4 Kirchhoff’s Laws and Circuit Analysis

Why Kirchhoff’s laws are useful

Kirchhoff’s laws provide a systematic way to find unknown currents and voltages in circuits that cannot be reduced using series- and parallel-resistor rules alone. They follow from conservation of electric charge and energy. Used with Ohm’s law, they turn a circuit diagram into a set of equations that can be solved for unknowns.

Current at a node

Kirchhoff’s current law (KCL) states that the total current entering any node equals the total current leaving it. A node is a point where circuit branches connect. If currents entering a node are assigned one sign and currents leaving it the opposite sign, their algebraic sum is zero:

For example, if currents $I_1$ and $I_2$ enter a node and $I_3$ leaves, then:

KCL expresses conservation of charge: in a steady circuit, charge does not accumulate at an ordinary junction.

Voltage changes around a loop

Kirchhoff’s voltage law (KVL) states that the signed voltage changes around any closed path, or loop, sum to zero:

This expresses conservation of energy: a charge returning to its starting point has no net change in electric potential energy.

Use a consistent sign convention while tracing a loop:

• Across a resistor in the direction of the assumed current, the potential changes by $-IR$; against the current, it changes by $+IR$.

• Across an ideal source from its negative terminal to its positive terminal, the change is $+\mathcal{E}$; from positive to negative, it is $-\mathcal{E}$.

For a single source $V$ driving current $I$ through series resistors $R_1$ and $R_2$, KVL gives:

Therefore, $I=\frac{V}{R_1+R_2}$, consistent with Ohm’s law and the equivalent resistance of series resistors.

A systematic method

Use a consistent set of current directions and voltage polarities to form independent equations. For networks made of ideal voltage sources and resistors, the equations are linear, and enough independent equations are needed to determine the unknowns.

1. Label nodes and components. Mark source polarities and resistor values. Components connected by ideal wire belong to the same node.

2. Assign unknown currents or node voltages. Choose directions for branch currents that are not yet known. These choices are assumptions, not requirements for the final answer.

3. Write independent KCL equations. For a network with $N$ nodes, at most $N-1$ node equations are independent.

4. Write independent KVL equations. Choose loops that together include the branches to analyze. Avoid equations that merely repeat combinations of other equations.

5. Use component relations. For a resistor, apply Ohm’s law, $V=IR$, with voltage polarity and current direction defined consistently. For an ideal voltage source, use its specified voltage and polarity.

6. Solve and check. A negative solved current means that the actual current flows opposite to the assumed direction. Check that the result satisfies the original node and loop equations; where appropriate, verify that total power supplied equals total power absorbed.

Worked example with parallel branches

A $12\text{ V}$ source supplies a $2\,\Omega$ resistor in series with parallel $6\,\Omega$ and $3\,\Omega$ resistors. Let the source’s negative terminal be $0\text{ V}$, and call the junction voltage $V_A$. The current through the $2\,\Omega$ resistor toward the junction is $\frac{12-V_A}{2}$; the currents leaving the junction through the $6\,\Omega$ and $3\,\Omega$ branches are $\frac{V_A}{6}$ and $\frac{V_A}{3}$, respectively.

Applying KCL at the junction:

Solving gives:

The currents through the resistors are:

KCL checks because $3=1+2\text{ A}$. KVL also checks: the source’s $12\text{ V}$ rise equals the $6\text{ V}$ drop across the $2\,\Omega$ resistor plus the $6\text{ V}$ across either parallel branch. This example uses a node equation and Ohm’s law to analyze the network without first guessing its total resistance.