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Courses/Electrical Fundamentals/Kirchhoff's Laws

lesson · 12min · Lesson 18 of 37

Mesh and nodal analysis

Course syllabusCourse overview
01Atoms, Electrons & Electric Charge
  1. ReadWhat is electricity?: atomic model
  2. ReadElectric charge and the coulomb
  3. ReadConductors, insulators, and semiconductors
  4. ReadConventional current vs electron flow
  5. quizModule quiz
02Voltage, Current & Resistance
  1. ReadPotential difference: the driving force
  2. ReadCurrent: the rate of charge flow
  3. ReadResistance and resistivity
  4. ReadOhm's Law: derivation and examples
  5. exerciseWorked examples: Ohm's Law problems
03Series & Parallel Circuits
  1. ReadSeries circuits: characteristics and rules
  2. ReadParallel circuits: characteristics and rules
  3. ReadCombined series-parallel networks
  4. ReadVoltage dividers and current dividers
  5. exerciseCircuit analysis practice
04Kirchhoff's Laws
  1. ReadKCL: Kirchhoff's Current Law
  2. ReadKVL: Kirchhoff's Voltage Law
  3. ReadMesh and nodal analysis
  4. exerciseKirchhoff's law problems set
05Power & Energy
  1. ReadElectrical power: watts and horsepower
  2. ReadEnergy: kilowatt-hours and joules
  3. ReadEfficiency and power loss in cables
  4. quizPower quiz
06Alternating Current Fundamentals
  1. ReadAC vs DC: why AC won
  2. ReadSinusoidal waveforms: peak, RMS, average
  3. ReadFrequency and period
  4. ReadPhase relationships
  5. ReadAC circuit analysis introduction
07Capacitors & Inductors
  1. ReadCapacitor construction and capacitance
  2. ReadCapacitors in AC circuits: reactance
  3. ReadInductor construction and inductance
  4. ReadInductors in AC circuits: reactance
  5. ReadRC and RL circuits: time constants
08Measurement & Instruments
  1. ReadMultimeters: AC/DC voltage and current
  2. ReadClamp meters and measuring live current
  3. ReadOscilloscopes: reading waveforms
  4. quizFinal assessment
Lesson · 12min
KCL: CURRENT AT A NODEI₁I₂I₃I₁ = I₂ + I₃KVL: VOLTAGE ROUND A LOOPVV₁V₂V = V₁ + V₂ (ΣV = 0)
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In this lesson

PurposeCore theoryWorked exampleKnowledge checkSources

In this lesson

PurposeCore theoryWorked exampleKnowledge checkSources
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ElectraCore lesson handout

Mesh and nodal analysis

Electrical Fundamentals · Lesson 18 · Beginner

BeginnerReview: professional review pending

Purpose

Choose mesh or nodal equations for small linear networks and understand what each unknown represents.

Before you beginKCL · KVL · Simultaneous equations

Learning objectives

  • Distinguish node voltage from mesh current
  • Choose an efficient method
  • Form a two-equation system
  • Check the solved network

Mesh and nodal analysis organise Kirchhoff equations so complex networks can be solved systematically rather than by trial reductions.

MESH AND NODAL VARIABLESMESH AND NODAL VARIABLESincomingbranch 1branch 2choose node V or mesh I

Core theory

Nodal analysis assigns voltages relative to a reference node and applies KCL; it is often efficient when a circuit has few essential nodes.

Mesh analysis assigns loop currents to planar meshes and applies KVL; a shared resistor drop depends on the difference between adjacent mesh currents.

Both methods describe the same physical circuit and should agree on branch quantities.

Terms, symbols, and units
TermMeaningSymbolUnit
Reference nodeNode assigned zero voltsNot applicableNot applicable
Node voltagePotential relative to the reference nodeNot applicableNot applicable
Mesh currentMathematical loop current assigned to one meshNot applicableNot applicable
Nodal: Σ((Vn−Vk)/R)=0; Mesh: ΣV=0

Use Ohm's law to express branch quantities in the chosen unknowns.

V, A, Ω
Worked exampleA node connects to 10 V through 5 Ω and to ground through 5 Ω. Find its voltage.

Assumptions: No other branch; Ideal 10 V source.

  1. KCL: (Vn−10)/5 + Vn/5 = 0.
  2. Collect: 2Vn − 10 = 0.
  3. Solve: Vn = 5 V.

The node is 5 V above reference.

Reasonableness check: Equal resistances place the node halfway between 10 V and 0 V.

Common mistakes
  • Confusing mesh currents with physical shared-branch current
  • Omitting the reference node
  • Writing a resistor current without both endpoint voltages

Where this appears in practice

These methods underpin circuit simulation, control-panel analysis, electronics, and systematic fault models.

SafetyAnalytical node voltage is not proof of isolation or touch safety. Verify actual conductors, sources, and stored energy with appropriate test equipment.
Local code checkThese analysis laws are universal. Installation design, conductor selection, protection, testing, and permitted work must follow current local standards and manufacturer data.

Knowledge check

Which law is applied directly at each non-reference node in nodal analysis?

KCL. Branch currents are expressed from node-voltage differences.

Answer: KCL. Branch currents are expressed from node-voltage differences.

Practical exercise

Write, but do not yet solve, node equations for a two-node resistive network and label every current direction.

Summary

  • Nodal uses KCL and node voltages
  • Mesh uses KVL and loop currents
  • Validate results with conservation laws

Sources and review

  • University Physics Volume 2: Kirchhoff's Rules: OpenStax; Current web edition; Physics reference
  • SI Units: Electric Current: NIST; Current web edition; United States / SI reference
  • The International System of Units (SI Brochure): BIPM; 9th edition, updated 2026; International

Editorial review date: 2026-08-21. Professional electrical review is pending.

Educational material for learning and preliminary checks. Verify current local requirements and exact equipment instructions. This lesson does not replace competent professional work.

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