Help in deriving Zin small-signal expression

[promach]

Could anyone guide me in deriving the Zin expression below ? How do I model the two transconductance block in small-signal model including their output impedance ?

[The Electrician]

A model for a single transconductance would be like this:



This is a two-port: https://en.wikipedia.org/wiki/Two-port_network

Its two-port Y parameters can be written like this:



But you have another transconductance going in the reverse direction.  Its admittance matrix looks like this:



A gyrator is two transconductances going in opposite directions like you have.  See: https://en.wikipedia.org/wiki/Gyrator

The Y matrix (admittance matrix) for a gyrator with infinite output impedances for your circuit would be:

[The Electrician]

If you add a capacitor CL at the right side of the gyrator, that load can be added to the admittance matrix of the gyrator in the 2,2 element of the admittance matrix:



Now to find the input impedance of the gyrator with that CL load on the output, we need only invert our admittance matrix:



The 1,1 element of the inverse matrix is your input impedance.  Compare that to what your book gives for the first case.

To add output impedances to the two transconductances we need to add a couple of terms to the admittance matrix:



Now we need to add the capacitor  CL to the output:



Finally, to get the input impedance, we invert the admittance matrix:



The desired input impedance is the red 1,1 element of the inverse matrix.

[promach]

You made things too complicated. Please see the derivation process below:

The first gm stage plus CL forms an integrator. What is it integrating? The voltage at node X. The second gm stage completes a feedback loop which tries to keep the voltage at node X held constant at the point where you get no output current from the first gm stage. If you inject a current into node X, the feedback loop will respond to adjust the voltage on CL until the 2nd gm stage absorbs all of that current. This gives you a low impedance at low frequencies. How low? Just 1/(gm1rout1gm2). I.e. 1/total_gm. What happens at high frequencies? CL becomes low impedance effectively removing the effect of the feedback loop. Now the impedance you see is high. How high? Ignoring capacitances associated with the gm stages, Rout2. So you have a low impedance at low frequencies that becomes high impedance at high frequencies. That is an inductor. For the two break frequencies think about it this way. At very low frequencies, Rout1 has a lower impedance than CL and it isn't until Rout1 = |1/(jwCL)| that you start to see appreciable effects from CL. That is just w = 1/(Rout1CL). At the high frequency end, you can solve for when the inductive part equals Rout2, or you can disconnect node X from all but the gm stages and realize that CL sees a resistance of 1/(gm2Rout2gm1) and so the pole frequency is 1/(gm2Rout2*gm1). Note that this analysis and the plot you show both ignore the effects of capacitances associated with the transistors in the gm stages. Those will be important at some point.

Regarding the quote above, How do I derive the equation "Just 1/(gm1rout1gm2). I.e. 1/total_gm" ?

[The Electrician]

I don't think I made it too complicated.  The image you show has everything all together on one page.  I showed everything in linear fashion, one step at a time.

If you want my solution to look less complicated here's the version without all the steps shown in excruciating detail:



As to your question, the large quote you show apparently was posted on another forum I haven't seen, and I believe there are some errors in it.

Look at the Bode plot associated with the original problem statement.  It shows an upper corner frequency at (Gm1 Gm2 Rout2)/CL due to the pole in the expression for Zin.  It is not exactly correct, but it is a good approximation if you can neglect the (+1) term in the denominator of the Zin expression.

The denominator of the Zin expression is (Rout1 CL s + Gm1 Rout1 Gm2 Rout2 + 1).  If this is solved for s, the result is:

(1+Gm1 Rout1 Gm2 Rout2)/(Rout1 CL) which is the exactly correct pole frequency.

In order to get the pole frequency shown in the Bode plot we have to neglect the (+1) term
and solve (Rout1 CL s + Gm1 Rout1 Gm2 Rout2) for s.  Then we get a pole frequency of (Gm1 Gm2 Rout2)/CL

The large comment says "If you inject a current into node X...This gives you a low impedance at low frequencies. How low? Just 1/(gm1rout1gm2)"

That is not the result I get for the resistance at node X.  To get the resistance at node X, simply remove the +s CL term from the admittance matrix and invert it.  The resistance seen at node X is the red element of the inverse:



In order to get the expression in the large comment (from another forum) you show, it's necessary to ignore the (1+) term in the resistance expression I show in red.  I believe the expression in the large comment is in error.

The expression in the large comment for the resistance seen by CL suffers from the same neglect of the (1+) term.  The commenter simply asserts: "...you can disconnect node X from all but the gm stages and realize that CL sees a resistance of 1/(gm2Rout2gm1) and so the pole frequency is 1/(gm2Rout2*gm1)"

He doesn't show how he calculated that "CL sees a resistance of 1/(gm2Rout2gm1)", and then he makes a mistake when he says "the pole frequency is 1/(gm2Rout2*gm1)"; the pole frequency would be gm2 Rout2 gm1/CL if the expression for the resistance seen by CL were correct.  Perhaps he didn't actually calculate the resistance seen by CL, but just asserted that it is 1/(gm2Rout2gm1) because that would match up with what is shown on the Bode plot.

[promach]

To get the resistance at node X, simply remove the +s CL term from the admittance matrix

Could you explain why so ?

[The Electrician]

To get the resistance at node X, simply remove the +s CL term from the admittance matrix

Could you explain why so ?

You left out the last part of the sentence; the complete sentence is:

"To get the resistance at node X, simply remove the +s CL term from the admittance matrix and invert it."

I'm not sure exactly what you're asking.  Are you asking why remove the +s CL term?  Or are you asking why invert the admittance matrix?

The +s CL term is removed because we want the resistance at low frequencies.  The lowest possible frequency is DC, and we get that result by removing CL.

We invert the admittance matrix because that converts it into an impedance matrix and the diagonal elements of the Z (impedance) matrix of a two port are the impedances of the ports.

Another way to get the impedance at low frequencies is to take the limit of the Zin expression as s approaches zero:



This gives the same result.

[promach]

Why total_gm = gm1rout1gm2 ?

Besides, if you look at https://cloudfront.escholarship.org/dist/prd/content/qt1pg6p6nb/qt1pg6p6nb.pdf?t=o3sama#page=57 which is the thesis "RF Synthesis without Inductors", there is another rewritten expression of Zin. Could you elaborate/explain on the interpretation of this expression without deriving it step-by-step using small-signal circuit model ?

Zin = [CLs/(Gm1Gm2)+(Gm1Gm2Rout1)-1] || Rout2

[The Electrician]

Why total_gm = gm1rout1gm2 ?

This is a matter of interpretation.  I can't explain why the product of the two transconductances and rout1 would be considered some sort of total transconductance.

Besides, if you look at https://cloudfront.escholarship.org/dist/prd/content/qt1pg6p6nb/qt1pg6p6nb.pdf?t=o3sama#page=57 which is the thesis "RF Synthesis without Inductors", there is another rewritten expression of Zin. Could you elaborate/explain on the interpretation of this expression without deriving it step-by-step using small-signal circuit model ?

Zin = [CLs/(Gm1Gm2)+(Gm1Gm2Rout1)-1] || Rout2

If you consider this circuit:



The impedance expression for the circuit would be:

(s L + Rs) || Rp

You can see how Rs is the series resistance and Rp is the parallel resistance.  Compare to the expression you're asking about.

[danmc]

I don't think I made it too complicated.  The image you show has everything all together on one page.  I showed everything in linear fashion, one step at a time.

If you want my solution to look less complicated here's the version without all the steps shown in excruciating detail:



As to your question, the large quote you show apparently was posted on another forum I haven't seen, and I believe there are some errors in it.

Look at the Bode plot associated with the original problem statement.  It shows an upper corner frequency at (Gm1 Gm2 Rout2)/CL due to the pole in the expression for Zin.  It is not exactly correct, but it is a good approximation if you can neglect the (+1) term in the denominator of the Zin expression.

The denominator of the Zin expression is (Rout1 CL s + Gm1 Rout1 Gm2 Rout2 + 1).  If this is solved for s, the result is:

(1+Gm1 Rout1 Gm2 Rout2)/(Rout1 CL) which is the exactly correct pole frequency.

In order to get the pole frequency shown in the Bode plot we have to neglect the (+1) term
and solve (Rout1 CL s + Gm1 Rout1 Gm2 Rout2) for s.  Then we get a pole frequency of (Gm1 Gm2 Rout2)/CL

The large comment says "If you inject a current into node X...This gives you a low impedance at low frequencies. How low? Just 1/(gm1rout1gm2)"

That is not the result I get for the resistance at node X.  To get the resistance at node X, simply remove the +s CL term from the admittance matrix and invert it.  The resistance seen at node X is the red element of the inverse:



In order to get the expression in the large comment (from another forum) you show, it's necessary to ignore the (1+) term in the resistance expression I show in red.  I believe the expression in the large comment is in error.

The expression in the large comment for the resistance seen by CL suffers from the same neglect of the (1+) term.  The commenter simply asserts: "...you can disconnect node X from all but the gm stages and realize that CL sees a resistance of 1/(gm2Rout2gm1) and so the pole frequency is 1/(gm2Rout2*gm1)"

He doesn't show how he calculated that "CL sees a resistance of 1/(gm2Rout2gm1)", and then he makes a mistake when he says "the pole frequency is 1/(gm2Rout2*gm1)"; the pole frequency would be gm2 Rout2 gm1/CL if the expression for the resistance seen by CL were correct.  Perhaps he didn't actually calculate the resistance seen by CL, but just asserted that it is 1/(gm2Rout2gm1) because that would match up with what is shown on the Bode plot.

Unfortunately what was missing in that quoted comment (from me) was the context.  It was in response to promach asking if there was any simple way to understand the basic operation of the circuit (Why is it a gyrator? Approximately what should be a result?)  without getting bogged down in algebra.  So yes, I intentionally ignored a number of 1+ terms.  Like the DC resistance at node X.  As you noted it is Rout2 || (1/(gm1*rout1*gm2) which is hopefully dominated by the gm1*rout1*gm2 part.  Same thing for the exact pole frequency.  It ends up being a similar thing with an output resistance in parallel with a 1/gm as you noted.  So you are correct that there are 1+'s missing but it wasn't so much a mistake/error as it was trying to give a really simplified view point.  There was a long thread basically starting with why his circuit didn't simulate as expected and I had given a bunch of suggestions on simplifying to something small with known results to prove out things like the test bench and post processing.  Some of the values in the original circuit made it questionable (CL pretty small to where transistor capacitances could affect things).  The test bench hadn't been fully proven (do you get the right result with an inductor instead of the gyrator?  Do you get the right answer with ideal gm stages?  Are you getting the gm you expected from the real transistors?).    There were questions about how to actually simulate some of the individual stages like making sure the test circuit was biased and correctly processing the simulated data.  Lots of stuff.  Towards the end was the question of how do you look at the circuit and internalize why it should look inductive as opposed to doing a bunch of algebra and coming up with an answer of it being inductive over some frequency range.  So what I was giving was the minimal-math version of about what you should see.  So, that is the context.  It wasn't meant to be an exact result.

-Dan

[danmc]

Why total_gm = gm1rout1gm2 ?

This is a matter of interpretation.  I can't explain why the product of the two transconductances and rout1 would be considered some sort of total transconductance.

[/quote]

Again, context.  It was from the viewpoint of you wiggle the voltage going into the first gm stage and see what current comes out of the 2nd gm stage at low frequencies.  First stage has dc gain gm1*rout1.  So first stage (low freq) output voltage is Vx * gm1*rout1, multiply by gm2 to get 2nd gm stage output current (ignoring Rout2), and you have I2= Vx*gm1*rout1*gm2.  So it looks like a transconductance whose gm is gm1*rout1*gm2.  Then DC resistance is Rout2 || 1/(gm1*rout1*gm2).   To the extent that gm1*rout1*gm2*rout2 >> 1 this 1/(gm1*rout1*gm2).  The exact expression of course is a little more complicated.

[The Electrician]

Unfortunately what was missing in that quoted comment (from me) was the context.  It was in response to promach asking if there was any simple way to understand the basic operation of the circuit (Why is it a gyrator? Approximately what should be a result?)  without getting bogged down in algebra.  So yes, I intentionally ignored a number of 1+ terms.  Like the DC resistance at node X.  As you noted it is Rout2 || (1/(gm1*rout1*gm2) which is hopefully dominated by the gm1*rout1*gm2 part.  Same thing for the exact pole frequency.  It ends up being a similar thing with an output resistance in parallel with a 1/gm as you noted.  So you are correct that there are 1+'s missing but it wasn't so much a mistake/error as it was trying to give a really simplified view point.  There was a long thread basically starting with why his circuit didn't simulate as expected and I had given a bunch of suggestions on simplifying to something small with known results to prove out things like the test bench and post processing.  Some of the values in the original circuit made it questionable (CL pretty small to where transistor capacitances could affect things).  The test bench hadn't been fully proven (do you get the right result with an inductor instead of the gyrator?  Do you get the right answer with ideal gm stages?  Are you getting the gm you expected from the real transistors?).    There were questions about how to actually simulate some of the individual stages like making sure the test circuit was biased and correctly processing the simulated data.  Lots of stuff.  Towards the end was the question of how do you look at the circuit and internalize why it should look inductive as opposed to doing a bunch of algebra and coming up with an answer of it being inductive over some frequency range.  So what I was giving was the minimal-math version of about what you should see.  So, that is the context.  It wasn't meant to be an exact result.

-Dan

Without the context you have given, I could only answer promach's questions stand alone.

I found promach's thread on the Designer's Guide forum.  Over there he wanted an intuitive understanding of Zin.  Here he just asked to be shown how to derive Zin, with no mention of intuition.

He has posted this question on three forums.

The image in the first post showing a Bode plot neglected the +1 term, so when I derived Zin for him and there was a +1 term in it, I assumed that it might be negligible and I said so; to determine if it's really negligible, one would have to plug in some numbers for the variables.

I think when somebody gives expressions for things like impedance and breakpoints, they should mention if they are neglecting parts of them and deriving a simpler approximation.  For guys like you and me it might be obvious but for the newbie when they go through the algebra and get a different result, they may be wondering why.

For a question like "Regarding the quote above, How do I derive the equation "Just 1/(gm1rout1gm2). I.e. 1/total_gm" ?", promach should really be asking you to explain.

[The Electrician]

Again, context.  It was from the viewpoint of you wiggle the voltage going into the first gm stage and see what current comes out of the 2nd gm stage at low frequencies.  First stage has dc gain gm1*rout1.  So first stage (low freq) output voltage is Vx * gm1*rout1, multiply by gm2 to get 2nd gm stage output current (ignoring Rout2), and you have I2= Vx*gm1*rout1*gm2.  So it looks like a transconductance whose gm is gm1*rout1*gm2.  Then DC resistance is Rout2 || 1/(gm1*rout1*gm2).   To the extent that gm1*rout1*gm2*rout2 >> 1 this is 1/(gm1*rout1*gm2).  The exact expression of course is a little more complicated.

It's promach that asked for an explanation of this.  I couldn't provide it, so I hope this satisfies him.

-------------------------------------------------------------------------------------------------------------------

promach seems to me to lack the easy facility with algebra you and I have, so when you say:

"Then DC resistance is Rout2 || 1/(gm1*rout1*gm2).   To the extent that gm1*rout1*gm2*rout2 >> 1 this is 1/(gm1*rout1*gm2)."

Showing more intermediate steps might preclude his asking for further explanation:

[promach]

Do you get the right answer with ideal gm stages?

YES

Are you getting the gm you expected from the real transistors?

NO


@danmc

What can I do in this case ? I am planning to derive the Rout2 algebra as well as measure the output impedance in simulation.


@The Electrician

This is the development branch of the active inductor implementation that I am still working on https://github.com/promach/frequency_trap/tree/development


What do you guys have in mind ?
Besides, I am still a bit confused how  Zin = [CLs/(Gm1Gm2)+(Gm1Gm2Rout1)-1] || Rout2  is interpreted with simple intuition and without much algebra maths

[The Electrician]

Besides, I am still a bit confused how  Zin = [CLs/(Gm1Gm2)+(Gm1Gm2Rout1)-1] || Rout2  is interpreted with simple intuition and without much algebra maths

The -1 in red above should really be an exponent of -1 applied to the expression (Gm1 Gm2 Rout1)

Consider the circuit consisting of an inductor with a series resistor and a parallel resistor:



The impedance of this is: Z = (s L + Rs) || Rp.  Do you see why?

Now compare that term by term with the expression you're asking about:

Zin = [CLs / (Gm 1Gm2) + (Gm1 Gm2 Rout1)^-1] || Rout2
 Z  =  [         s L             +              Rs                ] ||    Rp

So the expression CL / (Gm1 Gm2) is the inductance of the synthetic inductor, 1/(Gm1 Gm2 Rout1) is the series resistance of the synthetic inductor, and Rout2 is the resistance in parallel.

[The Electrician]

promach, you have asked about intuitive understanding.  Have a look at the Wikipedia page on the gyrator: https://en.wikipedia.org/wiki/Gyrator

Go down the page to the part titled: Application: a simulated inductor

The circuit shown only uses a single opamp, and it's not a true gyrator, but read the explanation; you might find this circuit easier to intuitively understand.

------------------------------------------------------------------------

Also, when you say "I am planning to derive the Rout2 algebra as well as measure the output impedance in simulation.", just what is it that you do to "derive the Rout2 algebra"?  danmc may know what you mean, but I don't.

[promach]

I am trying to determine suitable mosfet sizing for the following transconductance circuit block.

It seems to me that I need to minimize 1/(Gm1Gm2Rout1)  as well as maximize Rout2 ?

This is the small signal model I have drawn up to derive the Rout2 algebra. I need some help in formulating the small-signal mesh/nodal equations to obtain Rout2

       

[promach]

Why is   sC/(Gm1Gm2) placed in series with 1/(Gm1Gm2Rout1) ?

[The Electrician]

Why is   sC/(Gm1Gm2) placed in series with 1/(Gm1Gm2Rout1) ?

Read this: https://en.wikipedia.org/wiki/Equivalent_series_resistance

Real inductors and capacitors are not loss free.  When a voltage or current is applied to them there are parasitic loss mechanisms that cause a loss of energy.  Those losses can be modeled as resistors in series, or in parallel, or both sometimes.

[The Electrician]

I am trying to determine suitable mosfet sizing for the following transconductance circuit block.

It seems to me that I need to minimize 1/(Gm1Gm2Rout1)  as well as maximize Rout2 ?

This is the small signal model I have drawn up to derive the Rout2 algebra. I need some help in formulating the small-signal mesh/nodal equations to obtain Rout2

Just above the fourth image down in your thread at the Designer's Guide forum I see nodal equations and a solution for gm obtained using Maxima.  To obtain Rout2 inject 1 amp into the node where you want to get Rout2 and solve for the voltage Vout there.  Then Rout2 will be Vout/1 amp; the expression obtained for Vout will also be the expression for Rout2.

You can expect that expressions you derive for various voltages, currents, and impedances will be huge, nearly intractable expressions just like the one you already obtained for gm.

[promach]

Let me try deriving Rout2 on my own then.

Besides, I have attached a book chapter (which you can also find in public domain without any paywall) detailing the lossy characteristics of the gyrator, leading to explanation on ESR and parallel resistance of the active inductor.

[promach]

In order to determine suitable transistor size, I need to determine the Rout2 equation with respect to ro of each mosfets.

Gate of mosfet M6 is not driven and connected to GND during Rout2 small-signal calculation.

But generally, how many nodal equations do I need since voltage at source node of M6 is not known to me at the moment ?

[The Electrician]

In order to determine suitable transistor size, I need to determine the Rout2 equation with respect to ro of each mosfets.

Gate of mosfet M6 is not driven and connected to GND during Rout2 small-signal calculation.

But generally, how many nodal equations do I need since voltage at source node of M6 is not known to me at the moment ?

In the image in response #16 above it looks like M5 is the one with the grounded gate.  Is that the model you will be using?

[promach]

M5 is the one with the grounded gate

I suppose gate of M5 has to be connected to AC ground ?

[The Electrician]

M5 is the one with the grounded gate

I suppose gate of M5 has to be connected to AC ground ?

I don't know if it "has to be" connected to AC ground, but isn't that what is shown in the image in response #16?  Which model do you want to use for your analysis?