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DC: generated EMF and torque

E_a = K_a·Φ·ω, T = K_a·Φ·I_a — one machine constant, two equations, four numbers in lockstep.

Freshman ~8 min

Step 1 — Generated EMF: E_a = K_a · Φ · ω

Animation speed 0.55×

E_a — T — P —

Reference notes

Use Next → on the narrator above to step through the two key DC-machine equations and the single machine constant that binds them together.

Generated EMF (per brush pair)

From the commutator lesson: when an N-turn coil rotates in a flux Φ at angular velocity ω, the induced EMF (after commutation) is a near-flat DC. Generalising to a real armature with P poles, Z conductors total, and A parallel paths between the brushes:

E_a = (P · Φ · Z · N_rpm) / (60 · A) = K_a · Φ · ω

where ω is the mechanical angular speed in rad/s and:

K_a = P · Z / (2 · π · A) ← the machine constant

Torque (same K_a)

By energy conservation, the mechanical power output (motor mode) or input (generator mode) equals the electrical power converted at the brushes:

E_a · I_a = T · ω

Substitute E_a = K_a·Φ·ω and divide both sides by ω:

T = K_a · Φ · I_a

The same K_a appears in both equations. One number characterises the machine; the only operator variables are Φ (field current) and I_a (armature current).

Lap vs wave winding

Lap winding: A = P (one parallel path per pole). For a P-pole machine, the current per path is small but the parallel-path EMFs are summed → low voltage, high current. Common in large machines where current is the constraint.
Wave winding: A = 2 (only two parallel paths regardless of pole count). Higher per-path voltage, lower current. Used in small to medium high-voltage machines.

K_a = P·Z / (2π·A) handles both: increase P (more poles) → K_a grows; A also grows (lap) or stays at 2 (wave) — different balance, same formula.

Power flow

Generator: shaft input mechanical power T·ω is converted to E_a·I_a at the brushes (minus internal I²R losses).
Motor: applied DC delivers E_a·I_a internally; the armature converts it to mechanical power T·ω (minus the same losses).

Worked feel

For a typical 1 kW DC motor: P = 4 poles, Z = 240 conductors, A = 4 (lap). Then K_a = 4·240/(2π·4) ≈ 38.2. With Φ = 0.005 Wb and ω = 157 rad/s (1500 rpm): E_a = 38.2 · 0.005 · 157 ≈ 30 V. With I_a = 35 A: T = 38.2 · 0.005 · 35 ≈ 6.7 N·m, mechanical power 6.7 · 157 ≈ 1050 W. ✓

Take-away. A DC machine is a torque-converter to an electrical engineer: you control flux (Φ via field current) and armature current (I_a via supply voltage and load). The torque and EMF you get are then perfectly linear functions of those two knobs. Linear, predictable, easy to control — that's why DC machines dominated industrial drives for nearly a century.

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