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Help in deriving Zin small-signal expression

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The Electrician:

--- Quote from: promach on May 23, 2018, 01:30:52 am ---
--- Code: ---node_1: V1/ro4 + gm4*V1 + (V1-V2)/ro6 + gm6*(0-V2) = 0

node_2: (V2-V1)/ro6 + (V2-V3)/Rs - gm6*(0-V2) = 0

node_3: (V3-V2)/Rs + (V3-Vx)/ro5 + gm5*V3 = 0

output_node: Ix - gm3*V1 - Vx/ro3 + gm5*V3 - (Vx-V3)/ro5 = 0
--- End code ---

Note: Vx is the test voltage at the output node at drain of M5 , while Ix is the test current source going into the output node

--- End quote ---

It took me a while to realize that you are using P channel FETs for M3 and M4.  The text you linked to seems to be using only N channel FETs.  Given that, your equations look correct, but I have a few comments.

Since you will probably be using these equations for calculating other quantities, such as various gains, I wouldn't use a variable name Vx just for the purpose of calculating Rout2.  Leave the 4th node as vo.

You used the convention that current out of a node is positive for your first 3 equations, and used the opposite for the last equation.  Since you can just multiply the left side by -1 and that won't change the sign of the zero on the right side, it doesn't matter when you go to solve the equations.  But, I think that's a bad practice; choose a convention and stick to it.

If you later need to calculate gains, you will need to include Vin in the equations.  You might as well leave that in there now and just set Vin = 0 for the Rout2 calculation to avoid forgetting where Vin goes later.

Just set Ix equal to 1, and then the impedance at the vo node will be equal to (vo)/(1 amp).  In other words, when you inject 1 amp into the vo node, the expression for the voltage there is the same as the expression for the impedance there.

If I make the above changes, here are the equations as I have them, ready to solve:



Here's the solution:



What do you get for the expression at vo if you plug in some numbers for the various transconductances and resistances?  Does it make sense?

bson:

--- Quote from: The Electrician on May 14, 2018, 06:19:35 am ---



--- End quote ---

You know there's a function TeXForm[] in Mathematica?  E.g.:

--- Code: ---%21 // TeXForm
\frac{\text{Rout2}}{\text{Gm1} \text{Gm2} \text{Rout1} \text{Rout2}+1}

--- End code ---

Copy and paste, with $ $ (no space) before and after, e.g.:

$$ \text{Limit}\left[\frac{\text{Rout1} \text{Rout2} s C_L+\text{Rout2}}{\text{Rout1} s C_L+\text{Gm1} \text{Gm2} \text{Rout1} \text{Rout2}+1},s\rightarrow 0\right] $$

$$ \frac{\text{Rout2}}{\text{Gm1} \text{Gm2} \text{Rout1} \text{Rout2}+1} $$

No need to fuss with screen captures...

Or, inline: \$ \frac{\text{Rout2}}{\text{Gm1} \text{Gm2} \text{Rout1} \text{Rout2}+1} \$

The Electrician:
I'm aware that Mathematica can produce TeX, but I prefer the appearance of the native Mathematica format.  The generated TeX has no spaces between variables, and it insists on having "\text" in front of every little thing.

You chose a fairly simple example, but see what Mathematica generates for my image in reply #30:



I don't mind pasting images.

promach:
However, for gm and ro calculation for each mosfets, how do I obtain some transistor characteristic parameter such as uo and Cox from BSIM4 mosfet models ?

http://ngspice.sourceforge.net/external-documents/models/BSIM480_Manual.pdf

https://github.com/promach/frequency_trap/blob/development/modelcard.nmos

https://github.com/promach/frequency_trap/blob/development/modelcard.pmos

The Electrician:

--- Quote from: promach on May 24, 2018, 11:30:20 pm ---However, for gm and ro calculation for each mosfets, how do I obtain some transistor characteristic parameter such as uo and Cox from BSIM4 mosfet models ?

http://ngspice.sourceforge.net/external-documents/models/BSIM480_Manual.pdf

https://github.com/promach/frequency_trap/blob/development/modelcard.nmos

https://github.com/promach/frequency_trap/blob/development/modelcard.pmos

--- End quote ---

I can't help you here.  Perhaps danmc can help.

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