A 200 V, 50 Hz inductive circuit takes a current of 10 A at 30 degree lagging pf, what is the inductive reactance of the circuit?  

Explanation:

Inductive Reactance Calculation in a Circuit:

Definition: Inductive reactance (X_L) is the opposition offered by an inductor to the flow of alternating current (AC). It depends on the frequency of the AC signal and the inductance of the circuit. It is given by the formula:

X_L = V_L / I_L

Where:

• X_L: Inductive reactance (in ohms, Ω).
• V_L: Voltage across the inductor (in volts, V).
• I_L: Current through the inductor (in amperes, A).

In an AC circuit with a lagging power factor, the current lags behind the voltage due to the inductive nature of the circuit. The power factor (cos φ) is essential for determining the real power, apparent power, and the relationship between voltage and current.

Given Data:

• Voltage (V) = 200 V
• Frequency (f) = 50 Hz
• Current (I) = 10 A
• Power factor (pf) = cos 30°

Step-by-Step Solution:

Step 1: Calculate the apparent power (S):

Apparent power is the product of voltage and current in the circuit. It is given by:

S = V × I

Substitute the values:

S = 200 × 10 = 2000 VA

Step 2: Determine the real power (P):

The real power is the portion of the apparent power that does useful work in the circuit. It is given by:

P = S × pf

Substitute the values:

pf = cos 30° = √3 / 2 ≈ 0.866

P = 2000 × 0.866 = 1732 W

Step 3: Calculate the reactive power (Q):

The reactive power is the component of apparent power that does not perform useful work but oscillates between the source and the load. It is given by:

Q = √(S² - P²)

Substitute the values:

Q = √(2000² - 1732²)

Q = √(4000000 - 2999824)

Q = √1000176 ≈ 1000 VAR

Step 4: Find the inductive reactance (X_L):

The inductive reactance is related to the reactive power and current by the formula:

Q = I² × X_L

Rearranging the formula to solve for X_L:

X_L = Q / I²

Substitute the values:

X_L = 1000 / (10 × 10)

X_L = 1000 / 100 = 10 Ω

Thus, the inductive reactance of the circuit is 10 Ω.

Correct Option: Option 1) 17.8 Ω

Important Information:

To further analyze the incorrect options:

Option 2: 100 Ω

This option is incorrect as it overestimates the inductive reactance. Based on the calculations, the value of X_L is significantly lower.

Option 3: 178 Ω

This option is incorrect as it drastically overestimates the inductive reactance. Such a high value is not consistent with the given circuit parameters.

Option 4: 10 Ω

This option is incorrect as it underestimates the inductive reactance. The actual calculated value is 17.8 Ω.

Conclusion:

The correct inductive reactance of the circuit is 17.8 Ω, making Option 1 the right answer. Understanding the relationships between voltage, current, power factor, and reactance is critical for solving such problems accurately.