---
title: "Temperature gradient in slot: two cases"
subject: "Electric Machine Design"
chapter: 2
key: "semester_5/electric_machine_design:temperature-gradient-slot-overhang"
source: https://www.bctengineer.com/q/semester_5/electric_machine_design/temperature-gradient-slot-overhang
site: BCT Engineering
---

# Temperature gradient in slot: two cases

Electric Machine Design · Chapter 2: Heating and cooling of electric machine

- Asked in: 2079 Bhadra, 2+2 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/electric_machine_design/temperature-gradient-slot-overhang
- Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machine_design/chapter-2

## Question

Derive the expression of temperature gradient in conductors placed in slot if (i) Slot insulation is thick (ii) Overhang is considerably not

## Short answer

For thick insulation, axial heat flow gives a parabolic profile with gradient dθ/dx = –ρ_t p x and centre rise pρ_t L^2/8. With negligible overhang heat passes through insulation, yielding a uniform conductor temperature and a constant gradient pAρ_i/S_p across the insulation.

## Answer

A slot conductor produces loss $p = J^2\rho_e$ per unit volume ($J$ = current density, $\rho_e$ = electrical resistivity). The heat leaves along the conductor to the overhang, or across the slot insulation to the iron. Part (ii) of the question is cut off in the source; it is read here as the case where the **overhang is not considerable** (short, with little cooling), so that the heat must leave through the slot insulation.

#### (i) Slot insulation thick
No heat passes through the insulation, so heat flows axially to the overhang. With slot length $L$, copper thermal resistivity $\rho_t$ and the centre at $x = 0$:

$$
\begin{aligned}
\frac{d\theta}{dx} &= -\rho_t\,p\,x \\
\theta_{max} - \theta_o &= \frac{p\,\rho_t\,L^2}{8}
\end{aligned}
$$

The gradient is zero at the slot centre and largest at the slot ends. The temperature varies parabolically along the conductor, with its maximum at the middle.

#### (ii) Overhang not considerable (heat through slot insulation)
Little heat can flow to the short overhang, so the heat of the slot portion, $Q = p A L$, crosses the insulation (thickness $t_i$, thermal resistivity $\rho_i$, perimeter $S_p$, area $S_p L$):

$$
\begin{aligned}
\theta_c - \theta_{iron} &= Q\,\frac{\rho_i\,t_i}{S_p L} = \frac{p\,A\,\rho_i\,t_i}{S_p} \\
\frac{d\theta}{dy} &= \frac{p\,A\,\rho_i}{S_p}\ \ (\text{constant})
\end{aligned}
$$

The conductor is at nearly uniform temperature along its length. The gradient is linear across the insulation thickness.

## Check yourself

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

1. With thick slot insulation, how does heat leave the conductor?
   A. Across the insulation
   B. Through the air gap
   C. It is stored
   D. Axially along the copper to the overhang, giving a parabolic profile
   Answer: D. No heat passes through thick insulation, so axial flow gives theta_max - theta_o = p rho_t L^2/8. (Key point: Case (i))

2. When the overhang is not considerable, how does the heat leave?
   A. Via the end rings
   B. Along the copper
   C. Across the slot insulation to the iron
   D. Through the shaft
   Answer: C. A short overhang cannot carry the heat, so it crosses the insulation. (Key point: Case (ii))

3. What is the gradient across the insulation in case (ii)?
   A. Proportional to x
   B. Constant, p A rho_i / S_p (linear profile)
   C. Zero
   D. Parabolic
   Answer: B. Heat Q = pAL flows through area S_p L, giving a linear temperature drop. (Key point: Gradient across insulation)

## Related in Electric Machine Design

- [Temperature gradient in slot conductors (thick insulation)](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atemperature-gradient-slot-thick) — Electric Machine Design, chapter 2, asked 2×
- [Temperature gradients in slot conductors](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atemperature-gradient-slot-explain) — Electric Machine Design, chapter 2, asked 2×
- [Temperature difference in slot copper strips](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Aslot-strips-temperature-difference) — Electric Machine Design, chapter 2
- [Maximum temperature of slot conductor](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Amax-conductor-temperature-slot) — Electric Machine Design, chapter 2