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)
beta is length in radians

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)

Admittance Form

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
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