Why is the same k used for torque and back-emf?

For an ideal energy-consistent model in SI units, the same flux linkage per ampere and per rad/s ties electrical and mechanical power: roughly P_elec ≈ E_b I ≈ τω, which forces k to appear in both relations when losses are lumped into R and b.

Where is the commutator?

Commutation is implicit: the model assumes torque and induced emf stay aligned so the rotor keeps turning in motor mode and delivers DC polarity into the resistive load in generator mode, as in a simplified textbook brushed machine.

DC Motor & Generator

A lumped brushed DC machine model uses one constant k linking magnetic flux linkage to both back-emf (E_b = kω) and torque (τ = kI). Motor mode integrates Jω̇ = kI − bω with V = I R_a + E_b; generator mode fixes shaft speed and computes E = kω and load current from R_a + R_load.

Who it's for: Introductory electromagnetism and energy conversion; complements the moving-magnet induction lab with a rotating-coil picture.

Key terms

back emf
DC motor
generator
Faraday induction
Lorentz torque
Ohm’s law

Live graphs

How it works

A rectangular coil in a uniform magnetic field illustrates the same energy conversion in both directions. With a DC supply, current produces torque and the rotor accelerates until back-emf balances the circuit. When you crank the rotor, motion generates emf and drives current through a load — a dynamo.

Key equations

E_b = kω · τ = kI

Motor: V = IR_a + E_b , Jω̇ = kI − bω

Generator: E = kω , I = E / (R_a + R_load)

Frequently asked questions

Why is the same k used for torque and back-emf?: For an ideal energy-consistent model in SI units, the same flux linkage per ampere and per rad/s ties electrical and mechanical power: roughly P_elec ≈ E_b I ≈ τω, which forces k to appear in both relations when losses are lumped into R and b.
Where is the commutator?: Commutation is implicit: the model assumes torque and induced emf stay aligned so the rotor keeps turning in motor mode and delivers DC polarity into the resistive load in generator mode, as in a simplified textbook brushed machine.

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