How to isolate two variables, Vout and Vin to obtain transfer function

Hi this is my following code:

syms Vg Vin Vout Rs ro Rd CGS CGD CDB CL s gm;

eq1 = ((Vg-Vin)/Rs) + Vg*s*CGS + (Vg-Vout)*s*CGD == 0;

eq2 = gm*Vg + Vout/Rd + Vout/ro + Vout*s*CDB + Vout*s*CL + s*CGD*(Vout-Vg) == 0;

solVg = solve(eq2,Vg);

eq3 = ((solVg-Vin)/Rs) + solVg*s*CGS + (solVg-Vout)*s*CGD == 0;

So at this point, I have Vg in terms of, Vout and Vin. I'm having difficulty isolating the two variables, Vout and Vin to have the following transfer function: Vout/Vin.

Answer 1

[Star Strider]

You need to solve for ‘Vout’ first, then divide that expression by ‘Vin’:

Vout = solve(eq3, Vout);
H = Vout/Vin
[Hn, Hd] = numden(H);
Hn = collect(Hn, s)
Hd = collect(Hd, s)

with ‘H’ being your transfer function, ‘Hn’ is the numerator, ‘Hd’ the denominator:

H =
-1/(Rs*(CGD*s*((CDB*s + CGD*s + CL*s + 1/Rd + 1/ro)/(gm - CGD*s) + 1) + (CDB*s + CGD*s + CL*s + 1/Rd + 1/ro)/(Rs*(gm - CGD*s)) + (CGS*s*(CDB*s + CGD*s + CL*s + 1/Rd + 1/ro))/(gm - CGD*s)))

Hn =
(CGD*Rd*ro)*s - Rd*gm*ro

Hd =
(CDB*CGD*Rd*Rs*ro + CDB*CGS*Rd*Rs*ro + CGD*CGS*Rd*Rs*ro + CGD*CL*Rd*Rs*ro + CGS*CL*Rd*Rs*ro)*s^2 + (CGD*Rd*Rs + CGS*Rd*Rs + CDB*Rd*ro + CGD*Rd*ro + CL*Rd*ro + CGD*Rs*ro + CGS*Rs*ro + CGD*Rd*Rs*gm*ro)*s + Rd + ro

[1582251394]

Thank you very much!

[Star Strider]

My pleasure!

I use the Symbolic Math Toolbox extensively in circuit design and analysis.

Answer 2

[Walter Roberson]

solVin = solve(eq3, Vin);
Vin_over_Vout = simplify( expand( SolVin/Vout ) );

[Walter Roberson]

Opps, yes, I did Vin/Vout instead of Vout/Vin, but you can use

Vout_over_Vin = simplify( expand( Vout/SolVin ) );

[1582251394]

Thank you very much!