Resistance switching numerical (220 kV, 600 Ω)

Resistance switching means connecting a resistor across the CB contacts (or arc) so that the restriking voltage transient is damped, its frequency is lowered and RRRV is reduced.

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 = 8 Ω, C = 0.025 μF, R = 600 Ω, f = 50 Hz.

L = 8 / 314.16 = 25.46 mH

(i) Natural frequency

1/LC = 1/(25.46×10⁻³ × 0.025×10⁻⁶) = 1.5708×10⁹
f_n  = √(1.5708×10⁹) / 2π
     = 6308 Hz ≈ 6.31 kHz

(ii) Damped frequency

1/(4R²C²) = 1/(4 × 600² × (0.025×10⁻⁶)²) = 1.1111×10⁹
f_d = √(1.5708×10⁹ − 1.1111×10⁹) / 2π
    = 3412 Hz ≈ 3.41 kHz

(iii) Critical resistance

R_c = ½ √(25.46×10⁻³ / 0.025×10⁻⁶)
    = ½ × 1009.25 = 504.6 Ω

(iv) Resistance for f_d = f_n/4

R = R_c / √(1 − 1/16) = 504.6 / 0.9682
  = 521.2 Ω
Check: f_d = 1577 Hz = 6308/4 ✓

Answer: (i) f_n = 6.31 kHz; (ii) f_d = 3.41 kHz; (iii) R_c = 504.6 Ω; (iv) R = 521.2 Ω.