An RLC series circuit consists of a 50-Ωresistor, a 200-μF capacitor, and a 120-mH inductor whos

An RLC series circuit consists of a 50-Ωresistor, a 200-μF...

An RLC series circuit consists of a $50-\Omega$ resistor, a $200-\mu \mathrm{F}$ capacitor, and a $120-\mathrm{mH}$ inductor whose coil has a resistance of $20 \Omega$. The source for the circuit has an ms emf of $240 \mathrm{V}$ at a frequency of $60 \mathrm{Hz}$. Calculate the ms voltages across the (a) resistor, (b) capacitor, and (c) inductor.

Key Concepts

Voltage Division in AC Circuits

Voltage division in AC circuits is the method of determining the individual voltage drops across components in a series configuration. By applying the ratio of each element's impedance to the total circuit impedance, one can calculate the voltage across resistors, capacitors, and inductors even when they have different phase relationships.

Series Circuits

In a series circuit, all elements share the same current since they are connected end-to-end. The total voltage from the source is divided among the individual components based on their impedances, making it possible to determine the voltage across each element using the concept of voltage division.

Phasor Representation

Phasor representation converts time-varying sinusoidal functions into complex numbers, which simplifies the addition of voltages and currents in AC circuits. By representing magnitudes and phase relationships as complex vectors, it enables straightforward analysis of circuits with multiple reactive components.

Reactance

Reactance is the component of impedance that arises from capacitors and inductors. It is frequency-dependent, with inductive reactance increasing with frequency and capacitive reactance decreasing with frequency. Reactance determines how much a capacitor or inductor opposes the flow of AC current, affecting voltage drops across these elements.

Impedance

Impedance is the measure of a circuit element's opposition to AC current, combining resistance (the real part) and reactance (the imaginary part). It plays a crucial role in determining the circuit’s overall behavior, influencing both the amplitude and phase of voltages and currents.

AC Circuit Analysis

AC circuit analysis is the study of circuits driven by alternating current sources, where voltages and currents vary sinusoidally over time. It involves the use of phasor techniques and complex numbers to simplify the analysis of circuits containing resistors, inductors, and capacitors, and to understand how these components interact with the time-varying source.

Transcript

00:01 We are given that this rlc circuit has a resistance of 50 oms, a capacitance of 200 times 10 to the negative 6th power.

00:19 That's micro -farids.

00:22 An inductance of 0 .1 -20 henry.

00:34 And the resistance of the coil, r sub l.

00:46 I'm not sure i like that, but anyway, resistance of the coil is 20.

00:57 The rms voltage is 240.

01:06 The frequency is 60 hertz.

01:10 Well, that means that the angular frequency is 2 pi times 60, so that would be 120 pi radiance per second.

01:26 So initially we need to calculate the rms voltage across the resistor.

01:31 The equivalent resistance across both resistors is 70 oms, 50 plus 20.

01:45 So to solve a, first i'm going to write down that z is the square root of re, q squared plus x sub l minus x sub c but x sub l is omega l x x sub c is one over omega c squared um i rms is v rms over z and then that will give me the rms current.

02:29 So let me put this in the calculator, the square root of the equivalent resistance squared, which is 70, plus omega -l 120 pi 1 -2, minus 1 over omega -c, 120, high, 200 times 10 to the negative 6 power.

03:10 Don't forget to square it.

03:14 That gives me 77 oms.

03:24 So i take my voltage divided by that, 240, divided by the 77, 7, 3 .1 amps.

03:46 Let's go back to the question.

03:51 The rms voltage across the resistor.

03:56 So voltage across the resistor is going to be the current times the resistance, which is 50 oms.

04:14 So i'll take the current times, and that gives me 156 to two significant digits...

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