I am studying how the lossless T-Junction divider works. I am going to assume that the susceptance in the junction is 0 for simplicity, and it is not relevant to my question.

The structure of a T-Junction consists of a feeding line, with characteristic impedance Z0, and two output lines with characteristic impedances Z1 and Z2 respectively.

We want the feeding line to be matched, so the condition necessary for matching is $$Y_{0} = Y_{1} + Y_{2}$$

Now suppose that we want a 1:1 power dividing ratio at the output. For this to happen we need to have $$Z_{1} = Z_{2} = 2Z_{0}$$ This second condition arises from the supposition that both output ports are matched. Only with that condition, the incident power from the feeding line is equal to the power of the forward traveling wave of output line 1, plus the power of the forward traveling wave of output line 2. In general, the sum of input powers is equal to the sum of output powers. In this particular case, where the are no inputs from lines 1 and 2 and no output from line 0, we get the mentioned result.

Now when we calculate the scattering parameters, I get lost. The result $$S_{11}=0$$ is understandable since we imposed that the feeding line is matched. However, we do NOT have $$S_{22}=0$$ or $$S_{33}=0$$.

Mathematically this is correct by the definition of the S-parameters. I am not discussing that. However, from a physical and logical point of view, this is something I really do not understand. How are these two parameters non-zero if we assumed that there were no reflected waves from the interface in lines 1 and 2?


1 Answer 1


The scattering matrix of reciprocal, lossless blocks are unitary. This means that there are no reciprocal lossless three ports that are matched at all ports

see. for example: physics.stackexchange.com/questions/487465/impossibility-of-lossless-matched-3-port-combiner

or if you just google "3 port lossless scattering matrix" you will get lots of explanations


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