Kirchhoff's Current and Voltage Laws (KCL, KVL)

Ohm's law handles a single element. Kirchhoff's laws let you analyze full networks. They are not optional tricks; they are direct consequences of conservation of charge and conservation of energy.

Learning objectives

After this lesson, you will be able to:

• write KCL equations at nodes with a consistent sign convention;
• write KVL equations around loops with a consistent traversal direction;
• solve unknown branch currents and voltage drops in simple networks;
• explain why Kirchhoff's laws still underlie nodal and mesh analysis;
• catch common sign, polarity, and topology mistakes before they propagate.

Kirchhoff's Current Law (KCL)

At any node, the algebraic sum of currents is zero:

$$
\sum I = 0
$$

This is equivalent to:

$$
\sum I_{in} = \sum I_{out}
$$

KCL follows from conservation of charge. Charge does not accumulate indefinitely at an ordinary circuit node in steady lumped-circuit analysis.

KCL example

If 5 A enters the node and 2 A leaves through one branch:

$$
5 = 2 + I_3
$$

$$
I_3 = 3\ \text{A}
$$

Interactive KCL calculator

Use positive values for currents defined as entering the node and negative values for currents defined as leaving, or stay consistent with the calculator's sign prompt.

Kirchhoff's Voltage Law (KVL)

Around any closed loop, the algebraic sum of voltages is zero:

$$
\sum V = 0
$$

KVL is a consequence of energy conservation in lumped-element circuits. As you traverse a loop, voltage rises and drops must balance.

KVL example

BT1: Device:Battery value="12 V source" rotate=0
R1: Device:R value="5 V drop" rotate=270
R2: Device:R value="4 V drop" rotate=270
R3: Device:R value="3 V drop" rotate=270

layout direction=LR gap=65
group LOOP label="Single closed loop" direction=LR gap=65 {
  BT1 R1 R2 R3
}

BT1.1 --> R1.1 color=#b91c1c
R1.2 --> R2.1 color=#2563eb
R2.2 --> R3.1 color=#2563eb
R3.2 --> BT1.2 color=#334155

Start at the battery positive terminal, move left to right through the three voltage drops, and follow the lower return wire back to the negative terminal.

Traversing the loop:

$$
12 - 5 - 4 - 3 = 0
$$

The equation balances, so KVL is satisfied.

Interactive KVL calculator

Enter voltage rises and drops with a consistent sign convention.

A systematic way to use them together

Worked example: one source, two series resistors

A 10 V source drives R_1 = 2 kΩ and R_2 = 3 kΩ in series.

Step 1: use KVL to find current

$$
10 - I(2000) - I(3000) = 0
$$

$$
10 - 5000I = 0
$$

$$
I = 2\ \text{mA}
$$

Step 2: find element voltages

$$
V_{R1} = IR_1 = 0.002 \times 2000 = 4\ \text{V}
$$

$$
V_{R2} = IR_2 = 0.002 \times 3000 = 6\ \text{V}
$$

Check:

$$
10 - 4 - 6 = 0
$$

Worked example: one node splitting into two resistors

Suppose 9 mA enters a node and splits into two branches. One branch current is 4 mA. By KCL:

$$
9 = 4 + I_2
$$

$$
I_2 = 5\ \text{mA}
$$

If both branches return to the same reference node, KCL is enough to determine the missing branch current.

Sign conventions that keep you out of trouble

For KCL

Choose one convention and stick to it:

• currents into the node positive, out negative; or
• currents leaving the node positive, entering negative.

Either works if used consistently.

For KVL

Choose a loop direction:

• clockwise or counterclockwise.

Then decide whether each step is a voltage rise or drop according to the polarity you cross.

Where these laws are used later

Limits of the simple model

For the lumped circuits used in introductory electronics, Kirchhoff's laws are reliable and foundational. At very high frequencies, over long interconnects, or in distributed electromagnetic systems, the ideal lumped-circuit assumptions need refinement. That is not a reason to distrust KCL/KVL here; it is simply a reminder that every model has a scope.

Common mistakes

• Mixing sign conventions halfway through a solution.
• Writing KCL at a point that is not actually one node.
• Forgetting that a loop must return to its starting point.
• Treating resistor voltage drops as arbitrary signs instead of using current direction and passive sign convention.
• Assuming a negative result means the method failed. It usually means the true direction or polarity is opposite to your initial guess.

Summary

• KCL expresses conservation of charge at nodes.
• KVL expresses conservation of energy around closed loops.
• Consistent current directions, polarity labels, and loop directions matter more than memorizing slogans.
• These laws combine with Ohm's law to solve real circuits.
• Negative answers are often informative, not erroneous.

Next: Network Theorems: Thevenin, Norton, Superposition.

Further reading

• OpenStax: Kirchhoff's Rules
• MIT OpenCourseWare: Circuits and Electronics
• All About Circuits: Kirchhoff's Laws