R L C circuit such as that of Fig. 31-7 has R=5.00 Ω, C=20.0 μF, L=1.00 H, and ℰm=30.0 V .(a) At

step 1 Fundamentals of Physics, Chapter 31, Question 47. For this problem, we'll determine some charact

R L C circuit such as that of Fig. 31-7 has R=5.00 Ω, C=20.0...

Key Concepts

Half-Power Frequencies and Bandwidth

Half-power frequencies are the angular frequencies at which the power in the circuit falls to half of its maximum value, corresponding to a reduction in current amplitude to 1/?2 of its peak value. These frequencies define the bandwidth of the circuit, and the fractional half-width is a normalized measure of this bandwidth relative to the resonance frequency. This concept is critical for characterizing the selectivity and performance of resonant circuits.

Maximum Current Amplitude

The maximum current amplitude in a series RLC circuit is achieved at the resonance frequency, where the reactive components cancel out. With only the resistive element affecting the impedance, this maximum amplitude facilitates optimal power transfer and is a key concept in understanding how tuned circuits operate.

Series RLC Circuit

A series RLC circuit is composed of a resistor, an inductor, and a capacitor connected in series with an AC voltage source. It is a fundamental circuit in electronics that exhibits both oscillatory and damping behavior. Its analysis involves understanding how resistance, inductive reactance, and capacitive reactance interact to determine the overall impedance and response of the circuit.

Resonance in AC Circuits

Resonance in a series RLC circuit occurs when the inductive reactance equals the capacitive reactance, causing them to cancel each other out. At this specific frequency, known as the resonance frequency, the circuit's impedance is minimized (limited only by the resistance), resulting in a maximum current amplitude. This principle is crucial for designing circuits that selectively respond to or filter out specific frequencies.

Angular Frequency

Angular frequency, denoted by ? and measured in radians per second, is used to describe the rate of oscillation of the AC signal. It is a central parameter in the analysis of AC circuits as it defines the behavior of both reactances (LX and 1/?C) and is essential for determining resonance conditions and the frequency response of the circuit.

Transcript

00:01 Fundamentals of physics, chapter 31, question 47.

00:04 For this problem, we'll determine some characteristic values for a very simple lrc circuit.

00:09 The circuit itself is a serial loop with four components, the voltage source, a resistor, a capacitor, and an inductor, with values as shown.

00:17 This problem is broken into five parts.

00:20 For part a, we'll calculate the resonance frequency, which is where the effects of the inductor and capacitor balance each other out, allowing for the greatest possible current flow in the circuit.

00:30 The formula from the text is simply the resonance frequency, gamma sub d, and it's equal to 1 over the square root of l times c.

00:40 When we plug in values for l and c, l and c, be careful to set all your units to their base values.

00:47 Using microferods here instead of ferrads will result in an answer that's literally a thousand times too small.

00:54 Finally, we can actually calculate the value for part a, and that's 224 radiance per second, is the resonance frequency for this particular circuit.

01:07 For part b, we'll use the resonant frequency to find what the maximum current flow actually is.

01:14 We start simply enough with oms law, but because this is an ac circuit, we'll use resistivity instead of just resistance.

01:22 To calculate the resistivity, we actually need to account for all the components in the system.

01:29 And we use this formula to do that.

01:30 The resistivity is going to, here marked with a z, is going to be the square root of the resistance squared plus the resistivity of the inductor minus the resistivity of the capacitor quantity squared.

01:45 And those resistivities are the resistivity of the inductor is omega d l, and the resistivity of the capacitor is omega d, or one over omega d c.

01:59 So we plug those back in, and at this point we can we can plug some numbers in, and if you do that, you will get, depending on how much your calculator retained, you'll end up getting a number pretty close to five or maybe even exactly five.

02:16 The answer itself is actually exactly five.

02:19 And this is because this is an unusual situation.

02:21 Normally, there will be some resistance provided by the inductor and capacitor portions of the circuit.

02:30 But because we have an inductor that has an inductance of one henry, things cancel out.

02:39 For those feeling ambitious, you can actually substitute the original formula for the resonant frequency in this part of the calculation, cancel out the inductance values of one, and you'll find out that this whole term goes to zero.

02:56 So our final resistivity, is just 5 oms.

03:02 And if we plug that into oms law, we find our maximum current is 6 amps.

03:07 This situation is unusual, and in fact the next couple steps we'll see where the resistivity is definitely not 5 oms.

03:18 Which brings us to part c and d.

03:23 This part actually uses as looking for a maximum and minimum values of frequency where the current is half of the maximum current.

03:34 And it uses the same formulation right up until the very end.

03:38 So we're going to go ahead and solve them together.

03:43 So we're looking for the upper and lower frequencies where the current is one half of the maximum current.

03:49 So we know what that is, one half of the maximum current is from before.

03:55 And we're actually going to end up solving for z because that's where we need that, where we're going to find the frequencies that we ultimately need to solve for.

04:05 If we rearrange to solve for z, plug in the numbers, we end up with a total resistivity equal to 10 oms.

04:15 And now it's algebra time.

04:17 We're looking for two frequencies.

04:22 We have a single formula...

Please give Ace some feedback