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A voltage having a crest value of 3000 KV is travelling on the line of 750 KV. The protective level is 1700 kV and the surge impedance of the line is 300 ohm.

Subject: Power System Analysis

Topic: Power System Transients

Difficulty: High

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## 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.

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.