# Transmission Line

In electrical engineering , a transmission line is a special cable or other structure designed to conduct electromagnetic waves in an implicit manner. The term is applied when the conductors are so long that the wave nature of the transmission must be taken into account. This is particularly applicable to radio-frequency engineering because the short wavelength means that wave events are generated over very short distances (this can be as small as a millimeter, depending on the frequency). However, the principle of transmission lines has historically been used for very long telegraphs .was developed to interpret events on lines, in particular submarine telegraph cables .

Transmission lines are used to connect radio transmitters and receivers with their antennas (then called feed lines or feeders), distributing cable television signals, trunkline routing calls between telephone switching centers , computer network connections, and high -speed communication. Speed ​​computer data bus . RF engineers typically use small pieces of transmission line, usually in the form of printed planar transmission lines , arranged in some pattern to form a circuit such as a filter . These circuits, known as distributed-element circuits , are discreteThere are an alternative to conventional circuits that use capacitors and inductors .

Ordinary power cables are sufficient to carry low frequency alternating current ( AC) and audio signals . However, they cannot be used to carry currents in the radio frequency range above about 30 kHz , as the energy radiates down the cable in the form of radio waves , causing power loss. RF currents are also reflected from imbalances such as connectors and joints in the cable, and travel back down the cable toward the source. These reflections act as barriers preventing the signal power from reaching the destination. Transmission lines of special construction to carry electromagnetic signals with minimal reflectance and power loss, anduses impedance matching . The distinguishing feature of most transmission lines is that they have uniform cross-sectional dimensions along their length, giving them a uniform impedance , called characteristic impedance , to prevent reflections. The higher the frequency of electromagnetic waves traveling through a given cable or medium, the shorter the wavelength of the waves . Transmission lines become necessary when the wavelength of the transmitted frequency is sufficiently short that the length of the cable becomes a significant portion of the wavelength. At microwave frequencies and above, power losses in transmission lines are excessive, and waveguides instead act as “pipes” to confine and guide electromagnetic waves. At even higher frequencies in the terahertz , infrared and visible ranges, waveguides in turn become lossy, and optical methods (such as lenses and mirrors) are used to direct electromagnetic waves.

## Observation

Ordinary power cables are sufficient to carry low-frequency alternating current (AC), such as mains power , which reverses direction 100 to 120 times per second, and audio signals . However, they cannot be used to carry currents in the radio frequency range, [1] above about 30 kHz, because the energy radiates down the cable in the form of radio waves , causing power loss. Radio frequency currents are also reflected from imbalances such as connectors and joints in the cable, and travel up the cable back toward the source. [1] [2]These reflections act as barriers, preventing signal power from reaching the destination. Transmission lines use special construction, and impedance matching, to carry electromagnetic signals with minimal reflectance and power loss. The distinguishing feature of most transmission lines is that they have uniform cross sectional dimensions along their length, giving them a uniform impedance , called characteristic impedance, [2] [3] [4] to prevent reflections. . Types of transmission line include parallel line (ladder line, twisted pair), coaxial cable, and planar transmission lines such as stripline and microstrip. [5] [6]The higher the frequency of electromagnetic waves traveling through a given cable or medium, the shorter the wavelength of the waves. Transmission lines become necessary when the wavelength of the transmitted frequency is sufficiently short that the length of the cable becomes a significant portion of the wavelength.

At microwave frequencies and above, power losses in transmission lines are excessive, and waveguides are used instead, [1] to act as “pipes” to confine and guide electromagnetic waves. [6] Some sources define a waveguide as a type of transmission line; [6] However, they will not be covered in this article. At even higher frequencies in the terahertz, infrared and visible ranges, waveguides in turn become lossy, and optical methods (such as lenses and mirrors) are used to direct electromagnetic waves.

## History

The mathematical analysis of the behavior of power transmission lines developed from the work of James Clerk Maxwell, Lord Kelvin and Oliver Heaviside. In 1855, Lord Kelvin developed the diffusion model of 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 paper describing his analysis of propagation in cables and a modern form of telegrapher’s equations. [7]

## Four terminal model

For the purposes of analysis, an electrical transmission line can be modeled as a two-port network (also known as a quadripole) as follows:

In the simplest case, the network is assumed to be linear (i.e. the complex voltage at any port is proportional to the complex current when there is no reflection), and the two ports are considered to be interchangeable. If the transmission line is uniform along its length, its behavior is largely described by a parameter called the characteristic impedance , symbol Z 0 . It 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 Z 0 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 transmissions.

When sending power down a transmission line, it is generally desirable that as much power as possible will be absorbed by the load and as little as possible is reflected back to the source. This can be ensured by making the load impedance Z equal to 0 , in which case the transmission line is said to match .

Some of the power being put into the transmission line is lost due to its resistance. This effect is called ohmic or resistive loss (see ohmic heating). At high frequencies, another effect called dielectric loss becomes important, which adds to the losses due to resistance. Dielectric loss occurs when the insulating material inside a transmission line absorbs energy from the alternating electric field and converts it into heat (see dielectric heating). The transmission line is designed to have a capacitance (C) and conduction (G) in series with a resistance (R) and inductance (L) in series. Resistance and conductance contribute to losses in a transmission line.

The total loss of power in a transmission line is often specified in decibels per meter (dB/m), and usually depends on the frequency of the signal. The manufacturer often provides a chart showing the loss in dB/m over a range of frequencies. A loss of 3 dB roughly corresponds to half of the power.

High-frequency transmission lines can be designed to carry electromagnetic waves whose wavelength is less than or comparable to the length of the line. Under these conditions, estimates useful for calculations at low frequencies are no longer accurate. It is often accompanied by signals found in radio, microwave and optical signals, metal mesh optical filters, and high-speed digital circuits.

## Telegrapher’s equations

Telegrapher’s equations (or simply telegraph equations ) are a pair of linear differential equations which describe voltage ( ) and current ( ) on a power transmission line over distance and time. They were developed by Oliver Heaviside who created the transmission line model , and are based on Maxwell’s equations. VI

The transmission line model is an example of a distributed-element model. It represents the transmission line as an infinite series of two-port primary components, each representing an infinitely small segment of the transmission line:

• Distributed resistance conductors are represented by a series resistor (expressed in ohms per unit length). R
• Distributed inductance (due to magnetic fields around wires, self-inductance, etc.) is represented by a series inductor (in henries per unit length). L
• The capacitance is represented by a shunt capacitor (in farads per unit length) between the two conductors. C
• Conductance The dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (in Siemens per unit length).G

The model consists of an infinite series of elements shown in the figure , and the values ​​of the components are specified per unit length so that the picture of the component can be misleading. , , , and can also be functions of frequency. An alternative notation is to use , , and to emphasize that the values ​​are derived with respect to length. These quantities may also be referred to as primary line constants to distinguish them from the secondary line constants derived from them, these are the diffusion constant, the attenuation constant, and the phase constant.

RLCGR'L'C'G'

Line voltage and current can be expressed in the frequency domain:

V(x)I(x)

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

{\displaystyle {\frac {\partial I(x)}{\partial x}}=-(G+j\,\omega \,C)\,V(x)~\,.}


(See differential equation, angular frequency and the imaginary unit J )

### The special case of a lossless line

When the elements and are negligibly small, the transmission line is considered to be a lossless structure. In this hypothetical case, the model relies only on and elements that greatly simplify the analysis. For a lossless transmission line, the equations for a second-order steady-state telegrapher are: R G L C

{\displaystyle {\frac {\partial ^{2}V(x)}{\partial x^{2}}}+\omega ^{2}L\,C\,V(x)=0}
{\displaystyle {\frac {\partial ^{2}I(x)}{\partial x^{2}}}+\omega ^{2}L\,C\,I(x)=0~\,.}


These are wave equations in which plane waves are the solution with the same propagation speed in forward and opposite directions. Its physical significance is that electromagnetic waves propagate down transmission lines and, in general, have a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.

### Common case of loss line

In the general case the loss terms, and , are both included, and the full form of the telegrapher’s equations becomes: RG

{\displaystyle {\frac {\partial ^{2}V(x)}{\partial x^{2}}}=\gamma ^{2}V(x)\,}
{\displaystyle {\frac {\partial ^{2}I(x)}{\partial x^{2}}}=\gamma ^{2}I(x)\,}

where is (complex) is the diffusion constant. These equations are fundamental to transmission line theory. They are also wave equations, and have similar solutions to the special case, but which are a mixture of sine and cosine with exponential decay factors. Solving for the diffusion constants in terms of the elementary parameters , , , and gives:

\gamma\gamma~RLGC
{\displaystyle \gamma ={\sqrt {(R+j\,\omega \,L)(G+j\,\omega \,C)\,}}}


and the characteristic impedance can be expressed as

{\displaystyle Z_{0}={\sqrt {{\frac {R+j\,\omega \,L}{G+j\,\omega \,C}}\,}}~\,.}


There are more solutions for : V(x) I(x)

{\displaystyle V(x)=V_{(+)}e^{-\gamma \,x}+V_{(-)}e^{+\gamma \,x}\,}
{\displaystyle I(x)={\frac {1}{Z_{0}}}\,\left(V_{(+)}e^{-\gamma \,x}-V_{(-)}e^{+\gamma \,x}\right)~\,.}

The constant must be determined from the boundary conditions. For a voltage pulse , at the beginning and running in the positive  direction, then can be obtained by computing the Fourier transform on the transmitted pulse position , attenuating each frequency component by , , moving its phase , and Taking the inverse Fourier transform. can be calculated as the real and imaginary part of

{\displaystyle V_{(\pm )}}V_{\mathrm{in}}(t) \,x=0xV_{\mathrm{out}}(x,t) \,x\tilde {V} (\omega)V_{\mathrm{in}}(t) \,{\displaystyle e^{-\operatorname {Re} (\gamma )\,x}\,}{\displaystyle -\operatorname {Im} (\gamma )\,x\,}\gamma
{\displaystyle \operatorname {Re} (\gamma )=\alpha =(a^{2}+b^{2})^{1/4}\cos(\psi )\,}
{\displaystyle \operatorname {Im} (\gamma )=\beta =(a^{2}+b^{2})^{1/4}\sin(\psi )\,}


With

{\displaystyle a~\equiv ~R\,G\,-\omega ^{2}L\,C\ ~=~\omega ^{2}L\,C\,\left[\left({\frac {R}{\omega L}}\right)\left({\frac {G}{\omega C}}\right)-1\right]}

{\displaystyle b~\equiv ~\omega \,C\,R+\omega \,L\,G~=~\omega ^{2}L\,C\,\left({\frac {R}{\omega \,L}}+{\frac {G}{\omega \,C}}\right)}

The right hand expressions hold when neither , nor , nor is zero , and with LCW

{\displaystyle \psi ~\equiv ~{\tfrac {1}{2}}\operatorname {atan2} (b,a)\,}

where atan2 is the everywhere defined form of the two-parameter arctangent function, when both arguments are zero, the arbitrary value is zero.

Alternatively, the complex square root can be evaluated algebraically:

{\displaystyle \alpha ={\frac {\pm b}{\sqrt {2\left(-a+{\sqrt {a^{2}+b^{2}}}\right)~}}},}


And

{\displaystyle \beta =\pm {\sqrt {{\tfrac {1}{2}}\left(-a+{\sqrt {a^{2}+b^{2}}}\right)~}},}

With a plus or minus sign chosen opposite to the direction of motion of the wave through the conducting medium. (Note that a is usually negative, since and is usually much smaller than and , respectively, so −a is usually positive. b is always positive.) GRwCwL

### Special, less damage case

For small losses and high frequencies, the general equations can be simplified to: if and then

{\displaystyle {\tfrac {R}{\omega \,L}}\ll 1}{\displaystyle {\tfrac {G}{\omega \,C}}\ll 1}
{\displaystyle \operatorname {Re} (\gamma )=\alpha \approx {\tfrac {1}{2}}{\sqrt {L\,C\,}}\,\left({\frac {R}{L}}+{\frac {G}{C}}\right)\,}

{\displaystyle \operatorname {Im} (\gamma )=\beta \approx \omega \,{\sqrt {L\,C\,}}~.\,}


Since an advance in step by step is equal to a time delay , can simply be calculated as

\omega  \delta  \delta V_{out}(t)
{\displaystyle V_{\mathrm {out} }(x,t)\approx V_{\mathrm {in} }(t-{\sqrt {L\,C\,}}\,x)\,e^{-{\tfrac {1}{2}}{\sqrt {L\,C\,}}\,\left({\frac {R}{L}}+{\frac {G}{C}}\right)\,x}.\,}

### State of heaviness

The Heaviside condition is a special case where the wave is down the line without any dispersion distortion. condition to be

{\displaystyle {\frac {G}{C}}={\frac {R}{L}}}


## Input impedance of transmission line

The characteristic impedance of a transmission line is the ratio of the amplitude of a single voltage wave to its current waveform. Since most transmission lines also have a reflected waveform, the characteristic impedance is usually not the impedance that is measured on the line. Zo

The impedance is measured at a certain distance from the load impedance can be expressed as l ZL

{\displaystyle Z_{\mathrm {in} }\left(\ell \right)={\frac {V(\ell )}{I(\ell )}}=Z_{0}{\frac {1+{\mathit {\Gamma }}_{\mathrm {L} }e^{-2\gamma \ell }}{1-{\mathit {\Gamma }}_{\mathrm {L} }e^{-2\gamma \ell }}}},

where is is the propagation constant and is the voltage reflection coefficient measured at the load end of the transmission line. Alternatively, the above formula can be rearranged to express input impedance in terms of load impedance rather than load voltage reflection coefficient: Y

{\displaystyle {\mathit {\Gamma }}_{\mathrm {L} }={\frac {\,Z_{\mathrm {L} }-Z_{0}\,}{Z_{\mathrm {L} }+Z_{0}}}}
{\displaystyle Z_{\mathrm {in} }(\ell )=Z_{0}\,{\frac {Z_{\mathrm {L} }+Z_{0}\tanh \left(\gamma \ell \right)}{Z_{0}+Z_{\mathrm {L} }\,\tanh \left(\gamma \ell \right)}}}.

### Input impedance of lossless transmission line

For a lossless transmission line, the propagation constant is purely imaginary , so the above formulas can be rewritten as

{\displaystyle \gamma =j\,\beta }

{\displaystyle Z_{\mathrm {in} }(\ell )=Z_{0}{\frac {Z_{\mathrm {L} }+j\,Z_{0}\,\tan(\beta \ell )}{Z_{0}+j\,Z_{\mathrm {L} }\tan(\beta \ell )}}}


where is wavenumber.

{\displaystyle \beta ={\frac {\,2\pi \,}{\lambda }}}


In calculations the wavelength is usually separated in free-space inside the transmission line. Consequently, the velocity factor of the material of the transmission line must be taken into account when making such calculations.

.\beta ,

### Special cases of lossless transmission lines

#### Half wave length

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

{\displaystyle \beta \,\ell =n\,\pi }
{\displaystyle Z_{\mathrm {in} }=Z_{\mathrm {L} }\,}


For all this includes the case when , which means that the length of the transmission line is negligibly small compared to the wavelength. Its physical significance is that the transmission line can be ignored (ie treated as a wire) in any case. n = 0

#### Quarter wave length

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

{\displaystyle Z_{\mathrm {in} }={\frac {Z_{0}^{2}}{Z_{\mathrm {L} }}}~\,.}


Another special case is when the load impedance is equal to the line’s characteristic impedance (ie the line is matched ), in which case the impedance reduces the line’s characteristic impedance so that

{\displaystyle Z_{\mathrm {in} }=Z_{\mathrm {L} }=Z_{0}\,}


For all and all

\ell\lambda

low

In the case of short load (i.e. ), the input impedance is purely imaginary and is a periodic function of position and wavelength (frequency) ZL=0

{\displaystyle Z_{\mathrm {in} }(\ell )=j\,Z_{0}\,\tan(\beta \ell ).\,}

#### Open

In the case of open loads (ie ), the input impedance is once again imaginary and periodic

{\displaystyle Z_{\mathrm {L} }=\infty }
{\displaystyle Z_{\mathrm {in} }(\ell )=-j\,Z_{0}\cot(\beta \ell ).\,}


### Phased transmission line

A simple example of a stepped transmission line consisting of three sections.

A phased transmission line is used for wide range impedance matching. It can be thought of as several transmission line segments connected in series, with the characteristic impedance of each individual element . [8] The input impedance can be obtained by successive application of the series relation

{\displaystyle Z_{\mathrm {0,i} }}
{\displaystyle Z_{\mathrm {i+1} }=Z_{\mathrm {0,i} }\,{\frac {\,Z_{\mathrm {i} }+j\,Z_{\mathrm {0,i} }\,\tan(\beta _{\mathrm {i} }\ell _{\mathrm {i} })\,}{Z_{\mathrm {0,i} }+j\,Z_{\mathrm {i} }\,\tan(\beta _{\mathrm {i} }\ell _{\mathrm {i} })}}\,}


where is is the wavenumber of the -th transmission line segment and is the length of this segment, and is the front-end impedance that loads the -th segment.

{\displaystyle \beta _{\mathrm {i} }}{\displaystyle \mathrm {i} }{\displaystyle \ell _{\mathrm {i} }}{\displaystyle Z_{\mathrm {i} }}{\displaystyle \mathrm {i} }

Because the characteristic impedance of each transmission line segment is often different from the impedance. Fourth, the input cable (shown only as the arrow marked on the left side of the diagram above), the impedance change cycle is off-centered along the axis of the Smith chart. whose impedance representation is usually . is normalized against .

A stepped transmission line is an example of a distributed-element circuit. A large variety of other circuits can also be built along transmission lines, including filters, power dividers and directional couplers.

## Practical type

### coaxial wire

Coaxial lines confine almost all electromagnetic waves to the region inside the cable. Therefore coaxial lines can be bent and bent (subject to limitation) without negative effects, and can be tied to conductive supports without inducing unwanted currents in them. In radio-frequency applications up to a few gigahertz, the wave propagates only in the transverse electric and magnetic mode (TEM), meaning that both the electric and magnetic fields are perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential) ). However, at frequencies for which the wavelength (in the dielectric) is much shorter than the circumference of the cable, other transverse modes can propagate. These modes are classified into two groups, the transverse electric (TE) and transverse magnetic (TM) waveguide modes. When more than one mode may be present,

The most common use for coaxial cables is for television and other signals with bandwidths of several megahertz. In the middle of the 20th century they took long distance telephone connections.

### Planar lines

#### Microstrip

A microstrip circuit uses a thin flat conductor that is parallel to a ground plane. Microstrips can be made by placing a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate, while the other side has a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating layer determine the characteristic impedance. Microstrip is an open structure whereas coaxial cable is a closed structure.

#### Stripline

A stripline circuit uses a flat strip of metal that is sandwiched between two parallel ground planes. The insulating material of the substrate forms a dielectric. The width of the strip, the thickness of the substrate and the relative permeability of the substrate determine the characteristic impedance of the strip which is a transmission line.

#### Coplanar waveguide

A coplanar waveguide consists of a center strip and two adjacent outer conductors, all three flat structures deposited on the same insulating substrate and thus located in the same plane (“coplanar”). The width of the center conductor, the distance between the inner and outer conductors, and the relative permittivity of the substrate determine the characteristic impedance of the coplanar transmission line.

### Balanced lines

A balanced line is a transmission line consisting of two conductors of the same type, and of equal impedance to ground and other circuits. There are several forms of balanced lines, the most common being twisted pair, star quad, and twin-lead.

#### Twisted pair

Twisted pairs are commonly used for terrestrial telephone communication. In such cables, several pairs are grouped together in a single cable, from two to several thousand. [9] The format is also used for data network distribution inside buildings, but cable is more expensive because transmission line parameters are tightly controlled.

The star quad is a four-conductor cable with all four conductors twisted together around the cable axis. It is sometimes used for two circuits, such as in 4-wire telephony and other telecommunications applications. In this configuration each pair uses two non-adjacent conductors. Other times it is used for a single, balanced line, such as in audio applications and 2-wire telephony. In this configuration two non-adjacent conductors are terminated together at both ends of the cable, and the other two conductors are also terminated together.

When used for two circuits, crosstalk is reduced relative to cables with two different twisted pairs.

When used for a single, balanced line, the magnetic interference picked up by the cable comes in the form of an almost perfect common mode signal, which is easily removed by a transformer.

The combined benefits of twisting, balanced signaling and a quadruple pattern provide excellent noise immunity, especially beneficial for low signal level applications such as microphone cables, even when installed very close to power cables. [10] [11] [12] [13] [14] The disadvantage is that the star quad, in combination with two conductors, typically doubles the capacitance of the same two-conductor twisted and shielded audio cable. Distortion increases as distance increases due to higher capacitance and higher frequencies suffer more losses. [15] [16]

Twin-lead consists of a pair of conductors held apart by a continuous insulator. By spacing the conductors a known distance apart, the geometry is fixed and the line characteristics are reliably consistent. It is less loss than coaxial cable because the characteristic impedance of twin-lead is usually higher than that of coaxial cable, thereby reducing resistive losses due to low current. However, it is more susceptible to interference.

#### Lecher lines

Latcher lines are a form of parallel conductor that can be used to create resonant circuits in UHF. They are a convenient practical format that fills the gap between lumped components (used in HF/VHF) and resonant cavities (used in UHF/SHF).

### Single-wire line

Unbalanced lines were previously used extensively for telegraph transmission, but this form of communication has now become unusable. Cables are similar to twisted pair in that multiple cores are bundled into a single cable but only one conductor is provided per circuit and no twisting takes place. All circuits in the same route use a common path for return current (earth return). There is a single-wire earth return version of the power transmission in use in many places.

## Common applications

### Signal transfer

Power transmission lines are very widely used to transmit high frequency signals over long or short distances with minimal power loss. A familiar example is the down lead from the TV or radio aerial to the receiver.

### Pulse generation

Transmission lines are also used as pulse generators. By charging the transmission line and then discharging it to a resistive load, a rectangular pulse equal to twice the electrical length of the line can be obtained, albeit with half the voltage. A Blumlein transmission line is a related pulse forming device, which overcomes this limitation. These are sometimes used as pulsed power sources for radar transmitters and other equipment.

### Stub filter

If a short-circuit or open-circuit transmission line is wired parallel to the line used to transfer the signal from point A to point B, it will act as a filter. The method for making stubs is similar to the method for using letcher lines for crude frequency measurements, but it is ‘working backwards’. One method recommended in the RSGB’s Radiocommunication Handbook is to take an open-circuit length of transmission line wired in parallel with the feeder delivering the signal from an aerial. The minimum can be found in the strength of the signal seen at a receiver, by cutting off the free end of the transmission line. At this stage the stub filter will reject this frequency and odd harmonics, but if the free end of the stub is shortened the stub will become a filter that will reject even harmonics.

## Sound

The principle of sound wave propagation is mathematically similar to that of electromagnetic waves, so the techniques of transmission line theory are also used to construct structures for conducting acoustic waves; And these are called acoustic transmission lines.