Circuit variables and Kirchhoff’s laws

Every circuit-analysis technique taught later in EE11001 — node-voltage, mesh-current, superposition, Thevenin's theorem — is just repeated bookkeeping built on three small tools: Ohm's law and Kirchhoff's two laws. Ohm's law ties a resistor's voltage to its current. Kirchhoff's Current Law (KCL) says charge cannot pile up at a junction, so currents in must balance currents out. Kirchhoff's Voltage Law (KVL) says a charge carried all the way around a closed loop and back to its start gains no net energy, so voltage rises must balance voltage drops. This node sets up the sign conventions carefully, because a shaky grip on signs here is the single biggest source of errors in every later network-analysis method. Two fully worked examples — a series loop and a current-splitting node — anchor the ideas in numbers you can check by hand before you meet larger networks.

The core insight

At every node the currents balance (KCL, from charge conservation) and around every closed loop the voltages balance (KVL, from energy conservation), with Ohm's law $V = IR$ linking voltage and current inside each resistor.

The core idea

Ohm's law relates the voltage across a resistor to the current flowing through it:

V = IR

where: $V$ is voltage in volts, $I$ is current in amperes, $R$ is resistance in ohms ( $\Omega$ ).

Kirchhoff's Current Law (KCL) says that at any node (junction) of a circuit, the algebraic sum of currents is zero:

\sum I = 0

where: currents entering the node are taken with one sign (say positive) and currents leaving with the other (negative), consistently. This is nothing but a statement that charge cannot accumulate at a point — whatever current arrives at a junction must leave it.

Kirchhoff's Voltage Law (KVL) says that around any closed loop, the algebraic sum of voltage rises and drops is zero:

\sum V = 0

where: you pick one direction to traverse the loop, and as you cross each element you add a rise as positive and a drop as negative (or the reverse, again consistently). KVL follows from energy conservation: a charge that returns to its starting point after going around a loop has done zero net work, so the voltage gains and losses along the way must cancel. To use these laws on a network with $b$ branches and $n$ nodes: assign a current variable (with an assumed direction) to every branch, write KCL at $(n-1)$ independent nodes, write KVL around enough independent loops to supply the remaining equations, then solve the resulting simultaneous linear equations. A negative result for an assumed current simply means the true direction is opposite to what you guessed — the magnitude is still correct.

Figure

A single loop circuit with a battery and two resistors showing assumed current direction and KVL sign convention

Derivation AKVL solves a single-loop series circuit

The goal is to find the loop current and each resistor's voltage in a single-loop DC circuit using nothing but Ohm's law and KVL, and to check the result by confirming the voltage drops sum back to the source EMF, the way any correctly solved loop must. The circuit is a 12 V battery in series with $R_1 = 4\ \Omega$ and $R_2 = 8\ \Omega$ .

Step 1. Assume a single loop current $I$ flowing clockwise, leaving the positive terminal of the battery. This is an arbitrary choice — if it is wrong, the algebra will simply return a negative $I$ .

Step 2. Apply KVL by traversing the loop in the direction of $I$ , treating a rise through the source as positive and a drop across a resistor as negative: starting at the battery, we rise by $12\ \text{V}$ , then drop $IR_1$ across $R_1$ , then drop $IR_2$ across $R_2$ , arriving back at the start with the sum equal to zero:

12 - IR_1 - IR_2 = 0

Step 3. Substitute the resistor values and solve for $I$ : $12 - 4I - 8I = 0$ , so $12 = 12I$ , giving $I = 1\ \text{A}$ .

Step 4. Use Ohm's law to find each resistor's voltage: $V_1 = IR_1 = 1 \times 4 = 4\ \text{V}$ and $V_2 = IR_2 = 1 \times 8 = 8\ \text{V}$ .

Step 5. Check by KVL: $V_1 + V_2 = 4 + 8 = 12\ \text{V}$ , exactly equal to the source EMF, confirming the solution is consistent with energy conservation around the loop.

Worked example

Problem. A single-loop circuit has a 12 V battery in series with $R_1 = 4\ \Omega$ and $R_2 = 8\ \Omega$ (solved fully above: $I = 1\ \text{A}$ , $V_1 = 4\ \text{V}$ , $V_2 = 8\ \text{V}$ ). Now extend it: three branches meet at a node A. Branch 1 carries a known 5 A into node A. Branch 2 carries a known 2 A out of node A. Branch 3 has unknown current $I_3$ . Find $I_3$ and its direction.

Step 1. Take currents entering the node as positive and currents leaving as negative, and assume $I_3$ leaves node A — again an arbitrary assumed direction.

Step 2. Apply KCL at node A: currents in minus currents out sum to zero, so

5 - 2 - I_3 = 0

Step 3. Solve: $I_3 = 5 - 2 = 3\ \text{A}$ .

Step 4. Since $I_3$ came out positive, the assumed direction was correct: $I_3 = 3\ \text{A}$ , flowing out of node A through branch 3.

Step 5. Physical check: total current in ( $5\ \text{A}$ ) equals total current out ( $2\ \text{A} + 3\ \text{A} = 5\ \text{A}$ ) — charge is conserved at the node, exactly as KCL requires.

Competitive stretch

If instead branch 2 carried 2 A into node A (not out), KCL becomes $5 + 2 - I_3 = 0$ , so $I_3 = 7\ \text{A}$ out of the node — a reminder that every branch current's sign must be re-checked against the actual arrow you drew, not assumed from the previous problem.

Common traps

Mixing sign conventions mid-solution — calling current "entering positive" at one node and "leaving positive" at another within the same equation set. Fix one convention per problem and use it everywhere.
Treating a negative answer for an assumed current or loop direction as an error. It only means the real direction is opposite to the guess; the magnitude is still correct.
Applying KVL around a path that never returns to its starting node. The law holds only for genuinely closed loops.
Double-counting a source's EMF — adding it once as the ideal source voltage and again as a separately tracked terminal-voltage drop. Track each circuit element's contribution exactly once.

Ask the tutor: if a sign-convention step or a bigger multi-loop network is tripping you up, ask the AI tutor to walk through it with you.

Why does KCL say the sum of currents at a node is exactly zero rather than approximately zero?

KCL is a direct statement of charge conservation: charge cannot appear or vanish at a plain wire junction, and it cannot pile up there either, because a growing charge would build an ever-increasing voltage that circuit elements cannot sustain in steady state. So whatever current arrives must leave, current in equals current out, and writing entering currents as positive and leaving currents as negative makes that balance read as a sum equal to exactly zero, not approximately zero.

Does it matter which direction I assume for the loop current before applying KVL?

No. You may assume any direction for any branch current, and any traversal direction for any loop, as long as you apply the same sign rule consistently while writing the equation. If your assumed direction happens to be correct, the algebra returns a positive value; if it is opposite to the real direction, it returns a negative value of the same magnitude. Either way the physics is captured correctly — the assumption only affects the sign, never the correctness of the final magnitude.

How do I know how many independent KCL and KVL equations I need for a bigger network?

For a network with $b$ branches and $n$ nodes, you need $b$ independent equations in total to solve for all $b$ branch currents. KCL supplies $(n-1)$ independent equations — one node is always redundant because its equation follows from the others. The remaining $b - (n-1)$ equations come from KVL, written around that many independent loops (loops that each include at least one branch not used in a previous loop). This node-and-loop counting is exactly the setup you will reuse in the node-voltage and mesh-current methods later in the DC-networks unit.

Circuit variables and Kirchhoff’s laws

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads