--- title: "Limitations of Townsend theory; Paschen's law" subject: "High Voltage Engineering" chapter: 6 key: "semester_8/high_voltage_engineering:townsend-limitations-paschen" source: https://www.bctengineer.com/q/semester_8/high_voltage_engineering/townsend-limitations-paschen site: BCT Engineering --- # Limitations of Townsend theory; Paschen's law High Voltage Engineering · Chapter 6: Dielectric Breakdown - Asked in: 2077 Chaitra, 8 marks - Video: [DERIVATION OF PASCHEN'S LAW|PASCHEN'S LAW OF GASEOUS DIELECTRIC BREAKDOWN|PASCHEN'S THEORY| HVE Lect](https://www.youtube.com/watch?v=Aq5sNCq0SIw) — Simplified EEE Studies - 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-limitations-paschen - Whole chapter: https://www.bctengineer.com/md/semester_8/high_voltage_engineering/chapter-6 ## Question What are the limitations of Townsend's theory? Derive the relation given by Paschen law. ## Short answer Townsend's theory fails at high pd: it predicts large time lags, cathode independence, diffuse glow, and neglects space charge, not explaining streamers. Paschen's law V_b = B pd / ln[Apd / ln(1 + 1/γ)] is derived by substituting α = Ap e^{-Bpd/Vb} into αd = ln(1+1/γ), giving breakdown voltage as a function of pd only. ## Answer #### Limitations of Townsend's theory Townsend's theory (avalanche growth by $\alpha$, sustained by cathode secondary emission $\gamma$, breakdown when $\gamma e^{\alpha d} = 1$) explains breakdown well at low $pd$, but fails in several respects at higher $pd$ (above about 1000 torr·cm, e.g. long gaps in atmospheric air): 1. **Time lag:** Townsend predicts formative times of the order of $10^{-5}$ s (time for ions to cross the gap many times). Observed times in long gaps are about $10^{-8}$–$10^{-7}$ s, far too short for ion transit. 2. **Cathode independence:** At high pressures breakdown voltage is nearly independent of cathode material, which should not happen if $\gamma$ at the cathode controls breakdown. 3. **Channel shape:** Townsend predicts a diffuse glow-type discharge, but actual sparks at high $pd$ are thin, bright, branched and zig-zag channels. 4. **Space charge ignored:** The theory neglects the distortion of the field by the space charge of the avalanche, which becomes significant when the avalanche contains about $10^8$ electrons. 5. **Non-uniform fields:** It does not explain corona and breakdown in non-uniform fields, or polarity effects. 6. **Breakdown with no cathode involvement:** Discharges can start mid-gap and grow towards both electrodes, which Townsend's mechanism cannot explain. These limitations led to the **streamer (Raether–Meek) theory** for high $pd$. #### Derivation of Paschen's law Townsend's breakdown criterion: $$ \gamma\left(e^{\alpha d} - 1\right) = 1 \;\Rightarrow\; \alpha d = \ln\left(1 + \frac{1}{\gamma}\right) \qquad (1) $$ Experimentally, the first coefficient depends on $E/p$: $$ \frac{\alpha}{p} = A\, e^{-Bp/E} \qquad (2) $$ where $A$ and $B$ are gas constants. In a uniform field $E = V/d$, so at breakdown $E = V_b/d$: $$ \alpha = Ap\, e^{-Bpd/V_b} \qquad (3) $$ Substitute (3) in (1): $$ \begin{aligned} Apd\, e^{-Bpd/V_b} &= \ln\left(1 + \frac{1}{\gamma}\right) = K \\ e^{-Bpd/V_b} &= \frac{K}{Apd} \\ \frac{Bpd}{V_b} &= \ln\frac{Apd}{K} \\ V_b &= \frac{B\,pd}{\ln (A\,pd) - \ln K} \end{aligned} $$ $$ \boxed{V_b = \frac{B\,(pd)}{\ln\left[\dfrac{A\,(pd)}{\ln(1 + 1/\gamma)}\right]} = f(pd)} $$ Since $A$, $B$ and $\gamma$ are constants of the gas (and electrode), the breakdown voltage of a uniform-field gap is a function of the product $pd$ only. This is **Paschen's law**. For air at temperature other than 20 °C, $p$ is replaced by gas density ($Nd$ or $\delta d$). #### Minimum sparking voltage Setting $\dfrac{dV_b}{d(pd)} = 0$: $$ (pd)_{min} = \frac{e}{A}\ln\left(1 + \frac{1}{\gamma}\right), \qquad V_{b,min} = \frac{eB}{A}\ln\left(1 + \frac{1}{\gamma}\right) $$ ($e = 2.718$.) For air, $V_{b,min} \approx 327$ V at $pd \approx 0.57$ torr·cm. ```text Vb |\ / | \ / | \ / | \_______ / | min ~327 V (air) +---------------------------> pd ``` Left of the minimum, too few collisions; right of the minimum, too little energy per collision; in both cases $V_b$ rises. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why does Townsend's theory fail for long gaps at high pd? A. It predicts too short a time lag B. Predicted formation time (~10^-5 s) is far longer than the observed ~10^-8 s C. It predicts breakdown at too low a voltage in all gases D. It ignores the ionization coefficient alpha Answer: B. Ion transit across the gap takes too long to explain the observed fast sparks. (Key point: Time lag) 2. Which observation contradicts Townsend's mechanism at high pd? A. Breakdown voltage depends strongly on cathode material B. The spark is a diffuse glow C. Current grows as e^(alpha d) D. Breakdown voltage is nearly independent of cathode material Answer: D. If cathode emission (gamma) controlled breakdown, the cathode material should matter. (Key point: Cathode dependence) 3. Which two relations are combined to derive Paschen's law? A. gamma(e^(alpha d) - 1) = 1 and alpha/p = A e^(-Bp/E) with E = V/d B. I = I0 e^(alpha d) and Ohm's law C. E_r = E and alpha x = 17.7 + ln x D. W = sigma E^2 and the heat balance equation Answer: A. Substituting alpha from the empirical E/p relation into the breakdown criterion gives Vb as a function of pd. (Key point: Paschen derivation) 4. What does Paschen's law state? A. Breakdown voltage is proportional to d only B. Breakdown voltage falls steadily as pd increases C. Breakdown voltage of a uniform gap is a function of the product pd D. Breakdown voltage depends on p and d separately Answer: C. Since A, B and gamma are constants, Vb = f(pd). 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