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Help in deriving Zin small-signal expression
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danmc:

--- Quote from: The Electrician on May 14, 2018, 04:13:57 pm ---
--- Quote from: promach on May 14, 2018, 10:43:26 am ---Why total_gm = gm1rout1gm2 ?
--- End quote ---

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.


--- End quote ---

[/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:

--- Quote from: danmc on May 14, 2018, 04:16:19 pm ---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

--- End quote ---

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:

--- Quote from: danmc on May 14, 2018, 04:22:54 pm ---
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.

--- End quote ---

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:

--- Quote ---Do you get the right answer with ideal gm stages?
--- End quote ---

YES


--- Quote --- Are you getting the gm you expected from the real transistors?
--- End quote ---

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:

--- Quote from: promach on May 15, 2018, 12:50:34 am ---Besides, I am still a bit confused how  Zin = [CLs/(Gm1Gm2)+(Gm1Gm2Rout1)-1] || Rout2  is interpreted with simple intuition and without much algebra maths

--- End quote ---

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.
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