Transmission Line

Components of transmission lines include Wire s, Coaxial Cable s, Dielectric slabs, Optical Fibre s, electric power lines, and Waveguide s.

HISTORY

Mathematical analysis of the behaviour of electrical transmission lines grew out of the work of James Clerk Maxwell , Lord Kelvin and Oliver Heaviside . In 1855 Lord Kelvin formulated a diffusion model of the current in a submarine cable. This law correctly predicted the poor performance of the 1858 trans-Atlantic Submarine Telegraph cable. In 1885 Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the telegrapher's equations.

THE FOUR TERMINAL MODEL

For the purposes of analysis, an electrical transmission line can be modelled as a two-port network (also called a quadrupole network), as follows:

In the simplest case, the network is assumed to be linear (i.e. the Complex voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable. If the transmission line is uniform along its length, then its behaviour is largely described by a single parameter called the '' Characteristic Impedance '', symbol Z₀. This is the ratio of the complex voltage to the complex current at any point on the line. Typical values of Z₀ are 50 or 75 Ohm s for a Coaxial Cable , about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission.

When sending power down a transmission line, it is usually desirable that all the power is absorbed by the load and none of it is reflected back to the source. This can be ensured by making the source and load impedances equal to Z₀, in which case the transmission line is said to be '' Matched ''.

Some of the power that is fed into a transmission line is lost because of its resistance. This effect is called ''ohmic'' or ''resistive'' loss. At high frequencies, another effect called ''dielectric loss'' becomes significant, adding to the losses caused by resistance. Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to Heat .

The total loss of power in a transmission line is often specified in Decibel s per Metre , and usually depends on the frequency of the signal. The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies. A loss of 3 dB corresponds approximately to a halving of the power.

High-frequency transmission lines can be defined as transmission lines that are designed to carry electromagnetic waves whose Wavelength s are comparable to the length of the line. Under these conditions, the approximations useful for calculations at lower frequencies are no longer accurate. This often occurs with Radio , Microwave and Optical signals, and with the signals found in high-speed Digital Circuit s.

TELEGRAPHER'S EQUATIONS

Oliver Heaviside developed the ''transmission line model'', also known as the telegrapher's equations, that describes how electrical Voltage and Current vary along a Conductor .

The theory applies to (x, t) and the other one for I (x, t). The model demonstrates that the electrical current can be reflected on the wire, and that wave patterns can appear along the line.

The equations

The telegrapher's equations can be understood as a simplified case of Maxwell's Equations . In a more practical approach, one assumes that the conductor is composed out of an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

The resistivity of the conductor (expressed in ohms per unit length) is represented by a series resistor R.

The distributed inductance (due to the Magnetic Field around the wire, Self-inductance , etc.) is represented by a series Coil ( Inductance per unit length L).

The capacitive behaviour of the insulation between the signal wire and the return wire is represented by a Shunt Capacitor C (capacitance per unit length).

The conductivity of the insulation material is accounted for by a resistor G shunted between the signal wire and the return wire (conductance per unit length).

When the elements R and G are very small, their effects can be neglected, and the transmission line is considered as an idealized, lossless, structure. In this case, the model depends only on the L and C elements, and we obtain a pair of first-order partial differential equations, one function describing the voltage V along the line and the other the current I, both function of position x and time t:

:

rac{\partial}{\partial x} V(x,t) =
-L rac{\partial}{\partial t} I(x,t)

rac{\partial}{\partial x} I(x,t) =
-C rac{\partial}{\partial t} V(x,t)

These equations may be combined to form either of two exact wave equations:

:

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Transmission Line