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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: -

enter image description here

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

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  • \$\begingroup\$ this dodgy looking site appears to have kidnapped my answer. \$\endgroup\$
    – Andy aka
    Commented Jan 11, 2022 at 15:54
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    \$\begingroup\$ and the amusing thing is, they've left the markup as is, so it's broken on their site \$\endgroup\$
    – Neil_UK
    Commented Feb 9, 2023 at 8:07
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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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What is the difference between characteristic impedance and input impedance in transmission lines?

When solving the telegrapher's equations you will find that $$I_+=\frac{V_+}{\cfrac{\gamma}{Y}}=\frac{V_+}{Z_0}$$ where we have defined $$Z_0=\frac \gamma Y=\frac{\sqrt{\left(R'+j \omega L'\right)\left(G'+j \omega C'\right)}}{\left(G'+j \omega C'\right)}$$ as the characteristic impedance of the transmission line and where \$I_+\$ and \$V_+\$ are the complex constants relative to the incident wave of, respectively, the current and the voltage difference phasors in the transmission line.

As you can see the characteristic impedance depends only on the \$R'\$, \$L'\$, \$G'\$, \$C'\$ values, which depend on various factors such as the geometry of the line, the materials used, etc... So the characteristic impedance is an intrinsic property of the transmission line. You can also see that the characteristic impedance defines the ratio between the \$V_+\$ and \$I_+\$ complex values inside the transmission line.

You can think of the characteristic impedance as the ratio between the voltage difference and current phasors if there was only an incident wave, and no reflected wave (so for example in an hypotetical infinite length transmission line or one with a reflection coefficient of 0): $$\frac{V(-l)}{I(-l)}=\frac{V_+e^{j\beta l}}{I_+e^{j\beta l}}=Z_0\frac{V_+e^{j\beta l}}{V_+e^{j\beta l}}=Z_0$$ (in this example \$\beta\$ is the imaginary part of the propagation constant, assuming we are working with a lossless transmission line, assumption I will mantain from here onwards)

Now what is the input impedance?

The input impedance in a transmission line is the ratio between the voltage difference phasor and the current phasor at a given point \$-l\$ (usually that's the point where you have attached an impedance matching network): $$Z_{in}=\frac {V(-l)}{I(-l)}=Z_0 \frac{Z_L+jZ_0\tan{\beta l}}{Z_0+jZ_L\tan{\beta l}}$$ where \$Z_L\$ is the load impedance.

As you can see the input impedance depends on the point \$-l\$ where you measure it, on the load impedance, etc.., so it is not an intrinsic property of the transmission line.

Also beware that the definition of the input impedance seems very similar to the example above, where we've seen that the ratio between the voltage difference and current phasors can be equal to the characteristic impedance if there is no reflected wave; but here we made no such assumption on the presence or absence of that type of wave, which in general will be present in a transmission line.

Indeed let's write the voltage difference phasor and current phasor functions: $$V(-l)=V_+e^{j\beta l}+V_-e^{-j\beta l}$$ $$I(-l)=I_+e^{j\beta l}+I_-e^{-j\beta l}=\frac{V_+}{Z_0}e^{j\beta l}-\frac{V_-}{Z_0}e^{-j\beta l}$$ where I used another consequence of the telegrapher's equations: \$I_-=-\cfrac{V_-}{Z_0}\$. Pay attention to the sign! (which appears in the reflected current phasor as a consequence of the relationship between the direction of the voltage difference vector, the current vector and the wave vector \$\hat{k}\$)

Given that the reflected wave function of the current has a different sign from the reflected wave function of the voltage difference in their corresponding phasors, it is no longer true that the ratio between the two phasors, which, as we said, is our input impedance, is equal to the characteristic impedance of the transmission line.

Indeed usually, in impedance matching, you will try to make the input impedance equal to the intrinsic impedance of the transmission line, which as we've seen is not generally already true.

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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
    Commented Dec 5, 2019 at 14:30

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