vision electricity: TRANSMISSION LINE

A transmission line is the material medium or structure that forms all or part of a path from one place to another for directing the transmission of energy, such as electromagnetic waves or acoustic waves, as well as electric power transmission. Types of transmission line include wires, coaxial cables, dielectric slabs, striplines, optical fibers, electric power lines, and waveguides.

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.

Applicability

In many electric circuits, the length of the wires connecting the components can for the most part be ignored. That is, the voltage on the wire at a given time can be assumed to be the same at all points. However, when the voltage changes in a time interval comparable to the time it takes for the signal to travel down the wire, the length becomes important and the wire must be treated as a transmission line. Stated another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to or less than the length of the wire.
A common rule of thumb is that the cable or wire should be treated as a transmission line if the length is greater than 1/10 of the wavelength. At this length the phase delay and the interference of any reflections on the line become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.

The four terminal model

Variations on the schematic electronic symbol for a transmission line.

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 of a given wave to the complex current of the same wave at any point on the line. Typical values of Z₀ are 50 or 75 ohms 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 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 Z₀, in which case the transmission line is said to be matched. Ensuring the source impedance matches Z₀ will maximize power transfer from the source to the transmission line, but has no other effect on the behavior of the line.
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 (see ohmic heating). 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 (see dielectric heating).
The total loss of power in a transmission line is often specified in decibels per metre (dB/m), 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 those designed to carry electromagnetic waves whose wavelengths are shorter than or 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 circuits.

Telegrapher's equations

The Telegrapher's Equations (or just Telegraph 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.

Schematic representation of the elementary component of a transmission line.

The transmission line model 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 $R$ of the conductors is represented by a series resistor (expressed in ohms per unit length).
The distributed inductance $L$ (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
The capacitance $C$ between the two conductors is represented by a shunt capacitor C (farads per unit length).
The conductance $G$ of the dielectric material separating the two conductors is represented by a conductance G shunted between the signal wire and the return wire (siemens per unit length).

The model consists of an infinite series of the elements shown in the figure, and that the values of the components are specified per unit length so the picture of the component can be misleading.

R

L

C

, and

G

may also be functions of frequency. An alternative notation is to use

R'

L'

C'

and

G'

to emphasize that the values are derivatives with respect to length. These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the propagation constant, attenuation constant and phase constant.

The line voltage

V (x)

and the current

I (x)

can be expressed in the frequency domain as

$\frac{\partial V(x)}{\partial x} = -(R + j \omega L)I(x)$

$\frac{\partial I(x)}{\partial x} = -(G + j \omega C)V(x)$

When the elements

R

and

G

are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the

L

and

C

elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

$\frac{\partial^2V(x)}{\partial x^2}+ \omega^2 LC\cdot V(x)=0$

$\frac{\partial^2I(x)}{\partial x^2} + \omega^2 LC\cdot I(x)=0$

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.
If

R

and

G

are not neglected, the Telegrapher's equations become:

$\frac{\partial^2V(x)}{\partial x^2} = \Gamma^2 V(x)$

$\frac{\partial^2I(x)}{\partial x^2} = \Gamma^2 I(x)$

where

$\Gamma = \sqrt{(R + j \omega L)(G + j \omega C)}$

and the characteristic impedance is:

$Z_0 = \sqrt{\frac{R + j \omega L}{G + j \omega C}}$

The solutions for

V (x)

and

I (x)

are:

$V(x) = V_+ e^{-\Gamma x} + V_- e^{\Gamma x} \,$

$I(x) = \frac{1}{Z_0}(V_+ e^{-\Gamma x} - V_- e^{\Gamma x}) \,$

The constants $V_\pm$ and $I_\pm$ must be determined from boundary conditions. For a voltage pulse $V_{\mathrm{in}}(t) \,$ , starting at

x = 0

and moving in the positive

x

-direction, then the transmitted pulse $V_{\mathrm{out}}(x,t) \,$ at position

x

can be obtained by computing the Fourier Transform, $\tilde{V}(\omega)$ , of $V_{\mathrm{in}}(t) \,$ , attenuating each frequency component by $e^{\mathrm{-Re}(\Gamma) x} \,$ , advancing its phase by $\mathrm{-Im}(\Gamma)x \,$ , and taking the inverse Fourier Transform. The real and imaginary parts of

Γ

can be computed as

$\mathrm{Re}(\Gamma) = (a^2 + b^2)^{1/4} \cos(\mathrm{atan2}(b,a)/2) \,$

$\mathrm{Im}(\Gamma) = (a^2 + b^2)^{1/4} \sin(\mathrm{atan2}(b,a)/2) \,$

where atan2 is the two-parameter arctangent, and

$a \equiv \omega^2 LC \left[ \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right]$

$b \equiv \omega^2 LC \left( \frac{R}{\omega L} + \frac{G}{\omega C} \right)$

For small losses and high frequencies, to first order in

R / ω L

and

G / ω C

one obtains

$\mathrm{Re}(\Gamma) \approx \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) \,$

$\mathrm{Im}(\Gamma) \approx \omega \sqrt{LC} \,$

Noting that an advance in phase by

- ωδ

is equivalent to a time delay by

δ

V o u t (t)

can be simply computed as

$V_{\mathrm{out}}(x,t) \approx V_{\mathrm{in}}(t - \sqrt{LC}x) e^{- \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) x } \,$

Input impedance of lossless transmission line

The characteristic impedance

Z 0

of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.
For a lossless transmission line, it can be shown that the impedance measured at a given position

l

from the load impedance

Z L

$Z_\mathrm{in} (l)=Z_0 \frac{Z_L + jZ_0\tan(\beta l)}{Z_0 + jZ_L\tan(\beta l)}$

where $\beta=\frac{2\pi}{\lambda}$ is the wavenumber.
In calculating

β

, the wavelength is generally different inside the transmission line to what it would be in free-space and the velocity constant of the material the transmission line is made of needs to be taken into account when doing such a calculation.

Half wave length

For the special case where

β l = n π

where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that

$Z_\mathrm{in}=Z_L \$

for all

n

. This includes the case when

n = 0

, meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.

Quarter wave length

For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes

$Z_\mathrm{in}=\frac{Z_0^2}{Z_L} \$

Matched load

Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that

$Z_\mathrm{in}=Z_L=Z_0\$

for all

l

and all

λ

Short

For the case of a shorted load (i.e.

Z L = 0

), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)

$Z_\mathrm{in} (l)=j Z_0 \tan(\beta l) \,$

Open

For the case of an open load (i.e. $Z_L=\infty$ ), the input impedance is once again imaginary and periodic

$Z_\mathrm{in} (l)=-j Z_0 \cot(\beta l) \,$

vision electricity

Saturday, October 3, 2009

TRANSMISSION LINE

History

Applicability

The four terminal model

Telegrapher's equations

Input impedance of lossless transmission line

Half wave length

Quarter wave length

Matched load

Short

Open

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