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What is the difference between characteristic impedance and input impedance in transmission lines? When are these quantities are equal?

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What is the difference between characteristic impedance and input impedance in transmission lines?

Characteristic impedance (\$Z_0\$) depends on the transmission line and its physical properties. Mathematically it can be shown that if you know the inductance (L), capacitance (C), resistance (R) and conductance (G) per unit length, \$Z_0\$ is: -

\$\sqrt{\dfrac{R+j\omega L}{G+j\omega C}}\$

And of course these quantities can be deduced from the physical dimensions, dielectric properties (including dielectric losses) and conductivities of the materials used.

when these quantities are equal?

  • If you have an infinite line then input impedance = \$Z_0\$
  • If you terminate a non-infinite line in \$Z_0\$ they are equal
  • If you terminate the line in an impedance not equal to \$Z_0\$ then, providing you choose the correct line length, the impedances can be made equal.

This last bullet point makes use of the relationship between input impedance, load impedance and \$Z_0\$ in the following way: -

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\$V_P\$ is velocity of propagation as a ratio to speed of light and, for normal coax cables is about 0.7 but is \$Z_0\$ value dependent.

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The characteristic impedance is a function of the line only.

The input impedance of a line is a function not only of its characteristic impedance, but also of its loading impedance and electrical length (or physical length and frequency).

They are equal when the line is loaded in its characteristic impedance.

A quarter-wave line will present an input impedance of \$\frac{Z_{char}^2}{Z_{load}}\$

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The input impedance of the line is the ratio of the voltage and current at any point. These current and voltage values are the result of interference between incident and reflected signal.

If the line is infinitely long, the reflected signal from the load will take some time to go back to the beginning of the line. Hence, the current and voltage values will only be due to the incident signal. If you take the ratio of this voltage and current due to only the incident signal, you will get the characteristic impedance of the line.

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  • \$\begingroup\$ "If the line is infinitely long, the reflected signal from the load will take some time to go back to the beginning of the line": If the line is infinitely long (and show no discontinuity), the incident signal will never reach the load, and therefore won't return to the source. \$\endgroup\$ – mins Dec 5 '19 at 14:30

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