---
title: "Townsend breakdown condition with secondary ionization"
subject: "High Voltage Engineering"
chapter: 6
key: "semester_8/high_voltage_engineering:townsend-breakdown-condition"
source: https://www.bctengineer.com/q/semester_8/high_voltage_engineering/townsend-breakdown-condition
site: BCT Engineering
---

# Townsend breakdown condition with secondary ionization

High Voltage Engineering · Chapter 6: Dielectric Breakdown

- Asked in: 2067 Mangsir (old course), 10 marks
- Video: [Current Growth Equation in Presence Secondary Ionization Process - High Voltage Engineering](https://www.youtube.com/watch?v=wtZkttLKeXs) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/high_voltage_engineering/townsend-breakdown-condition
- Whole chapter: https://www.bctengineer.com/md/semester_8/high_voltage_engineering/chapter-6

## Question

How is the condition for breakdown obtained in Townsend discharge due to primary and secondary ionization?

## Short answer

Primary ionization alone gives I=I0 e^(αd). With secondary processes of coefficient γ, I=I0 e^(αd)/[1−γ(e^(αd)−1)]. Breakdown occurs when denominator zero, i.e. γ(e^(αd)−1)=1, making the discharge self-sustaining.

## Answer

In Townsend's theory, breakdown of a gas in a uniform field is the point where the discharge becomes **self-sustaining**: each avalanche produces, through secondary processes, at least one new electron at the cathode to start the next avalanche, so the current continues even without the external source of initial electrons.

#### 1. Primary ionization

Let $n_0$ electrons per second leave the cathode due to an external agency (UV light). Each electron makes $\alpha$ ionizing collisions per cm in the field direction ($\alpha$ = Townsend's first ionization coefficient, a function of $E/p$). In distance $dx$:

$$
\begin{aligned}
dn &= \alpha\, n\, dx \\
n &= n_0\, e^{\alpha x} \\
I &= I_0\, e^{\alpha d}
\end{aligned}
$$

Each electron leaving the cathode produces $e^{\alpha d} - 1$ new electrons and the same number of positive ions in the gap. With primary ionization alone the current is finite for any $d$; it stops if $I_0$ is removed. This cannot explain breakdown.

#### 2. Secondary ionization processes

Experiments show the current rising faster than $e^{\alpha d}$ at higher voltages. This is due to secondary electrons from:

- **Positive ions** striking the cathode and releasing electrons ($\gamma_i$)
- **Photons** from excited atoms falling on the cathode (photo-emission, $\gamma_{ph}$)
- **Metastable atoms** diffusing to the cathode ($\gamma_m$)
- Photo-ionization in the gas (minor in Townsend's range)

All are combined into one coefficient $\gamma = \gamma_i + \gamma_{ph} + \gamma_m$ = number of secondary electrons released from the cathode per positive ion produced (Townsend's second coefficient).

#### 3. Current growth with secondary ionization

Let $n_0'$ = total electrons leaving the cathode, $n_+$ = secondary electrons, $n$ = electrons reaching the anode:

$$
\begin{aligned}
n_0' &= n_0 + n_+ \\
n &= n_0'\, e^{\alpha d} \\
n_+ &= \gamma\,(n - n_0')
\end{aligned}
$$

Eliminating $n_0'$ and $n_+$:

$$
\begin{aligned}
n_0' &= \frac{n_0 + \gamma n}{1 + \gamma} \\
n(1 + \gamma) &= (n_0 + \gamma n)\, e^{\alpha d} \\
n &= \frac{n_0\, e^{\alpha d}}{1 - \gamma\left(e^{\alpha d} - 1\right)}
\end{aligned}
$$

$$
I = \frac{I_0\, e^{\alpha d}}{1 - \gamma\left(e^{\alpha d} - 1\right)}
$$

#### 4. Condition for breakdown

As the voltage (or gap) is raised, $\alpha$ and $e^{\alpha d}$ increase, and the denominator decreases. When

$$
\boxed{\gamma\left(e^{\alpha d} - 1\right) = 1}
$$

the current becomes theoretically infinite; in practice it is limited only by the external circuit, and a spark forms. Since normally $e^{\alpha d} \gg 1$:

$$
\gamma\, e^{\alpha d} \approx 1 \qquad \text{or} \qquad \alpha d = \ln\left(1 + \frac{1}{\gamma}\right)
$$

**Physical meaning:** $\gamma(e^{\alpha d} - 1)$ is the number of secondary electrons produced at the cathode by one primary avalanche.

| Value of $\gamma(e^{\alpha d} - 1)$ | Discharge | Behaviour |
|---|---|---|
| $< 1$ | Non-self-sustaining | Stops if $I_0$ is removed |
| $= 1$ | Breakdown threshold | Self-sustaining; sparking voltage |
| $> 1$ | Self-sustaining, growing | Current rises rapidly to an arc |

```text
 ln I
  |                       |  breakdown
  |                      /   (gamma term)
  |                   _/
  |              ___/
  |         ___/  slope = alpha
  |    ___/
  +-------------------------> d (or V)
```

#### 5. Consequences

- **Sparking voltage:** the voltage at which the criterion is just satisfied. Combining it with $\alpha/p = Ae^{-Bp/E}$ gives **Paschen's law**:
$$
V_b = \frac{B\,pd}{\ln\left[A\,pd/\ln(1 + 1/\gamma)\right]} = f(pd)
$$
- **Effect of attachment:** in electronegative gases (SF₆, O₂) electrons are captured with coefficient $\eta$; the criterion becomes
$$
\frac{\alpha}{\alpha - \eta}\left[e^{(\alpha - \eta)d} - 1\right]\gamma = 1
$$
so breakdown needs a higher field, explaining the high strength of SF₆.
- **Validity:** the Townsend criterion holds for low $pd$ (below about 1000 torr·cm); for long gaps at high pressure the streamer criterion applies.

## Check yourself

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

1. Why can primary ionization alone (I = I0 e^(alpha d)) not explain breakdown?
   A. Alpha becomes negative at high fields
   B. The current is finite for any gap and stops if I0 is removed
   C. It predicts the current falls as the gap grows
   D. It only applies to DC voltages
   Answer: B. With only e^(alpha d) multiplication the current stays finite and depends on the external source, so the discharge is not self-sustaining. (Key point: Primary ionization alone)

2. What is Townsend's second coefficient gamma?
   A. Number of ionizing collisions per cm by an electron
   B. Probability that an electron attaches to a gas molecule
   C. Number of secondary electrons released from the cathode per positive ion produced
   D. Ratio of anode current to cathode current
   Answer: C. gamma = gamma_i + gamma_ph + gamma_m combines ion impact, photo-emission and metastable effects, per positive ion produced. (Key point: Meaning of gamma)

3. Which expression for the current results from including secondary ionization?
   A. I = I0 e^(alpha d) / [1 - gamma(e^(alpha d) - 1)]
   B. I = I0 e^(alpha d) [1 - gamma(e^(alpha d) - 1)]
   C. I = I0 e^(alpha d) / [1 + gamma(e^(alpha d) - 1)]
   D. I = I0 e^(gamma d) / [1 - alpha(e^(gamma d) - 1)]
   Answer: A. Eliminating n0' and n+ from n = n0' e^(alpha d) and n+ = gamma(n - n0') gives this form. (Key point: Current with secondary ionization)

4. What is the Townsend breakdown condition and what does it mean?
   A. gamma e^(alpha d) = 0; no secondary electrons are produced
   B. gamma(e^(alpha d) - 1) = 1; one avalanche yields one new starting electron, so the discharge is self-sustaining
   C. alpha d = 1; each electron makes one ionization
   D. gamma(e^(alpha d) - 1) < 1; the discharge stops without I0
   Answer: B. When the denominator reaches zero, each avalanche produces at least one replacement electron at the cathode and the current is limited only by the circuit. (Key point: Breakdown criterion)

## Related in High Voltage Engineering

- [Townsend current growth with secondary ionization](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Atownsend-current-growth-secondary) — High Voltage Engineering, chapter 6, asked 2×
- [Townsend's primary ionization coefficient (0.4 cm to 0.1 cm)](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Atownsend-alpha-0-1cm) — High Voltage Engineering, chapter 6, asked 6×
- [Townsend's primary ionization coefficient (0.4 cm to 0.1 mm)](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Atownsend-alpha-0-1mm) — High Voltage Engineering, chapter 6, asked 6×
- [Solid breakdown due to actual working conditions](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Asolid-breakdown-working-conditions) — High Voltage Engineering, chapter 6
- [Townsend current growth with two coefficients](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Atownsend-two-coefficients) — High Voltage Engineering, chapter 6
- [Townsend's first and second coefficients from data](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Atownsend-first-second-coefficients) — High Voltage Engineering, chapter 6

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