An R L C circuit has resistance R=225 Ωand inductive reactance XL=175 Ω. (a) Calculate the circu

An R L C circuit has resistance R=225 Ωand inductive...

An $R L C$ circuit has resistance $R=225 \Omega$ and inductive reactance $X_{L}=175$ \Omega. (a) Calculate the circuit's capacitive reactance $X_{C}$ if its power factor is $\cos \phi=0.707 .$ Repeat the
calculation for $(b) \cos \phi=1.00$ and $(c) \cos \phi=1.00 \times 10^{-2}$

Key Concepts

Phase Angle

The phase angle is the measure of the time difference between the peaks of the voltage and current waveforms in an AC circuit, which is determined by the ratio of net reactance to resistance. This angle is crucial for calculating the power factor and understanding the balance between the energy stored in the magnetic field of the inductor and the electric field of the capacitor.

RLC Circuit

An RLC circuit is an electrical network comprising resistors (R), inductors (L), and capacitors (C). Its behavior under alternating current (AC) conditions is determined by the interplay between resistance, inductance, and capacitance, which together affect both the magnitude and phase angle of the circuit’s current and voltage.

Reactance

Reactance refers to the opposition offered by the inductors and capacitors in an AC circuit, which results in phase shifts between current and voltage. Inductive reactance (X_L) and capacitive reactance (X_C) are frequency-dependent and contribute to the imaginary part of the circuit's impedance.

Impedance

Impedance is the overall opposition that an AC circuit presents to current flow and is a complex quantity combining resistance (the real part) with net reactance (the imaginary part). It is expressed as Z = R + j(X_L - X_C) and plays a vital role in determining both current magnitude and phase relationships in the circuit.

Power Factor

The power factor is the cosine of the phase angle between the voltage and current in an AC circuit and represents the fraction of the electrical power that is used to do useful work. It indicates how effectively the circuit converts electrical power into work, with values closer to 1 meaning efficient power utilization.

Transcript

00:01 In this question, we have this rfc circuit with the following components.

00:05 So we want to find the inductive reactance, capacitive reactants, impedance, maximum current, phase angle between the currents and the voltage, and the individual maximum voltages across the resistor, inductor and capacitor.

00:20 So we just follow through the question part by part.

00:28 So for part a, we want to find the inductive reactance, xl is equal to omega -l, which is 2 pi f times l, which is 2 pi times 50 times 0 .25.

00:43 And the answer is 78 .5 oms.

00:58 Okay, then in part b, we want to find a capacitive reactance.

01:05 Xc is equal to 1 over omega -c, which is 1 over 2 -5 -f.

01:11 So we put in the numbers.

01:21 Using a calculator you get 1 .59 kilo oms.

01:30 And then in part c you want to find the impedance.

01:36 So impedance c is equal to r square plus xl minus xc square and square root.

01:45 So you put in the numbers 150 square plus 1590 minus 78 .550.

01:54 Square and this whole thing take the square root and the answer is 1 .52 kilooms.

02:11 Then in part d we want to find the phase of the maximum current.