2.2 kW cage motor designed for best power factor

Data: 2.2 kW, 440 V, 50 Hz, 1430 rpm (4 poles, $n_s = 1500$ rpm $= 25$ rps), $\eta = 0.8$, $\cos\phi = 0.85$, $B_{av} = 0.44$ Wb/m², $ac = 21000$ A/m. Assumptions: $K_w = 0.955$; delta-connected stator ($E_{ph} = 440$ V); tooth flux density limit 1.7 Wb/m²; iron stacking factor 0.9.

(i) Main dimensions

$$\begin{aligned} Q &= \frac{2.2}{0.8 \times 0.85} = 3.235\ \text{kVA} \\ C_0 &= 11 B_{av}\, ac\, K_w \times 10^{-3} = 11 \times 0.44 \times 21000 \times 0.955 \times 10^{-3} = 97.07 \\ D^2L &= \frac{Q}{C_0 n_s} = \frac{3.235}{97.07 \times 25} = 1.333 \times 10^{-3}\ \text{m}^3 \end{aligned}$$

Best power factor: $\tau = \sqrt{0.18L}$, so $L = \tau^2/0.18 = (\pi D/4)^2/0.18$:

$$\begin{aligned} D^2 \cdot \frac{\pi^2 D^2}{16 \times 0.18} &= 1.333\times10^{-3} \\ D^4 &= 3.889 \times 10^{-4} \Rightarrow D = 0.1404\ \text{m} \\ L &= \frac{1.333\times10^{-3}}{0.1404^2} = 0.0676\ \text{m} \end{aligned}$$

Check: $\tau = \pi \times 0.1404/4 = 0.1103$ m $= \sqrt{0.18 \times 0.0676}$.

(ii) Stator turns per phase

$$\begin{aligned} \Phi &= B_{av}\tau L = 0.44 \times 0.1103 \times 0.0676 = 3.281 \times 10^{-3}\ \text{Wb} \\ T_s &= \frac{440}{4.44 \times 50 \times 3.281\times10^{-3} \times 0.955} = 632.6 \end{aligned}$$

(iv) Number of stator slots

Slot pitch 12–15 mm. With $q = 3$: $S_s = 3 \times 4 \times 3 = 36$, slot pitch $= \pi \times 140.4/36 = 12.26$ mm (OK). 36 slots. Conductors per slot $= 6 \times 632.6/36 = 105.4$, take 106, so $Z_s = 3816$ and $T_s = 636$.

(iii) Area of stator conductor

$$I_{ph} = \frac{3235.3}{3 \times 440} = 2.451\ \text{A}, \qquad a_s = \frac{2.451}{4} = 0.613\ \text{mm}^2$$

(Bare diameter $\approx 0.88$ mm.)

(v) Area and dimensions of each slot

$$A_{slot} = \frac{106 \times 0.613}{0.4} = 162.4\ \text{mm}^2$$

$d/w = 4$: $4w^2 = 162.4$, $w = 6.37$ mm, $d = 25.5$ mm.

(vi) Minimum width of stator teeth

$L_i = 0.9 \times 0.0676 = 0.0608$ m, teeth per pole $= 36/4 = 9$:

$$W_{ts(min)} = \frac{\Phi}{1.7 \times 9 \times L_i} = \frac{3.281\times10^{-3}}{1.7 \times 9 \times 0.0608} = 3.53\ \text{mm}$$

(vii) Stator slot loading

$$I_{ph} \times \text{cond/slot} = 2.451 \times 106 = 259.8\ \text{A-conductors}$$

(Well below the usual limit of about 1000–1200.)

Answer: $D = 14.04$ cm, $L = 6.76$ cm; $T_s = 636$; $a_s = 0.613$ mm²; 36 slots; slot 162.4 mm² ($6.37 \times 25.5$ mm); $W_{ts(min)} = 3.53$ mm; slot loading 260 A-conductors.