Transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances. However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.
Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses. RF engineers commonly use short pieces of transmission line, usually in the form of printed planar transmission lines, arranged in certain patterns to build circuits such as filters. These circuits, known as distributed-element circuits, are an alternative to traditional circuits using discrete capacitors and inductors.
Overview
Ordinary electrical cables suffice to carry low frequency alternating current, such as mains power, which reverses direction 100 to 120 times per second, and audio signals. However, they are not generally used to carry currents in the radio frequency range, above about 30 kHz, because the energy tends to radiate off the cable as radio waves, causing power losses. Radio frequency currents also tend to reflect from discontinuities in the cable such as connectors and joints, and travel back down the cable toward the source. These reflections act as bottlenecks, preventing the signal power from reaching the destination. Transmission lines use specialized construction, and impedance matching, to carry electromagnetic signals with minimal reflections and power losses. The distinguishing feature of most transmission lines is that they have uniform cross sectional dimensions along their length, giving them a uniform impedance, called the characteristic impedance, to prevent reflections. Types of transmission line include parallel line, coaxial cable, and planar transmission lines such as stripline and microstrip. The higher the frequency of electromagnetic waves moving through a given cable or medium, the shorter the wavelength of the waves. Transmission lines become necessary when the transmitted frequency's wavelength is sufficiently short that the length of the cable becomes a significant part of a wavelength.At frequencies of microwave and higher, power losses in transmission lines become excessive, and waveguides are used instead, which function as "pipes" to confine and guide the electromagnetic waves. Some sources define waveguides as a type of transmission line; however, this article will not include them.
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. The model 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, as follows:In the simplest case, the network is assumed to be linear, 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 two parameters called characteristic impedance, symbol and propagation delay, symbol is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of are for a coaxial cable, about for a twisted pair of wires, and about for a common type of untwisted pair used in radio transmission. Propagation delay is proportional to the length of the transmission line and is never less than the length divided by the speed of light. Typical delays for modern communication transmission lines vary from to.
When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the load impedance equal to 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 transmission line is modelled with a resistance and inductance in series with a capacitance and conductance in parallel. The resistance and conductance contribute to the loss in a transmission line.
The total loss of power in a transmission line is often specified in decibels per metre, and always 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 corresponds approximately to a halving of the power.
Propagation delay is often specified in units of nanoseconds per metre. While propagation delay usually depends on the frequency of the signal, transmission lines are typically operated over frequency ranges where the propagation delay is approximately constant.
Telegrapher's equations
The telegrapher's equations are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's equations.Image:Transmission line element.svg|thumb|right|250px|Schematic representation of the elementary component of a transmission line
The transmission line model is an example of the distributed-element model. It represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:
- The distributed resistance of the conductors is represented by a series resistor.
- The distributed inductance is represented by a series inductor.
- The capacitance between the two conductors is represented by a shunt capacitor.
- The conductance of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire.
The line voltage and the current can be expressed in the frequency domain as
Special case of a lossless line
When the elements and are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the and elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.
General case of a line with losses
In the general case the loss terms, and, are both included, and the full form of the Telegrapher's equations become:where is the propagation constant. These equations are fundamental to transmission line theory. They are also wave equations, and have solutions similar to the special case, but which are a mixture of sines and cosines with exponential decay factors.
Solving for the propagation constant in terms of the primary parameters,,, and gives:
and the characteristic impedance can be expressed as
The solutions for and are:
The constants must be determined from boundary conditions. For a voltage pulse, starting at and moving in the positive direction, then the transmitted pulse at position can be obtained by computing the Fourier Transform,, of, attenuating each frequency component by, advancing its phase by, and taking the inverse Fourier Transform. The real and imaginary parts of can be computed as
with
the right-hand expressions holding when neither, nor , nor is zero, and with
where atan2 is the everywhere-defined form of two-parameter arctangent function, with arbitrary value zero when both arguments are zero.
Alternatively, the complex square root can be evaluated algebraically, to yield:
and
with the plus or minus signs chosen opposite to the direction of the wave's motion through the conducting medium.
Special, low loss case
For small losses and high frequencies, the general equations can be simplified: If and thenSince an advance in phase by is equivalent to a time delay by, can be simply computed as