Ohm's & Kirchhoff's Laws

Ohm's law, Kirchhoff's current and voltage laws, and the series/parallel rules they produce. Learn to solve any resistor network with systematic nodal and mesh analysis.

Three laws solve almost every circuit you will meet: Ohm's law relates voltage, current and resistance in a single component; Kirchhoff's current law (KCL) says current is conserved at every junction; Kirchhoff's voltage law (KVL) says voltages around any closed loop sum to zero.

From these come the series and parallel rules, voltage and current dividers, and the systematic methods — nodal and mesh analysis — that circuit simulators like SPICE use internally.

An analogy

KCL is like traffic at a roundabout: every car that enters must leave. KVL is like a hiking loop: however much you climb and descend, you finish at the same altitude you started.

Ohm's law

For a resistor, current is proportional to voltage: I = V/R. Resistance depends on material and shape, R = ρL/A — longer wires resist more, thicker wires less. Components that obey it are 'ohmic'; diodes, lamps and transistors are not.

Series and parallel

Series components share the same current and their voltages add. Parallel components share the same voltage and their currents add.

• Series: R_eq is larger than the largest resistor.
• Parallel: R_eq is smaller than the smallest resistor.
• Two in parallel: R_eq = R₁R₂ / (R₁ + R₂) ('product over sum').

Kirchhoff's laws

KCL comes from conservation of charge, KVL from conservation of energy (the electric field is conservative in lumped circuits). Pick current directions arbitrarily — a negative answer just means the current flows the other way.

Nodal analysis (the systematic method)

Choose a ground node, label every other node voltage, write KCL at each node using Ohm's law for each branch current, and solve the simultaneous equations. For n nodes you get n − 1 equations. SPICE builds exactly this as a matrix, G·V = I.

Mesh analysis and when to use each

Mesh analysis assigns a circulating current to each window of a planar circuit and writes KVL around it. Use nodal when there are fewer nodes than meshes or current sources; use mesh when there are fewer meshes or voltage sources. Supernodes and supermeshes handle sources between two unknowns.

Where lumped laws break down

KVL assumes no changing magnetic flux links the loop and that circuit dimensions are much smaller than the wavelength. At RF, or with large loops near transformers, induced voltages and transmission-line effects mean you need field theory or distributed models.

Formulas

• Ohm's law: V = I · R
• KCL (any node): Σ I_in = Σ I_out
• KVL (any loop): Σ V = 0
• Series / parallel: R_s = R₁ + R₂ + … 1/R_p = 1/R₁ + 1/R₂ + …
• Current divider: I₁ = I_total · R₂ / (R₁ + R₂)

Worked example: Series–parallel network

Given: V = 24 V; R₁ = 4 Ω in series with (R₂ = 6 Ω ∥ R₃ = 12 Ω)

Find: Total current and every branch current and voltage.

• R₂ ∥ R₃ = 6 × 12 / 18 = 4 Ω
• R_eq = 4 + 4 = 8 Ω → I = 24 / 8 = 3 A
• V_R1 = 3 × 4 = 12 V; V_parallel = 24 − 12 = 12 V (KVL)
• I₂ = 12/6 = 2 A, I₃ = 12/12 = 1 A; KCL check: 2 + 1 = 3 A ✓

Answer: I = 3 A, I₂ = 2 A, I₃ = 1 A, 12 V across each section

Worked example: Nodal analysis with two sources

Given: 10 V source through R₁ = 2 Ω to node N; R₂ = 4 Ω from N to ground; 5 V source through R₃ = 1 Ω to node N

Find: Node voltage V_N.

• KCL at N (currents leaving): (V_N − 10)/2 + V_N/4 + (V_N − 5)/1 = 0
• Multiply by 4: 2V_N − 20 + V_N + 4V_N − 20 = 0
• 7V_N = 40 → V_N = 5.71 V
• Currents: from 10 V source 2.14 A in, from 5 V source 0.71 A out, into R₂ 1.43 A; KCL: 2.14 = 0.71 + 1.43 ✓

Answer: V_N ≈ 5.71 V

Common mistakes

• Wrong: Adding parallel resistances directly. — Right: Add conductances (1/R), then invert.
• Wrong: Mixing sign conventions halfway through a KVL loop. — Right: Pick a direction, treat rises and drops consistently, and let negative answers tell you the true direction.

Applications

• LED resistors: KVL sizes every LED's current-limiting resistor: R = (V_supply − V_f) / I_LED.
• House wiring: Sockets and lights are in parallel, so each sees full mains voltage and works independently.
• Circuit simulators: SPICE formulates modified nodal analysis automatically for circuits with millions of nodes.

Key takeaways

• V = IR for ohmic components; R = ρL/A.
• KCL: currents into a node = currents out. KVL: voltages around a loop sum to zero.
• Series adds resistance; parallel adds conductance.
• Nodal and mesh analysis solve any linear network systematically.

Going further

• Modified nodal analysis (MNA): SPICE extends nodal analysis with extra unknowns for voltage-source currents, giving a sparse matrix it solves with LU decomposition. Non-linear devices are linearised each Newton–Raphson iteration.
• Tellegen's theorem: For any circuit obeying KCL and KVL, Σ vₖ·iₖ = 0 over all branches — power is conserved. It holds even for non-linear and time-varying elements and underpins sensitivity analysis.
• Resistor tolerances in design: A 1 % divider can have a ratio error of nearly 2 % in the worst case. Precision designs use matched resistor networks, whose ratio tracks over temperature much better than two separate parts.