A Unified Formula for Five Basic Forms of Discharge in an Electric Field Under Short Pulses L. Zhao Abstract— This study proposes a unified formula for five basic forms of discharge—gas, liquid, solid, and vacuum breakdown, and vacuum surface flashover—under short pulses in an electric field. This formula considers the effects of the number of dimensions and pulsewidth on the electric field. It is verified by using the results of experiments reported at the Aldermaston Weapon Research Establishment (AWRE) and the Northwest Institute of Nuclear Technology. The ranges of application of this formula to different discharge forms are also summarized. The proposed formula can be used to transform experimental data at a small scale under a known pulsewidth into those at a large scale under the application pulsewidth and is thus important for designing insulation. Index Terms— Gas, insulation design, liquid, pulsed power, solid, vacuum, vacuum surface. SEVERAL high-voltage (HV) accelerators that can generate short pulses at a high current have been constructed, such as the Z/ZR, Saturn, Magpie, Angara-5-1, PST, Yang, Flash-II, and Qiangguang-I. High-power microwave generators have also been manufactured to generate HV short pulses and include the Sinus- and Radan-series generators in Russia as well as the TPG-, CKP-, and CHP-series generators in China. The construction of these pulsed power generators has led to rapid developments in HV insulation technology under short pulses. Starting in 1960, Martin [1], [4] at the Aldermaston Weapon Research Establishment (AWRE) in U.K. made significant contributions to the field of pulsed power. He proposed several formulas for the breakdown of gases, liquids, and solids as well as fast-pulsed vacuum surface flashover in an electric field. The breakdown of a gas in a quasi-unified field can be given by Eg = 24.6 p + 6.7(p/g)^0.5 (1) where Eg is the electric field during gas breakdown (kV/cm), p is the gas pressure (atm), and g is the electrode gap (cm). Gas breakdown in a nonunified field [6] is given by Eg * te^(1/6) * g^(1/6) = kg * p^n (2) where te is the effective duration of the electric field (μs) and denotes the time for which the field exceeds 0.63 Eg. kg is a constant, and p ranges from 1 to 5 atm. The formula for liquid breakdown is El * te^(1/3) * Al^(1/10) = kl (3) where El is the electric field during the breakdown of liquid, te is in units of μs, Al is the effective area of the electrode in the liquid that corresponds to the area in which the field exceeds 0.9 El, and kl is a constant. Vacuum surface flashover is given by Evf * te^(1/6) * Avf^(1/10) = kvf (4) where Evf is the electric field during surface flashover, Avf is the surface area of the insulator, and kvf is a constant. Solid breakdown is given as follows: Es * Vs^(1/10) = ks (5) where Es is the electric field during solid breakdown, Vs is the volume of the solid dielectric that sustains a field exceeding 0.9 Es, and ks is a constant. Since 1966, a large number of experiments under nanosecond pulses have been conducted at the Tomsk Polytechnic University (TPU) in Russia. A formula for the value of Es of solid dielectrics was proposed [10]: lg Es = K1 - K2 lg d - (K3 - K4 lg d) lg τ (6) where d is thickness in mm, τ is pulsewidth in ns, and K1–K4 are constants. In 1996, the North Star Research Corporation (NSRC) published: Ev = kv * gv^(-0.3) (7) In 1999, Sandia National Laboratories (SNL) reported [12]: Evf * (te * Cb)^(1/10) * exp(-0.27/d) = 224 (8) Vitkovitsky [14] reported thresholds for radial low-inductance insulators based on Martin's formula: ES ≤ 5 × 10^5 * te^(-1/6) * A^(-1/8.5) ET ≤ 1.6 × 10^6 * te^(-1/6) * A^(-1/8) EP ≤ 5 × 10^6 (9) Research at the Institute of Electrical Engineering (IEE) reported: ρ * td = A1 * (Eg / ρ)^(-B1) (10) Northwest Institute of Nuclear Technology (NINT) proposed: Es = ks * ζ^(-1/b) (11) Es = kτ * τ^(-1/r), (τ ≤ tf_max) (12) Ev = 140 * gv^(-0.35) (13) Ev = kv * Av^(-1/6) (14) A. Review on the Discharge Mechanisms The basic process of gas, liquid, and solid breakdown involves primary electron injection (T-emission, T-F emission, or F-emission), acceleration, impact ionization, and electron avalanche formation. Vacuum breakdown involves primary electron emission from cathode, anode bombardment creating positive ions and metallic vapor, cathode bombardment, and vapor ionization. Vacuum surface flashover occurs via secondary electron emission avalanche (SEEA) along the insulator surface coupled with gas desorption and ionization. B. Similarities in Mechanism for Five Forms of Discharge All five forms require two criteria: (1) source of primary electrons/carriers, and (2) low-density region providing a sufficient mean free path for impact ionization, followed by a positive feedback mechanism. C. Simplified Model Assuming je electrons emitted per emission site with nm sites, and multiplication n times with factor δ: jn = nm * je * δ^n (15) Breakdown occurs when jn ≥ jc, defining critical field Ec (16). Ionization coefficient determines n = αe * l (17). A. Effects of Number of Dimensions and Time Breakdown probability follows two-parameter Weibull distribution: F(E) = 1 - exp(-(E / η)^b) (18) Time-dependent Weibull distribution: F(E, t) = 1 - exp(-c * t^a * E^b) (19) At fixed time tc: F(E)|t=tc = 1 - exp(-c * tc^a * E^b) (20) Dimensional scaling with N subsystems: 1 - F(E)|Ω2 = [1 - F(E)|Ω1]^N (22) E_Ω2 = E_Ω1 * N^(-1/b) (23) Time dependence with fixed dimensions: Et2 = Et1 * (t2 / t1)^(-a/b) (25) B. Transition From State (Ω1, t1) to (Ω2, t2) Combining spatial and temporal transitions yields: E_Ω2,t2 = (Ω1 / c)^(1/b) * Ω2^(-1/b) * t2^(-a/b) (30) Defining k = (Ω1 / c)^(1/b), β = b, and α = b/a gives the unified formula: E * t^(1/α) * Ω^(1/β) = k (31) Formulas supported by AWRE and NINT: - Gas breakdown (AWRE): Eg * te^(1/6) * g^(1/6) = kg,p (32), where α = 6, β = 6. - Liquid breakdown (AWRE): El * te^(1/3) * Al^(1/10) = kl (3), where α = 3, β = 10. - Solid breakdown (NINT): Es * te^(1/5) * ζ^(1/8) = ks (34), where α = 5, β = 8. - Vacuum breakdown by gap (NINT): Ev * te^(1/6) * gv^(1/3) = kvg (36), where α = 6, β = 3. - Vacuum breakdown by area (NINT): Ev * te^(1/6) * Av^(1/6) = kvA (37), where α = 6, β = 6. - Vacuum surface flashover (AWRE): Evf * te^(1/6) * Avf^(1/10) = kvf (4), where α = 6, β = 10. Application ranges: - Gas breakdown: non-unified fields, g up to 10 cm, te from 0.1 ns to 10 μs. - Liquid breakdown: te from 0.1 to 10 μs, Al from 0.1 to 10^5 cm^2. - Solid breakdown: te < 100 ns, V from 10^-6 to 10^4 cm^3, A up to 10^4 cm^2, d from nm to cm. - Vacuum surface flashover: single insulator te from 10 ns to 10 μs, Avf up to 40 cm^2; insulator stacks te from 30 ns to 1 μs, Avf up to 4 × 10^4 cm^2. - Vacuum breakdown: te from 1 ns to 10 μs, gv from 0.4 to 400 cm, Av from 2 × 10^3 to 7 × 10^5 cm^2. Comparisons with historical literature confirm that the differences in reported power exponents arise from limited experimental ranges or specific geometry approximations. Application to Engineering Design: Transformation from small-scale (Ω1, t1) to large-scale (Ω2, t2): E_Ω2,t2 = E_Ω1,t1 * (t1 / t2)^(1/α) * (Ω1 / Ω2)^(1/β) (46) Reliability R at operating field Eop: R(E) = exp(-(E / E63.2%)^β) (47) E63.2% = (ln 2)^(-1/β) * E50% (48) Practical application demonstrated for a 660-kV, 45-ns multifunctional HV vacuum insulator (profile λ nylon) separating 5-atm SF6 from vacuum in a coaxial line with backward-wave oscillator (BWO). Conclusions: 1) A unified formula E * t^(1/α) * Ω^(1/β) = k is established for gas, liquid, solid, vacuum breakdown, and vacuum surface flashover under short pulses. 2) Key parameter pairs (α, β) and dimension definitions Ω are identified: - Gas breakdown: (6, 6), Ω = g - Liquid breakdown: (3, 10), Ω = Al - Solid dielectric breakdown: (5, 8), Ω = d, As, or Vs - Vacuum surface flashover: (6, 10), Ω = Avf - Vacuum breakdown (gap): (6, 3), Ω = gv - Vacuum breakdown (area): (6, 6), Ω = Av 3) The formula successfully enables scaling small-scale test data to large-scale high-voltage insulation systems.