Resistance switching numerical (132 kV, 350 Ω)

A resistance across the CB contacts damps the restriking voltage oscillation; if it is at or below the critical value, the voltage rises without oscillation.

Formulas (resistance R across the contacts, parallel R–L–C circuit)

Natural frequency  f_n = 1 / (2π √(LC))
Damped frequency   f_d = (1/2π) √(1/LC − 1/(4R²C²))
Critical resistance R_c = ½ √(L/C)
   (R ≤ R_c gives no oscillation)
For f_d = f_n / k:
   1/(4R²C²) = (1/LC)(1 − 1/k²)
   R = R_c / √(1 − 1/k²)

Inductance from reactance: L = X/(2π × 50).

Data

X = 5 Ω, C = 0.035 μF, R = 350 Ω, f = 50 Hz.

L = 5 / 314.16 = 15.915 mH

(a) Natural frequency

1/LC = 1/(15.915×10⁻³ × 0.035×10⁻⁶) = 1.7952×10⁹
f_n  = √(1.7952×10⁹) / 2π
     = 6743 Hz ≈ 6.74 kHz

(b) Damped frequency

1/(4R²C²) = 1/(4 × 350² × (0.035×10⁻⁶)²) = 1.6660×10⁹
f_d = √(1.7952×10⁹ − 1.6660×10⁹) / 2π
    = 1809 Hz ≈ 1.81 kHz

(c) Critical resistance

R_c = ½ √(15.915×10⁻³ / 0.035×10⁻⁶)
    = ½ × 674.34 = 337.2 Ω

Since 350 Ω > 337.2 Ω, the given resistor still allows a (heavily damped) oscillation, as found in (b).

(d) Resistance for f_d = f_n/3

R = R_c / √(1 − 1/9) = 337.2 / 0.9428
  = 357.6 Ω
Check: f_d = 2248 Hz = 6743/3 ✓

Answer: (a) f_n = 6.74 kHz; (b) f_d = 1.81 kHz; (c) R_c = 337.2 Ω; (d) R = 357.6 Ω.