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

# Temperature gradients in slot conductors

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

- Asked in: 2076 Chaitra, 6 marks; 2071 Chaitra, 6 marks (asked 2 times)
- 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-explain
- Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machine_design/chapter-2

## Question

Explain the temperature gradients in the conductor placed in the slot with necessary figures and expressions.

## Short answer

Thick slot insulation forces heat axially, giving a parabolic temperature profile along the conductor with peak rise (J^2 rho_e rho_t L^2)/8 above the ends. Thin insulation lets heat flow radially, creating a linear gradient across the insulation and a nearly uniform conductor temperature.

## Answer

A conductor in a slot produces copper loss $p = J^2\rho_e$ per unit volume. This heat can leave by two paths: **across the slot insulation into the iron** (then to the cooling air), or **along the conductor to the overhang**, where it is dissipated to the air. Which path dominates depends on the thickness of the slot insulation and the length of the overhang. This gives two limiting cases.

#### Case 1: slot insulation very thick (heat flows along the conductor)
**Assumptions**
- The conductor carries current density $J$. Loss per unit volume is $p = J^2\rho_e$, where $\rho_e$ is the electrical resistivity.
- The slot insulation is **very thick** (a poor conductor of heat). No heat flows sideways into the iron; all heat produced in the slot portion flows **along the conductor** to the overhang (end connections), where it is dissipated.
- $\rho_t$ = thermal resistivity of the conductor (copper), $A$ = conductor cross-section, $L$ = embedded (slot) length, $\theta_o$ = temperature of the conductor at the slot ends (overhang).

```text
   overhang      slot portion (length L)      overhang
  ~~~~~~~~~ |============================| ~~~~~~~~~
            |<--- heat    hottest  heat --->|
           x=-L/2         x = 0          x=+L/2
               temperature: parabola, max at centre
```

**Derivation**
By symmetry the hottest point is at the centre of the slot length ($x = 0$). Heat produced between the centre and a section at distance $x$ flows through that section:

$$
Q_x = p\,A\,x
$$

Fourier's law along the conductor gives

$$
\frac{d\theta}{dx} = -\frac{\rho_t\,Q_x}{A} = -\rho_t\,p\,x
$$

Integrating from the centre ($\theta = \theta_m$ at $x = 0$):

$$
\theta = \theta_m - \frac{\rho_t\,p\,x^2}{2}
$$

At the slot ends, $x = L/2$, $\theta = \theta_o$:

$$
\theta_m - \theta_o = \frac{p\,\rho_t\,L^2}{8} = \frac{J^2\rho_e\,\rho_t\,L^2}{8}
$$

So the temperature varies **parabolically along the conductor**. The gradient at any point is $\rho_t p x$, which is zero at the centre and largest at the slot ends.

#### Case 2: slot insulation thin (heat flows across the insulation)
If the insulation is thin, almost all the heat of the embedded part flows radially through it to the teeth and core. The conductor temperature is then nearly uniform along its length, and the temperature drop occurs **across the insulation**.

```text
  iron | insulation | copper (theta_c)
       |  t_i       |
  theta_i <-- heat --
```

Heat produced in length $L$: $Q = p A L$. It flows through insulation of thickness $t_i$, thermal resistivity $\rho_i$ and area $S_p L$, where $S_p$ is the perimeter of the slot insulation:

$$
\theta_c - \theta_{iron} = Q \times \frac{\rho_i\,t_i}{S_p L} = \frac{p\,A\,\rho_i\,t_i}{S_p}
$$

The gradient is now **linear across the insulation**: $\dfrac{d\theta}{dy} = \dfrac{p A \rho_i}{S_p}$.

#### Summary
| Case | Heat path | Temperature profile | Max. rise |
|---|---|---|---|
| Thick insulation | Along conductor to overhang | Parabolic along length | $p\rho_t L^2/8$ above ends |
| Thin insulation | Across insulation to iron | Linear across insulation | $pA\rho_i t_i/S_p$ above iron |

In practice both paths act together. Thin, high-conductivity insulation (mica-epoxy) and ventilating ducts keep the conductor hot spot low.

## Check yourself

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

1. By which two paths can heat leave a conductor in the slot?
   A. Only through the overhang
   B. Across the slot insulation into the iron, or along the conductor to the overhang
   C. Through the shaft or through the rotor
   D. Only through the air gap
   Answer: B. The dominant path depends on slot insulation thickness and overhang. (Key point: Two heat paths)

2. What temperature profile exists along the conductor when insulation is thick?
   A. Parabolic along its length with maximum at the middle
   B. Constant
   C. Exponential decrease
   D. Linear across the insulation
   Answer: A. Heat flows axially, giving a parabolic profile with the peak at the centre. (Key point: Thick insulation profile)

3. For thin slot insulation, where does the main temperature drop occur?
   A. In the overhang only
   B. In the core
   C. Along the conductor length
   D. Across the insulation, with the conductor nearly uniform in temperature
   Answer: D. Heat flows radially through the insulation, so the drop is there. (Key point: Thin insulation)

4. What is the temperature difference across thin insulation?
   A. p L^2 / (8 rho_i)
   B. p A S_p / (rho_i t_i)
   C. p A rho_i t_i / S_p
   D. p rho_t L^2 / 8
   Answer: C. Q = pAL flows through area S_p L, giving p A rho_i t_i/S_p, and the gradient is linear. (Key point: Thin insulation formula)

## 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×
- [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
- [Temperature gradient in slot: two cases](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atemperature-gradient-slot-overhang) — Electric Machine Design, chapter 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
- [True/False: ampere conductors and temperature rise](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atf-ampere-conductor-temperature) — Electric Machine Design, chapter 5

## Related in other subjects

- [Voltage matrix for two nearby conductors](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Avoltage-matrix-two-conductors) — High Voltage Engineering, chapter 5, asked 2×