State one use for a resonant circuit.
As we get started, let’s remind ourselves what a resonant circuit is. Because we’re talking about resonance, that means, for one thing, that our circuit can’t be direct current. It can’t be a DC circuit. That’s because resonance is inextricably associated with frequency. And it’s only an AC or alternating current circuit that has a frequency associated with it. Knowing that, let’s start sketching out an example circuit that could be at resonance. We know that the power supply in this circuit will be AC. As we said, we need to supply a frequency to the circuit. And beyond this, we can build up our circuit with an inductor, a capacitor, and a resistor. Over on the left, we have our inductor. We’ll say it has a self-inductance 𝐿. Then, up top, we have our resistor. And on the right-hand side, we have our capacitor.
Notice our capacitor has a diagonal arrow moving through it. That means this is a variable capacitor. In other words, its capacitance can be tuned or changed. In a circuit such as this, it’s possible for our circuit to be in resonance. As we said, it requires a certain oscillation frequency 𝑓 in this circuit. And beyond that, it requires particular values for our circuit inductor and capacitor. To see why this is so, let’s consider an AC circuit concept, known as impedance. The impedance of an AC circuit is similar to the resistance of a DC circuit. That is, it summarizes the overall circuit resistance to the flow of current.
If our AC circuit, or AC circuits in general, only ever had resistors and no inductors or capacitors, then impedance, which is often symbolized using the letter 𝑍, would always be equal to simply the resistance of the circuit 𝑅. But whenever we do have an inductor and/or a capacitor in our circuit, impedance is no longer this simple. In the case of an 𝐿𝑅𝐶 AC circuit, like we have here, we write impedance 𝑍 as the vector sum of resistance plus the difference between what are called the reactances of our inductor 𝑋 sub 𝐿 and the reactance of our capacitor 𝑋 sub 𝐶. We bring up this topic of impedance because it’s a critical concept to understand circuit resonance. When our circuit is resonating, one physical sign of that is that the resistance to current flow is the least that it will ever be.
Let’s think practically about how we would detect resonance in our circuit. If resonance happens when the resistance to current flow is the smallest it will ever be, then that means that the flow of current, under the condition of resonance, is the largest it will ever be. One way to monitor this would be to open up our circuit and then place an ammeter, a device for measuring current, in series with all the other components. Going back to impedance for a moment, remember, we said that resonance occurs in a circuit when impedance is at its minimum value. What is this minimum value though? Well, we can get an idea for it by looking at the form of this equation.
There are really three parts to the right-hand side of this equation, the resistance 𝑅, the inductive reactance 𝑋 sub 𝐿, and the capacitive reactance 𝑋 sub 𝐶. Given a fixed resistance for our circuit 𝑅, we know that that value won’t be able to change. So to make 𝑍 as small as possible, we would achieve this by making 𝑋 sub 𝐿 and 𝑋 sub 𝐶 equal to one another. When these two reactances are equal to one another, then 𝑋 sub 𝐿 minus 𝑋 sub 𝐶 is zero. So our impedance reduces to the square root of 𝑅 squared or simply 𝑅. In this condition, impedance being equal to circuit resistance is the condition signifying resonance. This is what happens when the resistance of the current flow in our circuit is minimized. And therefore, as we look at the output of our ammeter, we see that this value is reaching a peak when 𝑍 is equal to 𝑅.
We said that this resonance is achieved only when the inductive reactance and capacitive reactance are equal to one another. Typically, with any old inductor and any old capacitor placed in our circuit, it won’t be the case that these reactances are equal. But remember that we have a tunable or a changeable capacitor here. If you or I were tuning this circuit to reach resonance then, what we might do is keep an eye on the ammeter output. And as we watch the current go up or down, we would adjust the capacitance of our capacitor. When we change the capacitance of our capacitor, that also changes the capacitive reactance. And if we’re able to change it in such a way that we bring it in line with the inductive reactance, then at that point we achieve resonance. So then, at resonance, our ammeter would record a maximum current value. And we could observe this maximum by looking at the readout of this ammeter. So let’s say we’re there. We’ll say we’ve tuned capacitor until we observe a current maximum, indicating that our impedance in the circuit is equal to 𝑅, i.e., were at resonance.
To see how this resonating circuit might be useful, let’s consider another object that resonates and the use of that. For the musically inclined, we may be familiar with an object known as a tuning fork. These forks are designed so that when they’re given a gentle bump, they vibrate at a very particular resonant frequency. This frequency can then be used to tune a musical instrument, such as a guitar or a piano. By design, a tuning fork resonates at a very particular frequency. Say we were to send a powerful high amplitude sound wave across the tuning fork. And say that this wave had a frequency 𝑓. If 𝑓, the frequency of the sound wave, did not match the resonant frequency of the fork, then the fork wouldn’t respond with motion to the sound wave. But then say that we change the frequency of our sound wave to a new frequency. We’ll call it 𝑓 sub R, the resonant frequency of the fork. In this case, because of the match between our sound wave and our tuning fork, the fork would respond to the wave by starting to vibrate. In other words, our resonating fork has effectively picked out an ambient sound wave frequency. And it turns out that a resonance circuit can do something similar.
Imagine that our resonant AC circuit was coupled to an antenna, a device for receiving radio waves. We know that, in general, a radio antenna is subject to waves of all sorts of frequencies. That’s because there are many different radio waves being broadcast all the time over a range of frequencies. Some frequencies apply to stations that play music. Other frequencies are for news broadcasts and so on. Of course, if we’re listening to our radio, there’s often a particular station that we have in mind that we want to listen to. If we have a resonant circuit tuned to resonate at the frequency that we’re interested in receiving, then, through coupling this frequency to our antenna through mutual inductance, we’re able to make our radio receiver, our antenna, sensitive to that particular frequency of interest.
This means that if a radio wave with a different frequency, we’ll call it 𝑓 sub one, is incident on our antenna, then the antenna doesn’t respond to that frequency. That’s because the only current allowed, so to speak, to run through the circuit is current which oscillates at the resonant frequency 𝑓. Any other frequency is disallowed. But then, if a wave with the desired frequency 𝑓 is incident on our antenna, the antenna is receptive or sensitive to that wave. In response to a wave of this resonant frequency, current in our circuit then increases. And that current is directed towards a speaker system, where it’s converted and output as sound waves.
This tells us then one use for a resonant circuit. We can put it this way in writing. The tuning circuit is used in radio receivers to pick up a particular broadcast station. That’s one use for a resonating circuit.