# Mesh current analysis - with AC in s-domain

I have the following problem:

Consider the circuit below, with

• $$\v_1(t)=5\text{V} \ \sin(\omega_0t)\$$

• $$\v_2(t)=1.2\text{V} \, \cos(\omega_0t)\$$

• $$\V_3=9 \text{V}\$$

• $$\i_s(t)=0.5\text{A} \, \cos(\omega_0t+50^\circ)\$$

The bottom of the arrow indicates the $$\+\$$-pole of each voltage source.

At $$\f_0 = 1.25 \text{MHz}\$$ the circuit is in steady state.

What is $$\ |\mathbf{I_{out}}(s)| \$$ at $$\ f_0\$$?

What is the phase of $$\ \mathbf{I_{out}}(s)\$$ at $$\f_0\$$ referenced to $$\ \mathbf{V_2}\$$?

EDIT

I tried to simulate with LT spice and got the following result:

But I cannot confirm if what I'm getting is a correct result. That is why I would really like some help with this problem.

• Are you assuming that the voltage across the current source is zero? If so, why? – Elliot Alderson Apr 11 '20 at 16:29
• Oh you are right, there is no reason that there shouldn't be a voltage drop across the current source. I will call that voltage drop $v_s$ and edit the question. – Carl Apr 11 '20 at 17:46
• The Kirchhoff voltage law states that the sum of the voltages around a loop is zero. The current law states that the sum of the currents into a node is zero. Maybe you should begin by marking the loops as voltages. Also mark the currents of all the sources as a starting point - calculating the current in L3. – skvery Apr 11 '20 at 18:51

You set up the voltages wrong: for V1 it should be AC 5 because it's a sine, thus zero phase shift, for V2 it should be AC 1.2 90 (cosine), V3 is correct, and I1 should be AC 0.5 140 (= 90 + 50). With these changes, I get these results:

I have also explicitly zeroed out all the parasitics for inductances and capacitors, since they could be influencing the result.

I've also ran a .TRAN analysis, as @VerbalKint's coment, and got these results:

The magnitude isn't exactly as in .AC, but I attribute that to the finite timestep and the differences between the way .AC is calculated behind the scene. The phase agrees, though.

• Oh, I actually thought that the "small signal AC analysis" was cosine, but now I know better! I hope that $\mathbf{I}_{out}(s)=0.594A$ and that $\angle \mathbf{I}_{out}(s) = -133.71^\circ$, because the fact that my instructor wanted the answer in the s-domain really confused me. Thanks a lot for your answer. – Carl Apr 14 '20 at 8:40
• @Carl Internally, yes (even if the results are complex), but when you set up a sine source, it has zero phase shift, so then the .AC phase should be zero. – a concerned citizen Apr 14 '20 at 8:41
• I've never tried to run ac analyses at a single frequency with different active ac sources. If I use transient generators with adjusted delays for sin and cos waveforms, I find an rms current around 760 mA with a 63° phase shift with respect to $V_2$. – Verbal Kint Apr 14 '20 at 9:44
• @VerbalKint I just recreated the schematic and ran it with 1k points over 125kHz...12.5MHz, and the readings are 594.66835mA and -133.71411deg with the cursor spot on 1.25MHz. Then I tried .TRAN and got 0.594648V and -133.715deg (with .MEAS). I have to say you gave me quite a scare. Maybe you're using -cos, instead of cos (delay of 90 deg, plus)? Or maybe you are reading I(V2) (as seen in the picture)? I'll update my answer to reflect my method. – a concerned citizen Apr 14 '20 at 11:59
• @a concerned citizen, I had the wrong delay polarity on the cos sources. Once adjusted properly, I read 419.53 mA rms which translates to 593.3 mA peak and the signal is delayed by 136° with respect to $V_2$ so I confirm your numbers : ) – Verbal Kint Apr 14 '20 at 13:19

Well, let's solve this mathematically. We have the following circuit:

simulate this circuit – Schematic created using CircuitLab

When we use and apply KCL, we can write the following set of equations:

$$\begin{cases} \text{I}_1=\text{I}_4+\text{I}_5\\ \\ \text{I}_3=\text{I}_2+\text{I}_4\\ \\ \text{I}_8=\text{I}_\text{k}+\text{I}_5\\ \\ \text{I}_8=\text{I}_\text{k}+\text{I}_7\\ \\ \text{I}_6=\text{I}_3+\text{I}_7\\ \\ \text{I}_6=\text{I}_1+\text{I}_2 \end{cases}\tag1$$

When we use and apply Ohm's law, we can write the following set of equations:

$$\begin{cases} \text{I}_1=\frac{\text{V}_\text{x}-\text{V}_1}{\text{R}_1}\\ \\ \text{I}_2=\frac{\text{V}_\text{y}-\text{V}_1}{\text{R}_2}\\ \\ \text{I}_3=\frac{\text{V}_6}{\text{R}_3}\\ \\ \text{I}_3=\frac{\text{V}_5-\text{V}_6}{\text{R}_4}\\ \\ \text{I}_3=\frac{\text{V}_3-\text{V}_5}{\text{R}_5}\\ \\ \text{I}_6=\frac{\text{V}_1-\text{V}_2}{\text{R}_6}\\ \\ \text{I}_7=\frac{\text{V}_3-\text{V}_7}{\text{R}_7}\\ \\ \text{I}_7=\frac{\text{V}_7-\text{V}_4}{\text{R}_8}\\ \\ \text{I}_8=\frac{\text{V}_4}{\text{R}_9} \end{cases}\tag2$$

Now, we also know that $$\\text{V}_3-\text{V}_2=\text{V}_\text{z}\$$.

I used Mathematica to solve your problem. The code is used is:

In[1]:=FullSimplify[
Solve[{Vz == V3 - V2, I1 == I4 + I5, I3 == I2 + I4, I8 == Ik + I5,
I8 == Ik + I7, I6 == I3 + I7, I6 == I1 + I2, I1 == (Vx - V1)/R1,
I2 == (Vy - V1)/R2, I3 == (V6)/R3, I3 == (V5 - V6)/R4,
I3 == (V3 - V5)/R5, I6 == (V1 - V2)/R6, I7 == (V3 - V7)/R7,
I7 == (V7 - V4)/R8, I8 == (V4)/R9}, {I1, I2, I3, I4, I5, I6, I7,
I8, V1, V2, V3, V4, V5, V6, V7}]]

Out[1]={{I1 -> (-Ik R2 (R3 + R4 + R5) R9 + (R5 R6 + R5 R7 + R6 R7 + R5 R8 +
R6 R8 + (R5 + R6) R9 + R3 (R6 + R7 + R8 + R9) +
R4 (R6 + R7 + R8 + R9)) (Vx - Vy) +
R2 (R3 + R4 + R5 + R7 + R8 + R9) (Vx + Vz))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
I2 -> (-Ik R1 (R3 + R4 + R5) R9 - R4 R6 Vx - R5 R6 Vx - R4 R7 Vx -
R5 R7 Vx - R6 R7 Vx - R4 R8 Vx - R5 R8 Vx - R6 R8 Vx -
R4 R9 Vx - R5 R9 Vx - R6 R9 Vx -
R3 (R6 + R7 + R8 + R9) (Vx - Vy) + R1 R3 Vy + R1 R4 Vy +
R1 R5 Vy + R4 R6 Vy + R5 R6 Vy + R1 R7 Vy + R4 R7 Vy +
R5 R7 Vy + R6 R7 Vy + R1 R8 Vy + R4 R8 Vy + R5 R8 Vy +
R6 R8 Vy + R1 R9 Vy + R4 R9 Vy + R5 R9 Vy + R6 R9 Vy +
R1 (R3 + R4 + R5 + R7 + R8 + R9) Vz)/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
I3 -> (Ik (R2 R6 + R1 (R2 + R6)) R9 + (R7 + R8 + R9) (R2 (Vx + Vz) +
R1 (Vy + Vz)))/(R1 (R4 R6 + R5 R6 + R4 R7 + R5 R7 + R6 R7 +
R4 R8 + R5 R8 + R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
I4 -> (Ik (R2 R6 + R1 (R2 + R3 + R4 + R5 + R6)) R9 + R4 R6 Vx +
R5 R6 Vx + R2 R7 Vx + R4 R7 Vx + R5 R7 Vx + R6 R7 Vx +
R2 R8 Vx + R4 R8 Vx + R5 R8 Vx + R6 R8 Vx + R2 R9 Vx +
R4 R9 Vx + R5 R9 Vx + R6 R9 Vx +
R3 (R6 + R7 + R8 + R9) (Vx - Vy) - R1 R4 Vy - R1 R5 Vy -
R4 R6 Vy - R5 R6 Vy - R4 R7 Vy - R5 R7 Vy - R6 R7 Vy -
R4 R8 Vy - R5 R8 Vy - R6 R8 Vy - R4 R9 Vy - R5 R9 Vy -
R6 R9 Vy - R1 (R4 + R5) Vz + R2 (R7 + R8 + R9) Vz -
R1 R3 (Vy + Vz))/(R1 (R4 R6 + R5 R6 + R4 R7 + R5 R7 + R6 R7 +
R4 R8 + R5 R8 + R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
I5 -> (-Ik (R2 (R3 + R4 + R5 + R6) +
R1 (R2 + R3 + R4 + R5 + R6)) R9 + (R3 + R4 +
R5) (R2 (Vx + Vz) + R1 (Vy + Vz)))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
I6 -> (-Ik (R1 + R2) (R3 + R4 + R5) R9 + (R3 + R4 + R5 + R7 + R8 +
R9) (R2 (Vx + Vz) + R1 (Vy + Vz)))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
I7 -> (-Ik (R2 (R3 + R4 + R5 + R6) +
R1 (R2 + R3 + R4 + R5 + R6)) R9 + (R3 + R4 +
R5) (R2 (Vx + Vz) + R1 (Vy + Vz)))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
I8 -> (Ik (R1 (R4 R6 + R5 R6 + R4 R7 + R5 R7 +
R6 R7 + (R4 + R5 + R6) R8 + R2 (R3 + R4 + R5 + R7 + R8) +
R3 (R6 + R7 + R8)) +
R2 (R5 R6 + R5 R7 + R6 R7 + (R5 + R6) R8 +
R3 (R6 + R7 + R8) + R4 (R6 + R7 + R8))) + (R3 + R4 +
R5) (R2 (Vx + Vz) + R1 (Vy + Vz)))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
V1 -> (Ik R1 R2 (R3 + R4 + R5) R9 + (R5 R6 + R5 R7 + R6 R7 + R5 R8 +
R6 R8 + (R5 + R6) R9 + R3 (R6 + R7 + R8 + R9) +
R4 (R6 + R7 + R8 + R9)) (R2 Vx + R1 Vy) -
R1 R2 (R3 + R4 + R5 + R7 + R8 + R9) Vz)/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
V2 -> -Vz + ((R3 + R4 +
R5) (Ik (R2 R6 + R1 (R2 + R6)) R9 + (R7 + R8 +
R9) (R2 (Vx + Vz) + R1 (Vy + Vz))))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
V3 -> ((R3 + R4 +
R5) (Ik (R2 R6 + R1 (R2 + R6)) R9 + (R7 + R8 +
R9) (R2 (Vx + Vz) + R1 (Vy + Vz))))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
V4 -> (R9 (Ik (R1 (R4 R6 + R5 R6 + R4 R7 + R5 R7 +
R6 R7 + (R4 + R5 + R6) R8 +
R2 (R3 + R4 + R5 + R7 + R8) + R3 (R6 + R7 + R8)) +
R2 (R5 R6 + R5 R7 + R6 R7 + (R5 + R6) R8 +
R3 (R6 + R7 + R8) + R4 (R6 + R7 + R8))) + (R3 + R4 +
R5) (R2 (Vx + Vz) + R1 (Vy + Vz))))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
V5 -> ((R3 +
R4) (Ik (R2 R6 + R1 (R2 + R6)) R9 + (R7 + R8 +
R9) (R2 (Vx + Vz) + R1 (Vy + Vz))))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
V6 -> (R3 (Ik (R2 R6 + R1 (R2 + R6)) R9 + (R7 + R8 +
R9) (R2 (Vx + Vz) + R1 (Vy + Vz))))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9))),
V7 -> (Ik ((R3 + R4 + R5) (R2 R6 +
R1 (R2 + R6)) + (R2 (R3 + R4 + R5 + R6) +
R1 (R2 + R3 + R4 + R5 + R6)) R7) R9 + (R3 + R4 + R5) (R8 +
R9) (R2 (Vx + Vz) + R1 (Vy + Vz)))/(R1 (R4 R6 + R5 R6 +
R4 R7 + R5 R7 + R6 R7 + R4 R8 + R5 R8 +
R6 R8 + (R4 + R5 + R6) R9 +
R2 (R3 + R4 + R5 + R7 + R8 + R9) + R3 (R6 + R7 + R8 + R9)) +
R2 (R5 R6 + R5 R7 + R6 R7 + R5 R8 + R6 R8 + (R5 + R6) R9 +
R3 (R6 + R7 + R8 + R9) + R4 (R6 + R7 + R8 + R9)))}}


Now, applying your given values we get:

In[2]:=R1 = 1/(s*30*10^(-9));
R2 = s*360*10^(-9);
R3 = 18/10;
R4 = 1/(s*27*10^(-9));
R5 = s*(9/10)*10^(-6);
R6 = 3/10;
R7 = 1/(s*100*10^(-9));
R8 = 100*10^(-3);
R9 = s*100*10^(-9);
Vx = LaplaceTransform[5*Sin[\[Omega]0*t], t, s];
Vy = LaplaceTransform[(12/10)*Cos[\[Omega]0*t], t, s];
Vz = LaplaceTransform[9, t, s];
Ik = LaplaceTransform[(1/2)*Cos[\[Omega]0*t + (50*(Pi/180))], t, s];
\[Omega]0 = 2*Pi*f0;
f0 = (125/100)*10^6;
FullSimplify[
Solve[{Vz == V3 - V2, I1 == I4 + I5, I3 == I2 + I4, I8 == Ik + I5,
I8 == Ik + I7, I6 == I3 + I7, I6 == I1 + I2, I1 == (Vx - V1)/R1,
I2 == (Vy - V1)/R2, I3 == (V6)/R3, I3 == (V5 - V6)/R4,
I3 == (V3 - V5)/R5, I6 == (V1 - V2)/R6, I7 == (V3 - V7)/R7,
I7 == (V7 - V4)/R8, I8 == (V4)/R9}, {I1, I2, I3, I4, I5, I6, I7,
I8, V1, V2, V3, V4, V5, V6, V7}]]

Out[2]={{I1 -> (60000000 s (31250000000000000000000000000000000000 \[Pi] +
625000000000000000000000000 (-4800 + \[Pi] (4835 +
10287 \[Pi])) s +
6250000000000000000 (-46416 + \[Pi] (410225 +
41553 \[Pi])) s^2 +
18750000000000 (49052 + 27 \[Pi] (209 + 270 \[Pi])) s^3 +
60750000 (608 + 625 \[Pi]) s^4 + 21141 s^5) +
67500000 \[Pi] s^4 (10000000000000000 +
243 s (2000000 + s)) Cos[(2 \[Pi])/9] -
27 s^5 (10000000000000000 + 243 s (2000000 + s)) Sin[(2 \[Pi])/
9])/(2 (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
I2 -> (10000000 (3 (595312500000000000000000000000000000000000000 \
\[Pi]^2 +
15625000000000000000000000000000000 \[Pi] (-2000 +
1539 \[Pi]) s +
3125000000000000000000000000 (35504 + \[Pi] (-967 +
4050 \[Pi])) s^2 -
6250000000000000000 (-744096 + 181625 \[Pi]) s^3 -
356250000000000 (-6748 + 135 \[Pi]) s^4 -
60750000 (-76 + 125 \[Pi]) s^5 + 729 s^6) +
312500000000000 \[Pi] s^2 (10000000000000000 +
243 s (2000000 + s)) Cos[(2 \[Pi])/9] -
125000000 s^3 (10000000000000000 + 243 s (2000000 + s)) Sin[(
2 \[Pi])/9]))/((6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
I3 -> (40500000 (20 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (100000000000000 +
s (1000000 + s)) -
2500000 \[Pi] s^3 (2500000000000000 +
3 s (1000000000 + 9 s)) Cos[(2 \[Pi])/9] +
s^4 (2500000000000000 + 3 s (1000000000 + 9 s)) Sin[(2 \[Pi])/
9]))/((6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
I4 -> (500000 (120 (-234375000000000000000000000000000000000000000 \
\[Pi]^2 -
15625000000000000000000000000000000 \[Pi] (-1000 +
729 \[Pi]) s -
312500000000000000000000000 (140800 + \[Pi] (-4835 +
16038 \[Pi])) s^2 +
3125000000000000000 (-707376 + \[Pi] (230225 +
2187 \[Pi])) s^3 +
9375000000000 (-104308 + 27 \[Pi] (101 + 27 \[Pi])) s^4 +
151875000 (-8 + 35 \[Pi]) s^5 + 729 s^6) -
2500000 \[Pi] s^2 (25000000000000000000000000 +
81 s (17500000000000000 + 3 s (3500000000 + 9 s))) Cos[(
2 \[Pi])/9] +
s^3 (25000000000000000000000000 +
81 s (17500000000000000 + 3 s (3500000000 + 9 s))) Sin[(
2 \[Pi])/9]))/((6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
I5 -> (60000000 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (10000000000000000 +
243 s (2000000 + s)) +
2500000 \[Pi] s^2 (25000000000000000000000000000000 +
27 s (52500000000000000000000 +
s (41500000000000000 + 81 s (7000000 + 3 s)))) Cos[(
2 \[Pi])/9] -
s^3 (25000000000000000000000000000000 +
27 s (52500000000000000000000 +
s (41500000000000000 + 81 s (7000000 + 3 s)))) Sin[(
2 \[Pi])/9])/(2 (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
I6 -> (600000000 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (1270000000000000 +
27 s (1900000 + s)) +
2500000 \[Pi] s^2 (2500000000000000 +
27 s^2) (10000000000000000 + 243 s (2000000 + s)) Cos[(
2 \[Pi])/9] -
s^3 (2500000000000000 + 27 s^2) (10000000000000000 +
243 s (2000000 + s)) Sin[(2 \[Pi])/
9])/(2 (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
I7 -> (60000000 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (10000000000000000 +
243 s (2000000 + s)) +
2500000 \[Pi] s^2 (25000000000000000000000000000000 +
27 s (52500000000000000000000 +
s (41500000000000000 + 81 s (7000000 + 3 s)))) Cos[(
2 \[Pi])/9] -
s^3 (25000000000000000000000000000000 +
27 s (52500000000000000000000 +
s (41500000000000000 + 81 s (7000000 + 3 s)))) Sin[(
2 \[Pi])/9])/(2 (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
I8 -> (2000000 (15 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (10000000000000000 +
243 s (2000000 + s)) -
2500000 \[Pi] (625000000000000000000000000000000000000 +
s (60437500000000000000000000000000 +
9 s (5753125000000000000000000 +
3 s (89425000000000000 +
81 s (331250000 + 3 s))))) Cos[(2 \[Pi])/9] +
s (625000000000000000000000000000000000000 +
s (60437500000000000000000000000000 +
9 s (5753125000000000000000000 +
3 s (89425000000000000 +
81 s (331250000 + 3 s))))) Sin[(2 \[Pi])/
9]))/((6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
V1 -> (37500000 s (8000000000000000000000000000 (10000000000 +
967 s) -
135000000000000000000000 \[Pi]^2 (1270000000000000 +
27 s (1900000 + s)) +
3 \[Pi] s (3000000000000000000000000000000 +
s (-100000000000000000000 (-2901 +
1000 Cos[(2 \[Pi])/9]) +
3 s (-5000000000000 (-7265 + 324 Cos[(2 \[Pi])/9]) +
81 s (3 s -
1000000 (-19 + 10 Cos[(2 \[Pi])/9]))))) +
4 s^2 (-20000000 (306575000000000000 +
81 s (152000000 + 87 s)) +
3 s (10000000000000000 + 243 s (2000000 + s)) Sin[(
2 \[Pi])/9])))/((6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +

27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
V2 -> -((33750000000000000000000 \[Pi]^2 (1270000000000000 +
27 s (1900000 + s)) (2500000000000000 +
3 s (1000000000 + 9 s)) +
600000000 s (-100000000000000000000000000000000000000 +
s (22715000000000000000000000000000 +
s (31965650000000000000000000 +
243 s (9170000000000000 +
s (2951300000 + 27 s))))) +
7500000 \[Pi] s^2 (10000000000000000 +
243 s (2000000 + s)) (-900 s (1000000 + s) +
2500000000000000 (-36 + Cos[(2 \[Pi])/9]) +
3 s (1000000000 + 9 s) Cos[(2 \[Pi])/9]) -
3 s^3 (10000000000000000 +
243 s (2000000 + s)) (2500000000000000 +
3 s (1000000000 + 9 s)) Sin[(2 \[Pi])/
9])/(20 (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 +
81 s (19000000 + 3 s)))))))),
V3 -> (3 (10000000000000000 +
243 s (2000000 +
s)) (20 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (100000000000000 +
s (1000000 + s)) -
2500000 \[Pi] s^3 (2500000000000000 +
3 s (1000000000 + 9 s)) Cos[(2 \[Pi])/9] +
s^4 (2500000000000000 + 3 s (1000000000 + 9 s)) Sin[(2 \[Pi])/
9]))/(20 s (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
V4 -> (s (15 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (10000000000000000 +
243 s (2000000 + s)) -
2500000 \[Pi] (625000000000000000000000000000000000000 +
s (60437500000000000000000000000000 +

9 s (5753125000000000000000000 +
3 s (89425000000000000 +
81 s (331250000 + 3 s))))) Cos[(2 \[Pi])/9] +
s (625000000000000000000000000000000000000 +
s (60437500000000000000000000000000 +
9 s (5753125000000000000000000 +
3 s (89425000000000000 +
81 s (331250000 + 3 s))))) Sin[(2 \[Pi])/
9]))/(5 (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
V5 -> (300000 (5000000000 +
243 s) (20 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (100000000000000 +
s (1000000 + s)) -
2500000 \[Pi] s^3 (2500000000000000 +
3 s (1000000000 + 9 s)) Cos[(2 \[Pi])/9] +
s^4 (2500000000000000 + 3 s (1000000000 + 9 s)) Sin[(2 \[Pi])/
9]))/(s (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
V6 -> (72900000 (20 (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (100000000000000 +
s (1000000 + s)) -
2500000 \[Pi] s^3 (2500000000000000 +
3 s (1000000000 + 9 s)) Cos[(2 \[Pi])/9] +
s^4 (2500000000000000 + 3 s (1000000000 + 9 s)) Sin[(2 \[Pi])/
9]))/((6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s))))))),
V7 -> (60 (1000000 + s) (46875000000000000000000000000 \[Pi]^2 +
6250000000000 (1360 + 81 \[Pi]^2) s^2 +
112500000 \[Pi] s^3 + 81 s^4) (10000000000000000 +
243 s (2000000 + s)) -
2500000 \[Pi] s (2500000000000000000000000000000000000000 +
3 s (72250000000000000000000000000000 +
3 s (22855000000000000000000000 +
9 s (105400000000000000 +
27 s (1318000000 + 9 s))))) Cos[(2 \[Pi])/9] +
s^2 (2500000000000000000000000000000000000000 +
3 s (72250000000000000000000000000000 +
3 s (22855000000000000000000000 +
9 s (105400000000000000 +
27 s (1318000000 + 9 s))))) Sin[(2 \[Pi])/
9])/(20 (6250000000000 \[Pi]^2 +
s^2) (2500000000000000000000000000000000000000000000 +
s (241750000000000000000000000000000000000 +
s (232112500000000000000000000000000 +
27 s (410200000000000000000000 +
s (148825000000000000 + 81 s (19000000 + 3 s)))))))}}

• Thanks for the contribution Jan, but could you maybe elaborate a bit? – Carl Apr 11 '20 at 16:55
• @Carl First of all you're welcome. You asked for the s-domain expression of the current I4 (in my circuit). So in my code, I solved for it. – Jan Apr 11 '20 at 16:57
• Yes, I can see that in your code $I_4$ is an expression of a lot of $\pi$, and $s$. However, I am expected to get a single number for this problem, so is there a way to convert your expression into that? A number. – Carl Apr 11 '20 at 17:53
• @Carl I will look at it when I have the time. Later today or tomorrow. – Jan Apr 11 '20 at 18:01
• Thank you for your help! – Carl Apr 11 '20 at 18:06