Output equation of three phase induction motor

The output equation relates the kVA input rating $Q$ of a 3-phase induction motor to its main dimensions ($D$ = stator bore diameter, $L$ = core length), its specific loadings ($B_{av}$, $ac$) and its synchronous speed $n_s$.

Symbols

• $P$ = number of poles, $f$ = frequency, $n_s$ = synchronous speed (rps)
• $\Phi$ = flux per pole (Wb), $T_{ph}$ = turns per phase, $K_w$ = winding factor
• $E_{ph}$, $I_{ph}$ = phase voltage and phase current
• $B_{av}$ = specific magnetic loading (Wb/m²), $ac$ = specific electric loading (A/m)

Step 1: kVA input

Neglecting the small drop, $V_{ph} \approx E_{ph}$:

$$Q = 3E_{ph}I_{ph}\times 10^{-3}\ \text{kVA}$$

Step 2: Induced emf

$$E_{ph} = 4.44\, f\, \Phi\, T_{ph} K_w, \qquad f = \frac{P n_s}{2}$$

Step 3: Magnetic loading

Total flux around the gap is $P\Phi$ over area $\pi D L$:

$$B_{av} = \frac{P\Phi}{\pi D L} \;\Rightarrow\; \Phi = \frac{B_{av}\pi D L}{P}$$

Step 4: Electric loading

Total stator conductors $= 3 \times 2T_{ph} = 6T_{ph}$, each carrying $I_{ph}$:

$$ac = \frac{6 I_{ph} T_{ph}}{\pi D} \;\Rightarrow\; I_{ph}T_{ph} = \frac{ac\,\pi D}{6}$$

Step 5: Substitute

$$\begin{aligned} Q &= 3 \times 4.44 \times \frac{P n_s}{2} \times \frac{B_{av}\pi D L}{P} \times K_w \times \frac{ac\,\pi D}{6} \times 10^{-3} \\ &= 1.11\,\pi^2 B_{av}\, ac\, K_w\, D^2 L\, n_s \times 10^{-3} \\ &= 11\, B_{av}\, ac\, K_w \times 10^{-3}\, D^2 L\, n_s \end{aligned}$$

So

$$Q = C_0 D^2 L n_s, \qquad C_0 = 11 B_{av}\, ac\, K_w \times 10^{-3}$$

$C_0$ is called the output coefficient. The kVA input is found from the rated shaft output:

$$Q = \frac{\text{kW output}}{\eta \cos\phi}$$

Meaning of the result

• $D^2L = \dfrac{Q}{C_0 n_s}$, so the volume of the active part is directly proportional to rating and inversely proportional to speed and specific loadings.
• Higher $B_{av}$ and $ac$ give a smaller, cheaper machine, but power factor, temperature rise and overload capacity limit them.
• A high-speed motor is smaller than a low-speed motor of the same rating.
• Typical ranges: $B_{av} = 0.3$–$0.6$ Wb/m², $ac = 5000$–$45000$ A/m, $K_w \approx 0.955$ for a full-pitch, 60° phase-spread winding.

The product $D^2L$ is then split into $D$ and $L$ using a chosen ratio $L/\tau$ (for example $L/\tau = 1$ for good overall design) and a check on peripheral speed ($\pi D n_s \le 30$ m/s for normal designs).