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# Pitch Factor, Distribution Factor, and Winding Factor for Harmonic Waveforms

When the flux density distribution in the alternator is non-sinusoidal, the induced voltage in the winding will also be non-sinusoidal. Thus, the *pitch factor* or *coil span factor, distribution factor and winding factor* will be different for each harmonic voltage.

## Pitch Factor for n^{th} Harmonic

As the electrical angle is directly proportional to the number of poles and the angle between the adjacent slots, i.e.,

$$\mathrm{𝜃_{𝑒} =\frac{𝑃}{2}𝜃_{𝑚} … (1)}$$

The chording angle increases with an increase in the order of the harmonics (n). In a short pitch coil, the chording angle is α° (electrical) for the fundamental flux wave. For the n^{th} harmonic, the chording angle becomes nα° electrical. Hence, the *pitch factor or the coil span factor* for the n^{th} harmonic is given by,

$$\mathrm{𝑘_{𝑐𝑛} = cos\frac{𝑛α}{2}… (2)}$$

The harmonic voltages decrease in a short pitch coil, thereby improving the waveform of the induced voltage in the winding. Actually, a certain harmonic can be completely eliminated from the winding voltage by choosing an appropriate pitch for the coils that makes the pitch factor zero for that harmonic. Therefore, to eliminate the n^{th} harmonic voltage, the coil pitch would be,

$$\mathrm{cos\left (\frac{𝑛𝛼}{2} \right)= 0\:or \:cos\left (\frac{𝑛𝛼}{2} \right)= cos\:90°}$$

$$\mathrm{\Rightarrow \:\frac{𝑛𝛼}{2}= 90°}$$

$$\mathrm{\Rightarrow \:α =\frac{180°}{𝑛}… (3)}$$

For example, to eliminate the 3^{rd} harmonic, the coil should be shorted by,

$$\mathrm{α=\frac{180°}{𝑛}=\frac{180°}{3}= 60°}$$

## Distribution Factor for n^{th} Harmonic

The phase difference between the n^{th} harmonic voltages of the adjacent coils is nβ. Hence, the *distribution factor* for the n^{th} harmonic voltage is given by,

$$\mathrm{𝑘_{𝑑𝑛} =\frac{sin\left (\frac{𝑛𝑚𝛽}{2} \right)}{m\:sin\left (\frac{𝑛𝛽}{2} \right)}… (4)}$$

## Winding Factor for n^{th} Harmonic

For the n^{th} harmonic voltage, the *winding factor *is given by,

$$\mathrm{𝑘_{𝑤𝑛} = 𝑘_{𝑐𝑛}\cdot 𝑘_{𝑑𝑛} … (5)}$$

Therefore, for the n^{th} order harmonic, the induced EMF per phase is given by,

$$\mathrm{𝐸_{𝑝ℎ𝑛} = 4.44𝑘_{𝑐𝑛}𝑘_{𝑑𝑛}(𝑛𝑓)𝜑_{𝑛}𝑇}$$

$$\mathrm{\Rightarrow\:𝐸_{𝑝ℎ𝑛}= 4.44𝑘_{𝑤𝑛}(𝑛𝑓)𝜑_{𝑛}𝑇 … (6)}$$

Where, the total flux per pole for n^{th} harmonic is,

$$\mathrm{𝜑_{𝑛} =\frac{2𝐷𝑙}{𝑛𝑃}𝐵_{𝑚𝑛} … (7)}$$

Where,

D is the diameter of armature, and

$l$ is the axial length of armature.

## Numerical Example

A 6-pole synchronous machine has a 3-phase winding wound in 90 slots. The coils are short-pitched in such a way that if one coil side lies in slot number 1, the other side of the same coil lies in slot number 14. Calculate the winding factor for (i) fundamental and (ii) third harmonic.

**Solution**

$$\mathrm{Slots\:per\:pole\:per\:phase, 𝑚 =\frac{slots}{poles × phases}=\frac{90}{6 × 3}= 5}$$

$$\mathrm{Slot\:angle,\:𝛽 =\frac{180° × poles}{slots}=\frac{180° × 6}{90}= 12°}$$

$$\mathrm{Slot\:per\:pole =\frac{90}{6}= 15}$$

For a full-pitch coil, the coil span is 15 slots. But, the given coil is short-pitched so that the coil span is

$$\mathrm{coil\:span = (14 − 1)β = 13β}$$

$$\mathrm{∴\:α = (15 − 13)β = 2β = 2 × 12° = 24°}$$

**Winding factor for fundamental component**

$$\mathrm{Coil\:span\:factor,\:𝑘_{𝑐1} = cos\frac{α}{2}=cos\frac{24°}{2}= 0.978}$$

$$\mathrm{Distribution\:factor,\:𝑘_{𝑑1} =\frac{sin\left (\frac{𝑚𝛽}{2} \right)}{m\:sin\left (\frac{𝛽}{2} \right)}=\frac{sin((5 × 12)/2)}{5 × sin(12/2)}= 0.957}$$

$$\mathrm{∴\:Winding\:factor, 𝑘_{𝑤1} = 𝑘_{𝑐1}\:𝑘_{𝑑1} = 0.978 × 0.957 = 0.936}$$

**Winding factor for 3**^{rd}harmonic component

$$\mathrm{Coil\:span\:factor,\:𝑘_{𝑐3} = cos\left(\frac{3α}{2} \right)= cos\left (\frac{3 × 24°}{2} \right)= 0.809}$$

$$\mathrm{Distribution\:factor,\:𝑘_{𝑑3} =\frac{sin\left (\frac{3𝑚𝛽}{2} \right)}{m\:sin\left (\frac{3𝛽}{2} \right)}=\frac{sin((3 × 5 × 12)/2)}{5 × sin((3 × 12)/2)}= 0.647}$$

$$\mathrm{∴\:Winding\:factor, 𝑘_{𝑤3} = 𝑘_{𝑐3}\:𝑘_{𝑑3} = 0.809 × 0.647 = 0.523}$$

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