1.2 Kirchhoff's Laws & Series-Parallel Network Analysis

Ohm's Law alone can solve simple series or parallel circuits, but real branch-circuit and control-panel wiring often forms networks with multiple loops and junctions that Ohm's Law cannot untangle by itself. Gustav Kirchhoff, a German physicist, published two laws in 1845 that, together with Ohm's Law, can solve any DC network no matter how complex.

Kirchhoff's Current Law (KCL)

Kirchhoff's Current Law (KCL) states that the sum of currents entering a node (junction) equals the sum of currents leaving that node. In other words, charge cannot accumulate at a junction — whatever current flows in must flow out.

I_in = I_out

At a junction where one 10 A supply conductor splits into three branch conductors carrying 4 A, 3 A, and 3 A, KCL confirms the split is valid: 4 + 3 + 3 = 10 A.

Exam tip: KCL applies at nodes (junction points), not around loops. If a question describes a wire splitting or joining, you're applying KCL.

Kirchhoff's Voltage Law (KVL)

Kirchhoff's Voltage Law (KVL) states that the sum of the voltage drops around any closed loop equals the source voltage supplying that loop — or equivalently, the algebraic sum of all voltage rises and drops around a closed loop equals zero.

V_source = V_drop1 + V_drop2 + V_drop3 + ...

This is the same rule verified in Section 1.1's series-circuit example, where 20 V + 30 V dropped across two resistors equaled the 50 V source. KVL simply generalizes that rule to loops that may contain multiple sources and branches.

Exam tip: KVL applies around closed loops (a complete path that returns to its starting point), not at single junctions. If a question describes tracing a path around a circuit, you're applying KVL.

Voltage-Divider Reasoning

Because voltage drops in a series string are proportional to resistance, you can find the voltage across any one resistor without first solving for current. The voltage-divider formula for a resistor R_x in a series string is:

V_x = V_source × (R_x / R_total)

This shortcut is common on trade exams that ask for the voltage across one resistor in a series string without asking for current.

Current-Divider Reasoning

In a parallel network, current splits between branches in inverse proportion to resistance — the lower-resistance branch carries more current. For exactly two parallel resistors, the current-divider formula for the current through R1 is:

I_1 = I_total × (R2 / (R1 + R2))

Notice the crossed pattern: the current through R1 uses R2 in the numerator, and vice versa.

Solving a Series-Parallel Network Step by Step

Worked Example — Multi-Resistor Network

A 60 V DC source feeds R1 = 5 Ω in series with a parallel combination of R2 = 12 Ω and R3 = 6 Ω, which is itself in series with R4 = 3 Ω.

Step 1 — Reduce the parallel section. R_23 = (R2 × R3)/(R2 + R3) = (12 × 6)/(12+6) = 72/18 = 4 Ω

Step 2 — Add the series resistors. R_T = R1 + R_23 + R4 = 5 + 4 + 3 = 12 Ω

Step 3 — Apply Ohm's Law for total current. I_T = V / R_T = 60 / 12 = 5 A (this current flows through R1 and R4, and is the total current entering the parallel section)

Step 4 — Find the voltage across each series element using KVL. V1 = I_T × R1 = 5 × 5 = 25 V V_23 = I_T × R_23 = 5 × 4 = 20 V V4 = I_T × R4 = 5 × 3 = 15 V Check (KVL): 25 + 20 + 15 = 60 V, matching the source

Step 5 — Find the branch currents through R2 and R3 using Ohm's Law and confirm with KCL. I2 = V_23 / R2 = 20 / 12 = 1.67 A I3 = V_23 / R3 = 20 / 6 = 3.33 A Check (KCL): I2 + I3 = 1.67 + 3.33 = 5 A, matching I_T

Every worked network problem should end with these two checks — KVL around the loop and KCL at the junction — because they catch arithmetic errors before they cost you exam points.

Common Trade-Exam Traps

• Confusing which law applies. KCL is about current at a point (a node); KVL is about voltage around a path (a loop). A question that mentions a junction splitting wires needs KCL; a question that asks you to trace a complete circuit path needs KVL.
• Sign errors in KVL. When tracing a loop, a voltage rise (through a source, moving from − to +) and a voltage drop (across a resistor, in the direction of current flow) have opposite signs. Skipping this bookkeeping is the single most common reason candidates get the right magnitude but the wrong direction on a KVL problem.
• Forgetting a branch. In networks with three or more parallel branches, it's easy to compute only two branch currents and forget the third exists — always recount the branches against the schematic before finalizing an answer, and use the KCL check as a safety net.
• Applying series rules to a parallel section, or vice versa. Before combining any two resistors, confirm whether they share the same two nodes (parallel) or carry the same single current path with no other branch (series).

Key Takeaways

• KCL: the sum of currents into a node equals the sum of currents out — apply it at junctions.
• KVL: the sum of voltage drops around a closed loop equals the source voltage — apply it around loops.
• Voltage dividers split voltage in proportion to resistance; current dividers split current in inverse proportion to resistance.
• Multi-resistor networks are solved by reducing parallel sections first, then applying Ohm's Law and KVL/KCL as checks — always verify your final answer against both laws.