I am doing basic circuit analysis, which involves solving multiple algebraic equations. solve() fails for a simple linear circuit (image does not work -- see circuit description below).
Vi, Vo, Vm, Vx, R1, R2, R3, C, A, s = var('Vi,Vo,Vm,Vx,R1,R2,R3,C,A,s') # Three Kirchoff equations at nodes Vm, Vx, Vo eq_at_Vm = (Vi-Vm)/R1 + (Vx-Vm)/R2 == 0 eq_at_Vx = (Vm-Vx)/R2 + (Vo-Vx)/R3 + (0-Vx)/(1/(C*s)) == 0 eq_at_Vo = Vo == -A*Vm # solving for Vo FAILS solve([eq_at_Vm,eq_at_Vx,eq_at_Vo],Vo)The output is just [].
Am I doing something wrong, or is this a limitation of solve()?
I did find a workaround, solving each equation individually, which in this case is easy, but in general is not:
Vi, Vo, Vm, Vx, R1, R2, R3, C, A, s = var('Vi,Vo,Vm,Vx,R1,R2,R3,C,A,s') # Three Kirchoff equations at nodes Vm, Vx, Vo eq_at_Vm = (Vi-Vm)/R1 + (Vx-Vm)/R2 == 0 eq_at_Vx = (Vm-Vx)/R2 + (Vo-Vx)/R3 + (0-Vx)/(1/(C*s)) == 0 eq_at_Vo = Vo == -A*Vm # eliminate Vm Vm_eq = solve(eq_at_Vo,Vm)[0] eq_at_Vm = eq_at_Vm.substitute(Vm_eq) eq_at_Vx = eq_at_Vx.substitute(Vm_eq) # eliminate Vx Vx_eq = solve(eq_at_Vm,Vx)[0] eq_at_Vx = eq_at_Vx.substitute(Vx_eq) # final transfer function print((solve(eq_at_Vx,Vo)[0]/Vi).simplify_full())The output is:
Vo/Vi == -(A*C*R2*R3*s + A*R2 + A*R3)/((C*R1 + C*R2)*R3*s + (A + 1)*R1 + R2 + R3)
Circuit Description: This is an inverting OpAmp with a R2-C-R3 Tee network as feedback. The OpAmp has gain=A with in+ grounded, so Vo=-A*in-. Nodes are labeled Vi, Vm, Vx, Vo: Vi is the input, Vo is the OpAmp output, Vm is the OpAmp in-, and Vx is inside the Tee. R1 is Vi to Vm, R2 is Vm to Vx, R3 is Vx to Vo, and C is Vx to ground.
