1000 kW generator: dimensions, poles, air gap

Assumptions: the speed is read as 300 rpm (the digit after "30" is unclear; 300 rpm is the usual value for this problem). The air-gap mmf is taken as 50% of the armature mmf per pole (usual design value, not stated), and the limits $f \le 50$ Hz and current per brush arm $\le 400$ A are used to choose the poles.

The main dimensions come from the output equation of a dc machine (A.K. Sawhney):

$$P_a = C_0 D^2 L n, \qquad C_0 = \pi^2 B_{av}\, ac \times 10^{-3}\ \text{kW/(m}^3\text{ rps)}$$

where $P_a$ = armature power (kW), $D$ = armature diameter, $L$ = core length (m), $n$ = speed (rps).

For a generator (Sawhney), $P_a = \dfrac{1+2\eta}{3\eta}P$.

Step 1: $D^2L$

$$\begin{aligned} P_a &= \frac{1+2\times0.92}{3\times0.92}\times1000 = 1029\ \text{kW} \\ C_0 &= \pi^2\times0.7\times40000\times10^{-3} = 276.3 \\ n &= 300/60 = 5\ \text{rps} \\ D^2L &= \frac{1029}{276.3\times5} = 0.7447\ \text{m}^3 \end{aligned}$$

Step 2: Number of poles

Armature current $I_a \approx 10^6/500 = 2000$ A. Lap winding, current per brush arm $= 2I_a/p \le 400 \Rightarrow p \ge 10$. Square pole face gives $L = 0.7\pi D/p$, so $D^3 = D^2L\,p/(0.7\pi)$.

$p = 8$ fails the brush current limit. $p = 10$ is the smallest number that satisfies it, with low frequency and armature mmf below 10 000 A. Choose $p = 10$.

Step 3: Main dimensions ($p = 10$)

$$\begin{aligned} D^3 &= \frac{0.7447\times10}{0.7\pi} = 3.386\ \text{m}^3 \\ D &= 1.502\ \text{m} \\ L &= \frac{0.7\pi\times1.502}{10} = 0.330\ \text{m} \end{aligned}$$

Peripheral speed $= \pi Dn = \pi\times1.502\times5 = 23.6$ m/s (safe).

Step 4: Air-gap length

$$\begin{aligned} AT_a &= \frac{ac\,\pi D}{2p} = \frac{40000\times\pi\times1.502}{20} = 9435\ \text{A} \\ AT_g &= 0.5\times9435 = 4718\ \text{A} \\ B_g &= \frac{B_{av}}{\psi} = \frac{0.7}{0.7} = 1.0\ \text{Wb/m}^2 \\ l_g &= \frac{AT_g}{800000\,B_g k_g} = \frac{4718}{800000\times1.0\times1.15} = 5.13\ \text{mm} \end{aligned}$$

Answer: Poles = 10, $D \approx 1.50$ m, $L \approx 0.33$ m, air gap $l_g \approx 5.1$ mm.