Minimum number of commutator segments

The number of commutator segments $C$ (= number of armature coils) is limited by the voltage between adjacent segments. If this voltage is too high, arcing between segments can lead to flashover.

Derivation

Let $E$ = terminal (induced) voltage, $P$ = poles. For a lap winding there are $a = P$ parallel paths, so the number of coils in series per path is:

$$\text{Coils per path} = \frac{C}{P}$$

Average voltage per coil, which is the average voltage between adjacent segments:

$$E_{c(av)} = \frac{E}{C/P} = \frac{EP}{C}$$

The flux density under the pole is not uniform: armature reaction distorts it, and the peak density is about twice the average. So the maximum voltage between adjacent segments is about:

$$E_{c(max)} \approx 2E_{c(av)} = \frac{2EP}{C}$$

For safe operation, the maximum voltage between segments is limited to about 30 V:

$$\frac{2EP}{C} \le 30$$

Hence

$$C_{min} = \frac{2EP}{30} = \frac{EP}{15}$$

Since each coil ends on two segments and each segment joins two coil ends, the number of commutator segments equals the number of coils.

Example

A 220 V, 4-pole machine:

$$C_{min} = \frac{220 \times 4}{15} = 58.7 \approx 59\ \text{segments}$$

A practical value is chosen above this, also matching the number of slots (coils per slot $\times$ slots).

Other limits on $C$

• Segment pitch (about 4 mm minimum) for mechanical strength.
• Commutator peripheral speed (not more than about 30 m/s).
• Reactance voltage: fewer turns per coil (more coils) gives better commutation.