**Reflection and Refraction Coefficient Problem**

*The given data -*

Voltage = $ V^{'} $ = **3000 kV**

Protective Level = $ V_a $ = **1700 kV**

Surge Impedance of Line = $ Z_c $ = **300 Ω**

*To find -*

i] Line current before reaching the Arrester = $ I^{'} $ = ?

ii] Current through the Arrester = $ I_a $ = ?

iii] Value of the Arrester Resistance for this Condition = **R** = ?

iv] Reflect Voltage = $ V^n $ = ?

v] Verification of Reflection and Refraction Coefficients

*Formulae -*

$$ Line\ Current = I^{'} = \frac {V^{'}}{Z_c} $$

$$ Current\ through\ Arrester = I_a = \frac {2V^{'} - V_a}{Z_c} $$

$$ Resistance\ of\ Arrester = R = \frac {V_a}{I_a} $$

$$ Reflected\ Voltage = V^n = V_a - V^{'} $$

$$ Reflection\ Coefficient = \frac {V^n}{V{'}} = \frac {R - Z_c}{R + Z_c} $$

$$ Refraction\ Coefficient = \frac {V_a}{V{'}} = \frac {2R}{R + Z_c} $$

*Solution -*

**i] Line current before reaching the Arrester =** $ I^{'} $

$$ I^{'} = \frac {V^{'}}{Z_c} $$

$$ I^{'} = \frac {3000 \times 10^3}{300} = 10^4\ A $$

**II] Current through the Arrester =** $ I_a $

$$ Current\ through\ Arrester = I_a = \frac {2V^{'} - V_a}{Z_c} $$

$$ I_a = \frac {2 \times 3000 \times 10^3 - 1700 \times 10^3}{300} $$

$$ I_a = \frac {4300000}{300} $$

$$ I_a = 14333\ A $$

**iii] Value of the Arrester Resistance for this Condition = R**

$$ Resistance\ of\ Arrester = R = \frac {V_a}{I_a} $$

$$ R = \frac {1700 \times 10^3}{14333} = 118.61\ Ω $$

**iv] Reflect Voltage =** $ V^n $

$$ Reflected\ Voltage = V^n = V_a - V^{'} $$

$$ V^n = 1700 - 3000 = -1300\ kV $$

**v] Verification of Reflection and Refraction Coefficients**

$$ Reflection\ Coefficient = \frac {V^n}{V{'}} = \frac {R - Z_c}{R + Z_c} $$

$$ L.H.S. = \frac {V^n}{V{'}} = - \frac {1300}{3000} = - 0.4333 $$

$$ R.H.S. = \frac {R - Z_c}{R + Z_c} = \frac {118.61 - 300}{118.61 + 300} = -0.4333 $$

$$ L.H.S. = R.H.S.$$

Hence, *Coefficient of Reflection is Verified.*

$$ Refraction\ Coefficient = \frac {V_a}{V{'}} = \frac {2R}{R + Z_c} $$

$$ L.H.S. = \frac {V_a}{V{'}} = \frac {1700}{3000} = 0.567 $$

$$ R.H.S = \frac {2R}{R + Z_c} = \frac {2 \times 118.61}{118.61 + 300} = 0.567 $$

$$ L.H.S. = R.H.S.$$

Hence, *Coefficient of Refraction is Verified.*

*Answer -*

i] Line current before reaching the Arrester = $ I^{'} $ = $10^4\ A$

ii] Current through the Arrester = $ I_a $ = **14333 A**

iii] Value of the Arrester Resistance for this Condition = **R** = **118.61 Ω**

iv] Reflect Voltage = $ V^n $ = **- 1300 kV**

v] *Reflection Coefficient = ***- 0.4333** is verified.

*Refraction Coefficient = ***0.567** is verified.