A capacitor, a coil, and two resistors of equal resistance are arranged in an AC circuit as show

step 1 I'm thinking that I better draw this figure. Figure P 33 .52. So we've got a voltage source, a c

A capacitor, a coil, and two resistors of equal resistance...

Key Concepts

AC Circuit Analysis

AC circuit analysis involves studying circuits where the current and voltage vary sinusoidally over time. This analysis uses concepts such as impedance and phase relationships between voltage and current to understand the behavior of various circuit elements under alternating current conditions.

Impedance

Impedance is the total opposition a circuit offers to the flow of alternating current and is a complex quantity that combines both resistance (the real part) and reactance (the imaginary part). It plays a key role in AC circuits, replacing resistance from DC analysis to account for frequency-dependent effects of capacitors and inductors.

Root Mean Square (RMS) Values

The RMS value represents the effective value of an alternating current or voltage; it is the equivalent constant value that would deliver the same power to a load as a corresponding DC value. RMS measurements are crucial in analyzing power in AC circuits.

Capacitive Reactance

Capacitive reactance is the opposition that a capacitor presents to a change in voltage in an AC circuit. It depends on the frequency of the AC source and the capacitance, decreasing as the frequency increases, and is an essential factor in determining the overall impedance in a circuit containing capacitors.

Inductive Reactance

Inductive reactance is the opposition that an inductor presents to changes in current in an AC circuit. It is directly proportional to both the inductance and the frequency, increasing as either of these increases. This reactance contributes to the phase difference between current and voltage in circuits with inductors.

Phasor Representation

Phasor representation facilitates the analysis of AC circuits by transforming sinusoidally varying voltages and currents into rotating vectors (phasors) in the complex plane. This simplifies the handling of phase differences and the algebra of addition and subtraction of sinusoidal functions.

Transcript

00:01 I'm thinking that i better draw this figure.

00:03 Figure p 33 .52.

00:09 So we've got a voltage source, a capacitor, got a resistor over here, got an inductor down here.

00:28 Good one.

00:33 We're going back to the voltage source.

00:36 It's telling us that delta v is 20 volts rms.

00:49 Told that the frequency is 60 hertz.

00:56 We're not given a, it's not in the drawing.

00:59 We're not given a capacitance.

01:02 And there's a switch that i'm not going to draw.

01:05 So we're told, it's arranged like that, when the switch is open, which means it's just this drawing, the current is 183m.

01:27 So let's figure out z is the square root of r squared plus omega -l minus 1 over omega -c squared.

01:57 Okay, delta vs over z is the current because we're talking about rms here.

02:44 So that is 0 .183 amps.

02:55 Okay.

02:57 But now it's telling us when the switch is closed in position 1 that the current changes.

03:10 Well, when it's closed in position 1, wait, position 1? oh, okay.

03:23 I didn't read it correctly.

03:24 When the switch is closed in position 1, the current is, and when it's closed in position 2, the current is something.

03:32 Now i get it.

03:33 I was confused.

03:35 Okay.

03:37 When it's closed in position 1, we basically have a resistor here, and then it connects right there.

03:54 So that's what it's going to look like when it's closed in position 1.

04:02 So as a matter of fact, i'm going to draw this in red, and then i'm going to do my work.

04:12 In red.

04:19 Well, 1 over req, equivalent resistance for these two resistors, is going to be 1 over r plus 1 over r.

04:29 So 1 over r plus another 1 over r is 2 over r.

04:35 And so, r eq, am i right here? one r, one of the 1 over r's plus another of the 1 over r is 2 is going to be 2.

04:50 Of the 1 over rs.

04:53 Okay, that's true.

04:55 So, req is going to be one -half of r.

05:04 So, the current, which is now looking for it, looking for it, 298 mll amps is now going to be delta vs over the square root of r squared, but now all is one half r plus omega l minus one over omega c squared.

05:54 Alright, i do want to write, oops, i want this in black.

06:07 So i have this equation and i have this equation.

06:14 All right, i'll circle the black.

06:18 Now there's a third instance that i'm going to put in blue.

06:22 So the third instance, let's go ahead and erase this.

06:31 See how much i can get here.

06:35 Oh, nice.

06:38 Nice.

06:39 Okay, so i'm going to put this in blue.

06:43 Let's look at the diagram.

06:45 In the third instance, goes, i hate it when that happens.

06:58 In the third instance, it goes right through here.

07:07 So it's just bypassing the inductor.

07:15 So i'm going to make sure that i'm correct.

07:17 It's just bypassing the inductor.

07:19 You'd have the same voltage on both sides of the, inductor, so we're really just skipping the inductor.

07:28 So, the new current becomes 137 millie amps, which is delta v source over the square root of r squared plus.

07:53 There is no omega -l anymore, so it's just going to be one over omega -c squared.

08:01 Square.

08:05 All right.

08:06 So we've got three equations.

08:11 So the question is, what are the values of r, l, and c? well, let's count up our variables.

08:26 Delta v at the source, we know.

08:31 All the currents, we know.

08:36 We know omega because we know f frequency.

08:40 So our unknowns are rl and and c, r l and c, rl and c.

08:51 So, three equations and three unknowns.

09:10 So i'm trying to think about how i'm going to solve this.

09:18 Because if i were to plot it on a graph like on desmos .com, it would be a three -dimensional graph and i'd have to look at the intersection.

09:31 And i'm not sure it has the functionality to do that.

09:37 And i'm not sure it would even show me anyway the intersection point, even if it could do.

09:47 You know what? i'm going to try it, though.

09:51 I'm going to try putting in 0 .137 equals delta v source, which is 20, over the square root, i'm going to let x equals r, y equals l, and z is c.

10:24 This would be really cool if it worked.

10:27 R squared, which would be x squared, three -dimensional here.

10:36 That would be cool.

10:38 Plus 1 over omega, which is 2 .5, f, and we know f is 60, 2 pi times 60 is just 120 pi.

10:55 So by the way, omega is 120 pi.

11:08 K, 1 over omega, c.

11:20 But c is z.

11:25 Darn, it isn't going to, it's not going to let me do z.

11:30 Let's see if i can set this up for three -dimensional.

11:34 Reverse contrast, grid, arrows, axes.

11:44 Oh, man, give me some three -dimensional graphing here.

11:51 Oh, i hate it when that happens.

11:53 All right, i guess it's not going to work.

11:59 So i'm going to solve this last equation for, uh, hmm, 1 over omega c.

12:15 Let's do that.

12:16 So i'm giving up on three -dimensional graphing, even though i thought it would be cool.

12:40 Okay, so, and you know what, delta vs is going to be easier to just write 20 than delta vs.

12:47 So i'm just going to erase this.

12:50 And i'm just going to write 20.

12:56 Okay.

13:01 Now if i square both sides, now i'm going to divide by 0 .137 squared, and subtract r squared.

13:53 On both sides and take the square root of it.

14:20 So there's an equation for one over omega -c, and it has to be positive because the one -over frequency times capacitance has to be a positive number.

14:32 So this, i don't need to write plus or minus it.

14:37 So now i'm thinking that i can take this and substitute it into the other equations, therefore bringing me down to two equations.

14:47 So now, well, let's try to rewrite that first equation.

14:55 0 .183 amps equals delta v source, which is 20, over the square root of r squared.

15:14 Oops, yeah, that's fine.

15:18 R squared plus omega -l minus 1 over omega -c.

15:34 But 1 over omega c is the square root of this.

15:56 That's 1 over omega c squared.

16:01 Okay, so that first, the black equation became this.

16:10 Now, the second equation, the red equation, and i probably should write it in red, 0 .298, it was 1 half r squared.

16:29 If i remember correctly, yep.

16:31 I'm going to write r over 2 this.

16:32 Time.

16:37 But the last part of it is the same.

16:57 Okay, is that, there we go.

17:06 Okay.

17:08 Now i can put these two into the calculator.

17:13 1 .183, or i could do substitution again, but i'm thinking this is going to be easier.

17:21 20 over.

17:22 I'm going to let rbx, so it's x squared, plus omega -l.

17:28 Omega's 120 pi.

17:35 Is going to be y minus square root 400 over 0 .137 squared minus r squared.

18:00 R is x.

18:04 Okay, so i've got one of the equations plotted.

18:08 Now i'm just going to copy that equation.

18:10 Unfortunately, i can't show you this because it slows my computer down way too much to show you this on the the graphing calculator on desmos .com.

18:24 But the only thing that change is, is that, well there's two things, the 0 .183 becomes 0 .298, and where we had r, it's now r over 2...

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