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 (more on nepers))
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 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

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

sinh(x) = (exp(x)-exp(-x))/2

for complex arguments, 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 and the functions turn into the conventional trig functions:

sinh( j b) = sin(b)

cosh( j b) = cos(b)

NEC TL and NT cards

On line NEC reference: Network (NT) and Transmission Line (TL), both from an online html version of the NEC Part III manual at: http://members.home.net/nec2/part_3/toc.html

The TL card is primarily for lossless lines (although, I suppose that if you had a real shunt admittance on either end, it would be lossy). Of more interest is the NT (arbitrary two port network) card. This card specifies the admittance matrix for an arbitrary two port. You only need 3 complex numbers, because the matrix is symmetric. Here's a quick summary of the format of the card:

NT, <tag1>,<seg1>,<tag2>,<seg2>, <Y11R>,<Y11I>,<Y12R>,<Y12I>,<Y22R>,<Y22I>

So all we need to do is convert the "admittance" form of the transmission line equation (above) into a matrix, and, hopefully, come up with some simple equations to allow cranking out NT cards from useful things about the transmission line: length, velocity factor, loss, characteristic impedance. The admittance matrix tells you the currents, given the voltages:

I = Y E

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

radio/tleqn.htm - Revised 2 November 2001, Jim Lux
radio home | Jim Lux home
email comments jimlux@earthlink.net