 So, very warm welcome to this first discussion or lecture session on the course IC 2-1. Professor Suryanarayan has already explained to you the purpose and the objectives of this course, all right. And today our objective is to discuss the first experiment that you shall be doing over the next week. What I mean by that is different batches will be doing experiments on different days, but essentially it would be the same objectives for all the batches. The background of the experiment that we are going to do is to look at an electrical circuit which can reduce noise. Now, noise is a common problem in most kinds of equipment, whether it is electronic, whether it is mechanical or any other, even hydraulic for that matter, right, noise. What does noise mean? Noise essentially is a random phenomenon, a phenomenon which cannot be captured by describing it clearly as a function of time. At every point in time for a function or for a signal, you could query the value of the signal. Suppose you record an audio signal, right, you could query the value of the audio signal at a point in time. If it is recorded as a voltage value, it could be, you could ask at time t equal to 0.3 seconds. What is the value of the audio signal? 0.3 volts, 0.6 volts, all right. So, a signal, a deterministic signal can clearly be specified as a value against every time. Unlike that, a noise signal can only be specified as what is called a random variable for every time. So, all that can be said for noise as an example is that when you record it at the value, at the point, let us say 0.3 seconds, the value might be between 0.2 volts and 0.3 volts with a probability of 10 percent, that is all that you can say, all of you understand probability, yeah. So, all that we can say is, well, you know, when you record this noise phenomenon between 0.2 and 0.3 seconds on the time scale, its value is likely, sorry, when you record it at 0.3 seconds on the time scale, its value is likely to be between 0.2 and 0.3 seconds or 0.3 volts with a probability of 10 percent. It can be, let us say, between 0.3 volts and 0.5 volts with a probability of 60 percent. So, this is all that can be said about noise. In short, a noise phenomenon is always characterized by a density for every time, a probability density function for every time, that is called a random, it is called a stochastic process. Now, those are technical terms. What we are trying to bring out here in this experiment is a methodology to reduce noise and one common assumption that one can reasonably make about noise is that it varies very quickly and it varies much faster than most reasonable physical phenomena. So, even for example, when you look at the noise that you encounter in a power supply, see that is the first example of noise that you might see. If you were to look at, you think the power supply is 50 hertz in frequency, but if you were to actually monitor the waveform on a, you know, oscilloscope on a display device, you would see that it is far from being a sine wave exactly. It has a lot of fluctuation about the sinusoidal pattern, alright, that fluctuation is of course due to noise. Now, that fluctuation occurs much faster than the 50 hertz that you are trying to monitor. This is an example, take any other example, when you make measurements, you know, you are trying to measure say a link, but you are trying to measure a quantity with a mechanical device. Now, you stabilize the pointers of the mechanical instrument, so as to take your measurement and there is some amount of motion which does not allow you to take a measurement precisely. Typically that motion is much faster than the rate at which you take measurements and therefore, that is another example of a situation where the noise phenomenon is much faster than the phenomenon which you are trying to observe. This is in general, this is very often, I would not say it is always the case, but in most situations that one would encounter at least as a beginner, one would see that the noise phenomenon is much faster than the so called signal phenomenon or the phenomenon which you are trying to measure, observe or study. Now, as electrical engineers, I mean that is my bias, incidentally I should have introduced myself. So, you know my name is Vikram M. Gadre, I am a faculty member in the department of electrical engineering and I am going to be supervising your first experiment which as you can see projected here relates to noise removal or noise reduction and as I said you know with the with an electrical engineer's bias, we tend to think of most signals as a linear combination of sine waves. So, we say as long as we have reasonable signals to deal with, by reasonable I mean most of the signals that you would encounter in practical life, in ordinary life, most reasonable signals can be thought of as a linear combination of sine waves. What does that mean? It means that I can think of them as instead of describing them as a function of time, I could describe it as follows. I could say well, so you know what I am trying to bring out in short, very briefly is the idea of a different domain in which you can describe a signal. So, one way to describe a signal is this, time, value, value could be voltage, current, pressure, temperature, whatever it is, there you are. The other way of describing a signal is what I am just going to describe namely the frequency domain. So, what I say is well, I talk about the frequency of the sine wave and of course it could be the angular frequency for convenience and for the moment let us start from 0 and go towards infinity and I say well, at every frequency, now here we have to understand things a little better. Suppose, you had a periodic phenomenon, a phenomenon which repeats every so much of time, many of you would probably be exposed to the idea of harmonics, right. So, for example, what really is a harmonic? Well, a harmonic is a multiple of the basic frequency. So, for example, if you have a periodic phenomenon which varies at 50 hertz, let us talk about the power supply as an example. The power supply is supposed to be a sine wave of frequency 50 hertz, is that right? Many of you know this, how many of you know that the power supply is a signal of 50 hertz frequency, yeah many of you do, good. Now, the fact that it is not quite a sine wave of 50 hertz frequency can be expressed in another way by saying that it is a sine wave of frequency 50 hertz plus another sine wave of frequency 100 hertz plus another of 150 hertz and all multiples of 50 hertz, in fact in principle all the way up to infinity. Now, these other frequencies 100, 150, 200 and so on are called harmonics. So, ideally the harmonics should have had an amplitude of 0, but they do not have an amplitude of 0. This is our way of describing a situation where there is a distortion in the basic periodic pattern, alright. So, any periodic function with a periodic frequency of f can be described as a combination of sine waves with frequency f 2, f 3, f and all multiples of f, alright. Now, suppose your phenomenon is not periodic, what is another way of looking at it? You could say the phenomenon is periodic, but with a period of infinity, is that right? So phenomenon which is not periodic can be thought of as periodic with a period of infinity. Now, if it is period with infinity, how much is the fundamental frequency? 0, is that right? So, that means that when you go from periodic to aperiodic phenomena, you are coming from fundamental frequencies which go towards 0. That means instead of dealing with a discrete set of frequencies f 2, f 3, f and so on, you are now dealing with the continuous frequency axis. Is that clear to everybody? Yeah, anybody to whom this is not clear? So, do not hesitate to raise your hands. So, I am saying that if you have an aperiodic phenomenon, phenomenon is not periodic, you could think of it as a sum of many sine waves, but the fundamental frequency tends to 0, which means the sine waves now have frequencies that are continuous. Is that right? So, instead of a sum, you have an integral, that is a technical detail, what it means is I must describe this phenomenon in terms of a continuous axis now, which starts from 0 goes towards infinity. Now, this axis actually there are two functions to be described here, there is a magnitude function and there is a phase function. So, let us say for example, this is the magnitude function and let us say this is the phase function. Of course, the phase function must lie between minus pi and pi over an angle of 2 pi. So, of course, I said this must be between pi and minus pi. So, you could have some kind of a phase function, does not matter what it is, I have just drawn something here. I am just trying to bring out the physical interpretation. What is the physical interpretation? For example, if I were to pick this frequency here, if I were to pick this frequency here, I am going to denote it as capital omega naught, then I mark the amplitude point and I mark the phase point for omega naught. Suppose this phase is 3 pi by 4, just as an example and this magnitude here is 4 units, all right. What is the interpretation? The interpretation is that this signal includes sine waves of frequencies around omega naught with an amplitude 4 phase 3 pi by 4. This is the way we interpret it. So, what I mean to bring out here is typically if you look at the frequency axis, most phenomena are going to be predominant in magnitude around the lower frequencies and if you have a significant presence of higher frequencies, it is due to noise, is that right. So, the simplest way to reduce noise is to emphasize the lower frequencies and to deemphasize the higher frequencies. So, what we are trying to do in this experiment is to look at one electrical circuit which would do exactly this, emphasize lower frequencies and deemphasize higher frequencies. Is the objective absolutely clear? Any questions with the objective, right, straightforward, yes, how do we do that? Now, to do that, one must first agree that I must be able to deal with different frequencies in a decoupled way. That means I must be able to say definitively what this particular circuit does to a sine wave of a particular frequency, you know, it should not depend on what other sine waves are present if you know what I mean. So, if I know there is a sine wave of frequency, let us say 50 hertz present with this amplitude and this phase and if I pass it through the circuit, I know exactly what is going to happen to the 50 hertz sine wave when it comes out of the circuit. I do not need to worry about whether there is a 100 hertz sine wave present or not or whether there is a 200 hertz sine wave present or not, all right. So, that means each frequency can be dealt with in a decoupled manner, that is the first thing that we need. What does it mean? It means that if I were to give a sine wave as an excitation, as an input voltage, the output in every part of the circuit must also be a sine wave of the same frequency. Is that clear? So, we must first build a circuit where if I gave a sinusoidal input of a certain frequency, the output must be sinusoidal everywhere in the circuit, the voltage current everywhere must be sinusoidal of the same frequency everywhere in the circuit. What could change at different places is the amplitude and the phase. Now, one very beautiful property of the three basic elements that we deal with in electrical circuits namely the resistance, I am going to show all the three symbols to you, the resistance, the inductance and the capacitance, is that they obey this property. So, if I were to give a sinusoidal input to it, the output would be sinusoidal. By input I mean if I were to make a sinusoidal current input to any of them, the voltage would also be sinusoidal with the same frequency. If I were to give a sinusoidal voltage of a certain frequency, the output would be a current of the same frequency, sinusoidal of course. So, the reason for that is easy to see. If you look at the voltage current descriptions of each of these, what are they? The resistance is described very simply by V is equal to i r or let me write everything as a function of time. So, V t is r times i t resistance, V t is l d i d t, inductance, V v t d t times c is i t for the capacitance, all right, simple. Now, what is the beautiful property that all these circuits enjoy? All of them relate the voltage and current either proportionally or through a derivative. Now, when you take the derivative of a sine wave of a certain frequency, it also results in a sine wave of the same frequency. When you add two sine waves of the same frequency, they become a sine wave of the same frequency, is that right? So, when you apply Kirchhoff's current law or Kirchhoff's voltage law, whereupon you add currents or add voltages and if each of them is a sinusoid of the same frequency, the resultant would also be a sinusoidal of the same frequency. If you take the derivative at any point, it is still a sinusoidal same frequency. This is the reason why if you construct a circuit out of resistances, inductance and capacitances and if you were to excite it with a sinusoidal source, the resultant is a sinusoid all over the circuit. So, this is the basic premise from which we begin. Now, suppose I were to conduct, suppose I were to connect a resistance and a capacitance in series like this, I wish to excite here, excite this pair of terminals with a sinusoidal source. So, let me write it down, let me call the source say v 0 cos omega naught t plus phi naught and my objective is to find out the output here. Now, the first thing to do is to establish a methodology for obtaining this output. How do we analyze a circuit like this? The first thing we are going to do is to go away from sine waves. So, what we are going to do is to agree that if you have a rotating complex number. So, now, pay careful attention. This is a slightly tricky principle, but not too difficult once you pay attention. If you have a complex number, so visualize this paper to be the complex plane here. This is the origin of the complex plane and if you were to rotate, right. So, a complex number with a magnitude of v naught and an initial angle of phi naught at an angular velocity of omega naught in the counter clockwise direction, then its projection on the real axis is indeed a cosine, a cos sinusoid, is not it? What is its projection? So, how is this complex number described? It is v naught e raised to the power j. So, all of you are familiar with this representation, right, the polar representation, I am sure. So, omega naught t plus phi naught. Everybody is familiar with this representation. Anybody not familiar with this representation? All right. Anybody not familiar? No. Now, of course, it has a real and imaginary part. The real part is what we started off with here, all right. Now, what we are going to do is to work with this instead of the original sine wave and you will see the reason soon why. When we take the derivative of a sine wave, it is indeed a sine wave of the same frequency, but the derivative is not proportional to the original sine wave. You see what I mean? Let me illustrate by contrast. On the other hand, if you were to take this complex number, so let us call this as a function of t, x of t. If you were to take dx t dt here, what does it give you? It gives you, I am sure all of you can work it out. In fact, it gives you j omega naught x t. You all agree? So, what does it mean? It means the derivative is proportional to the original function. So, taking the derivative can be replaced by an operation of multiplication here, all right. Now, this is very similar to what happens in a resistance. So, for example, suppose you had an inductor, suppose this where the current, suppose you had x t as the current in the inductor. Let us take an example. Suppose the, suppose the current of an, the inductor had a current of i naught e raised to the power j omega naught t plus phi naught. Of course, this is an imaginary situation. We are talking about complex numbers. I know that we cannot actually have complex currents, but let us imagine that for a minute. Suppose you could have complex values for a current. What would be the corresponding inductor voltage? It would be, can you see it be essentially j omega naught times L multiplied by the current, all right. So, this is as if this inductor, does everybody agree with this? Yeah, L d i d t. So, it is j omega naught L times the current. Now, what does it mean? It means the inductor in the sense of this current behaves as if it were a resistance of value j omega naught L. So, if you allow for complex resistances, so to speak, then you can accommodate inductors, capacitors and resistors all as generalized resistances. Similarly, what would, so therefore, if you, you know, if you want to generalize this idea of resistance to the context of inductor and capacitor, the first thing we should do is to use a different name, right. So, instead of calling these things resistances, we will call them impedances, right. So, we will talk about impedances now. In fact, the word impedance has the same broad connotation as the word resistance. Why do we call a resistance a resistance? Because in given a certain voltage, the resistance in some sense opposes the flow of current, the extent to which it opposes the flow of current is the resistance, is that right. Now, the word impedance, the word impede means the same thing to stop, to come in the way of, right. So, an impedance is the term used to denote how much of opposition this element offers to the flow of current when you apply a voltage, alright. So, therefore, the impedance is essentially the voltage by the current, but only in the context of rotating complex numbers. I am expecting that you are taking down, that is why I am writing it there. I am expecting that you are noting this down. Now, these rotating complex numbers are also given a name in electrical engine. They are called phasors. So, let us write down the impedances of all the three kinds of elements that we encounter. R has an impedance of R and inductor has an impedance of j omega naught L. A capacitor has an impedance of 1 by j omega naught c. Let me derive the case of a capacitance just for completeness. So, for a capacitance, you know that c d v d t, c v t d v t d t is i t, alright. Now, if i t is of the form i naught e raised to the power j omega naught t plus phi naught, then v t can essentially be obtained by integrating 1 by c integral. Let us take the indefinite integral for the moment and you could then work out, I mean I leave it to you to work out v t by i t and show that it is equal to 1 by j omega naught c. So, now, we have an easy situation to deal with. You have this resistance and inductance and resistance in capacitance in series, alright and you have a sine wave which you have applied. What I am going to do is to replace the sine wave by this rotating complex number. Why am I doing it? Because once I do that, then I can treat both the resistance and capacitance as an impedance and I could analyze the circuit as if I were analyzing a circuit comprised only of resistance, ok. What would be the so called impedance of this capacitor 1 by j omega c as a function of omega of course. Now, it is very easy to write down the output voltage by the input voltage here. The output voltage by the input voltage is how much? That is very easily done by using the voltage divider principle. It is essentially this impedance divided by this impedance plus this impedance. If it if they were resistors, you would write that down by inspection. So, the output by the input would be essentially 1 by j omega naught c divided by r plus 1 by j omega naught c and we can simplify that. Let us do that. So, this is this ratio of the output to the input is called the transfer function of the circuit. It is also called the frequency response of the circuit and we will understand in a minute why. What is the frequency response of the circuit? It is 1, 1 can simplify it. It is 1 by 1 plus j omega naught c r, alright. What does this physically mean? This physically means two things. So, this frequency response is a complex number, right. Let us give it a name. Let us call it capital H of it is a function of omega naught. Assuming that the value of the capacitance and the resistance are fixed, it is a value of capital omega naught. Now, let us plot the magnitude and the angle of H omega naught. In fact, it is really the magnitude in which we are interested. So, let us look at the magnitude, magnitude of H omega naught. Let us forget about the angle for the moment. How much is the magnitude? It is essentially 1 by 1 plus omega naught squared c squared r squared under root positive, alright. Let us plot this as a function of omega naught. So, now, you know visualize the physical situation here. I have this resistance and capacitance connected in series. I have the flexibility of connecting a voltage source which is of a sinusoidal nature at the input and I have a flexibility of moving the knob of frequency to change the frequency. That is the physical situation and I want to find out how the output amplitude varies as a function. I can assume that the input amplitude is held constant. So, at the input I have a sine wave generator which generates a sine wave of a fixed amplitude. So, I can ensure the amplitude remains constant and I change the frequency and I am interested in seeing what happens to the output amplitude as a function of frequency. Simple. So, let us plot that. It is very easy to see at omega equal to 0. Of course, the amplitude is 1. As omega naught tends to infinity, the amplitude is 0 and in fact, it decreases monotonically from 0 towards infinity. What is the physical meaning of this? It means that this circuit in effect emphasizes lower frequencies or in other words keeps lower frequencies more or less as they are, but it deemphasizes higher frequencies. So, it suppresses higher frequencies. This is the simplest example of a circuit which would reduce noise because this would deemphasize higher frequencies. And our objective in fact, in short in the laboratory is to set up this circuit and its variations and to verify that this is indeed the case. Now, a little more about this circuit. You know what we have drawn is only qualitative, a little more quantitative analysis of the circuit. You see, we see the quantity C r in this expression. So, the quantity C r. Now, many of you who would have dealt with these circuits, I am sure you might have dealt with these circuits in high school, is not it with resistive and capacitive circuits. In class 12, not everybody, all right, does not matter. This resistive, this quantity, the product of the capacitance and resistance has a significance. One thing which you can tell me is, what are the units of this quantity? Yes, what are the units? Time, that is correct. The units are units of time. So, in fact, this is called the time constant. C r is called the time constant of the circuit. Why it is called the time constant? We would need to understand by looking at a different kind of excitation. But at the moment, let us give this time constant. Let us abbreviate this time constant by tau naught. And let us rewrite the expression for H of omega naught in terms of tau naught. So, essentially H of omega naught is 1 by 1 plus omega naught squared, I am sorry. So, the question is, you see, the units of 1 by tau naught are going to be angular frequency, essentially radians per second. So, what happens at the frequency 1 by tau naught? So, at capital omega equal to capital omega naught equal to 1 by tau naught. What is the magnitude H omega naught? It is 1 by square root of 2. What is so special about 1 by square root of 2? Now, if I have a sine wave of amplitude A, suppose you thought of the sine wave as a current signal going through a resistance. How much is the power consumed by the resistance? Suppose the resistance is of value 1 ohm. How much is the power consumed by the resistance? The square of the sine wave, is that right? Now, what it means is that it is the square of the amplitude which is indicative of the power, is that right? Now, so you see, what is it that determines the power? How does the power relate to the amplitude? It relates the square of the amplitude, is that right? So, essentially if I were to multiply the amplitude by 1 by square root of 2, what would happen to the power? It would come down by a factor of 2. So, therefore, this point where the amplitude becomes 1 by square root of 2 has the significance of bringing the power down by a factor of half. So, it is called the half power point, alright? This is a very important point on the curve. So, let us mark the half power point. And in fact, one of the objectives in your experiment is to be able to decide the half power point in your circuit. So, somewhere here where you have 1 by tau naught, you would see that the amplitude falls to 1 by square root of 2. How much is 1 by square root of 2? About 70 percent, about 0.7 ohm something, right? So, one of the things that you need to do in your experiment is to check for the half power point. In other words, check the point where the amplitude falls to 50 percent of what its amplitude is at very low frequency, at almost 0 frequency. And ensure that that half power point agrees with what we calculate theoretically. That means, at the angular frequency of 1 by tau naught, is that right? So, you would know the resistance value, you would know the capacitance value and you would check that the half power point is indeed at the angular frequency 1 by r c. Now, I am going to put a challenge before you. And while you think about the challenge and you know towards the end of the lecture, we will have somebody of a good deed now, we might have somebody come up with an answer to the challenge. I want somebody to tell me draw a similar curve of magnitude of h omega naught for this circuit. So, let us see who comes up with it first. And while somebody does that, so he or she would be asked to come and discuss it here in front of the audience. Let us reward. But while you do that, I am going to discuss a few other things. I am going to show you some of the things you are going to use tomorrow. How do you set up such a circuit? So, to set up such a circuit, you are going to use something called a breadboard. This is a breadboard, a breadboard you know because it gives that impression of slices. Anyway, now notice in this breadboard, please look at the markings of terminals very carefully in the breadboard. You notice there are several apertures here like this. I am marking the apertures with my or maybe I should mark it with you know. So, what I will do is I will sort of mark it this way. So, you will see the apertures. These are all apertures, small holes and these holes have metal inside. Now, you must understand in the breadboard that this from here to here. Now, you see a W, do not you? It is not, maybe it is not visible to you. But you know in between, when you look at the breadboard closely, you will see a W here in between at this point right in the middle on the top. When you will also see a W here, maybe what I will do is I will blacken this. So, you will see it. Now, can you see it somewhat? Yes. So, similarly here say here W there. Now, on one side of the W, you have two horizontal lines and on the other side of the W, again you have two horizontal lines. All these four horizontal lines are equipotential lines. So, they are all at the same potential that means they are connected metallic, I mean with a metal, with a metal line, metallical. Is that clear to everybody? Now, in between, now there is a groove here, which you simply ignore. Can everybody see that? Can everybody see what I am pointing out? Here, between this groove and this groove, I have several vertical. Now, you must understand these to be vertical lines. So, each vertical line, you know if you look carefully, you can count the vertical line has five apertures. Can you see that? Every vertical line has five apertures. Now, all these five apertures in a vertical line form one equipotential surface. So, you have as many equipotential surfaces or as many equipotential lines as the number of vertical half lines. In fact, they are numbered on top, I think nearly 62 or something like that, you know rather I mean yeah about 62 lines or so on both sides. So, why do we have a configuration like this? Suppose I wish to connect, now I will show you how I would connect the resistance and the capacitance. I wish to connect the resistance and capacitance in series. So, what I would do is put one of the terminals of the resistance on one of the equipotential lines is a little hard yes and the other in another equipotential line. Why does the breadboard provide this little hard this resistance? So, you know you have to be little yes. And I would take the capacitance and connect it to the same equipotential line as this resistance and bring it to another equipotential line on the other side. Is that right? Now, that is now what would I do next? I would take a wire, I would allow this wire now you see I would connect the wire to the input of the resistance and I would take the other point of the capacitance also through a wire maybe I should. So, now, what do I have here? I am holding the yellow line and the green line. So, effectively if you look back at the circuit here, this is the yellow line here oh sorry not on this yeah all right let me draw it in another piece of paper. So, I have resistance and capacitance. So, this is the yellow line here and this is the green line. Is that clear? Does everybody associate the electrical connection properly now? Yeah everybody clear? So, what would I do? I would take this circuit which I have mounted on a breadboard and I would connect these two points to the output of what is called a signal generator. Now, you in the laboratory would actually have I believe the signal generator and the display device which is called the oscilloscope both built into one. In fact, I believe it would look something like this I am just putting it before you I cannot show you the system operate at the moment, but this is how it would look right. So, you would be using what is called the PC based oscilloscope. What is an oscilloscope? An oscilloscope is a device where a beam responds to the input voltage in such a way that the motion of the beam corresponds to the way in which that voltage varies in time right. So, in other way in other words it is a graphical conversion of an electrical voltage right essentially depicts an electrical voltage graphically. So, what would you do in your experiment? You would actually connect a signal here between these terminals a sinusoidal signal and you would hold its amplitude constant and vary its frequency and you would then study how now you need to take one more wire to look at the output. So, I will take a red line here. So, if you go back to this circuit this is the red line and the oscilloscope would be connected. So, what you see here is the output between this between the red line and the green line and to take the correspondence with the original circuit and I am spelling this out very clearly. So, that you have no mistakes tomorrow is that right? This is the electrical point at which we are looking. So, that is about how you connect these. So, it is a simple experiment. Now, this is the basic experiment. What is expected as you have seen in your write up is much more. I believe all of you must have received some instructions on model have you not? Yes, how many of you have received the model instructions can I just check? Yeah. So, you see in those instructions I have mentioned a few variations on this experiment. One of the variations is to give not a sinusoidal input to this circuit, but a square input. By a square input I mean a voltage which varies like this repeated periodically. What I would like you to study and as a challenge explain if possible is the waveform that you see at the output. So, if I were to give this square wave as an input now here it would of course, depend on the frequency to an extent. So, what I have suggested in the write up is that you try first a very low frequency. Now, if the frequency is very low it is almost as if there were no repetition here that is how one should interpret it. If the frequency is too high then the circuit does not get time to stabilize. So, one would like you to study the variation of the behavior of the circuit when you go from a very low frequency towards higher and higher frequencies all right and if possible even to interpret this. Now, I shall only give you a hint this is as I said a challenge to you. I will only give you a hint on how you would analyze what would happen in the context of a square wave. You see when you analyze the circuit the resistive and capacitive circuit in the context of a square wave you can no longer take recalls to phases right. You must then go back to the basic describing equation of the resistance and the capacitance. So, let us assume the input voltage is V in T and of course, the output voltage is the same as the voltage of across the capacitor which we shall call V C T. What can we write as an expression for V in T in terms of V C T? So, for that we need to obtain the voltage across the resistance. How do we obtain the voltage across the resistance? The current times the resistance. How would we obtain the current? The current is essentially C d V C T d T. If you were to multiply this by the resistance it is the drop across the resistance plus V C T gives you V in T. So, you have a differential equation describing V C T. Now, of course, many of you are familiar with how to solve a differential equation of this kind. This is a first is an example of a situation where a differential equation gives you beautiful insights into the behavior of the system. So, now the hint is solve this differential equation with V in T equal to the kind of waveform that you have. In fact, assume that that waveform does not repeat first. So, V C T is 0 initially and suddenly becomes a certain value after a point in time and study how V C T varies as a function of time with this input. So, that is the hint and try and correlate what you come out with as an answer to your differential equation with what you see on the oscilloscope screen when you do the experiment is that right. So, not only does this experiment teach you about the frequency response and how such a circuit can be used to suppress higher frequencies and emphasize lower frequencies. It also teaches you how to correlate the behavior of a system which is described by a differential equation with what you see is an input output pattern right. So, you have here an example of a system comprising of a resistance and a capacitance excited by a certain input function and the differential equation that describes a system is known and therefore, you are expected to also explain the output and all this is a part of measurement and instrumentation right. When you deal with systems in measurements and instrumentation, you must also be able to model the system and what we have here is a model for the system and when you have a model for the system you also sometimes know how the system would behave with a certain input you can predict the output and verify that behavior experiment all right. So, these are all the objectives of this experiment. I have also given you one more variation in the write up that I have sent on Moodle and that is a further challenge. Suppose, I were to make the circuit a little more complicated by putting two more elements what would happen right I have given that as a part of the that is all these are to test to stretch your imagination little. A few remarks about how one reads resistances. So, if I wait now you have this resistance in front of you yeah you connected this resistance here you want to find out the value of the resistance. So, how do you do it look carefully at the resistance that we put a white paper beneath it look carefully at the resistance. Do you see four bands on the resistance now you will have to look carefully you see three easily, but there is also one band here at the end which is either silver or gold in color is that right. So, you do not see it as easily from a distance, but can you all see it now four bands yes yeah now the way to interpret the resistance is start from. So, in fact I should read it this way think of each of these bands as a color code for a digit the digits can be between 0 and 9 is that right. So, you have three digits there let us call the three digits A B and C and then you have a band silver or gold band. Now, the value of the resistance is the number A B into 10 raise to the power of C. So, for example, suppose A has the value 1 B the value 0 and C corresponds to 2 then you have 1 0 into 10 raise to the 2 that means 1 kilo ohm all right. How do you decide the values A B and C there is a very simple mnemonic which allows you to remember the values B B Roy of greater Bombay a very good wallet come on tell me the colors I forget I have to check I think it is green all right. So, that is this is 0. So, you start from 0 here 1 there and so on. So, you can continue up to 9 all right. So, simple mnemonic to remember the values of the resistance to read a capacitance you will understand tomorrow there is a code on the capacitance which you must learn how to read all right. So, you learn how to read the capacitance tomorrow whenever you have your lab that is. Now, just to complete the discussion although you are not going to use integrated circuits in your experiment tomorrow, but I still would like to show them to you what you see here is what is called an integrated circuit the thing that you see in black. Now, I will just keep it you know may be can you all see that it has pins on the side yes you see pins on the side I tell it you can see pins on the side now these pins are electrical points there they are you know inside the circuit and if I turn this integrated circuit around little difficult to see, but if I now you can when I turn it can you see there is a notch here can everybody see a notch there now the notch when you turn it you can see it. In fact, there is a notch and there is also a dot there when you look at it closely on this side. So, you see your counting begins from this side of the notch. So, if I were to look towards the notch your counting begins from the left of the notch right and you count in circular fashion starting from this end and going back to the notch. So, pin number 1 2 3 4 and so on in a circular fashion starting from the left of the notch and reaching the right that is how you count the pins on an integrated chip. Now, what is normally given to you is the description of the inside of the chip this is called a chip informally or an integrated circuit. So, what is normally given to you in a data sheet is the inside of a chip what comes out on the pins. So, you must that is called a pin diagram. So, you must learn to read a pin diagram and use the pin diagram to connect your chip properly all right in a circuit. So, much so than for connection of circuits and for your RC circuit experiment tomorrow. Now, anybody who is game to the challenge that I posed now what happens that circuit how does that circuit behave yes good good good come. My name is Gautam I am Gautam mechanical department very good. So, now, he is derived the expression straight away. So, just just write just sketch the expression I do not spend too much of time in deriving it I am just writing this then I will write it all right good. Now, just sketch this is a function of omega naught no no do not go there sketch it here raise it little raise it little bit here. It is 0 at omega naught equal to 0 this thing will be infinity. So, 1 by infinity 0. So, when omega naught tends to infinity this thing will tend to 1 and when omega naught is 0 this tends to infinity. So, 1 by infinity 0. That is right. So, the graph will be something like this. Good very good that is correct. And the derivative I wrote here comes around like this. That is all right that is fine good very good. So, what does this circuit do? This is called the high pass filter unlike what we had earlier this one emphasizes higher frequencies and deemphasizes lower frequencies what application can it have? Well of course, if you look at it in the noise context it would amplify noise that is not what we want to do right. What it would typically do is to isolate the faster part of a phenomenon for you not necessarily noise right. For example, in a certain sense in a certain region this can be used as what is called a differentiator. So, you know in the low frequency region this can be used as a differentiator. So, it could give you approximately the derivative of a waveform in the low frequency region all right. So, this is you know in the in the low frequency region. So, in a certain frequency region it could act as a as a as a differentiator all right. That is nice. So, so much so then for the instructions are there any questions that you would like me to answer before we disperse. So, if there are no questions then we will conclude the lecture here. Thank you.