Design: 150 kVA single phase transformer

Standard method (A.K. Sawhney, A Course in Electrical Machine Design): $E_t = K\sqrt{Q}$, net core area ratio for the core type, output equation for the window, yoke depth = width of largest stamping. In a single-phase core type transformer half of each winding is on each limb, so the ampere-turns per limb are $I_{HV}T_{HV}/2$. $K$ is not given; $K$ = 0.75 is taken (single-phase core type). Winding data are given: LV 255/293 mm (inside/outside diameter), HV 314/351 mm, radial depths $b_1$ = 14 mm, $b_2$ = 18 mm, gap $a$ = 11 mm. Axial length taken as the mean of 268 and 276 mm, $L_c$ = 272 mm.

i) Dimensions of core, window and yoke

Voltage per turn, $E_t = K\sqrt{Q}$ ($Q$ in kVA):

$$\begin{aligned} E_t &= 0.75\sqrt{150} = 9.186\ \text{V}\\ \phi_m &= \frac{E_t}{4.44f} = \frac{9.186}{4.44\times50} = 0.04138\ \text{Wb}\\ A_i &= \frac{\phi_m}{B_m} = \frac{0.04138}{1.35} = 0.03065\ \text{m}^2 = 306.5\ \text{cm}^2 \end{aligned}$$

For a cruciform (two-stepped) core with stacking factor 0.9, $A_i = 0.56d^2$ (standard ratio, Sawhney):

$$\begin{aligned} d &= \sqrt{\frac{A_i}{0.56}} = \sqrt{\frac{0.03065}{0.56}} = 0.2339\ \text{m} = 233.9\ \text{mm}\\ a &= 0.85d = 198.9\ \text{mm}\\ b &= 0.53d = 124.0\ \text{mm} \end{aligned}$$

Gross core area $A_{gi} = A_i/0.9 = 0.03405$ m².

Output equation (single phase): $Q = 2.22fB_m\delta K_wA_wA_i\times10^{-3}$

$$\begin{aligned} A_w &= \frac{Q}{2.22fB_m\delta K_wA_i\times10^{-3}}\\ &= \frac{150}{2.22\times50\times1.35\times2.75\times10^6\times0.27\times0.03065\times10^{-3}}\\ &= 0.04399\ \text{m}^2\\ W_w &= \sqrt{\frac{A_w}{2.5}} = 132.6\ \text{mm},\quad H_w = 2.5W_w = 331.6\ \text{mm}\\ D &= d + W_w = 366.6\ \text{mm (distance between adjacent limb centres)} \end{aligned}$$

Check: $d$ = 234 mm < LV inside diameter 255 mm; HV outside diameter 351 mm < $D$ = 367 mm; coil height 276 mm < $H_w$ = 332 mm. The given windings fit.

Yoke:

Yoke area 20% more than core area (hot-rolled steel, to reduce yoke loss and magnetising current). Yoke depth = width of largest stamping:

$$\begin{aligned} A_y &= 1.2A_i = 0.03678\ \text{m}^2,\quad A_{gy} = \frac{A_y}{0.9} = 0.04087\ \text{m}^2\\ D_y &= a = 198.9\ \text{mm}\\ H_y &= \frac{A_{gy}}{D_y} = \frac{0.04087}{0.1989} = 205.5\ \text{mm} \end{aligned}$$

ii) Overall dimensions of frame

$$\begin{aligned} H &= H_w + 2H_y = 331.6 + 2(205.5) = 742.6\ \text{mm}\\ W &= D + a = 366.6 + 198.9 = 565.4\ \text{mm}\\ \text{Depth} &= D_y = a = 198.9\ \text{mm} \end{aligned}$$

iii) Leakage reactance

Currents: $I_{LV} = \frac{Q}{V_{LV}} = 375.00$ A; $I_{HV} = 22.727$ A.

$$\begin{aligned} T_{LV} &= \frac{V_{LV}}{E_t} = \frac{400.0}{9.186} = 43.55 \approx 44\\ E_t\ (\text{revised}) &= \frac{400.0}{44} = 9.091\ \text{V}\\ T_{HV} &= T_{LV}\frac{V_{HV}}{V_{LV}} = 726.0 \approx 726\\ a_{LV} &= \frac{I_{LV}}{\delta} = 136.36\ \text{mm}^2,\quad a_{HV} = \frac{I_{HV}}{\delta_{HV}} = 8.264\ \text{mm}^2 \end{aligned}$$

Radial depths: $b_1 = 14$ mm (LV), $b_2 = 18$ mm (HV), duct $a = 11$ mm.

Mean lengths of turn:

$$\begin{aligned} L_{mt,LV} &= \pi\frac{255.0 + 293.0}{2} = 0.8608\ \text{m}\\ L_{mt,HV} &= \pi\frac{314.0 + 351.0}{2} = 1.0446\ \text{m}\\ L_{mt} &= \pi\frac{255.0 + 351.0}{2} = 0.9519\ \text{m (both windings, for reactance)} \end{aligned}$$

Leakage reactance (per unit), with $AT$ per limb $= 8250$ and axial winding length $L_c = 272.0$ mm:

$$\begin{aligned} \varepsilon_x &= 2\pi f\mu_0\frac{AT}{L_c}\cdot\frac{L_{mt}}{E_t}\left(a + \frac{b_1+b_2}{3}\right)\\ &= 2\pi(50)(4\pi\times10^{-7})\frac{8250}{0.2720}\cdot\frac{0.9519}{9.091}\left(11 + \frac{14+18}{3}\right)\times10^{-3}\\ &= 0.0272\ \text{p.u.}\\ X_{HV} &= \varepsilon_x\frac{V_{HV}}{I_{HV}} = 0.0272\times\frac{6600.0}{22.73} = 7.889\ \Omega\ \text{(referred to HV)}\\ X_{LV} &= X_{HV}\left(\frac{T_{LV}}{T_{HV}}\right)^2 = 0.02898\ \Omega \end{aligned}$$

iv) Overall dimensions (sketch)

 +-----------------------+  ---
 |  yoke, Hy = 206 mm    |   ^
 +-----+-----------+-----+   |
 |limb |  window   |limb |   H = 743 mm
 |d=234| 133 x 332 |     |   |
 +-----+-----------+-----+   |
 |        yoke           |   v
 +-----------------------+  ---
 |<--------- W --------->|
 W = 565 mm, D = 367 mm (limb centres)
 Depth = 199 mm

Answer: $d$ = 234 mm; window 133 × 332 mm; yoke 206 × 199 mm; frame 743 × 565 × 199 mm; leakage reactance ≈ 7.89 Ω referred to HV (2.72%).