An ac source of voltage amplitude 100 V and frequency 1.0 kHz drives an R L C series circuit wit

An ac source of voltage amplitude 100 V and frequency 1.0...

An ac source of voltage amplitude $100 \mathrm{V}$ and frequency $1.0 \mathrm{kHz}$ drives an $R L C$ series circuit with $R=20 \Omega, L=4.0 \mathrm{mH},$ and $C=50 \mu \mathrm{F} .$ (a) Determine the rms current through the circuit. (b) What are the ms voltages across the three elements? (c) What is the phase angle between the emf and the current? (d) What is the power output of the source? (e) What is the power dissipated in the resistor?

Key Concepts

Power in AC Circuits

Calculating power in AC circuits involves distinguishing between real power, which is dissipated as heat in resistors, and reactive power, which is stored and then returned by capacitors and inductors. The power dissipated in the circuit, especially in resistive elements, is determined using the RMS values of current and voltage along with the power factor. This concept is essential for designing circuits that are both efficient and safe by ensuring that they operate within their power ratings.

RMS Values

Root mean square (RMS) values provide a measure of the effective voltage or current in AC circuits, equivalent to the value of a constant DC voltage or current that would deliver the same power to a resistor. RMS calculations are crucial for practical circuit analyses and power calculations, ensuring that AC measurements accurately reflect the energy-delivering capability of the circuit.

AC Circuit Analysis

AC circuit analysis involves studying circuits in which the voltages and currents vary sinusoidally with time. This analysis includes understanding how alternating currents are applied to circuits, how elements such as resistors, capacitors, and inductors respond to sinusoidal inputs, and how these components interact when combined. It is essential for determining circuit behaviors like current distribution, voltage drops, and phase relationships in systems powered by alternating current sources.

Impedance in RLC Series Circuits

In a series RLC circuit, the impedance is the overall opposition to the alternating current and is composed of a resistive part, and frequency-dependent inductive and capacitive parts. The total impedance is calculated as the square root of the sum of the squares of the resistance and the net reactance (which is the difference between the inductive and capacitive reactances). This concept is critical as it determines the magnitude of the current and influences the voltage distribution across each element.

Phase Angle and Power Factor

The phase angle in an AC circuit is the angle by which the current either leads or lags the voltage, resulting from the presence of reactive components (inductors and capacitors). This phase difference directly affects the power factor, which is the cosine of the phase angle and indicates the fraction of the power that is used effectively in the circuit. Understanding phase angle is vital for analyzing the performance and efficiency of AC circuits.

Transcript

00:01 Okay, we're given that v -0 is 100 volts, and we're told that the frequency is 1 ,000 hertz, which means that the angular frequency is 2 pi times that, which would be 2 ,000 pi radiance per second.

00:30 Resistance is 20 oms.

00:37 Impedence is 0 .0040 henry.

00:44 Capacitance is 50 times 10 to the negative 6 power ferrid.

00:53 We need to figure out i sub rms.

00:59 So first i'm going to figure out i sub 0.

01:02 And before i do that, i'm going to figure out z.

01:05 So z is going to be the square root of r squared plus xl minus xc squared.

01:17 Well, that's going to be omega -l minus 1 over omega -c squared.

01:25 So i sub -0 is just going to be v -0 over z, and i sub -r -m -m -s.

01:37 Is just going to be 1 over the square root of 2 times i .0.

01:50 And so, irms is 1 over the square root of 2 times v .0 over, just putting this all together, the square root of r squared plus omega l minus 1 over omega c squared.

02:11 I'll put that all in a calculator.

02:18 1 over square root of 2, v .0, 100 over the square root of r, r is 20, plus omega l.

02:43 Omega is 2 ,000 pi, l is 0 .004, minus 1 over omega over omega which is 2000 pi, c, 50 times 10 to the negative 6th power.

03:13 And then don't forget to square it, 2 .38.

03:19 See, we only have two significant digits, though, so it's 2 .4 amps.

03:34 Question number.

03:36 Okay.

03:46 Okay.

03:48 Okay.

03:51 Vrms across each.

03:53 Of the three elements.

03:58 So this is just going to be irms times, i guess let's go with the capacitor first, x sub c.