Important Transmission-line Equations

Author: Edmund A. Laport

The following equations, derived from transmission-line theory and proved in the classical literature on this topic, have frequent utility in the design of systems, and are grouped here for reference.

In general

At radio frequencies, when ω becomes very large with respect to other factors

When the field of the transmission line is entirely within an isotropic dielectric medium having an inductivity, or dielectric constant, €,

The propagation constant y is in general complex;

At radio frequencies and with lossless lines, 7 becomes essentially a phase angle per unit length.

The velocity of propagation of transverse electromagnetic waves in systems of parallel linear conductors with air dielectric is equal to c, which is the velocity of light in free space (3·108 meters per second).

For a line in an isotropic dielectric ε, the velocity of propagation is

When a radio-frequency line of length βl degrees (or radians) is terminated in a complex impedance Zt the input impedance Zin is, in general, complex, in accordance with the equation

When Zt <> Z0, there is reflection from the termination. The reflection factor is, in general, complex, and is specified as follows:

When Zt = 0 (short circuit),

When Zt = (open circuit),

For a line of length βl = π radians = 180 degrees (one-half wavelength)

also

For a line of length βl = π/2 = 90 degrees (one-quarter wavelength)

This is an impedance-inverting circuit with a 90-degree change in relative phase between input and output currents and potentials. When βl = 45 degrees (one-eighth wavelength) and Zt = Rt + j0,

and in general |Zin| = |Z0| for all values of Rt, positive and negative, from 0 to . (Only the angle of Zin varies with Rt.)

The standing-wave ratio Q on a transmission line increases with increasing inequality between Zt and Z0 both in phase and in magnitude.

This equation for Q is useful when transmission lines are used as high-Q resonant circuits.

When a section of transmission line is used as a transformer to match an impedance Zt = Rt ± jXt with another impedance Zin = Rin ± jXin, the characteristic impedance Z00 of the transforming section is

and its electrical length must be

in which

In all the preceding equations the following symbols apply:

 R = resistance per loop meter, ohms L = inductance per loop meter, henrys G = leakage conductance per meter of line, mhos γ = complex propagation constant per meter of line C = capacitance per meter of line, farads α0 = attenuation constant per meter of line, nepers β0 = phase constant per meter of line, radians Ω = 2 πf f = frequency, cycles f0 = frequency of resonance c = velocity of propagation of light in free space c = 3 · 108 meters per second v = velocity of propagation in the line, meters per second λ = free-space wavelength in meters for a wave of frequency f zin = input impedance, ohms (complex) zt = terminal (load) impedance, ohms (complex) K = reflection coefficient due to terminal mismatch of impedance (complex) ε = specific inductive capacity (dielectric constant) of the medium in which the field of the line is contained SWR = Q = Vmax/Vmin = Imax/Imin - the standing- wave ratio - and is the ratio of energy stored in the line to the energy dissipated per second in the line and in the load termination βl = the over-all electrical length of a line, radians or degrees l = the length of a line, meters ψ = 4πδ/λ d = distance, meters, from the terminal end of a line to the first voltage maximum (or current minimum)

 1) 1 neper = 1 hyperbolic radian = 8.686 decibels, corresponding to a current (or voltage) ratio of 2.7182+ = e.

Last Update: 2011-03-19