5.5.2. Multi-Loop Circuits

Table of Contents

Solving circuits with more than one loop

A multi loop circuit is a circuit that contains several closed paths for current. In these circuits, current can split at junctions and then recombine later. Because of this, a single equation is usually not enough to find all unknown currents or voltages.

The key idea is simple. We combine the rules for series and parallel behavior with Kirchhoff's laws. In a multi loop circuit, the unknowns are often the branch currents. We write equations for junctions and for loops, then solve them together.

Why multi loop circuits are different

In a single loop circuit, the same current flows everywhere in the loop. In a multi loop circuit, that is no longer true. Different branches can carry different currents.

For example, if a current reaches a junction, part of it may go through one resistor and part through another. This means we must carefully label currents in each branch and keep the signs consistent.

A typical multi loop problem includes resistors and one or more sources of electromotive force. The arrangement can be more complex than simple series or parallel combinations, so direct reduction is not always possible.

Kirchhoff's rules in multi loop circuits

There are two main rules.

The junction rule says that the total current entering a junction equals the total current leaving it. This expresses conservation of charge.

The loop rule says that the algebraic sum of potential differences around any closed loop is zero. This expresses conservation of energy.

Important rules for multi loop circuits:
At a junction,
$$\sum I_{\text{in}} = \sum I_{\text{out}}$$
Around any closed loop,
$$\sum \Delta V = 0$$
When moving through a resistor in the direction of the current, the change in potential is
$$\Delta V = -IR$$
When moving through a resistor opposite to the current, the change in potential is
$$\Delta V = +IR$$
When moving through an ideal battery from negative terminal to positive terminal,
$$\Delta V = +\mathcal{E}$$
When moving from positive terminal to negative terminal,
$$\Delta V = -\mathcal{E}$$

Choosing currents

To solve a multi loop circuit, begin by assigning a current direction to each branch or loop. The chosen directions do not need to be correct at first. If a current comes out negative in the final answer, that only means the real current flows opposite to the direction you assumed.

This is very useful, because it lets you start the problem without guessing the actual flow correctly.

A simple two loop example

Consider a circuit with two loops that share one resistor. Let the left loop contain a battery $\mathcal{E}_1$ and resistor $R_1$, the right loop contain a battery $\mathcal{E}_2$ and resistor $R_2$, and let the middle resistor $R_3$ be shared by both loops.

We choose loop currents $I_1$ in the left loop and $I_2$ in the right loop, both clockwise. In the shared resistor, the current depends on both loop currents. If the loop currents pass through the shared resistor in opposite directions, then the current in that resistor is $I_1 - I_2$ or $I_2 - I_1$, depending on the chosen reference direction.

Rendering scientific sketch…

Two-loop circuit with a shared resistor

Writing loop equations

For the left loop, suppose the shared resistor current is $I_1 - I_2$ in the direction of $I_1$. Then the loop rule gives

$$
\mathcal{E}_1 - I_1R_1 - (I_1 - I_2)R_3 = 0
$$

For the right loop, if we move clockwise with $I_2$,

$$
\mathcal{E}_2 - I_2R_2 - (I_2 - I_1)R_3 = 0
$$

These are two simultaneous equations in the two unknowns $I_1$ and $I_2$. Once those are solved, every voltage drop in the circuit can be found from Ohm's law.

Loop current method

One common method for multi loop circuits is the loop current method. In this method, you assign a current to each independent loop and use Kirchhoff's loop rule to generate equations.

This works especially well when the circuit is planar and easy to separate into loops.

The procedure is as follows. First, choose loop currents and their directions. Second, identify any shared elements. Third, write one loop equation for each independent loop. Fourth, solve the system of equations.

For a resistor shared by two loop currents $I_a$ and $I_b$, the voltage drop is based on the net current through that resistor. If the currents oppose each other in that resistor, the drop is based on $I_a - I_b$.

For a resistor shared by two loop currents, do not use just $IR$ with one current unless that resistor belongs to only one loop.
Use the net current through the resistor.
If the currents oppose each other,
$$V_R = (I_a - I_b)R$$
or
$$V_R = (I_b - I_a)R$$
Choose the form that matches your chosen direction of traversal.

Junction equations and branch currents

Another method is to assign currents directly to branches instead of loops. Then you use junction equations together with loop equations.

This method is often helpful when the current splits at several nodes. Each branch gets its own current, and the junction rule connects them.

As a simple example, if current $I$ reaches a junction and splits into $I_1$ and $I_2$, then

$$
I = I_1 + I_2
$$

You then combine this with loop equations for the different closed paths.

How many equations are needed

The number of independent equations must match the number of unknown currents. Some loop equations may repeat information already contained in others, so not every possible loop gives a new independent equation.

In practice, you write enough junction and loop equations to solve for all unknowns, but avoid redundant equations.

Sign conventions

Most mistakes in multi loop circuits come from sign errors. A careful, fixed convention prevents confusion.

The table below summarizes the most common choices.

Circuit element or movement	Potential change
Through resistor in direction of current	$-IR$
Through resistor opposite to current	$+IR$
Through battery from negative to positive terminal	$+\mathcal{E}$
Through battery from positive to negative terminal	$-\mathcal{E}$

If your final current is negative, do not panic. The physics is still correct. It only means the actual current goes opposite to the direction you assumed.

Example with equations

Suppose a two loop circuit has

$$
\mathcal{E}_1 = 12\ \text{V}, \quad \mathcal{E}_2 = 6\ \text{V}
$$

and

$$
R_1 = 2\ \Omega, \quad R_2 = 4\ \Omega, \quad R_3 = 3\ \Omega
$$

Choose clockwise loop currents $I_1$ and $I_2$. The equations are

$$
12 - 2I_1 - 3(I_1 - I_2) = 0
$$

$$
6 - 4I_2 - 3(I_2 - I_1) = 0
$$

Simplify them:

$$
12 - 5I_1 + 3I_2 = 0
$$

$$
6 + 3I_1 - 7I_2 = 0
$$

$$
5I_1 - 3I_2 = 12
$$

$$
3I_1 - 7I_2 = -6
$$

Solving gives

$$
I_2 = \frac{33}{26}\ \text{A}
$$

and

$$
I_1 = \frac{81}{26}\ \text{A}
$$

The current through the shared resistor is

$$
I_{R_3} = I_1 - I_2 = \frac{48}{26} = \frac{24}{13}\ \text{A}
$$

This result shows clearly that the shared branch current is not equal to either loop current alone.

Interpreting results

After solving, you can find voltage drops across each resistor using Ohm's law. For the example above,

$$
V_{R_1} = I_1R_1, \quad V_{R_2} = I_2R_2, \quad V_{R_3} = (I_1 - I_2)R_3
$$

These values let you check your work by substituting them back into the loop equations.

A correct solution should satisfy every junction rule and every independent loop rule.

Practical strategy for beginners

When you first meet multi loop circuits, it helps to work slowly and systematically. Redraw the circuit clearly. Mark battery polarities. Choose current directions. Then write equations one loop at a time.

Do not try to think about everything at once. The mathematics becomes manageable when each step is written carefully.

A reliable strategy is:

Label all known values.
Choose current directions.
Write junction equations if needed.
Write one equation for each independent loop.
Solve the simultaneous equations.
Interpret any negative current as a reversed actual direction.

Common mistakes

A frequent mistake is forgetting that a shared resistor carries the difference of loop currents, not one loop current by itself. Another common mistake is changing sign conventions halfway through a problem. Some students also write too many equations and then become confused by redundant ones.

A good habit is to check each term in a loop equation by asking, "As I move around the loop, does the potential rise or fall here?"

A three-loop picture

Larger circuits may contain three or more loops. The same principles still work. You assign loop currents, identify shared elements, and write one loop equation per independent loop.

Rendering scientific sketch…

Three-loop schematic idea

The algebra becomes larger, but the logic stays exactly the same.

Final idea

Multi loop circuits are solved by combining current conservation at junctions with energy conservation around loops. The main new feature is that different parts of the circuit can carry different currents, and shared components must be treated with care.

Once you can label currents consistently and write Kirchhoff equations correctly, even complicated circuits become a system of equations that can be solved step by step.

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