# solve() fails for 3 simple linear equations

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)
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)
eq_at_Vx = eq_at_Vx.substitute(Vx_eq)
# final transfer function
print((solve(eq_at_Vx,Vo)/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.

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Note that many asterisks got omitted in the output Vo/Vi, even after editing to indicate it is preformatted.

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By solving against Vo only, you do as if every other symbols involved in the equations were constants. However, Vi, Vm, Vx are also unknown.

Imagine that the solver would return [Vo == -A*Vm] as a solution, i bet you will not be very happy.

So, what you have to do is to put all unknowns, and let only the constants outside:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vi, Vm, Vx])
[[Vo == (A*R3*r1 + (A*C*R3*r1*s + A*r1)*R2)/(A*R2 - R3), Vi == -((C*R3*r1*s + A*r1 + r1)*R1 + (C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vm == -((C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vx == r1]]


Now, as you can see, there is one degree of freedom : Vx can take any value, so Sage added a free parameter r1 and gave you all the solutions using the constants and this free parameter r1.

Now, i see in your workaround that you want to solve for Vo/Vi, so let me assume that you want Vi being the free parameter, not Vx.

Hence, you can solve as if Vi is also a constant:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vm, Vx])
[[Vo == -((A*C*R3*s + A)*R2 + A*R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vm == ((C*R3*s + 1)*R2 + R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vx == -(A*R2 - R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3)]]

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