Subelement B: Electrical Math— Topic 17: Impedance Networks-2
Question 3-17B4
Element 3 (GROL)In polar coordinates, what is the impedance of a network composed of a 300-ohm-reactance capacitor, a 600-ohm-reactance inductor, and a 400-ohm resistor, all connected in series?
Explanation
In a series RLC circuit, the total reactance (X) is the difference between the inductive reactance (XL) and the capacitive reactance (XC). Here, X = XL - XC = 600Ω - 300Ω = 300Ω. Since XL is greater than XC, the circuit is inductively reactive.
The total impedance (Z) is the vector sum of the resistance (R) and the total reactance (X), calculated using the Pythagorean theorem:
Z = √(R² + X²) = √(400² + 300²) = √(160000 + 90000) = √250000 = 500Ω.
The phase angle (θ) represents the phase difference between the voltage and current. It's calculated using the arctangent of the ratio of total reactance to resistance:
θ = arctan(X/R) = arctan(300/400) = arctan(0.75) ≈ 36.87°, which rounds to 37 degrees. Since the total reactance is positive (inductive), the angle is positive.
Thus, the impedance in polar coordinates is 500 ohms with a phase angle of 37 degrees, commonly written as 500 ohms, /37 degrees.
Related Questions
3-17B2 In polar coordinates, what is the impedance of a network composed of a 100-ohm-reactance inductor in series with a 100-ohm resistor?3-17B3 In polar coordinates, what is the impedance of a network composed of a 400-ohm-reactance capacitor in series with a 300-ohm resistor?3-17B5 In polar coordinates, what is the impedance of a network comprised of a 400-ohm-reactance inductor in parallel with a 300-ohm resistor?3-17B6 Using the polar coordinate system, what visual representation would you get of a voltage in a sinewave circuit?3-18B1 What is the magnitude of the impedance of a series AC circuit having a resistance of 6 ohms, an inductive reactance of 17 ohms, and zero capacitive reactance?