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R3 200 Qsine kHzOutput loadR4 100R5 100 Q...

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R3 200 Qsine kHzOutput loadR4 100R5 100 Q

R3 200 Q sine kHz Output load R4 100 R5 100 Q



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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?

We are given that are is 100 OEMs El is to Henry. See is five, um, times 10 to the negative six. Power, Farid. Okay, The resident frequency. Um, Omega zero, which is two pi f sub zero is the square root of one over l C. So f sub zero is 1/2 pi square root of one over. L c. Put that in a calculator. El is too. See is five times 10 to the negative sixth power. So that gives me 50 hurts. It's look good. Part B. So v sub zero is 100 volts. The frequency is the, um, resonant frequency. So I'm going to write to hi times, 50 hertz. So I'm not gonna write the hurts. Actually, I'm not going to read any of that because two times 50 is 100 so I'm just going to write 100 pie readings per second. So we're trying to figure out the power, so cower average is 1/2 I subzero v Subzero Hussein data. Okay, to figure this out, I'm going to have to get the impedance r squared plus omega l minus one over omega C squared. But I put that in a calculator where route of our squared are is 100 lus, wait a minute. I'm not thinking ahead enough. If it's at its resonance frequency than the resistance is just are, which is 100. OEMs. I wasn't thinking that it's at its resident frequency. So forget about that. That would have worked. But that would have taken a lot longer A gay. The power output from the source. Well, the power we could get rid of this coup sign of data now is just gonna be 1/2. Um, I subzero v subzero. Um, but we know that v subzero equals I subzero. Oh, are. And so I subzero is v subzero Over are so we can substitute this up here. 1/2 Um v subzero squared over R, um, so I can put that into a calculator. 1/2 be subzero, which is 100 squared over our, which is also 100. So that would give me 50 volts. Oops. I wrote that in the wrong place and said the wrong thing was supposed to go up here 50 watts. Okay. See, we're trying to figure out what Q is the quality factor is omega subzero l over are, So I'll just put that into a calculator. Um, Omega sub zero is the square root of one over Elsie. So I'm just gonna put in the square root of one over l C times L over our, which is really going to be the square root of l oversee one over our simplifying that. So I'm going to put that into the calculator. Sq rt of l oversee one over R to over five times 10 to the negative. Sixth power one over R 100 6.3. That's not what's listed in the back of the book, but I don't see how I could have. I made a mistake. Here is just one over the square root of l see Times square root of El over our we know ours 100 l is too. Um, C is five times 10 to the negative. Sixth power. All right, well, let's just go ahead and figure out the But what the band with would be, um, the band with is just, uh, que equals omega sub zero over dealt Ah double. You were dealt a W. Is the band with, um looking, inspecting this l over our right there is the same as Delta W or one over dollar W So one over a Delta double you His l over are so Delta Double you would be are over l And so I'll just do our over l 100 divided by two is 50. Let's see here. Radiance per second. So I'm off by a factor of two in C and D. So I'm gonna think about this some more, but right now, I don't know. I'm gonna think about this.

Well, we equals 20 volt and we know the impotence said equals weighty body. My eye and they're equals are here. Impotence equals are right now odd equals We divided by I and we have we will be equals 20 and eyes 50 multiply by 10 to the power minus tree and therefore r equals 400. All rooms now Omega equals Dubai. Multiply by to multiply be 10 to the power three and it's equal to 12.6. Multiply by 10 to the power tree, ready in for a second. And also Omega not Megan articles to buy. Multiply by 10 to the power three. Right. And therefore it's equal to 6.3. Multiply by 10 to the power free Radian for a second. Okay, No, here we have. Ah, Omega equals one divided by one. Divided by route Chelsea and taking screwed on both sides we have We'll make a scrape is equal to one divided pal. See, from here, capacitance c can be written is one divided by leg a square times l so capacitance sees omega squared times out. No, it Ah, we'll make a note. I know it is equal to we divided by Zed. Right? So we have really which equals 20. Divided by Zet Loves it is Ah, our square bliss. Ah, we have We'll make her time. Or maybe not. Time cell, We'll make a not times l minus one. Divided by omega not ah See Old square right? And further, we have our square are square less omega, not l minus one. Divided by Well, maybe not. See, hold square is equal to 400. Divided by 15 multiplied by 15. Multiply by 10 to the power minus six. Right. And for their we have Ah 16 160 times in its one billion 60,000 less 6.3 multiply by 10 to the power tree multiply May l minus um, Omega Square l remain the square out divided by well, maybe not all square is equal to 400 divided by 15 multiplied by 15 Multiply by 10 to the power minus six And therefore from here l A is equal to 12721272 divided by like 18.8 points, 18.8 five multiplied. Returned to the power three, which is equal to a 67.5. Multiply me 10 to the power minus three. Henry, Right and also capacitance C is equal to 12 body by 12.616 month to blindly 12.6. Multiply by 10 to the power six. Multiply by 67.5. Multiply by 10 to the power minus tree and therefore capacitance C equals, um see the point. CTO 94 zero points. You don't nine for micro Iran. Thanks So R equals R 400 arms and l equals are 68 million Henry and capacitance C equals 94 men of their odd

All right. I wrote down my impertinent for my pertinent information. And then I wrote down that the impedance is the square root of R squared plus XL minus X c squared. Um, we know that our is 500 in this problem. X of l is Omega. L accepts. See? Is one over a mega C. When I put that into a calculator, I get 686 which rounds the 690 Oh, in part B, the, um, amplitude of the current. So, uh, current is voltage subzero over Z 100 over 690. And let's go ahead and get that 100 over 600 90 gives me 0.15 amps heart. See, how does the current vary with time? So, um, we need to know the phase shift to solve this part. And that is the inverse tangent of X ub l minus x subsea over our so except l just to remind you x ub x of l is Omega. L accepts. See is one over Omega Psi. We put those in and you into a calculator and you get an inverse tangent. Uh, let me see here, Teoh 0.7. Fine. So again, the question waas how does the current vary with time? Um, well, the current a function of time. His eyes zero times. Um, where did I write the original function? Let's see. Okay, so it's telling us that V as a function of t is 100 volts sign of 1000 pi t. So the current is going to be that many amps. Sign of 1000 high tea minus 0.7 five. Now, if we change the capacitance to 2/10 then, um, way back on the impedance. 2/10 of a micro Farid to change that. 2.2. That's going to give me an impedance off one. I guess rounded to two significant digits is 1100. OEMs put a point to in here. Ah, I went to in here. That's going to give me a phase shift of negative 1.1. Um, this is gonna have to be 1100 here. Ah, 100. Divided by that gives me Ah, 0.92 amps. And so the answer would be 0.92 amps. Sign of 1000 pi t plus, uh, 1.1

Hi they're giving problem in a series. Our L. C. Circuit. The value of resistance is 50 on the self independence is 0.1. Hi Henry capacitance is having a value of U. N. E. Micro ferret. Our Miss world page applied is 120 fold and the source frequency is 60. There's no in the first part of the problem we have to find the maximum possible current which is at resonance. That is that resonance. So it will come out to be ah max equals two. We are as divided by the impedance and we know that resonance impedance is equal to that is going to our only. So for we are most this is 124 are this is 50 so maximum current maximum possible current passing through the circuit comes out to be 2.4 ampere. But we have to find the percentage of arms current at frequency 60 Hertz percentage of imax means percentage of armies with respect to this maximum possible current. What which we will have to find army's current also. And that RMS current is given losing homes Law as the RMS divided by impedance when the circuit is not in resonance. So now to find this zet that will be given us the vector sum of net re attends eggs, l minus ecstasy to the whole square plus r squared net resistance. So first of all we find the values of these. The expenses. The inductive area tends to buy FL is equal to two in 23.14 into 60 hers into 0.15 Henry. And it comes out to be 56.5 home. Then we find capacity. Ria tense ecstasy that is given as one by two by Fc. Since this is one by 2 to 3.14 into 60 hearts. Again into the capacitance is 20 micro or 20 into 10 disbarment six patty. So it comes out to be 132 0.7. Mm But finally impedance here will be given by X A c minus excel as exercise more than excel. But this is 132.7 minus 56.5 to the whole square plus square of resistance which is 15. Or we can say this is 76.2 square less 50 square. So finally this value here comes out to be 91.1. Therefore arm is current passing through the circuit will be given by um slow as the army's buys. It is this is 124 divided by 91.1 mm And this is 1.32 I'm here. Therefore percentage of this. Our miss current with respect to maximum or civil current. That is I. RMS divided by I mix multiplied by hundreds. Yes, this is 1.32 invited by 2.4 to hunt it. Which finally comes out to be approximately 55 percent. Which is answer. Which is the answer for the first part of the problem? No. Similarly in the second part of the problem we have to find percentage of average power consumed in the circuit at a frequency of 60 herds with respect to the maximum power in june in the circuit maximum possible power so again or maximum possible average power in june in the circuit the circuit should be at resonance. The power factor should be equal to one. Therefore that average power consumed will be given by we are a mess. Multiplied by I armors into cost five. So here it is. We are a mess. And using warm slaw. This is we are missed by impedance. And at resonance impedance is only are. And that resonance power factor is one. Well this is the square of armies voltage divided by resistance. This is average power consume. But then there is no resonance means frequency 60 hertz. In that case average power is even as and that will be here. This will hear. This is maximum power can do so we mark for it as phoenix maximum possible power. And here for average power consumed. We are a mess into I. RMS into cause five. We are a mess as it is I RMS. We are divided by zero this time and plus five as our biceps. So it comes out to be square off the RMS divided by square off impedance into our therefore percentage of the average power consumed with respect to the maximum power will be given as P average divided by P max in 200 means this is we are reminiscent square into our invited by Zetti square and divided by P. Max, which was we are mrs square invited by off multiplied by country. So canceling is the automatic squad. The automatic squared. Finally, here it comes out to be are square by 30 square in 200. Or we can say this is our by zet the whole square in 200. Now for our this is 50 preserve. This is 91.1 into square in 200. So this percentage finally comes out to be approximate 30 or sent. Which is the answer for the second part of this problem. Thank you.


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