Circuit Analysis

Part of Basic Electrical Circuits

Systematic methods for predicting voltage, current, and power at every point in an electrical circuit.

Why This Matters

Building electrical systems without the ability to analyze circuits is like constructing a bridge without calculating loads—you might get lucky, but you will often waste materials, burn out components, and occasionally cause fires or injury. Circuit analysis gives you the ability to predict what a circuit will do before you build it, and diagnose why it misbehaves after.

In a post-collapse context where copper wire, batteries, and electrical components are scarce, wasting them on circuits that don’t work is costly. A person who can calculate current draw before connecting a load protects both the power source and the load. A person who can analyze a failed circuit identifies the faulty component quickly rather than replacing everything by trial and error.

Circuit analysis relies on two foundational principles—Ohm’s Law and Kirchhoff’s Laws—plus systematic methods for applying them to complex networks. These techniques require nothing more than arithmetic and careful accounting.

The Vocabulary of Circuit Analysis

Before analyzing any circuit, you need precise language:

Node: Any point where two or more components connect. Every node has a single voltage.

Branch: A single path between two nodes, containing one or more components.

Loop: Any closed path through a circuit.

Mesh: The smallest possible loop—one that contains no other loops inside it.

Reference node (ground): The node chosen as the zero-voltage reference. All other voltages are measured relative to this point.

Voltage rise/drop: Moving in the direction of current through a resistor is a voltage drop (energy consumed). Moving through a battery from minus to plus is a voltage rise (energy supplied).

Marking all nodes, naming all branch currents, and identifying all loops before starting analysis saves enormous confusion later.

The Node Voltage Method

Node voltage analysis is the most systematic approach for circuits with multiple parallel paths.

Procedure:

1. Identify all nodes and choose one as ground (usually the node connected to the negative terminal of the power source)
2. Label the unknown voltage at every other node: V₁, V₂, V₃, etc.
3. At each non-ground node, write a current balance equation: sum of currents leaving = 0
4. Express each current as (node voltage difference) / resistance
5. Solve the resulting system of equations

Example — two-node circuit: A 12V battery with internal resistance 1Ω feeds two parallel loads: R₁ = 4Ω and R₂ = 6Ω.

Node A is at the positive terminal (after internal resistance). Ground is at the negative terminal.

Current entering node A from battery: (12 - V_A) / 1 Current leaving through R₁: V_A / 4 Current leaving through R₂: V_A / 6

Setting sum to zero: (12 - V_A)/1 = V_A/4 + V_A/6

Solving: 12 - V_A = V_A(0.25 + 0.167) = 0.417 V_A 12 = 1.417 V_A V_A = 8.47V

Current through R₁ = 8.47/4 = 2.12A Current through R₂ = 8.47/6 = 1.41A Total current from battery = 2.12 + 1.41 = 3.53A

The Mesh Current Method

Mesh analysis works by assigning a circulating current to each mesh and writing voltage equations around each loop.

Procedure:

1. Identify all meshes (smallest loops)
2. Assign a mesh current to each, flowing in a consistent direction (clockwise is conventional)
3. For each mesh, write a KVL equation: sum of voltage drops = sum of voltage sources
4. Shared branches carry the difference of the two adjacent mesh currents
5. Solve the system of equations

Example — two-mesh circuit: Two batteries and three resistors in a ladder network:

• Mesh 1: 10V source, R₁ = 2Ω (left branch), R₂ = 4Ω (middle branch shared)
• Mesh 2: 6V source, R₃ = 3Ω (right branch), R₂ = 4Ω (middle branch shared)

Mesh 1: 10 = I₁(2) + (I₁ - I₂)(4) = 6I₁ - 4I₂ Mesh 2: 6 = I₂(3) + (I₂ - I₁)(4) = -4I₁ + 7I₂

Solving simultaneously: From equation 1: I₁ = (10 + 4I₂)/6 Substituting into equation 2: 6 = -4(10 + 4I₂)/6 + 7I₂ 36 = -40 - 16I₂ + 42I₂ 76 = 26I₂ I₂ = 2.92A, I₁ = (10 + 11.69)/6 = 1.95A

Current through shared R₂ = I₁ - I₂ = 1.95 - 2.92 = -0.97A (flows opposite to mesh 1 direction)

Superposition

When a circuit contains multiple independent voltage or current sources, superposition allows analyzing the effect of each source separately, then adding the results.

Procedure:

1. Kill all sources except one: replace voltage sources with short circuits, replace current sources with open circuits
2. Analyze the simplified circuit for currents and voltages
3. Repeat for each source
4. Add all contributions algebraically at each point

When to use superposition:

• When the circuit has two or more batteries with different voltages
• When a circuit mixes AC and DC sources
• When you want to understand how much each source contributes

Limitation

Superposition only works for linear circuits—those containing only resistors, capacitors, and inductors. It does not apply to circuits with diodes, transistors, or other non-linear elements.

Thevenin and Norton Equivalents

Any linear circuit, no matter how complex, can be reduced to a single voltage source and series resistor (Thevenin) or a single current source and parallel resistor (Norton). This is extraordinarily useful when you need to understand how a complex source circuit interacts with a load.

Finding the Thevenin equivalent:

1. Identify the two terminals where you want the equivalent
2. Remove the load (open circuit the terminals)
3. Calculate the open-circuit voltage between terminals: this is V_th
4. Kill all independent sources; calculate the resistance seen from the terminals: this is R_th

Finding the Norton equivalent:

• I_N = V_th / R_th (short-circuit current)
• R_N = R_th (same resistance)

Practical use: Connect any load to the Thevenin equivalent and analyze with simple series circuit arithmetic rather than the full complex network.

Practical Fault Finding

Circuit analysis methods apply directly to troubleshooting:

Open circuit fault: One branch carries no current. Node voltages shift dramatically. The node on the source side of the break rises toward supply voltage; the other side drops toward zero.

Short circuit fault: Two nodes that should be separate become connected. Current through the shorted branch spikes; voltage across it drops to near zero. Check for burned insulation, solder bridges, or crushed conductors.

High resistance connection: A corroded joint or poor splice shows up as an unexpected voltage drop. Measure voltage across each connection point; any drop more than a fraction of a volt indicates a resistance that should be near zero.

Systematic fault finding sequence:

1. Measure supply voltage at the source
2. Measure voltage at the first junction point
3. Follow the voltage division downstream until you find where it deviates from calculation
4. The fault is in the branch where voltage deviates

This half-split method—checking the midpoint of a circuit, then the quarter point of the faulty half—minimizes the number of measurements needed to locate any fault.