Electronics > Beginners
Help in deriving Zin small-signal expression
promach:
Do you guys have any comments on the KCL equations I have set up in order to find out Rout2 expression which is Vx/Ix ?
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
The Electrician:
One of the things I find most annoying about trying to help someone on a forum is when I ask a question whose answer is important to my ability to help, and the question is not answered.
I asked you in reply #22 and #24 what model you will be using, and you haven't answered. I'm guessing that it's the model in reply #16, but you should acknowledge if that is true.
There is also a schematic in reply #16 from which I presume the model is derived. There are aspects of the schematic that you haven't included in the model. For example, the symbol Vss appears in the schematic, but not in the model. Vss, or course, is AC ground for small signal purposes.
Look at your first equation; the last term in that equation is (Vx - Vss)/ro5. You need to make sure that every voltage you have in any of your equations appears in the model. Since Vss is small signal ground, it shouldn't appear in any equation anyway whether on not you show it on the model. Get rid of it everywhere you used it. It's connected to your reference node (ground), and it's value is zero.
Also, please label the voltages at the various nodes. Label the voltage at node 1 as v1, the voltage at node 2 as v2, the voltage at node 3 as v3, and the voltage at the output node as vo.
Your model shows the output node as grounded, with Iout flowing into ground. Nodal analysis calculates voltages at nodes that are not grounded. Show the output node driving a load resistance RL, If you later need the current into ground (a short, in other words), you can let RL be zero ohms.
Show a model with the nodes labeled v1, v2, etc,. and rewrite the equations using those voltages. For example, the first term in your node 2 equation is (Vss - Vs6)/Rs; this can't be right because neither end of Rs is connected to Vss on the schematic (never mind that you don't show Vss on the model as I already pointed out). Remember that on the schematic Vss is actually ground for small signal purposes. The current sources I1 and I2 don't provide node 2 or node 3 with a signal carrying connection to Vss (ground). The current sources (if ideal) are infinite impedances. They only provide the DC conditions, and they have no effect on small signal behavior.
Look at the last term of the equation for node 2. What is that item in the numerator? It looks like VDS6. Is that the voltage from drain to source of M6? If so, don't represent it like that. The voltage from drain to to source of M6 should be represented as v1-v2. Use the symbols v1, v2, v3, and vo for voltages in the numerators of terms in your equations.
Since M5's gate is grounded, you could replace M5 with a fixed impedance.
Update your model as I've suggested and rewrite your equations. Show the results here and we can continue from there.
promach:
--- Quote ---Since M5's gate is grounded, you could replace M5 with a fixed impedance.
--- End quote ---
What happen to the current source gm5*Vgs5 ?
--- Quote ---Look at your first equation; the last term in that equation is (Vx - Vss)/ro5.
--- End quote ---
No, it is Vs5 instead of Vss
--- Quote ---It looks like VDS6. Is that the voltage from drain to source of M6?
--- End quote ---
Yes, you are right that it is Vds6
--- Quote ---Your model shows the output node as grounded, with Iout flowing into ground.
--- End quote ---
That model is used for Gm2 transconductance
For output impedance Rout2, output node is not grounded. Note that 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
The Electrician:
Forget about replacing M5 with a resistor. I made that comment before I looked carefully at where the source of M5 is connected.
Please do a new model image with the nodes labeled v1, v2, v3, vo and new equations using those designators.
promach:
--- 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
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