Scattering Matrix parameters: Conceptual questions

[dalarev]

First off, this is not a request for a numerical answer. We've started discussing scattering matrices in my High frequency engineering class, and I'm having some trouble understanding wave mechanics. This got no attention in the homework forum, so I'll just copy paste from there.

In a 2x2 matrix, scattering parameter S₁₁ can be defined as the ratio of the reflected wave from port 1 to the incident wave on port 1. If it is a 2-port network, we imply the condition that the load at port 2 is matched to the characteristic impedance of the line.

Take this excerpt from another website:

"If the output port 2 is terminated, that is, the transmission line is connected to a matched load impedance giving rise to no reflections, then there is no input wave on port 2."

My confusion it the "no input wave on port 2" part. A matched load impedance at port 2 will give rise to no reflections. Fine, I believe that. That can be seen from the definition of the reflection coefficient, Gamma.

However, there is no input wave on port 2? If there is no incident wave on port 2 to begin with, then why does the load impedance need to be matched? There is no danger of reflecting anything anyway, so why match the load (at port-2) to the line?

 

[waht]

However, there is no input wave on port 2? If there is no incident wave on port 2 to begin with, then why does the load impedance need to be matched? There is no danger of reflecting anything anyway, so why match the load (at port-2) to the line?

Take S11 for example, it's the ratio of reflected wave over incident wave at port 1. Even though port 2 is not used in this measurement we still put a load on it. The main point of s-parameters is to describe a 2 port network as it behaves under 50 ohm source and load only. It's still possible to have some weird non-linear 2 port network where the measurement of S11 will depend on the load at port 2.

Another example, S21 measures the ratio of a transmitted wave (port 2) over incident wave (port 1) so the load must be terminated - to get reliable measurements.

an example of a network analyzer: the source is 50 ohms, and the receivers are 50 ohms.

http://www.tmworld.com/photo/136/136661-TMW0102F3FIG5.gif

 

[dalarev]

Another example, S21 measures the ratio of a transmitted wave (port 2) over incident wave (port 1) so the load must be terminated - to get reliable measurements.[/PLAIN]

Thanks for the reply.

In a 2 port network, S21 would measure the ratio of the reflected wave FROM port 2, to the incident wave ON port 1. We apply the condition that V⁺₂ = 0 , which implies that port 2 is terminated on a matched load, 50 Ohm.

Since the load is matched at port 2, would that mean there will be no reflected wave FROM port 2? We're saying the incident wave at port 2 is non-existant, so how can there be a reflected wave?In other words, is S₂₁ = 0 ?

 

[f95toli]

Thanks for the reply.

In a 2 port network, S21 would measure the ratio of the reflected wave FROM port 2, to the incident wave ON port 1.

It is probably less confusing to think of it as S21 being the wave TRANSMITTED from port 1 to 2.

Since the load is matched at port 2, would that mean there will be no reflected wave FROM port 2?

Yes, a wave impinging on a 50 Ohm will always ge totally absorbed(in a 50 ohm system). But you can still get power coming OUT from a 50 ohm port.

In other words, is S₂₁ = 0 ?

No, most microwave components will have both 50 ohm output and input impedance, that does not mean that you can't transmitt power through them.
The key here is to realize that the S parameters for a "black box" don't tell you anything about what is going on INSIDE the box;for active and non-linear components the S parameters are essentially independent and there is no relation at all between them, e.g. an ideal isolator would have S21=1, S12=0, S11=0 and S22=0

 

[dalarev]

It is probably less confusing to think of it as S21 being the wave TRANSMITTED from port 1 to 2.

S₂₁ = V^-₂ / V⁺₁

when V⁺₂ = 0.

The incident wave occurs at port 1, is this wave going to penetrate through port 1 and cause a reflection in port 2? V^-₂ is not 0, I'm not understanding why.

The key here is to realize that the S parameters for a "black box" don't tell you anything about what is going on INSIDE the box

I'm not exactly sure what you mean by that. Is this a way of saying "just apply the formulas and it'll become clear later"?

 

[f95toli]

The incident wave occurs at port 1, is this wave going to penetrate through port 1 and cause a reflection in port 2? V-2 is not 0, I'm not understanding why.

No, because there is no wave traveling inside the "black box" characterized by the S-parameter. Part of the energy hitting port 1 will be reflected (S11) and the rest will -in the model- immediately come out of port 2(S21); nothing can "happen" to the wave inside the "box". Of course this is not what really happens, but the S parameters allow us to "pretend" that this is the case.

I'm not exactly sure what you mean by that. Is this a way of saying "just apply the formulas and it'll become clear later"?

No, it means that there really IS nothing more to it than just "applying formulas". Remember that all we are doing is using a mathematical trick. There is no new physics involved.
It is a bit like modelling a transistor by using an ideal current source plus a few resistors, it is a model that works well but it doesn't tell us anything about what is happening inside a real transistor.

The reason why S parameters are so useful is that they allow us to simplify a complicated network to a simple two port. It doesn't matter the network e.g. consists of a long transmission line where the wave is being reflected back and forth because of some impedance mismatch at its ends. As long as the network connected by two ports it is still complettely characherized by the 4 S parameters (although they are of course in general all frequency dependent) and we don't have to worry about reflections INSIDE the "box".

 

[f95toli]

I forgot to mention something, although I am not sure it if helps.

Remember that you for passive networks always can change the representation from S-parameters to Z-parameters, the two representations contain exactly the same information.

In the Z-representation the complete network is characterized by four equivalent impedances, which in matrix form allow you to write V=ZI.

However, in most cases the S-representation is more useful.

 

[waht]

The incident wave occurs at port 1, is this wave going to penetrate through port 1 and cause a reflection in port 2? V^-₂ is not 0, I'm not understanding why.

Consider again how the s-parameter test set measures s-parameters.

First, a signal having 50 ohms (source) is applied to a 2-port network which is terminated to a 50 ohm (load) power meter.

The reflected wave at the input is immediately picked up by a power meter, and the ratio is taken by the CPU to get S11. While at same time, the transferred wave passes through the 2-port network and is measured by the another power meter. A second s-parameter S21 is quickly obtained in one single shot.

If there is another reflection at port 2, then it won't be picked up by the second power meter. The reflecting wave may or may constructively or destructively interfere with the reflected wave at port 1. But even if it does, you still have characterized the behavior of the 2-port network under 50 source/load.

To measure the other parameters, the internal relay switches the pathways and the process is repeated when the set up is in reverse. You get S12, and S22.

 

[f95toli]

Another unhelpful(probably) piece of information that might help you understand just how "general" the S-parameter approach is that you can calculate S-parameters for just about any system, not only microwave networks.

I've even published a couple of papers which contain plots of S-parameters that were calculated using models that are fully quantum mechanical (by first calculating the susceptibility using methods I "borrowed" from quantum optics, then the impedance- i.e. the Z matrix- and from that finally the S-parameters).
There is nothing stopping you from e.g. calculating the S-parameters of for example a single atom as long as you set up your measurement so that it can be treated as a two-port.

 

[dalarev]

I've attached a schematic to add some meat to my problem.

The incident waves are pictured going into the port, which I assume is how they should be. Imagining all other sources (at all other ports) off, the incident wave V1+ is the only source acting in this network.What's going to happen when the wave enters the system? They start from a source and go (transmit) through the terminal plane, t1 as depicted here. Do we know anything about the source impedance vs the terminal plane impedance (which would equal the input impedance)?

Aside from that, I'd imagine that once the wave has penetrated t1, it will travel right pass the other terminal planes, since they are all matched to the line. Is that what happens when a wave is incident to a perfectly matching load? There is no reflected wave, so does that mean that all the wave is transmitted or...absorbed?

 

[waht]

What's going to happen when the wave enters the system? They start from a source and go (transmit) through the terminal plane, t1 as depicted here. Do we know anything about the source impedance vs the terminal plane impedance (which would equal the input impedance)?

We don't know anything about DUT (device under test). The whole point of s-parameters is to probe a network and find out how waves are scattered from it so to speak.

Is that what happens when a wave is incident to a perfectly matching load?There is no reflected wave, so does that mean that all the wave is transmitted or...absorbed?

could be absorbed, could be transmitted, or could be reflected off by some other components inside a DUT, that's why it's sometimes called a black box. the weirdness usually occurs when dealing with non-linear components such as diodes and transistors.

 

[dalarev]

We don't know anything about DUT (device under test). The whole point of s-parameters is to probe a network and find out how waves are scattered from it so to speak.

I'll buy that.

could be absorbed, could be transmitted, or could be reflected off by some other components inside a DUT, that's why it's sometimes called a black box.

Alright, so when we have the ratio V2- / V1+, with V2+ = 0 , we're just assuming that whatever reflected wave we measure, in this configuration, is due to the source at port 1 alone?

Or more clearly in an example, what is the difference between V2- / V1+ and V1-/V1+ ? I think this is where I'm getting confused; in both cases V2+ is off. I'm failing to see how we can tell the difference between them analytically.

 

[waht]

could be absorbed, could be transmitted, or could be reflected off by some other components inside a DUT, that's why it's sometimes called a black box

in the case where the input and output impedances are matched there is no reflection.

Alright, so when we have the ratio V2- / V1+, with V2+ = 0 , we're just assuming that whatever reflected wave we measure, in this configuration, is due to the source at port 1 alone?

yes because you are actually introducing a signal source at port 1, and see what comes out in other ports, and what gets reflected in port 1.

then do this for other ports

Or more clearly in an example, what is the difference between V2- / V1+ and V1-/V1+ ? I think this is where I'm getting confused; in both cases V2+ is off. I'm failing to see how we can tell the difference between them analytically.

V1-/V1+ is S11, (what you put in port 1 and what comes out of port 1)

V2-/V1 is S21: or the gain (what you put in port 1 and what comes out of port 2)hopefully I didn't mix up the indices

 

[waht]

Another unhelpful(probably) piece of information that might help you understand just how "general" the S-parameter approach is that you can calculate S-parameters for just about any system, not only microwave networks

an example could be the scattering of electrons off of a finite potential barrier.

S21 would be due to quantum tunneling

 

[dalarev]

First off, thank you all for your patience :rofl:.

The problem is I was set on the fact that V2- had to be the reflected wave from V2+, because well, they have the same number.

I see now, however, that in the S21 case, where V2+ is off (matched load), the V1+ wave that penetrates port 1 will again transmit through port 2 unscathed (since port 2 is matched to the line). So even though it is due solely to V1+, we call it V2- just because it exits at port 2...

With that said, let's consider S11 again. Is it fair to say that if V1- exists, it will be due SOLELY to the mismatching between the generator and the input impedance (at the transmission plane t1) at port 1?
I believe that is the case. I say solely because S11 assumes that all other ports are matched, so there is no reflection due to any other ports, except possibly port 1 itself.

Again, thanks a bunch for the guidance, I really did add some new material to my arsenal.

 

[f95toli]

With that said, let's consider S11 again. Is it fair to say that if V1- exists, it will be due SOLELY to the mismatching between the generator and the input impedance (at the transmission plane t1) at port 1?

Yes that is correct, although remember that input impedance is an effective parameter in the sense that we don't know WHERE the mismatch is; the wave could could have been reflected from some place far inside the DUT and not from some point near the physical port (e.g. the connector on the DUT); we only know that we are sending in one wave and getting another wave back.

I believe that is the case. I say solely because S11 assumes that all other ports are matched, so there is no reflection due to any other ports, except possibly port 1 itself.

I am not sure what you mean here. S11 is always well-defined, regardless of what is happening at the other ports. Although if we want to use S11 to measure properties of the DUT we obviosly need to worry about matching etc at the other ports because otherwise we might get reflections from some part of the circuit that is not technically part of the DUT.

 

[waht]

Have a read through Agilent app notes:

http://contact.tm.agilent.com/data/static/downloads/eng/Notes/interactive/an-95-1/an-95-1.pdf