FAQ
TL;DR: 100% of nodal voltages collapse to zero when the Y (source) column is zero—"You can't have a zero Y matrix!" [Elektroda, Earl Albin, post #21665560]
Why it matters: This FAQ helps students and hobbyists fix AC nodal-analysis setups that yield all-zero node voltages.
Quick Facts
- Put the AC source j60 on the right-hand side; a zero Y column forces the trivial (all-zero) solution. [Elektroda, Earl Albin, post #21665560]
- Bottom capacitor+inductor branch combines to an equivalent j1 impedance in this thread’s circuit. [Elektroda, Earl Albin, post #21665558]
- Approximate check: total current ≈ 10 A ∠106°, giving V1 ≈ 6.20 V ∠~90°. [Elektroda, Earl Albin, post #21665554]
- Edge case: V1 can be 0 V only if that node is effectively a short. [Elektroda, Earl Albin, post #21665553]
- Algebra/sign mistakes are the most common cause of wrong complex results; recheck reductions. [Elektroda, Earl Albin, post #21665548]
Why do all my nodal voltages solve to zero?
Because your excitation vector is zero. If you leave the source term out, the matrix yields the trivial all-zero solution. Move the AC source (j60) to the right-hand side so the Y column is non-zero, then solve again. “You can’t have a zero Y matrix!” [Elektroda, Earl Albin, post #21665560]
Where should the AC source appear in my equations?
Place the source phasor on the right-hand side of your nodal equations. Treat it as the excitation vector, separate from conductance/impedance terms. This prevents a zero Y column and avoids the trivial solution. [Elektroda, Earl Albin, post #21665560]
Do I need a schematic with consistent node labels (V1, V2, V3)?
Yes. Share the schematic and ensure labels match your equations. Mismatched or missing node names cause incorrect setups and confusion during checking. Clear labeling also lets others verify KCL/KVL quickly. [Elektroda, Earl Albin, post #21665544]
Should I use KCL or KVL to solve this AC circuit?
Either works. Write KCL at the nodes or convert to KVL loop equations; both lead to the same node voltages when set up correctly. Choose the form you manipulate most accurately. [Elektroda, Earl Albin, post #21665551]
How do I write correct KCL nodal equations here?
Form KCL at each node using admittances to connected elements and sources. Keep source terms separate from impedance terms. Verify each branch current expression before assembling the matrix. [Elektroda, Earl Albin, post #21665550]
What’s the equivalent of the bottom capacitor and inductor branch?
In the posted circuit, the capacitor and inductor combine to an equivalent j1. Use that single impedance in your nodal equations to simplify the matrix. [Elektroda, Earl Albin, post #21665558]
What values should I expect if my setup is correct?
A sanity check reported total current ≈ 10 A at ∠106°. With the left branch impedance, V1 was ≈ 6.20 V near ∠90°. Differences suggest algebra or setup errors. [Elektroda, Earl Albin, post #21665554]
Could V1 actually be zero volts?
Yes, but only if the node is effectively shorted. Check branch currents around V1 to confirm. Frequency extremes can guide intuition: at ∞ Hz, V1 ≈ 22.5 V while V2 and V3 are near 0 V. [Elektroda, Earl Albin, post #21665553]
How can I catch sign and algebra mistakes in complex arithmetic?
Reduce equations carefully and audit every sign. Small sign errors dominate phasor results. Re-derive reduced forms step by step, then verify with a fresh pass before solving. [Elektroda, Earl Albin, post #21665548]
What does “trivial solution” mean in nodal analysis?
It means every node voltage solves to zero. This happens when your excitation vector is zero, such as leaving j60 off the right-hand side. [Elektroda, Earl Albin, post #21665560]
Can my reference node include an inductor?
Yes. You can still choose that node as reference. Combine series reactive elements correctly first; in this case, the capacitor and inductor reduce to j1. [Elektroda, Earl Albin, post #21665558]
Should I compute in polar or rectangular form?
Use whichever reduces errors for you. “Sometimes rectangular works better than Polar.” Convert between forms as needed for clarity and accuracy. [Elektroda, Earl Albin, post #21665545]
Is there a quick three-step way to fix my matrix setup?
- Move the source phasor j60 to the right-hand side.
- Build the nodal matrix with only admittances/impedances.
- Solve for V1, then back-substitute to get V2 and V3.
[Elektroda, Earl Albin, post #21665560]
How can I verify my equations with software?
Enter your corrected matrix into any basic linear solver or free circuit tool. Software helps confirm arithmetic, but it won’t fix wrong equations. Start with a correct setup. [Elektroda, Earl Albin, post #21665544]
Why did someone say my equations looked like KVL instead of nodal?
Because earlier, node labels didn’t match the posted equations, suggesting loop equations instead. Align your method and notation with the schematic to remove ambiguity. [Elektroda, Earl Albin, post #21665544]
