---
title: "Circular windings and two-stepped core"
subject: "Electric Machine Design"
chapter: 3
key: "semester_5/electric_machine_design:stepped-core-circular-windings"
source: https://www.bctengineer.com/q/semester_5/electric_machine_design/stepped-core-circular-windings
site: BCT Engineering
---

# Circular windings and two-stepped core

Electric Machine Design · Chapter 3: Transformer Design

- Asked in: 2075 Chaitra, 6 marks
- Video: [5  Design of Crucifrom Two Stepped Core](https://www.youtube.com/watch?v=Pz6_57zZ_60) — Sandeep Shelar
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/electric_machine_design/stepped-core-circular-windings
- Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machine_design/chapter-3

## Question

Why are windings of a transformer made into circular form? What is the advantage of using stepped cores in transformers? Derive the most economical dimension of a two-stepped core.

## Short answer

Circular windings handle short‑circuit forces as uniform hoop stress, minimise copper, and are easy to wind. Stepped cores fill the circular coil better, increasing net iron area and reducing mean turn length. For a two‑stepped (cruciform) core, optimum dimensions: a = 0.85d, b = 0.53d, giving maximum gross area 0.618d² (stacked net area ~0.56d²).

## Answer

#### 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
```text
          <---- 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the advantage of a stepped core?
   A. It fills the circle better so net iron area is larger for a given d
   B. It removes eddy current
   C. It increases leakage
   D. It needs no stacking factor
   Answer: A. For the same iron area, d and mean turn are smaller, saving copper. (Key point: Stepped core advantage)

2. Net iron area for a square core is 0.45d^2. For cruciform it is:
   A. 0.62d^2
   B. 0.56d^2
   C. 0.60d^2
   D. 0.71d^2
   Answer: B. Cruciform 0.56d^2, 3-step 0.60d^2. (Key point: Net iron areas)

3. Gross area of a two-stepped core in terms of d and angle theta is:
   A. d^2 (cos2theta + sin theta)
   B. 2d^2 sin theta
   C. d^2 (sin2theta - sin^2 theta)
   D. d^2 cos^2 theta
   Answer: C. Two rectangles ab minus the common square b^2 gives 2ab - b^2. (Key point: Gross area formula)

4. Maximising the area gives tan2theta = 2, so a and b are:
   A. a = 0.71d, b = 0.71d
   B. a = 0.85d, b = 0.53d
   C. a = 0.93d, b = 0.37d
   D. a = 0.53d, b = 0.85d
   Answer: B. theta = 31.72 degrees, giving cos = 0.85 and sin = 0.53. (Key point: Optimum angle)

## Related in Electric Machine Design

- [Three phase output equation and circular windings](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Aoutput-equation-circular-windings) — Electric Machine Design, chapter 3
- [Six-stepped core area and turns (1000 kVA)](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Asix-stepped-core-1000kva) — Electric Machine Design, chapter 3
- [Leakage reactance of core type transformer](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Aleakage-reactance-core-type) — Electric Machine Design, chapter 3, asked 3×
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- [Requirements of transformer core material](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atransformer-core-material-requirements) — Electric Machine Design, chapter 1

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