Air gap and rotor bars, 10 kW motor

Data: 10 kW, 400 V, 3-phase, 4 poles, delta, 50 Hz; $D = 0.16$ m, $L = 0.085$ m, $B_{av} = 0.44$ Wb/m², $\eta = 0.86$, pf $= 0.85$, $\delta = 5$ A/mm², $S_s = 36$, $S_r = 30$, bar length 0.15 m, end-ring mean diameter 0.12 m. Assumptions: $K_{ws} = 0.955$; resistivity taken as $0.02$ Ω per m per mm² (i.e. $0.02 \times 10^{-6}$ Ω·m; "0.020 Ω-m" is a misprint); same current density 5 A/mm² in bars and rings.

Flux per pole

$$\begin{aligned} \tau &= \frac{\pi D}{P} = \frac{\pi \times 0.16}{4} = 0.1257\ \text{m} \\ \Phi_m &= B_{av}\tau L = 0.44 \times 0.1257 \times 0.085 = 4.70\times10^{-3}\ \text{Wb} \end{aligned}$$

Length of air gap

$$l_g = 0.2 + 2\sqrt{DL} = 0.2 + 2\sqrt{0.16 \times 0.085} = 0.433\ \text{mm}$$

Turns per phase

$$T_s = \frac{E_{ph}}{4.44f\Phi K_w} = \frac{400}{4.44 \times 50 \times 4.70\times10^{-3} \times 0.955} = 401.4$$

Conductors per slot $= 6 \times 401.4/36 = 66.9 \approx 67$, so $Z_s = 2412$ and $T_s = 402$.

Stator phase current

$$Q = \frac{10}{0.86 \times 0.85} = 13.68\ \text{kVA}, \qquad I_s = \frac{13680}{3 \times 400} = 11.40\ \text{A}$$

Rotor bar current and area

Rotor mmf $\approx 0.85$ of stator mmf:

$$\begin{aligned} I_b &= 0.85 \times \frac{6 I_s T_s K_{ws}}{S_r} = 0.85 \times \frac{6 \times 11.40 \times 402 \times 0.955}{30} = 744.0\ \text{A} \\ a_b &= \frac{744.0}{5} = 148.8\ \text{mm}^2 \end{aligned}$$

End-ring current and area

$$\begin{aligned} I_e &= \frac{S_r I_b}{\pi P} = \frac{30 \times 744.0}{\pi \times 4} = 1776\ \text{A} \\ a_e &= \frac{1776}{5} = 355.2\ \text{mm}^2 \end{aligned}$$

Losses in bars

$$\begin{aligned} r_b &= \frac{\rho l_b}{a_b} = \frac{0.02 \times 0.15}{148.8} = 2.016\times10^{-5}\ \Omega \\ P_b &= S_r I_b^2 r_b = 30 \times 744.0^2 \times 2.016\times10^{-5} = 334.8\ \text{W} \end{aligned}$$

Losses in end rings

$$\begin{aligned} r_e &= \frac{\rho\,\pi D_e}{a_e} = \frac{0.02 \times \pi \times 0.12}{355.2} = 2.122\times10^{-5}\ \Omega \\ P_e &= 2I_e^2 r_e = 2 \times 1776^2 \times 2.122\times10^{-5} = 133.9\ \text{W} \end{aligned}$$

Total rotor copper loss $= 334.8 + 133.9 = 468.7$ W.