# Power waves and scattering matrix

I have some doubts about the concept of power waves used in the description of the scattering matrix of an N-port component. In particular, I was wondering what is the most general and correct definition of incident and reflected power wave.

In almost all the texts and sites on which I have searched, it is taken as the impedance of normalization the same characteristic impedance of the LdT connected to the port. However, from what I understand, in the more general case it is possible to assume a normalization impedance different from that characteristic of the transmission line connected to the considered port. In this case, how are the incident and the reflected power waves defined? Is there any text where I can find this argument?

However from what I understand, in the more general case it is possible to assume a normalization impedance different from that characteristic of the transmission line connected to the considered port.

That doesn't sound useful or right.

If you don't use the same impedance of the transmission line ($$\Z_0\$$) you can't properly define reflections based on reflection coefficient ($$\\Gamma\$$): -

$$\Gamma = \dfrac{Z_L - Z_0}{Z_L + Z_0}$$

And this means you can't define standing wave ratio: -

$$SWR = \dfrac{1 + |\Gamma|}{1 - |\Gamma|}$$

And you can't define return loss: -

$$RL = -20\log_{10}(|\Gamma|)$$

I'm at a loss to think how using a non-$$\Z_0\$$ value has any significant use or merit. I've been wrong before of course!

• You can calculate the S-parameters of a segment of (for example) 75 ohm line in a 50 ohm system. You might use it as an equivalent (in a certain band) of a series inductor in your system. Or you might want to know the effect if your PCB trace comes out as 45 ohms or 55 ohms due to manufacturing variations. Commented Jun 24, 2020 at 17:03
• but in that case if i chose the reference impedence Zo not equal to the Zc of the line connected to that port, when i have to find the parameter S11 and S21, i have to put as load for the port 2 an impedence equal to Zo e not equal to Zc, so the incidente power wave a2 will be zero, right? Commented Jun 24, 2020 at 17:23

The choice of Z0 is arbitrary and does not affect the actual V and I in a circuit.

The S parameters are defined w.r.t. to a Z0 in order to normalize V and I to a common "square-root of power" quantity. The magnitude of S depends on Z0.

Now careful...

S parameters for a section of a network describe reflection, absorption and transference as a constituent of super-imposed waves in each section. The real reflection is a sum of reflections etc... from various sections. So the choice of Z0 affects the magnitude of the constituent waves, not their sum. In a ZG and ZL matched system, and a choice of Z0 that equals neither, the reflections will cancel out regardless Z0.

If you choose a Z0=100Ohm in a 50Ohm system (coax, source, load...) there will be non-zero Gammas all abound, but the resultant sum of super-imposed waves, the absorption into a load etc... are unaffected by that choice, and will still "match".

Where the constituent parts DO matter is in transient systems. In order to match-up a calculated Gamma with any observed/real reflections (e.g. spikes/pulses refelecting in a long open-ended or mismatched wire) you will have to use Z0 equal to the nominal Z of your system. If you don't, your Gamma is different, still correct, but it only paints part of the picture, a layer to be super-imposed so to say.