For circuits with several loops, mesh analysis assigns a circulating current to each loop and solves the resulting simultaneous equations. Work through a real two-loop example below, one clear step at a time.
📐The rule: assign a clockwise mesh current to every independent loop, write one KVL equation per loop, then solve the simultaneous equations. The current in a shared branch is the difference between the two mesh currents passing through it.
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Watch it happen — one step at a time
Circuit: E1 = 20 V (R1 = 4 Ω) and E2 = 10 V (R3 = 3 Ω), sharing R2 = 6 Ω between the two loops. Click through each step below.
Step 0 · The circuit
Two loops, two mesh currents
We assign a mesh current to each loop — I1 for the left loop, I2 for the right loop — both drawn clockwise by convention. Where they overlap (through R2), the two currents combine.
What we want to find:The current through the shared resistor R2
Step 1 · Loop 1 KVL
Go round loop 1, sum the voltages to zero
Going clockwise round loop 1: E1 drives current in, R1 drops I1×R1, and R2 (shared) drops the difference between the two mesh currents, (I1−I2)×R2 — because I2 also flows through R2 in the opposite sense.
Loop 1 equation:20 = 10·I1 − 6·I2
Step 2 · Loop 2 KVL
Same idea, round loop 2
Going clockwise round loop 2: E2 drives current in the opposite sense to how we defined loop 1's direction through the shared branch, so it appears negative. R3 drops I2×R3, and R2 drops (I2−I1)×R2.
Loop 2 equation:−10 = 9·I2 − 6·I1
Step 3 · Solve the simultaneous equations
Two equations, two unknowns — elimination method
Eliminate I2: its coefficients are −6 and 9. The lowest common multiple of 6 and 9 is 18 — multiply the first equation by 3, the second by 2.
Eq 1 × 3
30·I1 − 18·I2 = 60
Eq 2 × 2
−12·I1 + 18·I2 = −20
Add — the I2 terms cancel:18·I1 = 40
I1 = 40 ÷ 18= 2.22 A
Substitute into Eq 1: 10(2.22) − 6·I2 = 20I2 = 0.37 A
Step 4 · The shared branch
Watch the two loop currents combine
The current through R2 isn't I1 or I2 alone — it's the difference, since the two mesh currents pass through it in opposite senses.
I2 through R2 = I1 − I2 = 2.22 − 0.37= 1.85 A
Step 1 of 5
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Worked Example 2 — different values, same method
New circuit: E1 = 30 V (R1 = 5 Ω) and E2 = 10 V (R3 = 5 Ω), sharing R2 = 10 Ω. Click through.
Step 0 · The circuit
Two loops, two mesh currents
I1 for the left loop, I2 for the right loop, both clockwise.
What we want to find:The current through the shared resistor R2
Step 1 · Loop 1 KVL
Go round loop 1
Loop 1 equation:30 = 15·I1 − 10·I2
Step 2 · Loop 2 KVL
Go round loop 2
Loop 2 equation:−10 = 15·I2 − 10·I1
Step 3 · Solve the simultaneous equations
Elimination method
Eliminate I2: coefficients are −10 and 15. LCM(10,15)=30 — multiply Eq 1 by 3, Eq 2 by 2.
Eq 1 × 3
45·I1 − 30·I2 = 90
Eq 2 × 2
−20·I1 + 30·I2 = −20
Add — I2 cancels:25·I1 = 70
I1 = 70 ÷ 25= 2.8 A
Substitute into Eq 1: 15(2.8) − 10·I2 = 30I2 = 1.2 A
Step 4 · The shared branch
Watch the two loop currents combine
I through R2 = I1 − I2 = 2.8 − 1.2= 1.6 A
Step 1 of 5
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Try it yourself
🎛️ Change the values — everything updates live
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Practice
Question 1 · Write the loop equations
A circuit has E1 = 12 V (R1 = 2 Ω), E2 = 8 V (R3 = 2 Ω), sharing R2 = 4 Ω. Write Loop 1's equation and find its right-hand coefficient of I1 (i.e. R1+R2).
Using the same circuit (E1=12V, R1=2Ω, R2=4Ω, E2=8V, R3=2Ω), the two equations are: 12 = 6·I1 − 4·I2 and −8 = 6·I2 − 4·I1. Find I1.
Eq1×3: 18I1 − 12I2 = 36. Eq2×2: −8I1 + 12I2 = −16. Add: 10I1 = 20 → I1 = 2 A (Rounding note: exact solve gives I1 = 2 A — check your elimination arithmetic if you got something close but different)
Question 3 · Find the shared branch current
From Question 2, I1 = 2 A. Substituting back into Eq 1 gives I2 = 0 A. Find the current through the shared resistor R2.
I through R2 = I1 − I2 = 2 − 0 = 2 A (I2 = 0 here just means, for these particular values, loop 2 contributes no net current of its own — R2 carries exactly I1.)
📜 The story behind Mesh Analysis
Mesh analysis is really just Kirchhoff's Voltage Law used systematically — a way of turning a tangle of branches and sources into a small, tidy set of simultaneous equations that any circuit, however complicated, can be reduced to.
🕰️In short: the underlying law is nearly 180 years old, but "mesh current" as a formal, systematic method — assigning one circulating current per loop rather than tracking every branch current separately — became the standard way to teach and compute multi-loop circuits once matrix methods made solving several equations at once practical.
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Where it came from
⚡ Built on Kirchhoff's foundations
1845
Gustav Kirchhoff's circuit lawsKirchhoff's Voltage Law — that the voltages around any closed loop must sum to zero — is the law mesh analysis applies to every loop in a circuit at once.
Late 1800s
Maxwell's loop currentsJames Clerk Maxwell, working on electromagnetic theory, popularised describing circuits in terms of circulating "loop currents" rather than tracking every individual branch current — the core idea behind the mesh current I1, I2 used in this lesson.
20th century
Matrix methods take overAs circuits grew more complex, engineers formalised mesh analysis into a set of simultaneous equations solved with matrices — exactly the same structure used in Step 3 and Step 4 of this lesson.
Today
A standard toolbox methodMesh analysis now sits alongside Thevenin, Norton, and superposition as one of the core systematic techniques for solving circuits with more than one loop.
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The logic — why it works
🔁 One equation per loop, automatically consistent
Instead of assigning an unknown current to every branch and writing both KVL and KCL equations to solve them, mesh analysis assigns one circulating current to each independent loop. Because a mesh current flows all the way around its loop, Kirchhoff's Current Law is automatically satisfied everywhere — you never need to write it out.
That leaves just one equation per loop: apply KVL around that loop, accounting for shared resistors where two mesh currents overlap (like R2 in this lesson, carried by both I1 and I2 in opposite senses). For a circuit with two loops, that's exactly two equations and two unknowns — solvable directly, or by matrix methods for anything larger.
This is what makes mesh analysis so efficient for planar circuits (ones that can be drawn flat with no wires crossing): the number of equations needed equals the number of loops, which is usually far fewer than the number of branches.
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Where it's actually used
🔧 Real applications
🖥️PCB & circuit simulationSPICE-style simulators build and solve mesh (or the closely related nodal) equations automatically for every circuit you draw.
⚡Power network analysisAnalysing current flow and losses through interconnected loops in local and regional power distribution networks.
📡Filter & amplifier designWorking out currents through shared components in multi-stage analogue circuits with several feedback loops.
🧮Foundation for matrix methodsMesh equations are usually the first place students meet solving simultaneous equations with matrices in an engineering context.
🎓A companion to Thevenin & superpositionWhere superposition solves source-by-source, mesh analysis solves loop-by-loop — a different systematic route to the same correct answer.
🟢 Key words on this page
Mesh / loop
An independent closed path in a circuit that doesn't contain any smaller loop inside it.
Mesh current
An imaginary current assumed to circulate all the way around one mesh — a bookkeeping tool, not necessarily the actual current in every branch.
Shared branch
A branch (like R2 in this lesson) that belongs to two meshes at once, so it carries the difference between the two mesh currents.
Simultaneous equations
Two or more equations that must all be true at the same time — solved together, usually with substitution or matrices, to find every unknown mesh current.
💡Worth remembering: mesh analysis only works cleanly on planar circuits — ones you can draw on a flat page with no wire crossing another. For non-planar circuits, or for very large networks, nodal analysis (working with node voltages instead of loop currents) is often the more practical choice.
