2 port networks

[Simon]

Well having finished one section of my course i start the next. 2 port networks. I have read a few pages of giberish now. It seems they are explained as purely mathemartical models and with no practical explanations I am totslly lost. The net and youtube seems full of stuff that is as expanded and unhelpful as my course material. does anyone know of any more practical explanations ?

[Kremmen]

The thing is that 2 port networks are not supposed to be "practical". They are a conceptual model (a black box if you will) to abstract some real process or device.

The port model is a formula in linear algebra such that you (often/usually?) model input voltage and current vs output voltage and current. The format is simply
y = Ax where x is the input vector, y is the output vector and A is the transformation matrix.
The calculation is done by the usual linear algebra rules y1 = A11*x1 + A12*x2;  y2 = A21*x1 + A22*x2

There are several different 2 port models that are used depending on the subject topic- Such as the hybrid model used to model e.g. a transistor.
Check this page: http://gradestack.com/Analog-Electronic/Transistor-Biasing-and/h-Parameter-Equivalent/19280-3866-36878-study-wtw
There in the middle picture you find all 4 terms of the hybrid 2 port network matrix A (they are the hybrid parameters h11, h12 ...). The physical meaning is somewhat obvious from the picture such as h11 is the port input impedance and h21 is the port current amplification (also identified as hFE in transistor data sheets).

Many other 2 port presentations exist but maybe you get the idea from this.

P.S. You need to select the variables above such that y1 = Vi, y2 = Io; x1 = Ii, x2 = Vo

[The Electrician]

This is worth looking at: https://en.wikipedia.org/wiki/Two-port_network

[Neganur]

circuit simulation is based on those models (e.g. Spice) and some models' usefulness is more appearant than others': (already mentioned) H-parameter, S-parameters etc.

They are useful to simplify complex systems, and you will most likely learn to use this as a continuation from circuit analysis courses.

Get a good cheat sheet that lists the models and get to know how you can transform from one to the other. With time you'll know the 3-4 most useful ones by heart anyway.

[Brutte]

A: Once you know the equations that govern each independent building blocks, you can understand how a compound connection of those blocks behave by calculating the combined equation. With two port networks this step from block to complete system is elementary (linear algebra, you can train a dog to solve that).

B: However a design of a real life project works exactly in the opposite direction: You know the equation of the overall system and now it is time to decompose that to elementary building blocks, pick a soldering iron and make it work.

You won't solve task B without deep understanding of task A. Not mentioning that the step from B to A is not unique (not sure how that goes in English, I mean 1:1) as there are many component arrangements that meet same equation.

[T3sl4co1l]

Yeah, it's a black box method.  Would it help to reflect on what it means, practically?

Suppose you have a black box with two BNCs.  What is it?

Plug it into a VNA.  This applies RF to one port (i.e., hooked up a 50 ohm amplifier, which transmits at a variable frequency), and measures the power transmitted (to the other port) and reflected (back to the amplifier; a carefully balanced network separates the transmitted and reflected waves).

The various frameworks (h, Y, s, ABCD...) are just different ways of expressing the same thing, usually with a purpose in mind.  h-parameters are easier to measure at low frequencies, s-parameters at high.  Matrix decomposition methods tease apart the different parameters of the system, so they can be more easily worked with (but, this probably doesn't interest you much, because math).

Maybe this is way ahead of where you're at (what's a VNA?).  Suppose you just probe the box with a multimeter.  Suppose it's a resistor network!  How can you tell what it is?  It could be some complicated mess, but all that can be reduced to a few resistors.  (To be exact: (N^2 - N) / 2 resistors, for N nodes.)  For the two-port box, if you connect the grounds together, it has N = 3, so needs 3 resistors.

You have 3 unknowns, so you need to collect at least as many parameters.  Note that resistance is a ratio of voltage to current, so you need four parameters (e.g., two voltages and two currents), but only end up with three in the end (one gets eliminated).

That's easy to see, because to measure a resistance, you have to apply some voltage (or current), but the exact amount doesn't matter, as long as it's nonzero: it's a free variable.

So, you might apply a voltage between nodes 1 and 2, and measure the resulting current flow, and measure the voltage on nodes 2 and 3.  You now have V12, I12 and V23.  Lastly, you can short nodes 2-3 and measure the current, to get I23.  Now you have a complete set of data.

Alternately, you could apply voltage to nodes 2-3 and measure voltage V12, to obtain 3 voltages and 1 current.  You can't measure only voltages, or only currents, because then you can't find the resistance.

The symmetries at work here, are very similar to those of geometry: you can define a triangle from three sides, or two sides and an angle, or one side and two angles; but not three angles, because then there's no length reference.  Learning how the underlying symmetries work, can save you from a lot of trouble manipulating equations you're not otherwise very prepared to work with (like because your eyes glaze over every time you look at it!) -- anywhere you see the wrong combination of numbers being multiplied and divided together, or different amounts being added or subtracted, you can be sure you've made a mistake.

Tim

[rstofer]

I have been wondering why you got to 2 port networks so fast.  We didn't do it until much later than DC and AC circuit analysis.  Then I tumbled to the idea that they are just taking a more formal approach to Thevenin and Norton equivalent circuits.  Making the internals 'active' by modeling a transistor isn't a huge stretch.  When I was in school, the idea of 2 port networks didn't come up until filters and transmission lines - I still have the book...

I haven't done Chapter 4 of "Real Analog" and but I have worked through the first three chapters and the lectures are very good.  In the spare time you don't have, why not take a look:

https://learn.digilentinc.com/classroom/realanalog/

I went through the first 3 chapters when you posted the analysis problem a couple of weeks ago.  Good review, expecially the bit about the Super Node.

OT:  You know, it is common for students to form "study groups", hire tutors and ask questions of the TAs (Teacher's Assistants).  Your study group just can't all get to Mickey Dees at the same time.  Remote learning at its best!

[Vtile]

Were these the CCVS. VCCS. CCCS. VCVS ? I'm forgotten so much.. 

[rstofer]

Were these the CCVS. VCCS. CCCS. VCVS ? I'm forgotten so much.. 

They are, as they are all forms of dependent sources.

It's good that Simon brings up these topics.  If I EVER knew the  material, it was a very long time ago and the review has been interesting as well as educational.

Between these problems and my grandson's Calculus course, I'm getting quite a re-education.  I doubt there is any science to the idea but a lot of folks believe that keeping the brain working in old age is a good way to stave off Alzheimer's.  So, Simon, keep it up!

[rstofer]

I haven't done Chapter 4 of "Real Analog" and but I have worked through the first three chapters and the lectures are very good.  In the spare time you don't have, why not take a look:

https://learn.digilentinc.com/classroom/realanalog/

It's not until video 1.13 that the instructor gets to VCVS using an Op Amp.  The previous videos were strictly Thevenin and Norton, something that you have already covered (I assume).

I don't know how far the instructor will get with dependent sources in this series.  He implies that the majority of the coverage would come in a future course.

[Simon]

Sorry it's taken me so long to reply.

Thank you all for the help and encouragement. I really wish I had a better understanding of maths and was able to remember more of it as at the end of the day everything relies on it. To whoever pointed out that normally the task is to take a system that needs designing and has a known input to output equation and then that needs breaking down into smaller blocks. That is exactly why I wish I had a better grasp of maths and was able to understand it better.

I've just had to take a short detour into matrices and using them to solve equations (as if the 1600 pages of this module were not enough already) so I'm now back onto the actual 2 port material.

[Simon]

I have been wondering why you got to 2 port networks so fast.  We didn't do it until much later than DC and AC circuit analysis.  Then I tumbled to the idea that they are just taking a more formal approach to Thevenin and Norton equivalent circuits.  Making the internals 'active' by modeling a transistor isn't a huge stretch.  When I was in school, the idea of 2 port networks didn't come up until filters and transmission lines - I still have the book...

I haven't done Chapter 4 of "Real Analog" and but I have worked through the first three chapters and the lectures are very good.  In the spare time you don't have, why not take a look:

https://learn.digilentinc.com/classroom/realanalog/

I went through the first 3 chapters when you posted the analysis problem a couple of weeks ago.  Good review, expecially the bit about the Super Node.

OT:  You know, it is common for students to form "study groups", hire tutors and ask questions of the TAs (Teacher's Assistants).  Your study group just can't all get to Mickey Dees at the same time.  Remote learning at its best!

Oh don't worry, my next module is a project. Lord knows what sort of project they expect people to produce having only studied maths some physics and elementary circuit analysis. Perhaps a battery a switch and a light bulb.

However in all seriousness I think the project module is supposed to be for people to go away and study something of their own and produce a project. Well if they want they can have 1 of my commercially available brushless fan controllers with full explanation of how it works.

[T3sl4co1l]

Congrats on your recent 10k posts, by the way.

[Simon]

Congrats on your recent 10k posts, by the way.

Oh wow, I hadn't even noticed. Last time I looked I was just over 9000, or so I thought.

[rstofer]

Oh don't worry, my next module is a project. Lord knows what sort of project they expect people to produce having only studied maths some physics and elementary circuit analysis. Perhaps a battery a switch and a light bulb.

However in all seriousness I think the project module is supposed to be for people to go away and study something of their own and produce a project. Well if they want they can have 1 of my commercially available brushless fan controllers with full explanation of how it works.

Post the problem when you get it!

[Simon]

Well if it's just a case of me coming up with a project then that is much easier. This is the whole conundrum of this course, while I can't memorise and understand the depth maths or do a lot of the elementary circuit analysis I don't have a problem designing a circuit board to accomplish a task and even writing a little C code. Maybe the grade I get for the project can balance out the shit grade I'm going to get on the current module.

[Simon]

Well as it happens yes they want me doing transmission lines, the 2 port networks was to chuck me into that so they got it over with in one lesson out of 4 and except one question the whole assignment to foloow is about transmissions lines.

[G0HZU]

A simple inductor (wound as an air spaced solenoid) is a great component to study/model as a two port network because it is not as simple (in reality) as a pure inductor. You can either model it using lumped components like many people do or you can use transmission line theory to get a better model across a wider frequency range or you can go even further and measure it as a two port network (using a VNA) up to your chosen upper frequency limit.

The lumped (LCR) model is the classic inductor model but it falls apart even before you reach resonance. A crude transmission line based model can model it better up past the first resonance but to really capture the inductor as a wideband 'model' the two port model  is usually the best providing you just need a small signal (i.e. linear) model of the inductor. The two port model is usually obtained using a VNA that can measure the inductor in a fixture and then the VNA can output the measurement data in s parameter format)

[T3sl4co1l]

The lumped (LCR) model is the classic inductor model but it falls apart even before you reach resonance. A crude transmission line based model can model it better up past the first resonance but to really capture the inductor as a wideband 'model' the two port model  is usually the best providing you just need a small signal (i.e. linear) model of the inductor. The two port model is usually obtained using a VNA that can measure the inductor in a fixture and then the VNA can output the measurement data in s parameter format)

For some in-depth background on the matter, see:
http://hamwaves.com/antennas/inductance.html
This guy has researched the subject extensively.  (Making a detailed RLC model, equivalent to the parameters calculated by the formulas, would be relatively trivial.)  I don't suggest reading this in detail -- but skimming over it to see some of the effects at work, and seeing what methods are used to solve them (sometimes theoretical, sometimes empirical), is fascinating.

Tim

[G0HZU]

Yes, I've seen a lot of that stuff before, but I  really wanted to suggest alternative/better methods to model an inductor across a wide frequency range. The lumped models are pretty lame in this respect.

There seems to be a lot of research going on by the people in your link(s) but for reasons that totally baffle me they still want to chase the elusive 'self capacitance' to ultimately use in a lumped model. Some of those guys appear to have been into this stuff for decades  but I would suggest that even a junior RF engineer would leapfrog that stuff fairly quickly as the lumped model is so limited in frequency range.

That's why I suggested Simon looks at a typical 2 port model of an inductor or even a simple tline based model as a practical example to show how these models usually outperform the old school lumped models for wideband design work involving inductors

[G0HZU]

A practical example would be a very simple highpass filter. Maybe for use with a scanner receiver where reception below (say 10MHz) may cause overloading. The ideal HPF would be flat from 10MHz up to maybe 1GHz or more to allow it to be used over the full scanner range.

But if someone built such a HPF using a big (air) solenoid coil then they would discover that there would be some dips or notches in the response up in the UHF region when tested. These sharp/narrow dips could look like 10-20dB holes in the receiver range and would cause deaf spots. I don't think any of the mainstream research by the chaps who are chasing the ultimate  'inductance/self capacitance/SRF equations' would help much here because these resonance modes are not related to the classic lumped model that they help produce.

A two port model of just the inductor will typically give a very good indication of where these dips will appear once the inductor is fitted to the HPF. It would also be able to predict the width and depth of them quite well. Simply use the 2 port model in an RF simulator and draw the complete HPF circuit and then simulate/predict where trouble will happen.

A simple transmission line based model would be almost as good, but it won't predict the notch depths or locations as accurately.

[T3sl4co1l]

At least the notches are (usually?) pretty modest.  The helical waveguide is dispersive and quite high impedance.

Roughly speaking: think of each pair of turns as a parallel pair of transmission lines.  The end-to-end impedance is approximately (N-1) times the transmission line impedance of that pair.  Nearby pairs of round wire, in air, are in the 100-200 ohm range, so a ten turn inductor can easily be kohms.

The first resonant mode is parallel, so the impedance peak is towards infinity (limited by Q; practical cases are in the 10s to 100s of kohms).

Subsequent valleys and peaks (they always alternate) have similar ratios above and below the mean impedance.  But again, because it's dispersive, because those parallel wires are so intimately linked together, there isn't really a mean impedance, nor do the peaks and valleys come at integer multiples of the first resonant frequency.

So... to do an HPF like that, yeah, there may be dips.  Even though the dips can be quite significant relative to the mean impedance, it can also be that the mean is so high (~kohms) that it doesn't have much effect (not 10-20dB, but maybe only 2dB or less).  That's been my -- rather limited -- experience, but I've not played with enough combinations of coils and sizes, and not over that wide a frequency range (2 decades).

Also, ferrite loading can cure a lot of ills: for helical and helicotoroidal* inductors, you only get a couple of modes before the whole thing kind of decays into mush (the equivalent loss resistance of the core dominates over the wave properties of the winding).  Ferrite is wonderful stuff..

*As in, a helix wound around a toroidal path.  A single layer toroid winding.  These have a first resonance that's one full wavelength (because of the continuity of magnetic field around the loop), and are also dispersive (non-harmonic).  The resonant modes are relevant to any kind of toroidal transformers (including CTs), RFCs and tuning coils.  Different winding and shielding designs can be used to short out the resonant modes.

The worst off-the-shelf example I know of is the Triad CST-206, which is rated for 100kHz bandwidth, and they mean it: the first resonant mode is at a paltry ~300kHz, so the square wave response is absolute shit for anywhere near rated bandwidth.  The response also varies with where the primary loop is placed: if it's at right angles to the secondary pins, the ringing is minimal.  So it varies with placement...

Tim

[G0HZU]

So... to do an HPF like that, yeah, there may be dips.  Even though the dips can be quite significant relative to the mean impedance, it can also be that the mean is so high (~kohms) that it doesn't have much effect (not 10-20dB, but maybe only 2dB or less).  That's been my -- rather limited -- experience, but I've not played with enough combinations of coils and sizes, and not over that wide a frequency range (2 decades).

You certainly talk a good game However, why not just dig out an air spaced solenoid design program to help design the inductor, then design a 3rd order HPF for about 10MHz and wind a large (high Q) air wound inductor in the way a typical radio ham would do. I'd expect a ham to go for fat wire and a L/D ratio close to 1.5.

Then sweep the HPF over the range of a typical scanner (2GHz?) and check out just how many suckout dips you see. You are going to see several big ones. Most will be between 10-20dB deep and not the <=2dB you seem to suggest.

You can wind different types of (smaller, more loss, powdered iron?) inductor to change the dip responses/frequencies and minimise the depth of any dips but that isn't the point. The point is that the dips are there in the first place and a 2 port model can be used to model the inductor BEFORE it gets fitted to the filter. The model should be able to show you where the dips will appear, how deep they are and how wide they are. All I'm trying to do is offer Simon a practical example where a two port network can be used to good effect.

[G0HZU]

Note that you can use the same 2 port model for the inductor in a LPF design. eg a 3rd order LPF in the SW band and the model will help predict where the stopband will degrade up in the UHF region. A LPF with a big air solenoid like this could have a stopband that degrades to less than 20dB in places due to (narrow) resonance modes in the solenoid up towards 1GHz. It should be possible to predict this quite well in a simulator once the 2 port model is created

[Simon]

Well I'm lost and a little bit more scared of the topic

[G0HZU]

Well I'm lost and a little bit more scared of the topic

That's possibly because all the complicated stuff Tim posted and linked to after I suggested the inductor studying wasn't really needed. The whole point of an n port model is that you don't 'need' to know anything about the device as long as it is a linear device and you collect VNA data from it in a steady state.

Here's a 1 port s parameter model of a shunt inductor at 11MHz. You can see that it is just a single reflection coefficient measurement. Once you start looking at transmission lines you will be looking at reflection coefficients quite a lot.

# MHZ S MA R 50
11 0.995 97.138

A 2 port model of the same inductor will look a bit more complicated because you need to measure and record 4 coefficients with the VNA (not just one) and it will be measured in series between two 50R test ports.

# MHZ S MA R 50
11    0.399 65.95     0.915 -23.957     0.915 -23.954     0.4 65.927

Normally the VNA would produce a 2 port model like this every MHz or so from 1MHz up to maybe 2GHz and store it all in a big data file.

But in the case of the high pass filter example you could just use a 1 port (shunt inductor) model made up of hundreds of VNA measurements across 0.3MHz up to 2GHz and use this data file as a model in a HPF circuit in an RF simulator to predict the passband response of the complete HPF. This should be able to predict where the unwanted resonance dips will be in the passband.
 

[G0HZU]

To provide a quick demo, I wound a 500nH inductor using about 9 turns (airspaced) with a 14mm diameter and length 26mm and some 0.7mm wire.

I then measured this as a 1 port device using a VNA up to 2GHz. This is just a crude 1 port model of the inductor consisting of 1600 separate measurements of the reflection coefficient of the inductor across 0.3MHz to 2GHz. The VNA doesn't know it is measuring an inductor, it will just provide me with a 1 port model of the inductor.

I then entered this 1 port model into a simulator and used it in a simple (50 ohm) 15MHz highpass filter.
The network was a series 270pF cap then shunt 500nH inductor (but I used the 1 port model for the 500nH inductor here) then series 270pF cap to make up the HPF.

This simulation predicted the high pass filter will have the red trace as below when built and measured. Note all the deep and nasty dips in the passband. Note how they are typically 10-20dB deep

Then I actually built a 15MHz highpass filter using this inductor and some decent quality 270pF caps and a decent layout and I measured it on the VNA. The trace showing the real performance of the real HPF is shown in blue. Note how the 'real' dips are also typically 10-20dB deep Note also how well the simulation with the model agrees with the 'real' circuit.

You can now see that I was able to model this inductor and predict the misbehaviour that will occur in the passband if the inductor were used in a HPF like this. This shows the benefit of using an n port network derived from a VNA. To try and produce a (very complicated) lumped model or even a complicated tline model to give this level of agreement would be very time consuming and difficult. The VNA can create the 1 port model data in a couple of seconds.

[T3sl4co1l]

Nice.

Yeah, I think that 500nH inductor is below the impedances I was working with, so you should indeed see deeper notches!

Tim

[G0HZU]

Yes, this classic big old airwound inductor type gives very deep suckout notches in a filter like this. There are better inductor options but I chose this one as it misbehaves badly here

Leapfrogging over 2 port networks into 4 port, I dug out an old 4 port model and simulation I did of some classic 300 ohm balanced ribbon feeder and posted up a short video on youtube showing a screenshot of the results. I'm lucky to have a decent 4 port VNA here at home and I used it to experiment with balanced transmission line analysis a while back. If each of the 4 wire ends of the ribbon feeder are connected to the 4 ports of the  VNA and the reference plane set right at the start of the feeder then it's possible to play with the model on a simulator to see what the insertion loss will be when in a 300R balanced environment. You can see that the loss is lowest at about 307 ohm which is very close to 300R. The loss is also very low.

All this investigation work is possible because I took a valid n port model of the ribbon feeder where n is 4 in this case because there are 4 connections in total. The VNA I have here can be configured for balanced ports to let it look at balanced feeders directly but I did this the old school way with a basic 4 port connection and then I put baluns in the simulator to keep the model balanced.

Sorry, but there's no sound I don't have a proper mic here and Camstudio is a pain to set up for audio on this Win7 PC anyway. But hopefully you get the idea. It probably won't make much sense to Simon yet but maybe when he has looked at transmission lines and n port networks it will make more sense as this is a classic transmission line that was once popular as a feeder for TV and radio reception in many countries. You can see that I compared the results with a Genesys library model of a 306 ohm transmission line of similar dimensions and it all agrees very closely. Check out that flat group delay!

This is a neat way to look for the characteristic impedance of an unknown transmission line over a wide bandwidth. You can see that I did this two ways. I tweaked the source and load impedance for lowest insertion loss at this was about 307 ohm.

But I also measured (at low frequencies)  the inductance of the line and also its capacitance. As we all know we can calculate Zo from (L/C)^0.5  where C was 7.61pF and L was 724nH. This works out the Zo to be 308 ohm. Pretty close agreement for both methods!



In a way I wish I'd done this up to 1GHz or more because the loss is so low. But a big exposed length of cable like this is prone to RFI pickup and you can see a slight blip at 200MHz in the results for group delay. This might have been caused by some external interference.

https://youtu.be/xHoQAZBUu2k