Special case of a lossless line

When the elements ${\displaystyle R}$ and ${\displaystyle G}$ are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the ${\displaystyle L}$ and ${\displaystyle C}$ elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

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

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

These are wave equations 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, ${\displaystyle R}$ and ${\displaystyle G}$, are both included, and the full form of the Telegrapher's equations become:

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

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

where ${\displaystyle \gamma }$ is the (complex) 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 ${\displaystyle \gamma }$ in terms of the primary parameters ${\displaystyle R}$, ${\displaystyle L}$, ${\displaystyle G}$, and ${\displaystyle C}$ gives:

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

and the characteristic impedance can be expressed as

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

The solutions for ${\displaystyle V(x)}$ and ${\displaystyle I(x)}$ are:

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

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

The constants ${\displaystyle V_{(\pm )}}$ must be determined from boundary conditions. For a voltage pulse ${\displaystyle V_{\mathrm {in} }(t)\,}$, starting at ${\displaystyle x=0}$ and moving in the positive ${\displaystyle x}$ direction, then the transmitted pulse ${\displaystyle V_{\mathrm {out} }(x,t)\,}$ at position ${\displaystyle x}$ can be obtained by computing the Fourier Transform, ${\displaystyle {\tilde {V}}(\omega )}$, of ${\displaystyle V_{\mathrm {in} }(t)\,}$, attenuating each frequency component by ${\displaystyle e^{-\operatorname {Re} (\gamma )\,x}\,}$, advancing its phase by ${\displaystyle -\operatorname {Im} (\gamma )\,x\,}$, and taking the inverse Fourier Transform. The real and imaginary parts of ${\displaystyle \gamma }$ can be computed as

${\displaystyle \operatorname {Re} (\gamma )=\alpha =(a^{2}+b^{2})^{1/4}\cos(\psi )\,}$

${\displaystyle \operatorname {Im} (\gamma )=\beta =(a^{2}+b^{2})^{1/4}\sin(\psi )\,}$

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 holding when neither ${\displaystyle L}$, nor ${\displaystyle C}$, nor ${\displaystyle \omega }$ is zero, and with

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

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:

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

and

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

with the plus or minus signs chosen opposite to the direction of the wave's motion through the conducting medium. (a is usually negative, since ${\displaystyle G}$ and ${\displaystyle R}$ are typically much smaller than ${\displaystyle \omega C}$ and ${\displaystyle \omega L}$, respectively, so −a is usually positive. b is always positive.)

Special, low loss case

For small losses and high frequencies, the general equations can be simplified: If ${\displaystyle {\tfrac {R}{\omega \,L}}\ll 1}$ and ${\displaystyle {\tfrac {G}{\omega \,C}}\ll 1}$ then

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

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

Since an advance in phase by ${\displaystyle -\omega \,\delta }$ is equivalent to a time delay by ${\displaystyle \delta }$, ${\displaystyle V_{out}(t)}$ can be simply computed as

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

Heaviside condition

The Heaviside condition is a special case where the wave travels down the line without any dispersion distortion. The condition for this to take place is

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

Input impedance of lossless transmission line

For a lossless transmission line, the propagation constant is purely imaginary, ${\displaystyle \gamma =j\,\beta }$, so the above formulas can be rewritten as

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

where ${\displaystyle \beta ={\frac {\,2\pi \,}{\lambda }}}$ is the wavenumber.

In calculating ${\displaystyle \beta ,}$ the wavelength is generally different inside the transmission line to what it would be in free-space. Consequently, the velocity factor of the material the transmission line is made of needs to be taken into account when doing such a calculation.

Special cases of lossless transmission lines

Short

Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Black dots represent electrons, and the arrows show the electric field.

Main article: stub

For the case of a shorted load (i.e. ${\displaystyle Z_{\mathrm {L} }=0}$), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)

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

Coaxial cable

Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them. In radio-frequency applications up to a few gigahertz, the wave propagates in the transverse electric and magnetic mode (TEM) only, which means that the electric and magnetic fields are both 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 significantly shorter than the circumference of the cable other transverse modes can propagate. These modes are classified into two groups, transverse electric (TE) and transverse magnetic (TM) waveguide modes. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another.

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

Planar lines

Planar transmission lines are transmission lines with conductors, or in some cases dielectric strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Several forms of planar transmission lines exist.

Microstrip

A type of transmission line called a cage line, used for high power, low frequency applications. It functions similarly to a large coaxial cable. This example is the antenna feed line for a longwave radio transmitter in Poland, which operates at a frequency of 225 kHz and a power of 1200 kW.

Main article: microstrip

A microstrip circuit uses a thin flat conductor which is parallel to a ground plane. Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate while the other side is 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.

Balanced lines

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

Single-wire line

Unbalanced lines were formerly much used for telegraph transmission, but this form of communication has now fallen into disuse. Cables are similar to twisted pair in that many cores are bundled into the same cable but only one conductor is provided per circuit and there is no twisting. All the circuits on the same route use a common path for the return current (earth return). There is a power transmission version of single-wire earth return in use in many locations.

Signal transfer

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

Transmission line circuits

A large variety of circuits can also be constructed with transmission lines including impedance matching circuits, filters, power dividers and directional couplers.

Stepped transmission line

See also: Waveguide filter § Impedance matching

A simple example of stepped transmission line consisting of three segments

A stepped transmission line is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be ${\displaystyle Z_{\mathrm {0,i} }}$.^[13] The input impedance can be obtained from the successive application of the chain relation

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

where ${\displaystyle \beta _{\mathrm {i} }}$ is the wave number of the ${\displaystyle \mathrm {i} }$-th transmission line segment and ${\displaystyle \ell _{\mathrm {i} }}$ is the length of this segment, and ${\displaystyle Z_{\mathrm {i} }}$ is the front-end impedance that loads the ${\displaystyle \mathrm {i} }$-th segment.

The impedance transformation circle along a transmission line whose characteristic impedance ${\displaystyle Z_{\mathrm {0,i} }}$ is smaller than that of the input cable ${\displaystyle Z_{0}}$. And as a result, the impedance curve is off-centred towards the ${\displaystyle -x}$ axis. Conversely, if ${\displaystyle Z_{\mathrm {0,i} }>Z_{0}}$, the impedance curve should be off-centred towards the ${\displaystyle +x}$ axis.

Because the characteristic impedance of each transmission line segment ${\displaystyle Z_{\mathrm {0,i} }}$ is often different from the impedance ${\displaystyle Z_{0}}$ of the fourth, input cable (only shown as an arrow marked ${\displaystyle Z_{0}}$ on the left side of the diagram above), the impedance transformation circle is off-centred along the ${\displaystyle x}$ axis of the Smith Chart whose impedance representation is usually normalized against ${\displaystyle Z_{0}}$.

Stub filters

If a short-circuited or open-circuited transmission line is wired in parallel with a line used to transfer signals from point A to point B, then it will function as a filter. The method for making stubs is similar to the method for using Lecher lines for crude frequency measurement, but it is 'working backwards'. One method recommended in the RSGB's radiocommunication handbook is to take an open-circuited length of transmission line wired in parallel with the feeder delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in the strength of the signal observed at a receiver can be found. At this stage the stub filter will reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the stub will become a filter rejecting the even harmonics.

Wideband filters can be achieved using multiple stubs. However, this is a somewhat dated technique. Much more compact filters can be made with other methods such as parallel-line resonators.

Pulse generation

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