Subelement B: Electrical Math— Topic 17: Impedance Networks-2
Question 3-17B3
Element 3 (GROL)In polar coordinates, what is the impedance of a network composed of a 400-ohm-reactance capacitor in series with a 300-ohm resistor?
Explanation
The impedance of a series RC circuit is a complex quantity, combining resistance and reactance. Resistance (R) is the real component, while capacitive reactance (Xc) is the imaginary component, always negative for capacitors.
1. **Rectangular Impedance:** For a series circuit, impedance (Z) is R - jXc.
Z = 300 ohms - j400 ohms.
2. **Magnitude:** The magnitude of impedance (|Z|) is calculated using the Pythagorean theorem:
|Z| = sqrt(R^2 + Xc^2)
|Z| = sqrt(300^2 + (-400)^2)
|Z| = sqrt(90,000 + 160,000) = sqrt(250,000) = 500 ohms.
3. **Phase Angle:** The phase angle (theta) is found using the arctangent function:
theta = arctan(Xc / R)
theta = arctan(-400 / 300) = arctan(-1.333...)
theta = -53.13 degrees (approximately). The negative sign indicates a capacitive circuit where current leads voltage.
Therefore, the impedance in polar coordinates is 500 ohms, /-53.1 degrees. This matches option C. Options A and B have incorrect magnitudes. Option D has the correct magnitude but an incorrect positive angle, which would represent an inductive circuit.
Related Questions
3-17B1 What is the impedance of a network composed of a 100-picofarad capacitor in parallel with a 4000-ohm resistor, at 500 KHz? Specify your answer in polar coordinates.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?3-17B4 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?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?