EMF and Torque in a DC Machine
The raw output of every DC machine is alternating — watch the commutator reverse it at exactly the right instant.
Skip to the animationA DC machine generates an alternating EMF internally and rectifies it mechanically with a commutator, giving E ∝ Φ·N and T ∝ Φ·I_a — two equations that combine into a speed expression containing every DC speed control method there is.
The commutator
A coil rotating in a field generates an EMF proportional to the rate at which it cuts flux — zero along the field, maximum across it, and reversing after half a revolution. The raw output of every DC machine is alternating.
The commutator is a split ring turning with the armature, with fixed brushes on it. It reverses the external connection at precisely the instant the EMF would reverse, so the terminal voltage keeps one polarity. It is mechanical rectification, performed inside the machine.
One coil gives badly rippled DC, so a real armature has dozens of coils and segments, each contributing near its own peak — which makes the ripple negligible.
The two equations
E = PΦZN/60A and T = PΦZI_a/2πA. Since P, Z and A are fixed at manufacture, what matters in service is `E ∝ Φ·N` and `T ∝ Φ·I_a` — and the two share Φ, which is what couples them.
The commutator serves the motor as it served the generator, reversing each coil's current as it passes the neutral plane so the torque stays unidirectional.
The speed equation, and the three levers
Combining V = E_b + I_aR_a with E_b ∝ ΦN gives N ∝ (V − I_aR_a)/Φ. Every DC speed control method is a term in that expression, and there are only three.
| Method | Range | Character |
|---|---|---|
| Armature voltage | Below base speed | Constant torque available; efficient |
| Field weakening | Above base speed | Constant power; torque falls as speed rises |
| Armature resistance | Below base speed | Simple, wasteful, obsolete |
There is a hazard in the same equation: as Φ → 0, N → ∞. Losing the field of a running shunt motor makes it accelerate until something fails, which is why field-loss protection exists.
Back-EMF, and why starting is dangerous
I_a = (V − E_b)/R_a, so the speed sets the current rather than the reverse. At standstill E_b = 0 and only the armature resistance — often a fraction of an ohm — limits the current.
Starting current can be ten times running current, which is why any DC motor above a small size needs a starter that inserts resistance and removes it as the machine accelerates.
The numbers you will be asked for
- EMF equation
E = P·Φ·Z·N / (60·A)
In service: E ∝ Φ·N.
- Torque equation
T = P·Φ·Z·I_a / (2π·A)
In service: T ∝ Φ·I_a.
- Armature circuit
V = E_b + I_a·R_a
Motor. For a generator, V = E − I_aR_a.
- Speed
N ∝ (V − I_a·R_a) / Φ
Contains all three control methods.
- Mechanical power
P = E_b · I_a = T·ω
The converted power, before friction.
Advantages and disadvantages
Advantages
- Smooth, wide-range speed control with a simple controller.
- Torque follows armature current directly, so control is straightforward.
- High starting torque with a series field.
- The same machine generates and motors without modification.
Disadvantages
- The commutator and brushes wear, spark and need maintenance.
- Brush sparking rules out hazardous atmospheres.
- Starting current is limited only by armature resistance, so a starter is mandatory.
- Field loss on a shunt motor causes runaway.
Watch it work
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