## Transmission Line Equations

#### Impedance transformations

If a load of complex impedance Zload is connected to the end of a transmission line of characteristic impedance Z0 (which may be complex), the impedance at the other end of the line is Zin, calculated by the following equation:

`Zin = Z0 * (Zload*cosh(gamma) + Z0*sinh(gamma))/ (Zload*sinh(gamma)+Z0*cosh(gamma))`

gamma is the complex propagation constant. gamma = alpha + j beta where:

alpha is attenuation in nepers (1 neper = 8.688 dB)

Some cases of interest are where alpha=0 (i.e. a lossless line) in which case the hyperbolic functions turn into their trig equivalents, and where beta is pi/2 or pi.

If beta = pi/2 (a half wavelength line), then cosh(gamma) = -1 and sinh(gamma) = 0, so you get:

`Zin = Zload`

If beta = pi/4 ( a quarter wavelength line), then cosh(gamma) = 0 and sinh(gamma) = 1, so you get:

`Zin = Z0 * Z0/Zload`

or, written another way:

`Zin * Zload = Z0*Z0`

This is the basis of the popular 1/4 wave impedance transformers... To transform Zin to Zload, you need only construct a quarter wave line with impedance = sqrt(Zin*Zload)

`Yin = Y0 * (Yload*cosh(gamma)+Y0*sinh(gamma))/(Y0*cosh(gamma)+Yload*sinh(gamma))`

#### Voltages and Currents

Some algebraic manipulations can be used to create other forms of the transmission line equation for voltages and currents, rather than impedances.

`Ein = Eload *cosh(gamma)+ Iload * Z0 * sinh(gamma)`
`Iin = Iload * cosh(gamma) + Eload * Y0 * sinh(gamma)`

or, in ABCD matrix (also called the chain matrix) form:

`E1 = A*E2 + B*I2`
`I1 = C*E2 + D*I2`

where

`A = cosh(gamma)B = Z0 * sinh(gamma)C = Y0 * sinh(gamma)D = cosh(gamma)`

Y0 is, of course, = 1 / Z0

#### Impedance of transmission line

`Z0 = 1/ Y0 = sqrt( (R + j*{omega}*L)/(G+j*{omega}*C))`

where R, L, G, and C are per unit length

#### Hyperbolic Functions

cosh and sinh are hyperbolic functions (if the arguments are complex, the exponential function below is the complex exponential):

`cosh(x) = (exp(x)+exp(-x))/2`
`sinh(x) = (exp(x)-exp(-x))/2`

Another form, useful for complex arguments, is:

sinh( a + j*b ) = sinh(a)*cos(b) + j*cosh(a)*sin(b)
cosh( a + j*b ) = cosh(a)*cos(b) + j*sinh(a)*sin(b)

if you expand the sinh() and cosh() in the above equations, the following may be useful:

`sinh(a + j b) = .5 * (exp(a)-exp(-a))*cos(b) + j * .5 *(exp(a)+exp(-a))*sin(b)`
`cosh(a + j b) = .5 * (exp(a)+exp(-a))*cos(b) + j * .5 * (exp(a)-exp(-a))*sin(b)`

note that if a = zero in the above equations then exp(a) and exp(-a)=1 (i.e. sinh(a)=0, cosh(a)=1) and the functions turn into the conventional trig functions:

sinh( j b) = j * sin(b)

cosh( j b) = cos(b)

A reversed form is:

sin(a +/- jb) = sin(a)*cosh(b) +/- j*cos(a)*sinh(b)
cos(a +/- jb) = cos(a)*cosh(b) -/+ j*sin(a)*sinh(b)

Visual Basic Module for hyperbolic functions (hyperbolic.bas)

VB module for complex math (Complex.bas)

VB module for doing transmission line equations (incomplete) (TransmissionLine.bas)

#### Some references on transmission line properties

Attenuation at any frequency = (K1 x SqRt(Fmhz) + K2 x Fmhz)

Attenuation of Coaxial Transmission Lines in the VHF/UHF/Microwave Amateur and ISM bands http://hydra.carleton.ca/articles/atten-table.html

tleqn.htm - Revised 21 September 1999, Jim Lux
revised 24 Jan 2003 - fixed error in the next to last equation (thanks to David Peale for catching it)
added some more versions of the equations for cutting and pasting