In an RLC circuit, if the resistance is 10Ω, the inductance is 0.1H, and the capacitance is 20μF, what is the total impedance at 50Hz? (2022)

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Q: In an RLC circuit, if the resistance is 10Ω, the inductance is 0.1H, and the capacitance is 20μF, what is the total impedance at 50Hz? (2022)

Solution: Calculate XL = 2πfL = 31.42Ω and XC = 1/(2πfC) = 159.15Ω. Z = √(R² + (XL - XC)²) = √(10² + (31.42 - 159.15)²) = 15.81Ω.

Steps: 13

Step 1: Identify the given values: Resistance (R) = 10Ω, Inductance (L) = 0.1H, Capacitance (C) = 20μF, Frequency (f) = 50Hz.
Step 2: Convert capacitance from microfarads to farads: 20μF = 20 x 10^-6 F = 0.00002 F.
Step 3: Calculate the inductive reactance (XL) using the formula: XL = 2πfL.
Step 4: Substitute the values into the formula: XL = 2π(50)(0.1) = 31.42Ω.
Step 5: Calculate the capacitive reactance (XC) using the formula: XC = 1/(2πfC).
Step 6: Substitute the values into the formula: XC = 1/(2π(50)(0.00002)) = 159.15Ω.
Step 7: Calculate the total impedance (Z) using the formula: Z = √(R² + (XL - XC)²).
Step 8: Substitute the values into the formula: Z = √(10² + (31.42 - 159.15)²).
Step 9: Calculate the difference: XL - XC = 31.42 - 159.15 = -127.73.
Step 10: Square the difference: (-127.73)² = 16319.52.
Step 11: Calculate R²: 10² = 100.
Step 12: Add R² and the squared difference: 100 + 16319.52 = 16419.52.
Step 13: Take the square root to find Z: Z = √(16419.52) = 128.14Ω.