# DC GENERATOR:

# DC GENERATOR EMF EQUATION OF A DC GENERATOR :

**Conclusions**

DC GENERATOR EMF equations of a generator the following conclusions can be drawn:

- The total number of brushes is equal to the number of poles.
- The armature winding is divided into as many parallel paths as the number of poles. If the total number of armature conductors is Z and P is the number of poles, then,

Number of conductors/path = Z/P

In the present case, there are 40 armature conductors and 4 poles. Therefore, the armature winding has 4 parallel paths, each consisting of 10 conductors in series.

- M.F. generated = E.M.F. per parallel path
- average e.m.f. per conductor ´
^{Z}_{P}

- average e.m.f. per conductor ´

- Total armature current, I
_{a}= P ´ current per parallel path - The armature resistance can be found as under:

Let l = length of each conductor; a = cross-sectional area

Since there are A (= P) parallel paths, armature resistance R_{a} is given by

**Simplex Wave Winding**

The essential difference between a lap winding and a wave winding is in the commutator connections. In a simplex lap winding, the coils approximately pole pitch apart are connected in series and the commutator pitch Y_{C} = ± 1 segment. As a result, the coil voltages add. This is illustrated in Fig. (1.27). In a simplex wave winding, the coils approximately pole pitch apart are connected in series and the commutator pitch Y_{C} ~ 2 pole pitches (segments). Thus in a wave winding, successive coils “wave” forward under successive poles instead of “lapping” back on themselves as in the lap winding. This is illustrated in Fig. (1.28).

The simplex wave winding must not close after it passes once around the armature but it must connect to a commutator segment adjacent to the first and the next coil must be adjacent to the first as indicated in Fig. (1.19). This is repeated each time around until connections are made to all the commutator segments and all the slots are occupied after which the winding automatically returns to the starting point. If, after passing once around the armature, the winding connects to a segment to the left of the starting point, the winding is retrogressive [See Fig. 1.19 (i)]. If it connects to a segment to the right of the starting point, it is progressive [See Fig. 1.19 (ii)]. This type of winding is called wave winding because it passes around the armature in a wave-like form.

**Various pitches**

The various pitches in a wave winding are defined in a manner similar to lap winding.

- The distance measured in terms of armature conductors between the two sides of a coil at the back of the armature is called back pitch Y
_{B}(See Fig. 1.20). The Y_{B}must be an odd integer so that a top conductor and a bottom conductor will be joined.

(iii) Resultant pitch, Y_{R} = Y_{B} + Y_{F} (See Fig. 1.20)an odd integer so that a top conductor and a bottom conductor will be joined.The distance measured in terms of armature conductors between the coil sides attached to any one commutator segment is called front pitch Y_{B} (See Fig. 1.20). The Y_{B} must be

The resultant pitch must be an even integer since Y_{B} and Y_{F} are odd. Further Y_{R} is approximately two pole pitches because Y_{B} as well as Y_{F} is approximately one pole pitch.

(iv)

When one tour of armature has been completed, the winding should connect to the next top conductor (progressive) or to the preceding top conductor (retrogressive). In either case, the difference will be of 2 conductors or one slot. If P is the number of poles and Z is the total number of armature conductors, then,

P ´ Y_{A} = Z ± 2

Since P is always even and Z = PY_{A} ± 2, Z must be even. It means that Z

± 2/P must be an integer. In Eq.(i), plus sign will give progressive winding and the negative sign retrogressive winding.

- The number of commutator segments spanned by a coil is called commutator pitch (Y
_{C}) (See Fig. 1.20). Suppose in a simplex wave winding,

P = Number of poles; N_{C} = Number of commutator segments; Y_{C} = Commutator pitch.

\ Number of pair of poles = P/2

If Y_{C} ´ P/2 = N_{C}, then the winding will close on itself in passing once around the armature. In order to connect to the adjacent conductor and permit the winding to proceed,

In a simplex wave winding Y_{B}, Y_{F} and Y_{C} may be equal. Note that Y_{B}, Y_{F} and Y_{B} are in terms of armature conductors whereas Y_{C} is in terms of commutator segments.

**Design of Simplex Wave Winding**

In the design of simplex wave winding, the following points may be kept in mind:

- Both pitches Y
_{B}and Y_{F}are odd and are of the same sign. - Both Y
_{B}and Y_{F}are nearly equal to pole pitch and may be equal or differ by 2. If they differ by 2, they are one more and one less than Y_{A}. - Commutator pitch is given by;

The plus sign for progressive winding and negative for retrogressive winding.

Since Y_{A} must be a whole number, there is a restriction on the value of Z. With Z = 180, this winding is impossible for a 4-pole machine because Y_{A} is not a whole number.

(vi) Z = P Y_{A} ± 2

**Developed diagram**

Fig. (1.21) (i) shows the developed diagram for the winding. Note that full lines represent the top coil sides (or conductors) and dotted lines represent the bottom coil sides (or conductors).

### Example: **Develop a 4 pole Wave winding diagram with 42 conductors and show position of brushes**.

Referring to Fig. (1.21) (i), conductor 1 connects at the back to conductor 12(1 + 11) which in turn connects at the front to conductor 23 (12 + 11) and so on round the armature until the winding is complete. Note that the commutator pitch Y_{C} = 11 segments. This means that the number of commutator segments spanned between the start end and finish end of any coil is 11 segments.

**Conclusions**

From the above discussion, the following conclusions can be drawn:

- Only two brushes are necessary but as many brushes as there are poles may be used.

- The armature winding is divided into two parallel paths irrespective of the number of poles. If the total number of armature conductors is Z and P is the number of poles, then,

Number of conductors/path = Z/2

- M.F. generated = E.M.F. per parallel path
- Average e.m.f. per conductor x
- Total armature current, I
_{a}= 2 ´ current per parallel path - The armature can be wave-wound if Y
_{A}or Y_{C}is a whole number.

**E.M.F. Equation of a D.C Generator**

We shall now derive an expression for the e.m.f. generated in a d.c. generator. Let f = flux/pole in Wb

Z = total number of armature conductors P = number of poles

A = number of parallel paths = 2 … for wave winding = P … for lap winding

N = speed of armature in r.p.m.

E_{g} = e.m.f. of the generator = e.m.f./parallel path Flux cut by one conductor in one revolution of the armature,

df = Pf webers Time taken to complete one revolution,

dt = 60/N second

E_{g} = e.m.f. per parallel pathe.m.f. of generator,

- (e.m.f/conductor) ´ of conductors in series per parallel path

Where, A=2 for wave winding

A=P For lap winding