dc machines have closed circuit windings

Alternator windings are open, no closed path for the currents in the winding itself

One end is joined to neutral and other brought out (Y-connected)

two sides of any coil should be under two adjacent poles, i.e. coil span = pole pitch

winding can either be single layer or double layer

Winding is so arranged in different armature slots, that it must produce sinusoidal emf.

the complete slot is containing only one coil side of a coil

also referred to as

Consider- single-layer, one-turn, full-pitch, 4-pole, 12-slots

3 slots/pole or 1 slot/pole/phase and pole-pitch = 3

For max. emf, 2-sides of a coil should be 1 pole-pitch apart (\(180^{\circ}\) electrical)

\(R\)-phase: slot no.1 \(\Rightarrow\) 4,7 \(\Rightarrow\) 10

\(Y\)-phase: slot no.3- \(120^{\circ}\) afterwards

(3 slots = \(180^{\circ}\) \(\Rightarrow\) 2 slots = \(120^{\circ}\))\(B\)-phase: slot no. 5 \(\Rightarrow\) 8,11 \(\Rightarrow\) 2

The ends of the windings are joined to form a \(Y\)-connection

Number of slots = 2 \(\times\) No. of coils = No. of coil sides

polar group of each phase is \(360^{\circ}\) (electrical) apart

necessary to use two different shapes of coils to avoid fouling of end connection

Since polar groups of each phase are \(360^{\circ}\) apart, all such groups are connected in the same direction

disadvantage- short-pitched coils cannot be used

either wave-wound or lap-wound

simplest and most commonly used

No. of slots is multiple of no. of poles and the phases. E.g. 4-pole, 3-phase may have 12, 24, 36, 48 etc. slots (multiple of 12 = 4 \(\times\) 3)

No. of slots = No. of coils (of the same shape).

Thus, each slot contains two coil sides, one at the bottom and other at top of the slot. The coils overlap each other.

4-pole, 24-slot; \(\Rightarrow\) pole-pitch = \(24/4=6\)

For max. emf, coils should be full-pitched

means one side of coil in slot no. 1 the other at 7, the two slots being one pole-pitch or \(180^{\circ}\) apart

Each phase has \(24/3=8\) coils

Winding for one phase is shown:

The complete wiring diagram for 3-phase:

So far full-pitched coils i.e one pole-pitch = \(180^{\circ}\) is discussed

Full-pitched: coil sides in slots 1 and 7

Short-pitched or fractional-pitched: placed in slots 1 and 6, i.e. coil-span = 5/6 of a pole-pitch

Falls short by 1/6 pole-pitch = \(180^{\circ}/6 = 30^{\circ}\)

save copper of end connections

improve the waveform of the generated emf (sinusoidal) and distorting harmonics are eliminated

Due to elimination of high frequency harmonics, eddy current and hysteresis losses are reduced thereby increasing efficiency

Total voltage is somewhat reduced

Induced voltage in two sides of short-pitched coils are slightly out of phase, their resultant vectorial sum is less than their arithmetical sum

is defined as \[\boxed{=\dfrac{\mbox{vector sum of the induced emf per coil}}{\mbox{arithmetic sum of the induced emf per coil}}} < 1\]

Let \(E_s\) be induced emf in each side of the coil. Then for full-pitched total induced emf = \(2E_s\)

If short-pitched by \(30^{\circ}\) then resultant emf \[\begin{aligned} E & =2E_{s}\cos30^{\circ}/2=2E_{s}\cos15^{\circ}\\ k_{c} & =\dfrac{\mbox{vector sum}}{\mbox{arithmetic sum}}=\dfrac{E}{2E_{s}}=\dfrac{2E_{s}\cos15^{\circ}}{2E_{s}}=\cos15^{\circ}=0.966 \end{aligned}\]

In general, if the coil span falls by an angle \(\alpha\) (electrical) then \(k_c=\cos \alpha/2\)

\(\alpha\) is known as chording angle and the winding employed short-pitched coils is called chorded winding

Note: The value of \(\alpha\) will usually given in question, it not, then assume, \(k_c=1\)

In each phase, coils are not concentrated or bunched in one slot, but are distributed in a number of slots to form polar groups under each pole

The coils/phases are displaced from each other by a certain angle

Result is emfs induced in coil sides constituting a polar group are not in phase but differ by angular displacement of the slots

3-\(\Phi\), 4-pole, 36 slots \(\Rightarrow\) 9 slots/pole \(\Rightarrow\) 3 slots/pole/phase

Here, angular displacement between any two adjacent slots = \(180^{\circ}/9=20^{\circ}\) (elect.)

If three coils were bunched in one slot, then total emf induced in three sides of the coil would be the arithmetic sum of the three emfs i.e. = \(3E_s\), where \(E_s\) is emf induced in one coil side

Since the coils are distributed, the individual emfs have a phase difference of \(20^{\circ}\) with each other

\[E=E_{s}\cos20^{\circ}+E_{s}+E_{s}\cos20^{\circ}=2.88E_{s}\]

The distribution factor \(\left(K_d\right)\) is defined as \[=\dfrac{\mbox{e.m.f. with distributed winding}}{\mbox{e.m.f. with concentrated winding}}\]

In the present case \[K_d = \dfrac{\mbox{e.m.f in 3 slots/pole/phase}}{\mbox{e.m.f in 1 slots/pole/phase}} = \dfrac{E}{3E_s} = \dfrac{2.88E_s}{3E_s} = 0.96\]

\[\begin{array}{cl} \beta= & \mbox{Angular displacement betw. slots}\\ n= & \mbox{No. of slots/pole}\\ m= & \mbox{No. of slots/pole/phase}\\ m\beta= & \mbox{phase spread angle} \end{array}\]

\[\boxed{\beta = \dfrac{180^{\circ}}{\mbox{No. of slots/pole}} = \dfrac{180^{\circ}}{n}}\]

\[\begin{aligned} K_{d} & =\dfrac{\mbox{vector sum of coils emfs}}{\mbox{arithmetic sum of coil emfs}} =\dfrac{E_{r}}{mE_{s}}\\ & =\dfrac{2r\sin m\beta/2}{m\times2r\sin\beta/2}\\ & \boxed{K_{d} =\dfrac{\sin \left(m\beta/2\right)}{m\sin\left(\beta/2\right)}} \end{aligned}\]