Resonance

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. We connect a resistor, an inductor and a capacitor in series, on a one hundred volt A C source. Then a voltmeter across just the inductor reads one thousand volts! That is ten times the source voltage. So is the meter broken? The answer is a special effect called resonance, which comes up often in JEE questions. So, let's find out how it works.

0:28When does resonance happen?

So, recall the series LCR circuit: its impedance, capital Z, is the square root of R squared, plus X L minus X C, the whole squared, where R is the resistance. X L, capital X with a small L below, is the inductor's reactance, and X C the capacitor's. The graph shows the current, capital I, against omega, the angular frequency: it rises to a peak, then falls. Okay? Now, let me ask you this. With the resistance fixed, what value of X L minus X C makes the impedance smallest? When X L minus X C is zero, yes, exactly! X L equals X C: this is called resonance. Then Z is just R, the least value it can have. Clear? At resonance we write omega nought, which is omega with a small zero below it, okay? So omega nought times L equals one over omega nought times C, and the double arrow means which gives: omega nought squared is one over L times C, right? Taking the root, omega nought is one over the square root of L times C, okay? And f nought, the ordinary frequency, is omega nought over two pi, where pi is the Greek letter. With me? Now, a question that students often get wrong. If we make the resistance bigger, does the resonant frequency change? No, it does not change at all! The formula has no resistance in it. On the graph, the solid curve is for a small resistance, the dashed curve for a larger one: a lower, broader peak, but at the same omega nought. Got it?

2:08What happens at resonance

Now, let us see what happens at resonance. Z equals R, its least value, so the current, I equals V over R, is the greatest: the peak we saw on the graph. Right? Next, tan phi is X L minus X C, over R, which is now zero. So phi, the Greek letter for the phase angle, is zero, and the power factor, cos phi, equals one. Clear? Now, V L, the voltage across the inductor, equals V C, the voltage across the capacitor. They are in antiphase, so they cancel, and the source voltage V equals V R, the resistor's voltage. So each of them must be small, right? No, no, careful, that is the trap! They cancel in the sum, but each one alone can be larger than the source voltage, even many times larger. That is our thousand volts! Got it? And now, away from resonance. Below omega nought, X C is greater than X L, and the circuit is capacitive; the pointed sign means greater than. Above omega nought, X L is bigger, and the circuit is inductive. With me?

3:13How sharp is the peak? Q and bandwidth

To measure the sharpness of the peak, we use the quality factor, written capital Q, okay? Q equals omega nought times L, over R, which also equals one over omega nought times C times R, and one over R, times the square root of L over C, right? And here is what it means: at resonance, Q equals V L over V, so it tells you how many times larger the inductor's voltage is than the source voltage. That is the key to our puzzle. Clear? Now, the grey dashed line is at I max over root two, where I max, with the small word max below, is the peak current. Where the solid curve crosses it, we get the half-power points. But the current there is not half the peak, so why half power? Because power goes as the current squared, yes! The sign here means proportional to. One over root two, the whole squared, is one half, so the power there is half the peak power. Okay? The gap in omega between the two half-power points is the bandwidth, delta omega; delta is the capital Greek letter, the little triangle. Delta omega equals R over L, which is also omega nought over Q, got it? So, for a given inductor and capacitor, a small resistance means a high Q and a small bandwidth: a sharp peak, like the solid curve. With me?

4:33Example 1: resonant frequency for L 0.1 H, C 10 microfarad

Now, example one. L equals zero point one henry, the capital H on screen, and C equals ten microfarads: the Greek letter mu means micro, and capital F is the farad. Find the resonant angular frequency and the resonant frequency. Pause the video and try it first. Done? Okay, let's check. First the units, where many students slip. How much is ten microfarads, in farads? Micro means ten to the power minus six, so ten microfarads is ten times ten to the power minus six, which gives ten to the power minus five farads. Put in just ten, and the answer is completely wrong. Okay? And zero point one henry is ten to the power minus one. So L times C is ten to the power minus six, and its square root is ten to the power minus three. Right? So omega nought is ten to the power three, that is one thousand radians per second. On screen, r a d means radians, and s to the power minus one means per second. Got it? But the question also asks for the frequency, which is not omega! So f nought is one thousand over two pi, about one hundred and fifty nine hertz; the wavy equals sign means approximately, and capital H, small z, is hertz. Clear?

5:57Example 2: current and VL at resonance, 100 V and R 10 ohm

Now the harder one, which finishes our puzzle. The same circuit has R equals ten ohms, with capital omega for ohms, on a one hundred volt source, capital V for volts, at the resonant frequency. Find the current, and the voltage across the inductor. Pause the video and try it first. Done? Okay, let's check. For the current, do we first need X L and X C, and the big formula for Z? No, there is no need at all, because at the resonant frequency X L equals X C, so Z is simply R, ten ohms. That is the shortcut. Okay? So the current, V over R, is one hundred over ten, which gives ten amperes, written capital A, right? Now the inductor. Its reactance, X L, is omega nought times L, so one thousand times zero point one, which gives one hundred ohms. With me? So V L, which is I times X L, is ten times one hundred, which gives one thousand volts, and V C is the same. That is our puzzle exactly! The meter was fine; V L and V C cancel each other, so the source only sees the resistor. Clear? And here is a neat check. Q is omega nought times L, over R, so one hundred over ten, which gives ten. And V L over V is one thousand over one hundred, also ten. Got it?

7:20Quick revision: XL = XC, Z = R, Q factor and bandwidth

So, quick revision. Resonance is when X L equals X C, at omega nought equals one over the square root of L times C, whatever the resistance. Then Z equals R, the current is greatest, phi is zero, and V L equals V C, though each can exceed the source voltage. Q is omega nought times L over R, which is V L over V, the bandwidth is R over L, and a small resistance gives a sharp peak. Now go and try the practice problems on this page. All the best!