations to homes and it is made up of aluminum because it is more abundant, cheaper and less dense than copper. It carries electromagnetic energy from one point to another point and it consists of two conductors that are used to transmit electromagnetic waves over a long distance between transmitter and receiver are called transmission lines. There are both AC (Alternating Current) and DC (Direct Current) transmission lines. The AC transmission lines are used to transmits alternating current over a long distance using three conductors and the DC transmission lines are uses two conductors to transmit direct current over a long distance.

### Transmission Line Equation

Let us take the equivalent circuit of the transmission line, for this we are going to take the simplest form of transmission line which is two wirelines. This two wireline is made up of two conductors separated by a dielectric medium usually air medium, which is shown in the below figure

If we pass a current (I) through the conductor-1, will find that there is a magnetic field around the current-carrying wire of a conductor-1 and the magnetic field can be illustrated using series inductor due to the current flow in the conductor-1, there should be a voltage drop across the conductor-1, which can be illustrated by a series of resistance and inductor. The setup of the two wireline conductor can be made to a capacitor. The capacitor in the figure will always be loosy to illustrate that we have added conductor G. The total setup i.e, series resistance an inductor, parallel capacitor, and conductor make up an equivalent circuit of a transmission line.

The inductor and resistance put together in the above figure can be called as series impedance, which is expressed as

**Z = R+jωL**

The parallel combination of capacitance and conductor n the above figure can be expressed as

**Y = G+jωc**

Where l – length

I_{s} – Sending end current

V_{s} – Sending end voltage

dx – element length

x – a distance of dx from sending end

At a point, ‘p’ take current(I) and voltage(v) and at a point, ‘Q’ take I+dV and V+dV

The change in voltage for the length PQ is the

**V-(V + dV) = (R + jωL) dx * I**

**V-V-dv = (R + jωL) dx * I**

**-dv/dx = (R + jωL) * I ………………. eq(1)**

**I-(I + dI) = (G + jωc)dx * V**

**I – I+dI = (G + jωc)dx * V**

**-dI/dx = (G + jωc) * V … ……………. eq(2)**

Differentiating eq(1) and (2) with respect to dx will get

**-d ^{2}v/dx^{2} = (R + jωL) * dI/dx ………………. eq(3)**

**-d ^{2}I/dx^{2} = (G + jωc) * dV/dx … ……………. eq(4)**

Substituting eq(1) and (2) in eq(3) and (4) will get

**-d ^{2}v/dx^{2} = (R + jωL) (G + jωc) V ………………. eq(5)**

**-d ^{2}I/dx^{2} = (G + jωc) (R + jωL) I … ……………. eq(6)**

**Let P ^{2 } = (R + jωL) (G + jωc) … ……………. eq(7)**

Where P – propogation constant

Substitute d/dx = P in eq(6) and (7)

**-d ^{2}v/dx^{2} = P^{2}V ………………. eq(8)**

**-d**

^{2}I/dx^{2}= P^{2}I … ……………. eq(9)General solution is

**V = Ae ^{px }+ Be^{-px} … ……………. eq(10)**

**I = Ce ^{px }+ De^{-px} … ……………. eq(11)**

Where A, B C and D are constants

Differentiating eq(10) and (11) with respect to ‘x’ will get

**-dv/dx = P (Aepx – Be-px ) ………………. eq(12)**

**-dI/dx = P (Cepx – De-px) … ……………. eq(13)**

Substitute eq(1) and (2) in eq(12) and (13) will get

**-(R + jωL) * I = P ( Ae ^{px }+ Be^{-px} ) ………………. eq(14)**

**-( G + jωc) * V = P (Ce**

^{px }+ De^{-px}) ………………. eq(15)Substitute ‘p’ value in eq(14) and (15) will get

**I = -p/ R + jωL * (Ae ^{px }+ Be^{-px})**

** = √G + jωc / R + jωL * (Ae ^{px }+ Be^{-px}) ………………. eq(16)**

**V = -p/ G + jωc * (Ce ^{px }+ De^{-px} )**

**= √R + jωL / G + jωc * (Ce ^{px }+ De^{-px} ) ………………. eq(17)**

**Let Z _{0}= √R + jωL / G + jωc**

Where Z_{0}is the characteristic impedenc

Substitute boundary conditions x=0, V=V_{S} and I=I_{S} in eq(16) and (17) will get

**I _{S} = A+B ………………. eq(18)**

**V _{S} = C+D ………………. eq(19)**

**I _{S}Z_{0}= -A+B ………………. eq(20)**

**V _{S} /Z_{0} = -C+D ………………. eq(21)**

From (20) will get A and B values

**A = V _{S} -I_{S} Z_{0}**

**B =V _{S} +I_{S} Z_{0}**

From eq(21) will get C and D values

**C = (I _{S} – V_{S} /Z_{0}) /2**

**D = (I _{S} + V_{S} /Z_{0}) /2**

Substitute A, B, C and D values in eq(10) and (11)

**V= (V _{S} -I_{S} Z_{0}) e^{px} + (V_{S} +I_{S} Z_{0})e^{-px}**

**= V _{S} (e^{px} +e-px/2) –I_{S} Z¬0(e^{px} -e^{-px}/2)**

**= V _{S} coshx – I_{S} Z_{0} sinhx**

Similarly

**I= (I _{S} -V_{S} Z_{0}) e^{px} + (V_{S} /Z_{0}+I_{S} / 2)e^{-px}**

**=I _{S} (e^{px}+e^{-px}/2) –V_{S} /Z_{0} (e^{px} -e^{-px}/2)**

**=I _{S} coshx – V_{S} /Z_{0} sinhx**

Thus V = V_{S} coshx – I_{S} Z_{0} sinhx

I = I_{S} coshx – V_{S} /Z_{0} sinhx

Equation of transmission line in terms of sending end parameters are derived

#### Efficiency of Transmission Lines

The efficiency of the transmission line is defined as a ratio of received power by transmitted power.

**Efficiency = received power (P _{r}) / transmitted power (P_{t}) * 100%**

### Types of Transmission Lines

The different types of transmission lines include the following.

#### Open Wire Transmission Line

It consists pair of parallel conducting wires separated by a uniform distance. The two-wire transmission lines are very simple, low cost and easy to maintain over short distances and these lines are used up to 100 MHz The another name of an open-wire transmission line is a parallel wire transmission line.

#### Coaxial Transmission Line

The two conductors placed coaxially and filled with dielectric materials such as air, gas or solid. The frequency increases when losses in the dielectric increases, the dielectric is polyethylene. The coaxial cables are used up to 1 GHz. It is a type of wire which carries high-frequency signals with low losses and these cables are used in CCTV systems, digital audios, in computer network connections, in internet connections, in television cables, etc.

#### Optic Fiber Transmission Line

The first optical fiber invented by Narender Singh in 1952. It is made-up of silicon oxide or silica, which is used to send signals over a long distance with little loss in signal and at the speed of light. The optic fiber cables used as light guides, imaging tools, lasers for surgeries, used for data transmission and also used in a wide variety of industries and applications.

#### Microstrip Transmission Lines

The microstrip transmission line is a Transverse Electromagnetic (TEM) transmission line invented by Robert Barrett in 1950.

#### Wave Guides

Waveguides are used to transmit electromagnetic energy from one place to another place and they are usually operating in dominant mode. The various passive components such as filter, coupler, divider, horn, antennas, tee junction, etc. Waveguides are used in scientific instruments to measure optical, acoustic ad elastic properties of materials and objects. There are two types of waveguides are Metal waveguides and dielectric waveguides. The waveguides are used in optical fiber communication, microwave ovens, space crafts, etc.

### Applications

The applications of transmission line are

- Power transmission line
- Telephone lines
- Printed circuit board
- Cables
- Connectors (PCI, USB)

The transmission line equations in terms of sending end parameters are derived, applications and classification of transmission lines are discussed and, Here is a question for you what are the constant voltages in AC and DC transmission lines?