Journeyman
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Kirchhoff's laws

Updated · Written for the 2023 NEC

Short answer

Kirchhoff's current law (KCL) says the current flowing into any junction equals the current flowing out of it. Kirchhoff's voltage law (KVL) says the voltage drops around any closed loop add up to the source voltage. KCL is why branch currents add in a parallel circuit and why a neutral carries the unbalanced current. KVL is why voltage divides among loads in a series circuit.

What Kirchhoff's laws are

Kirchhoff's laws are two bookkeeping rules for current and voltage that hold in every circuit, no matter how it is arranged. Ohm's law tells you what happens inside one resistor. Kirchhoff's laws tell you how the pieces fit together.

  • Current law (KCL): current cannot pile up at a point or vanish from it. Whatever flows into a node (a splice, a terminal bar, a junction of branches) must flow out.
  • Voltage law (KVL): if you trace any closed path from the source, through the loads and back, the voltage used up by the loads equals the voltage the source supplied.

On the exam you use them in every series circuit, parallel circuit and neutral current question, usually together with Ohm's law.

The formulas

KCL: I(in) = I(out)
KVL: E(source) = E1 + E2 + E3 + ...

How the laws apply to series and parallel circuits

KVL governs series circuits and KCL governs parallel circuits.

In a series circuit there is one path, so the same current flows through every load, and KVL says the drops add up to the source voltage. A bigger resistor takes a bigger share.

In a parallel circuit each branch sees the full source voltage, and KCL says the source current equals the sum of the branch currents. See series and parallel circuits for the full set of rules and resistance formulas.

How KCL explains neutral current in multiwire branch circuits

On a 120/240 V single-phase, 3-wire system, the neutral carries only the difference between the two line currents. The two line currents are 180 degrees apart. Apply KCL where the two 120 V loads join the neutral: the line 2 current cancels part of the line 1 current, and only the unbalance is left for the neutral.

Single-phase 3-wire: IN = IL1 − IL2

On a three-phase, 4-wire wye system the phase currents are 120 degrees apart, so they do not simply subtract. For loads at unity power factor with no harmonics, KCL works out to this formula:

Wye neutral: IN = √(A² + B² + C² − AB − BC − CA)

When all three phases carry equal current, the neutral current is zero. When only two phases of a wye are loaded equally, the neutral carries about the same current as each phase, not zero.

The NEC rules for multiwire branch circuits rest on this physics:

  • 210.4 covers multiwire branch circuits. 210.4(B) requires a means that simultaneously disconnects all ungrounded conductors at the point where the circuit originates, and 210.4(D) requires the conductors of each multiwire circuit to be grouped.
  • 300.13(B) requires that the neutral of a multiwire branch circuit not depend on device terminals, so removing a receptacle cannot open the neutral for the other loads.
  • 310.15(E) tells you when the neutral counts as a current-carrying conductor for ampacity adjustment. A neutral carrying only the unbalanced current of the other conductors does not count. It does count when it is the common conductor of a 3-wire circuit taken from a 4-wire wye system, and when the major portion of the load is nonlinear (harmonic currents add in the neutral instead of canceling).

Worked examples

Example 1: KCL at a junction

Three branch circuits in parallel at 120 V draw 8 A, 5 A and 3 A. The current entering from the source is 8 + 5 + 3 = 16 A.

If 20 A enters a splice and one conductor leaving it carries 12 A, the other carries 20 − 12 = 8 A.

Example 2: KVL in a series circuit

Resistors of 5, 10 and 15 ohms are in series across 120 V.

  1. Total resistance: 5 + 10 + 15 = 30 ohms.
  2. Current: 120 ÷ 30 = 4 A.
  3. Drops: 4 × 5 = 20 V, 4 × 10 = 40 V, 4 × 15 = 60 V.
  4. Check with KVL: 20 + 40 + 60 = 120 V.

Example 3: KVL and conductor voltage drop

A 120 V, 2-wire circuit carries 16 A, and each conductor (out and back) has 0.1 ohm of resistance. Each conductor drops 16 × 0.1 = 1.6 V. By KVL the load gets what is left: 120 − 1.6 − 1.6 = 116.8 V. This is the same loop reasoning behind the voltage drop calculator.

Example 4: neutral current on a 120/240 V multiwire circuit

Line 1 carries 15 A of 120 V load and line 2 carries 9 A of 120 V load. The neutral carries 15 − 9 = 6 A.

Example 5: what happens when that neutral opens

Take a 12 A load (10 ohms) on line 1 and a 6 A load (20 ohms) on line 2. If the shared neutral opens, the two loads end up in series across 240 V.

  1. Total resistance: 10 + 20 = 30 ohms.
  2. Current: 240 ÷ 30 = 8 A.
  3. Voltage across the 10-ohm load: 8 × 10 = 80 V.
  4. Voltage across the 20-ohm load: 8 × 20 = 160 V.

The smaller (higher-resistance) load gets 160 V and can be damaged, which is why 300.13(B) protects neutral continuity.

Example 6: three-phase wye neutral

A 208Y/120 V system has 30 A on phase A, 20 A on phase B and 10 A on phase C, all resistive.

IN = √(900 + 400 + 100 − 600 − 200 − 300) = √300 = 17.3 A.

Common exam mistakes

  • Adding line currents to get neutral current. On a 120/240 V system the neutral carries the difference, not the sum.
  • Subtracting on a three-phase wye. Phase currents are 120 degrees apart, so use the square-root formula.
  • Assuming two loaded phases of a wye leave the neutral idle. Two equal phases leave the neutral carrying about the same current as each phase.
  • Forgetting the conductors. In KVL, the wire drops are part of the loop, so the load always sees a little less than source voltage.

Practice questions

Question 1 · Electrical theoryKirchhoff's current law states that at any junction (node) in a circuit:
Show answer

Answer: C. The sum of currents entering equals the sum of currents leaving.

Charge cannot pile up at a point, so whatever current flows into a junction must flow out. This is the basis for adding branch currents to find total current in parallel circuits.

Question 2 · Electrical theoryKirchhoff's voltage law states that around any closed loop:
Show answer

Answer: B. The sum of the voltage drops equals the applied voltage.

As you travel around a closed loop, the voltage dropped across loads must add up to the voltage supplied by the source. This is why voltage divides among series resistors.

Question 3 · Electrical theoryA 120/240-volt, single-phase, 3-wire circuit has 30 amperes of 120-volt load on line 1 and 20 amperes of 120-volt load on line 2. What is the current in the neutral? Assume sinusoidal currents, unity power factor, and negligible harmonics.
Show answer

Answer: D. 10 A.

On a single-phase 3-wire system the neutral carries only the difference between the two line currents. 30 − 20 = 10 A.

  1. IN = IL1 − IL2
  2. IN = 30 − 20
  3. IN = 10 A
Question 4 · Electrical theoryA 208Y/120-volt, 4-wire wye system has 40 amperes of 120-volt load on phase A, 40 amperes on phase B, and nothing on phase C. What is the neutral current? Assume sinusoidal currents, unity power factor, and negligible harmonics.
Show answer

Answer: B. 40 A.

Phase currents in a wye are 120 degrees apart, so they do not simply subtract. Using IN = √(A² + B² + C² − AB − BC − CA) = √(1,600 + 1,600 + 0 − 1,600 − 0 − 0) = 40 A.

  1. IN = √(A² + B² + C² − AB − BC − CA)
  2. IN = √(1,600 + 1,600 + 0 − 1,600 − 0 − 0) = √1,600
  3. IN = 40 A
Question 5 · Electrical theoryAn electrician clamps an ammeter around both the hot and neutral conductors of a 2-wire, 120-volt circuit that is carrying 15 amperes. What will the meter read?
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Answer: A. Approximately 0 A.

The currents in the two conductors are equal and flow in opposite directions, so their magnetic fields cancel. To read load current, clamp around only one conductor.

Frequently asked questions

What is the difference between Kirchhoff's current law and voltage law?

The current law applies at a junction: current in equals current out. The voltage law applies around a closed loop: the voltage drops add up to the source voltage. Use the current law for parallel branches and neutrals, and the voltage law for series drops.

Why does the neutral carry only the unbalanced current?

On a 120/240 V system the two line currents are 180 degrees apart. Applying Kirchhoff's current law where the loads meet the neutral, the opposing currents cancel and only the difference returns on the neutral.

Why does a clamp meter read zero around both conductors of a circuit?

The hot and neutral of a 2-wire circuit carry equal current in opposite directions, so by the current law the net current through the clamp is zero. Clamp around one conductor to read load current.

Where are Kirchhoff's laws on the journeyman exam?

They show up inside series and parallel calculations, voltage drop problems and neutral current questions. Practice them alongside the other exam calculations and take a practice test to check your speed.

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