Circuit Laws and Network Theorems

Every circuit question starts from two laws and a handful of theorems that simplify a network before you solve it.

Basic laws

• Ohm's law: V = IR.
• KCL: the sum of currents leaving a node is zero (charge conservation).
• KVL: the sum of voltages around any closed loop is zero (energy conservation).
• Series resistors add; parallel: 1/R = Σ 1/Rᵢ. Two in parallel: R₁R₂/(R₁ + R₂).
• Voltage divider: V₁ = V R₁/(R₁ + R₂). Current divider: I₁ = I R₂/(R₁ + R₂).

Systematic analysis

• Nodal analysis: pick a reference node, write KCL at the other (n − 1) nodes in terms of node voltages. A voltage source between two non-reference nodes forms a supernode.
• Mesh analysis: write KVL around each mesh in terms of mesh currents (planar circuits). A current source shared by two meshes forms a supermesh.
• Number of independent equations: nodal n − 1; mesh b − n + 1 (b = branches).

Theorems

• Source transformation: voltage source V with series R ⇄ current source V/R with parallel R.
• Superposition (linear circuits): find the response to each independent source alone (voltage sources shorted, current sources opened) and add. Not valid for power.
• Thevenin: any linear two-terminal network = V_th in series with R_th. V_th = open-circuit voltage; R_th = resistance seen with independent sources killed, or V_oc/I_sc (needed when there are dependent sources).
• Norton: I_N = short-circuit current in parallel with R_N = R_th.
• Maximum power transfer: R_L = R_th, giving P_max = V_th²/(4R_th) at 50% efficiency.
• Star–delta: R_Y = (product of the two adjacent Δ resistors)/(sum of all three); for equal resistors R_Y = R_Δ/3.
• Reciprocity, Millman, Tellegen (Σ v i = 0 over all branches) and substitution theorems.

Worked example

A 12 V source with internal resistance 2 Ω feeds a 4 Ω resistor in parallel with a 4 Ω load. Find the Thevenin equivalent seen by the load.

• V_th = 12 × 4/(2 + 4) = 8 V. R_th = 2 ∥ 4 = 1.333 Ω.
• Load current = 8/(1.333 + 4) = 1.5 A; maximum power would be 8²/(4 × 1.333) = 12 W at R_L = 1.333 Ω.

Practice