A single-page reference of the equations and constants that recur throughout pulser engineering.
Ohm's law. V = I × R
Power. P = V × I
Capacitor energy. E = ½ × C × V²
Inductor energy. E = ½ × L × I²
Capacitor current. I = C × dV/dt
Inductor voltage. V = L × dI/dt
RC time constant. τ = R × C
LR time constant. Ï" = L / R
Rise time, bandwidth. t_rise ≈ 0.35 / BW (for single-pole Gaussian-like response)
Combined rise time. t_observed = sqrt(t_signal² + t_scope²) (for measurement chain analysis)
Capacitive load power. P = C × V² × F (for periodic pulses into capacitive loads)
Droop on flat top. ΔV = (I × Δt) / C (for capacitive energy store)
Compliance voltage required for inductive load. V = L × dI/dt + R × I
Characteristic impedance. Z = sqrt(L / C) where L and C are per-unit-length
Propagation velocity. v = 1 / sqrt(L × C) ≈ 0.66c for typical coaxial cable with PE dielectric
Round-trip delay. t_RT = 2 × length / v
Reflection coefficient at load. Γ = (Z_load − Z_cable) / (Z_load + Z_cable)
| Cable | Impedance | Velocity | Capacitance per ft |
|---|---|---|---|
| RG-58 | 50 Ω | 0.66c | 28.5 pF |
| RG-59 | 75 Ω | 0.66c | 21 pF |
| RG-62 | 93 Ω | 0.84c | 13.5 pF |
| RG-11 | 75 Ω | 0.66c | 17 pF |
| RG-213 | 50 Ω | 0.66c | 30.8 pF |
| Technology | Voltage class | Rise time | Repetition rate |
|---|---|---|---|
| Silicon MOSFET | 30 V to 1.7 kV | 10 to 100 ns | to MHz |
| IGBT | 600 V to 6.5 kV | 100 ns to 1 µs | to 100 kHz |
| SiC MOSFET | 600 V to 3.3 kV | 5 to 30 ns | to MHz |
| GaN HEMT | 100 V to 900 V | sub-1 ns | to 10 MHz |
| Thyratron | 5 to 50 kV | 10 to 100 ns | to 10 kHz |
| Spark gap | 10 kV to MV | 1 to 100 ns | to 1 kHz |
| Pseudospark | 10 to 50 kV | 5 to 50 ns | to 10 kHz |
| DSRD | 1 to 30 kV | sub-1 ns | to 100 kHz |
| PCSS (GaAs) | 1 to 100 kV | sub-100 ps | low (lifetime limited) |
| Medium | Breakdown field |
|---|---|
| Air at sea level | 30 kV/cm |
| SF6 at 1 atm | 80 kV/cm |
| Mineral oil | 200 kV/cm |
| Deionized water (microsecond pulse) | 250 kV/cm |
| Polyethylene | 200 kV/cm |
| Mylar | 280 kV/cm |
| Glass | 100 to 400 kV/cm |
| Polyimide (Kapton) | 250 kV/cm |
For uniform-field gaps in air, breakdown voltage as a function of pressure-distance product (p × d):
| p × d (Torr·cm) | Approximate breakdown voltage |
|---|---|
| 1 (Paschen minimum) | 330 V |
| 10 | 750 V |
| 100 | 4 kV |
| 760 (1 atm × 1 cm) | 30 kV |
| 7600 (1 atm × 10 cm) | 250 kV |
The Paschen curve has a minimum near p × d = 1 Torr·cm. Below the minimum, breakdown voltage rises again because the gas density is too low for collision ionization.
| Convention | Definition | Industries |
|---|---|---|
| Rise time 10%-90% | Time from 10% to 90% of amplitude | Most engineering, default in this book |
| Rise time 20%-80% | Time from 20% to 80% of amplitude | Some EMC standards |
| Rise time 0%-90% | Time from baseline to 90% of amplitude | Less common |
| Pulse width FWHM | Width at 50% of peak amplitude | Laser, medical |
| Pulse width 90%-90% | Width between 90% points on rising and falling edges | Some EMC and pulsed-power |
When in doubt, document which convention you are using. Disagreements between scope vendor defaults and instrument data sheets are a common source of measurement confusion.
| Constant | Value |
|---|---|
| Speed of light, vacuum | 3 × 10^8 m/s |
| Speed of light in PE-dielectric coax | 2 × 10^8 m/s |
| Permittivity of vacuum, ε₀ | 8.85 × 10^−12 F/m |
| Permeability of vacuum, µ₀ | 1.26 × 10^−6 H/m |
| Boltzmann constant, k | 1.38 × 10^−23 J/K |
| Charge of electron, e | 1.60 × 10^−19 C |
End of Appendix B.