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

Electronics & ElectricalElectronDigital Electronics

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 nth 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 nth harmonic, the chording angle becomes nα° electrical. Hence, the pitch factor or the coil span factor for the nth 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 nth 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 3rd harmonic, the coil should be shorted by,

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

Distribution Factor for nth Harmonic

The phase difference between the nth harmonic voltages of the adjacent coils is nβ. Hence, the distribution factor for the nth 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 nth Harmonic

For the nth harmonic voltage, the winding factor is given by,

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

Therefore, for the nth 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 nth 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 3rd harmonic component

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

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

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

raja
Updated on 13-Oct-2021 12:22:40

Advertisements