Design: 125 kVA 6600/400 V three 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. $K$ = 0.45 assumed (3-phase distribution transformer), $\rho = 0.021\ \Omega$-mm²/m. 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, total for 3 phases):

$$\begin{aligned} E_t &= 0.45\sqrt{125} = 5.031\ \text{V}\\ \phi_m &= \frac{E_t}{4.44f} = \frac{5.031}{4.44\times50} = 0.02266\ \text{Wb}\\ A_i &= \frac{\phi_m}{B_m} = \frac{0.02266}{1.35} = 0.01679\ \text{m}^2 = 167.9\ \text{cm}^2 \end{aligned}$$

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

$$\begin{aligned} d &= \sqrt{\frac{A_i}{0.56}} = \sqrt{\frac{0.01679}{0.56}} = 0.1731\ \text{m} = 173.1\ \text{mm}\\ a &= 0.85d = 147.2\ \text{mm}\\ b &= 0.53d = 91.8\ \text{mm} \end{aligned}$$

Gross core area $A_{gi} = A_i/0.9 = 0.01865$ m².

Output equation (three phase): $Q = 3.33fB_m\delta K_wA_wA_i\times10^{-3}$

$$\begin{aligned} A_w &= \frac{Q}{3.33fB_m\delta K_wA_i\times10^{-3}}\\ &= \frac{125}{3.33\times50\times1.35\times2.75\times10^6\times0.4\times0.01679\times10^{-3}}\\ &= 0.03012\ \text{m}^2\\ W_w &= \sqrt{\frac{A_w}{2.5}} = 109.8\ \text{mm},\quad H_w = 2.5W_w = 274.4\ \text{mm}\\ D &= d + W_w = 282.9\ \text{mm (distance between adjacent limb centres)} \end{aligned}$$

Check with the given windings: $d$ = 173 mm fits inside the LV (255 mm), but the HV outside diameter 351 mm > $D$ = 283 mm and the coil height 276 mm > $H_w$ = 274 mm. So the window is enlarged to suit the given coils (10 mm between adjacent HV coils), and its height is increased to keep the given ratio $H_w/W_w = 2.5$:

$$\begin{aligned} D &= 351 + 10 = 361\ \text{mm},\quad W_w = D - d = 187.9\ \text{mm}\\ H_w &= 2.5W_w = 469.7\ \text{mm} \end{aligned}$$

Yoke:

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

$$\begin{aligned} A_y &= 1.2A_i = 0.02014\ \text{m}^2,\quad A_{gy} = \frac{A_y}{0.9} = 0.02238\ \text{m}^2\\ D_y &= a = 147.2\ \text{mm}\\ H_y &= \frac{A_{gy}}{D_y} = \frac{0.02238}{0.1472} = 152.1\ \text{mm} \end{aligned}$$

ii) Overall dimensions of frame

$$\begin{aligned} H &= H_w + 2H_y = 469.7 + 2(152.1) = 773.9\ \text{mm}\\ W &= 2D + a = 2(361.0) + 147.2 = 869.2\ \text{mm}\\ \text{Depth} &= D_y = a = 147.2\ \text{mm} \end{aligned}$$

iii) Leakage reactance

Phase voltages: LV (star) $V_{LV} = 230.9$ V; HV (delta) $V_{HV} = 6600.0$ V.

Phase currents: $I_{LV} = \frac{Q}{3V_{LV}} = 180.42$ A; $I_{HV} = 6.313$ A.

$$\begin{aligned} T_{LV} &= \frac{V_{LV}}{E_t} = \frac{230.9}{5.031} = 45.90 \approx 46\\ E_t\ (\text{revised}) &= \frac{230.9}{46} = 5.020\ \text{V}\\ T_{HV} &= T_{LV}\frac{V_{HV}}{V_{LV}} = 1314.6 \approx 1315\\ a_{LV} &= \frac{I_{LV}}{\delta} = 65.61\ \text{mm}^2,\quad a_{HV} = \frac{I_{HV}}{\delta_{HV}} = 2.296\ \text{mm}^2 \end{aligned}$$

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 $= 8302$ and axial winding length $L_c = 272.0$ mm:

$$\begin{aligned} \varepsilon_x &= 2\pi f\mu_0\frac{AT}{L_c}\cdot\frac{L_{mt}}{E_t}\left(a + \frac{b_1+b_2}{3}\right)\\ &= 2\pi(50)(4\pi\times10^{-7})\frac{8302}{0.2720}\cdot\frac{0.9519}{5.020}\left(11 + \frac{14+18}{3}\right)\times10^{-3}\\ &= 0.0495\ \text{p.u.}\\ X_{HV} &= \varepsilon_x\frac{V_{HV}}{I_{HV}} = 0.0495\times\frac{6600.0}{6.313} = 51.75\ \Omega\ \text{(referred to HV)}\\ X_{LV} &= X_{HV}\left(\frac{T_{LV}}{T_{HV}}\right)^2 = 0.06332\ \Omega \end{aligned}$$

iv) Overall dimensions of frame (sketch)

 +-----------------------------+  ---
 |   yoke, Hy = 152 mm         |   ^
 +-----+-----+-----+-----+-----+   |
 |limb |     |limb |     |limb |   |
 |  R  | Ww  |  Y  | Ww  |  B  |  H = 774 mm
 |     |188  |     |     |     |   |
 +-----+-----+-----+-----+-----+   |
 |           yoke              |   v
 +-----------------------------+  ---
 |<------------ W ------------>|
 W = 869 mm, D = 361 mm (limb centres)
 Depth of core and yoke = 147 mm
 Window height Hw = 470 mm

Answer: $d$ = 173 mm; window 188 × 470 mm; yoke 152 × 147 mm; frame 774 × 869 × 147 mm; leakage reactance ≈ 51.7 Ω/phase referred to HV (4.95%).