 Hi, I'm Zor. Welcome to Unizor Education. I would like to put some formulas behind whatever we were talking about. Amplitude, modulation, AM radio. Basically, physics always contains some practical aspects and some theoretical aspects. So today we'll talk about theory. And it's really very, very easy material today. Basically, it's how you can express in mathematics the AM transmission, amplitude modulation. This lecture is part of the course called Physics for Teens, presented on Unizor.com. In this particular lecture, I would definitely encourage you to not only listen to this lecture, which you can find on YouTube, let's say, but go to the website, which contains obviously the link to the lecture itself. But also every lecture of this course has notes, very detailed notes. And in this particular case, there are a few graphs, which I will definitely try to reproduce here on the board, but it will not be as good as I had presented it in notes, because it was actually some kind of a graphical software, which I was using. There is also a prerequisite course called Math for Teens on the same website, which I again do encourage you to familiarize yourself with. I mean, if you are proficient in maths, including calculus, that's fine, you don't have to. But there are many different aspects in the Physics for Teens course, which require good mathematical knowledge, especially vector algebra, calculus, and some other things, algebra of course. So, let's talk about theory behind AM transmission. Well, what we know about right now is that at the heart of AM transmission, you have an LC circuit, which with proper treatment, there is some source of energy, et cetera, et cetera, it actually has its own oscillations. And oscillations are, the frequency of oscillation is based on the capacitance of the capacitor and inductance of the inductor. So, if this is given, there is also an internal, its own angular frequency of oscillations, and the electric current in this particular circuit is oscillating something according to this particular, as a function of time. So, omega is related to this thing, and A is amplitude of these oscillations. It also, obviously, depends on the amount of initial energy which is accumulated in the capacitor, et cetera. So, this is an amplitude, and it's fixed, and this is angular frequency of oscillations, which is also fixed. Well, sometimes you probably know that omega is also equal to 2 pi f, where f is a plane frequency, which means number of cycles per second. And because this is angular, it's in radians, and every cycle is 2 pi radians. So, usually in all theoretical articles, textbooks, et cetera, they're using angular frequency, but sometimes instead of this, they're using 2 pi phi, 2 pi phi f. Now, there is also a component which might or might not be included, which is a phase shift, plus some angle phi. But, again, it all depends, and it's not really important for this particular discussion. So, if you will attach antenna, transmitting antenna, to this circuit, it will transmit frequency, this particular frequency. And the graph will be of this function is obviously something like this, constant amplitude and constant frequency. Now, we were talking that AM transmission requires modulation, which means it's the amplitude which should actually change. If these are the sound waves, then the amplitude should change. It should be bigger here and smaller here, bigger here and smaller here. So, this type of transformation should be achieved by putting certain connections, certain other elements into a circuit. But right now, I'm interested in how in C-rates it can be expressed in the equation type of style. And here is how. So, for example, we have this, and this is an amplitude of my original carrier. And this is my angular frequency of this carrier LC circuit. And this is carrier. Okay. Now, let's assume that we have some source of sound, one particular note which sounds one tone, one note. Well, it's vibration of the air. Now, what does it mean vibration? It's change of pressure. So, when this pressure comes to a microphone, it does something inside the microphone. And as a result, we have some kind of a current which goes from the microphone which can influence this. And this current is also oscillating with certain amplitude and angular frequency which correspond to the note which is taken. I think I was talking about high A which is like 440 kHz, something like this. And the amplitude depends, obviously, on how strongly you really vibrate and how strongly you blow into a horn or pluck the violin string or something. So, we can assume that there is something which is basically a pressure of the air which also depends on the time. And it has its own, this is amplitude of sound and also angular frequency of the sound. So, this is basically how I describe the sound. And as a result, there is some kind of a current which influence this, but the question is how it influence. So, the answer is, I'll just write it down and you will see that it makes sense. So, I'm using exactly the same equation here. I'm just adding to the amplitude, constant amplitude, this function which actually represents the pressure of the air. So, in sync with changing of the pressure of the air, obviously through the microphone or whatever other device, we can influence this amplitude. Now, we do not change this, the main frequency, but we do change only the amplitude. Now, how can it be done? Well, in theory, you understand that if you have, for instance, some kind of connection like this and you have electric current I1 here, I2 here, you will have I1 plus I2, right? So, that's how it's done. I mean, some kind of circuit connection from output from the microphone should actually go into this circuit. Somehow, I'm not really discussing how, but it will end its own oscillations, these ones, to the amplitude of these. Now, the frequency is still related to properties of this LC circuit, but the amplitude is as long as we will properly connect it. And how properly is probably outside of this course is for professionals. Or if you want to be a radio amateur or something like this, you can learn how these circuits are done. But basically, this is the goal. Now, there are some much more complicated sounds. I mean, this is the simple sound. Now, what's the complicated? Well, when the whole orchestra is playing, it's not just one note. What is it then? Well, again, if you have two different sources of oscillation of the air pressure here and here, and they come to the same point somehow, they propagate, obviously. To every point, you have the input from both sources of sound. Well, they add, they superimpose one oscillation onto another, and the actual oscillation will be represented as sum of different oscillations, which means you will have this S1 plus AS2 cosine omega S2t plus S3, etc. So, all these sounds are combined together and they come to the microphone. And it's a very, very kind of chaotic graph if you will represent this function graphically. But no matter what it is, it all is combined together and concentrated in oscillation of the current which goes out from the microphone. So, this current properly connected to the circuit will influence, so no matter how complex this equation is, which represents the sound, it all will be hidden here. Now, let's see how it will look graphically. Because that's a very interesting aspect of this thing. I will try to reproduce it graphically. But again, in the notes for this lecture on Unisor.com, so you go to Unisor.com, choose Physics for Chains. Part of this course is called Waves, and then there is a chapter radio, and the lecture is in there. The lecture is called N equation. So, in that particular notes, I reproduce graphics much better, but I will try. So, what happens if you have very frequently oscillating main carrier waves, electromagnetic waves or current in this LC circuit? And you will add to this very, even maybe chaotic, but really not as frequently oscillating, because sound oscillating from 20 Hz to 20,000 Hz. While LC oscillating is in kilohertz, megahertz, etc. So, it's at least like 10, 20 times greater than even the... I think I calculated in the previous lecture that the ratio of the frequency of AM, a radio, even against the highest note heard by human ear is something like 400 or 500, something in this range. So, it's much more frequently oscillating than this. So, what happens if you add this to this? Well, you will basically have something like this. So, you see the amplitude of the high frequency oscillation will be more or less in sync with the oscillations of the sound. What happens if this frequency is not as high? Well, then you will not actually be able to inscribe the oscillation of the LC circuit precisely into each curve of the sound. So, it will be something like this. I mean, some representation will be, but you know exactly it will be up and down. So, there will be some resemblance, but the sum of two impulses will not be as clearly defining the sound waves as if it's a high frequency. And the higher the frequency, the better you will be able to inscribe this amplitude into the sound. And obviously, if it's better inscribed on the transmission, it will be much easier scrambled, it will be deciphered by the receiver, obviously. So, the more precisely you represent the input sound using the high frequency, the easier it is to receive the clear sound. And considering that AM has a certain upper limit of the frequency, I think it's like 1.5 or 1.6 MHz, it has its own limitation of precision how you really represent the sound. And that's why very high-quality, high-fi transmission is not done by AM radio. It's done using a different technology like frequency modulation, which we will talk about. So, basically, that's it I wanted to talk about. What's very important is to take a look at the graph of the sum of high frequency and sound frequency. No matter how sound is complex, if the high frequency is high enough, it will really represent it well. And again, because AM has a certain range of frequency and there is a maximum frequency, there is a maximum quality, which is not the high-fi. But that's a different story. So, in the notes for this lecture, I present the very good graphical representation of this type of addition of the sound wave on the top of high-frequency oscillation of the base LC circuit. So, I do suggest you to read these notes after you listen to this lecture and be prepared for the next part of this radio-related lectures about frequency modulation. This is something which I find for myself much more complex, but we will talk about this another time. So, thanks very much and good luck.