Circuit Laws

Part of Electrical Theory

The complete set of laws governing electrical circuits—from Ohm’s Law through Kirchhoff’s Laws to Faraday’s and Lenz’s Laws—and how they interconnect.

Why This Matters

Electrical engineering rests on a small number of fundamental laws. These laws are not empirical rules-of-thumb—they are consequences of the conservation of energy and charge, verified by experiment to extraordinary precision. Together they enable the complete prediction of any electrical circuit’s behavior from first principles.

A person who internalizes these laws rather than just memorizing their formulas can analyze any circuit they encounter. They can derive unfamiliar formulas when needed. They can identify which law applies to each aspect of a problem. They can check results by applying a different law and seeing if the answers agree.

The laws covered here form the complete theoretical foundation for Tier 7 electrical work. None of them require advanced mathematics—only arithmetic and some algebra.

Ohm’s Law

V = I × R (or I = V/R, or R = V/I)

Discovered by Georg Ohm in 1827. Holds for ohmic materials at constant temperature.

Physical basis: In a metal conductor, free electrons experience a retarding force from collisions with the lattice. The average drift velocity (and thus current) is proportional to the applied electric field (and thus voltage). The proportionality constant is determined by the material’s resistivity and geometry.

Validity: Applies to metals and many other materials at constant temperature. Does not apply to non-linear devices (diodes, transistors, arcs). Can be extended to AC using impedance: V = I × Z.

Kirchhoff’s Current Law (KCL)

The sum of currents entering any node equals the sum of currents leaving it.

Algebraic form: ΣI = 0 at every node.

Physical basis: Conservation of charge. Electric charge cannot accumulate at a node in a DC or steady-AC circuit. Every coulomb that arrives must depart.

Application: Write one KCL equation for each non-reference node in a circuit. Solve the system for node voltages.

Kirchhoff’s Voltage Law (KVL)

The sum of voltage rises and drops around any closed loop equals zero.

Algebraic form: ΣV = 0 around any closed loop.

Physical basis: Conservation of energy. Voltage is energy per unit charge. Traversing a closed path returns to the same potential—no net energy is created or destroyed by the path.

Application: Write one KVL equation for each independent mesh (minimum loop). Solve the system for mesh currents.

Faraday’s Law of Electromagnetic Induction

The EMF (voltage) induced in a circuit equals the negative rate of change of magnetic flux through that circuit.

EMF = -dΦ/dt

Where Φ = B × A × cos θ (magnetic flux = field strength × area × cosine of angle between field and area normal).

Physical basis: A changing magnetic field creates an electric field. This is the fundamental principle behind generators, transformers, and inductors.

Practical statements:

• Move a wire through a magnetic field → voltage is induced
• Change the current in a coil near another coil → voltage is induced in the second coil
• A changing magnetic flux through any loop induces EMF in that loop

Generator equation: For a coil of N turns rotating at angular velocity ω in a field of strength B: EMF = N × B × A × ω × sin(ωt)

This is a sine wave—which is why generators naturally produce sinusoidal AC.

Transformer equation: For a transformer with N₁ primary turns and N₂ secondary turns: V₂/V₁ = N₂/N₁

The voltage ratio equals the turns ratio, a direct consequence of Faraday’s Law applied to two coils sharing the same magnetic flux.

Lenz’s Law

The induced EMF always acts to oppose the change that caused it.

This is the physical statement of the negative sign in Faraday’s Law.

Practical consequences:

• A generator requires mechanical force to overcome the opposing electromagnetic force
• An inductor opposes current increases (rising current → increasing flux → back-EMF opposing the increase)
• An inductor also opposes current decreases (falling current → decreasing flux → forward-EMF trying to maintain the current)
• Back-EMF from a motor reduces the current drawn when running (compared to stall current)
• A transformer secondary winding opposes the primary flux, preventing arbitrary energy transfer

The back-EMF of a motor: A DC motor in operation generates a voltage opposing the supply. At stall, back-EMF = 0, and current = V/R_winding (very high—the “stall current”). As the motor speeds up, back-EMF rises, net voltage drops, current falls to the running level. If the mechanical load is suddenly increased, the motor slows, back-EMF falls, current rises—the motor automatically demands more power when loaded.

Superposition Theorem

In a linear circuit with multiple independent sources, the response (voltage or current) at any point is the sum of responses from each source acting alone (all others replaced by their internal impedances).

Voltage sources are replaced by short circuits (ideal voltage source has zero internal impedance). Current sources are replaced by open circuits (ideal current source has infinite internal impedance).

Validity: Only for linear circuits. Does not apply to circuits with diodes, transistors, or other non-linear elements.

Use: Particularly valuable when sources have different characters (AC and DC in the same circuit), or when understanding the contribution of each source separately is useful.

Thevenin’s Theorem

Any linear two-terminal network can be replaced by an equivalent circuit consisting of a single voltage source (V_th) in series with a single resistance (R_th).

V_th = open-circuit voltage at the terminals R_th = resistance seen at the terminals with all independent sources killed

Use: Drastically simplifies circuit analysis. Replace a complex source network with the Thevenin equivalent; connect the load and analyze a simple two-element series circuit.

Norton equivalent: The dual of Thevenin. Any linear network can also be represented as a current source I_N in parallel with resistance R_N. I_N = V_th / R_th; R_N = R_th

Joule’s Law of Electrical Heating

The power dissipated as heat in a conductor is proportional to the square of the current and the resistance.

P = I² × R (equivalently P = V²/R or P = V × I)

Physical basis: Each collision between a conduction electron and the lattice transfers energy to the lattice as heat. Rate of energy transfer (power) is proportional to current squared.

Practical implications:

• Doubling current quadruples heating in conductors—a critical factor in fuse and wire design
• Halving current reduces heating by 75%—the efficiency benefit of transmitting power at high voltage (low current)
• Fuses work by Joule heating: excessive current heats the fuse wire until it melts

Maximum Power Transfer Theorem

Maximum power is transferred from a source to a load when the load resistance equals the source’s Thevenin resistance (R_load = R_th).

At this condition:

• Power to load = V_th² / (4 × R_th)
• Efficiency = 50% (equal power dissipated in source and load)

When this applies: Signal circuits (radio, audio) where delivering maximum signal power to the load (antenna, speaker) is the goal. Impedance matching transformers are used to make load impedance equal source impedance.

When this does NOT apply: Power distribution systems where efficiency is paramount. Here, load resistance should be much greater than source resistance, even though less than maximum power transfer occurs—the efficiency is much higher.

The Laws Together

These laws are not independent—they are complementary tools for the same underlying physics:

• Conservation of charge → KCL
• Conservation of energy → KVL
• Electromagnetic induction → Faraday’s and Lenz’s Laws
• Material properties → Ohm’s Law
• Linear superposition → Thevenin, Norton, Superposition Theorem

Master all of them, and any electrical circuit—however complex—yields to systematic analysis.