The circuit
V1 feeds node A through R1; R2 runs from A to ground; R3 connects A to a second source V2. Neither source alone explains the currents, so the circuit needs a systematic method.
Mesh analysis (KVL)
Give each window of the circuit a loop current, both clockwise. Walk round each loop adding voltage drops (Kirchhoff’s voltage law). R2 is shared, so it carries the difference of the two loop currents. Two loops give two equations in I1 and I2:
A negative answer just means that loop current flows anticlockwise.
(R1 + R2)·I1 − R2·I2 = V1 −R2·I1 + (R2 + R3)·I2 = −V2 I_R2 = I1 − I2
Nodal analysis (KCL)
Pick ground as 0 V. Only node A is unknown. Write every current leaving A in terms of VA (Kirchhoff’s current law says they sum to zero), and solve one equation:
(VA − V1)/R1 + VA/R2 + (VA − V2)/R3 = 0 VA = (V1/R1 + V2/R3) / (1/R1 + 1/R2 + 1/R3)
Which method to choose
Both give the same currents. Count the unknowns: mesh needs one equation per window, nodal one per node apart from ground. Here that is two against one, so nodal is shorter. Circuits with many parallel branches favour nodal; long ladders with few nodes but many current sources can favour mesh.
Worked example (the simulator’s default values)
- V1 = 9 V, V2 = 5 V, R1 = 220 Ω, R2 = 470 Ω, R3 = 330 Ω.
- Mesh: 690·I1 − 470·I2 = 9 and −470·I1 + 800·I2 = −5; Δ = 690·800 − 470² = 331 100 Ω².
- I1 = 14.65 mA, I2 = 2.36 mA, so I_R2 = I1 − I2 = 12.29 mA.
- Node: VA·(4.545 + 2.128 + 3.030) mS = 40.909 + 15.152 mA, so VA = 56.06/9.703 = 5.78 V.
- Check: I_R1 = (9 − 5.78)/220 = 14.65 mA = I1, and I_R3 = (5.78 − 5)/330 = 2.36 mA = I2. The 5 V source absorbs power: it is being charged.
Common mistakes
- Forgetting that a shared resistor carries both loop currents: its drop is R2·(I1 − I2), not R2·I1.
- Mixing current directions in KCL. Write every current as leaving the node, and let the signs come out of the algebra.
- Reading a negative mesh current as an error. It only means the current flows opposite to the assumed direction.
Common questions
What is the difference between mesh and nodal analysis?
Mesh analysis uses loop currents and Kirchhoff’s voltage law, one equation per window. Nodal analysis uses node voltages and Kirchhoff’s current law, one equation per non-ground node. Both give the same answer.
When should I use nodal analysis instead of mesh analysis?
When the circuit has fewer non-ground nodes than windows, as with many parallel branches. Pick whichever gives fewer unknowns.
What does a negative mesh current mean?
That the loop current actually flows opposite to the direction you assumed, usually anticlockwise if you assumed clockwise. The working is still correct.