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Re: S11 ? if Zc complexe

 

Telecommunications engineers who work with transmission lines (telephone cable pairs) that exhibit complex characteristic impedances (Zo) generally use return loss and reflection loss to characterize those facilities, not SWR and reflection coefficient.

The reason might best be made clear by an example from voice frequency transmission where the angle of Zo is close to -45 degrees. An ideal case might be a cable pair that presents Zo = 600 - j600 ohms at 1000 Hz. To transmit maximum power to the load, we should make the load (Zl) the conjugate of the line, 600 + j600 ohms.

Then the reflection coefficient = rho = (Zl - Zo) / (Zl + Zo) = 0 - j1 ohms
SWR = (1 + |rho|) / (1 - |rho|) = infinity

If you use the replacement theorem I mentioned previously to replace a portion of the load's impedance with an equivalent voltage generator, you will find that it generates a voltage that is in quadrature with the current and satisifies the calculation of the reflection coefficient. An examination of the voltage waveform as a function of position along the line away from the load will show that the voltages are periodically out of phase and add vectorially to zero as is required by the denominator of the SWR expression.

So both the reflection coefficient and the SWR are calculated correctly for such an extreme case, but they are not meaningful as measures of performance as are return loss and reflection loss. If you deal with Zo complex, but with a smaller imaginary component, just be aware that for Zo other than real, the reflection coefficient and SWR will not be exact measures of what is happening in the line.

Chipman points out that, from an exact standpoint, SWR has meaning only for a losless line. When the line is lossy, adjacent peaks and nulls in voltage will not be measured (or calculated) from the same signal voltages and their ratios will not be a mathematically accurate measure of the performance of the line. We can ignore this in most practical cases. He points out that we often assume that Zo = sqrt(L / C) and, for the same line, assume some loss, which means that either R or G or both cannot be zero. The small errors caused by these assumptions can usually be ignored.

73,

Maynard
W6PAP

On 6/28/23 08:37, Maynard Wright, P. E., W6PAP wrote:
Everitt and Anner, "Communication Engineering," Third Edition, McGraw-Hill, 1956, provide a general method (pages 330-331) for examining a mismatch between a source and load, such as a transmission line feeding an antenna.? They factor the source and the load into series networks, one element of which in each case is Zo, the desired impedance.
They then use "Compensation Theorem A" from page 107, which states":
"Any impedance in a network (either linear or nonlinear) may be replaced by a generator of zero internal impedance, whose generated voltage at every instant is equal to the instantaneous potential difference produced across the replaced impedance by the current flowing through it."
Using this theorem, they replace the added network segments by voltage generators so that the resultant equivalent network consists of a matched source and load with added voltage generators to represent the voltages generated by the mismatch.
I applied this to the special case of the complex conjugate load in "More Octave for Complex Zo," QEX, September/October, 2019.? In that case, no modification of the source voltage or impedance is needed and the complex conjugate load is replaced by a matched load (image impedance) in series with a reactance of opposite sign that is twice the value of the reactance of the complex conjugate load's reactance.? That "extra" reactance may then be replaced by a complex voltage source of zero internal impedance that will be seen to be generating a voltage that corresponds to the value of the reflected signal.
73,
Maynard
W6PAP
On 6/26/23 23:01, Fran?ois wrote:
That's it, we have converged. In my application where I am looking for an equality between a value obtained by scanning and a target value, the distance between between the two points (in the complex plane) is a much better criterion for a good result than the ROS which only complicates by creating a non-monotonic function.

When the result of the calculation is equal to the target, the ROS is indeed ONE and the distance zero. It is in the search for the optimal that the ROS disrupts everything.
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