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Home Radiofrequency Transmission Lines Radiofrequency Currents in Linear Conductors Important Transmissionline Equations  
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Important Transmissionline EquationsAuthor: Edmund A. Laport
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 (3·10^{8} meters per second).
For a line in an isotropic dielectric ε, the velocity of propagation is
When a radiofrequency line of length βl degrees (or radians) is terminated in a complex impedance Z_{t} the input impedance Z_{in} is, in general, complex, in accordance with the equation
When Z_{t} <> Z_{0}, there is reflection from the termination. The reflection factor is, in general, complex, and is specified as follows:
When Z_{t} = 0 (short circuit),
When Z_{t} = (open circuit),
For a line of length βl = π radians = 180 degrees (onehalf wavelength) also For a line of length βl = π/2 = 90 degrees (onequarter wavelength)
This is an impedanceinverting circuit with a 90degree change in relative phase between input and output currents and potentials. When βl = 45 degrees (oneeighth wavelength) and Z_{t} = R_{t} + _{j}0,
and in general Z_{in} = Z_{0} for all values of R_{t}, positive and negative, from 0 to . (Only the angle of Z_{in} varies with R_{t}.) The standingwave ratio Q on a transmission line increases with increasing inequality between Z_{t} and Z_{0} both in phase and in magnitude.
This equation for Q is useful when transmission lines are used as highQ resonant circuits. When a section of transmission line is used as a transformer to match an impedance Z_{t} = R_{t} ± jX_{t} with another impedance Z_{in} = R_{in} ± jX_{in}, the characteristic impedance Z_{00} of the transforming section is
and its electrical length must be
in which
In all the preceding equations the following symbols apply:


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