Circular windings and two-stepped core

Why windings are circular

• Under short circuit, radial forces on a circular coil produce uniform hoop stress, so the coil keeps its shape. Rectangular coils would deform.
• For a given area enclosed, a circle has the smallest perimeter, so the mean length of turn, copper weight and copper loss are minimum.
• Circular coils are easy to wind on formers and give uniform clearances and oil ducts.

Advantages of stepped cores

A square core inside a circular coil leaves much space unused. A stepped core fills the circle better.

• For the same circumscribing diameter $d$, the net iron area is larger (square $0.45d^2$, cruciform $0.56d^2$, 3-step $0.60d^2$).
• For the same iron area, the diameter $d$ and so the mean turn of the windings are smaller, saving copper and copper loss.
• Less space wasted, so a smaller, cheaper transformer with lower leakage reactance.

Most economical dimensions of a two-stepped (cruciform) core

          <---- a ---->
          +-----------+
       +--+-----------+--+  ^
       |  |           |  |  b
       +--+-----------+--+  v
          +-----------+
     circle of diameter d

Let $a$ = width of the larger stamping, $b$ = width of the smaller one, and $\theta$ the angle that the diagonal makes with the side $a$. Then

$$a = d\cos\theta,\qquad b = d\sin\theta$$

Gross core area (two rectangles $a\times b$, minus the common square $b^2$):

$$\begin{aligned} A_{gi} &= 2ab - b^2 = 2d^2\sin\theta\cos\theta - d^2\sin^2\theta\\ &= d^2\left(\sin2\theta - \sin^2\theta\right) \end{aligned}$$

For maximum area:

$$\begin{aligned} \frac{dA_{gi}}{d\theta} &= d^2(2\cos2\theta - \sin2\theta) = 0\\ \tan2\theta &= 2 \;\Rightarrow\; \theta = 31.72^\circ \end{aligned}$$

So

$$\begin{aligned} a &= d\cos31.72^\circ = 0.85\,d\\ b &= d\sin31.72^\circ = 0.53\,d\\ A_{gi,max} &= d^2(\sin63.43^\circ - \sin^2 31.72^\circ) = 0.618\,d^2 \end{aligned}$$

With stacking factor 0.9, net iron area $A_i = 0.9 \times 0.618d^2 \approx 0.56\,d^2$. The cruciform core uses 0.618/0.785 ≈ 79% of the circle area, against 64% for a square core.