Multi-loop Circuits

Definition

A multi-loop circuit (also called a complex circuit or mesh circuit) is an electrical circuit that contains:

Multiple closed paths (loops)
Junctions where current splits or combines
Components that cannot be reduced to simple series or parallel combinations

These circuits require Kirchhoff's Rules for complete analysis.

Characteristics

What Makes a Circuit "Multi-loop"?

Feature	Description
Multiple Loops	Contains 2+ independent closed paths
Junctions	Points where 3+ components meet
Branch Currents	Different currents in different branches
Cannot Simplify	Resistors not purely series or parallel

Examples of Multi-loop Circuits

Circuits with multiple voltage sources
Bridge circuits (e.g., Wheatstone Bridge)
Ladder networks
Circuits with components arranged in a "mesh" pattern

Analysis Method

Using Kirchhoff's Rules

Multi-loop circuits are solved using both of Kirchhoff's laws:

KCL (Junction Rule): Conservation of charge at nodes
KVL (Loop Rule): Conservation of energy around loops

Counting Equations Needed

For a circuit with:

$b$ branches (unknown currents)
$n$ nodes

Number of equations needed: $b$

From KCL: $(n - 1)$ independent equations
From KVL: $b - (n - 1)$ independent equations

Systematic Analysis Procedure

Step 1: Identify and Label

Count branches, nodes, and loops
Assign current variables ($I_1$, $I_2$, ...) to each branch
Choose current directions (arbitrary — negative means opposite)

Step 2: Apply KCL

Write $(n-1)$ current conservation equations at junctions
Example: At junction A with currents $I_1$ entering and $I_2$, $I_3$ leaving: $$I_1 = I_2 + I_3$$

Step 3: Apply KVL

Choose independent loops
Write voltage conservation for each loop
Follow sign conventions

Step 4: Solve

You now have $b$ equations with $b$ unknowns
Use substitution or matrix methods (e.g., Cramer's rule)

Step 5: Verify

Check that KCL is satisfied at all junctions
Verify power balance: $P_{\text{supplied}} = P_{\text{dissipated}}$

Common Circuit Configurations

Two-Loop Circuit

    ┌─R₁─┬─R₂─┐
    │    │    │
   ℰ₁   R₃   ℰ₂
    │    │    │
    └────┴────┘

3 branches, 2 nodes
Need 3 equations: 1 KCL + 2 KVL

Bridge Circuit

    ┌─R₁─┬─R₂─┐
    │    │    │
   ℰ    R₅    
    │    │    │
    └─R₃─┴─R₄─┘

Cannot reduce using series/parallel
Requires full Kirchhoff analysis

Worked Example

Problem

Find all currents in this two-loop circuit:

        I₁        I₂
    ┌──→──┬──→──┐
    │     │     │
   12V   4Ω    9V
    │     │     │
    │    I₃     │
    │     ↓     │
    └───7Ω──────┘

Solution

Step 1: KCL at top junction: $$I_1 + I_2 = I_3$$

Step 2: KVL, left loop (clockwise): $$-12 + 4I_3 + 7I_1 = 0$$ $$7I_1 + 4I_3 = 12$$

Step 3: KVL, right loop (clockwise): $$-9 + 4I_3 + 8I_2 = 0$$ $$8I_2 + 4I_3 = 9$$

Step 4: Solve three equations simultaneously...

Related Concepts

Kirchhoff's Rules — the fundamental laws
Kirchhoff's Current Law (KCL) — junction analysis
Kirchhoff's Voltage Law (KVL) — loop analysis
Wheatstone Bridge — specific multi-loop circuit
FAD1022 L12 — Kirchhoff's Rules (Applications) — detailed examples
DC Circuits — general DC analysis

Mermaid Diagram: Multi-loop Circuit Analysis

graph TD
    A[Multi-loop Circuit] --> B[Count<br/>b = branches<br/>n = nodes]
    B --> C[Label Currents<br/>I₁, I₂, I₃...]
    C --> D[Apply KCL<br/>n-1 equations]
    D --> E[Apply KVL<br/>b-n+1 equations]
    E --> F[Total: b equations<br/>b unknowns]
    F --> G[Solve<br/>Simultaneous Equations]
    G --> H[Check<br/>Power Balance]
    H --> I[Final Answer]