 according to Sanskrit. If Hindi is derived from Sanskrit then that is wrong. This is sixth lecture in 110L. Today is 11194 and our topic would be signal waveform. What we are going to discuss today are the various kinds of signals that arise in electronic circuits. And the general topic is signal processing circuits. As part of this general topic, signal processing circuits, we shall first discuss some of the signal waveforms. It is important to understand what a signal is and what processing means so that we can device electronic circuits to perform the processing. A signal by definition in electrical engineering, a signal is either a voltage or a current. A signal is a voltage or a current which varies with time that is the signal is either a time varying voltage or a time varying current as simple as that which just any time varying voltage or current is not enough which carries some information that is called a signal. A signal is a voltage or a current which varies with time in such a manner as to convey an information, all right. For example, the current that you receive in a telephone receiver it varies with time and that is what is converted or transduced into sound and that is how you hear the telephone conversation. So that is a signal or the 1s and 0s of the dashes and dots in a typical telegraph system, the mode scored, these are time varying voltages or currents which carry information and therefore that is also a signal. Something interesting at the door? Okay why do not you open? So a signal is a time varying signal is a time varying voltage or current which carries some information that second part is important it has to carry an information, any time varying voltage or current is not enough. Also no voltage or current which does not vary with time can carry an information a steady current or steady voltage for example does not carry any information, so that is not a signal. It is important to distinguish between signals and non-signals, all voltages and currents are not signals, all right. In order to qualify as a signal it must be time varying, 1st qualification, 2nd qualification it must carry an information or a message. When I say the current in a telephone receiver or the mode scored that is transmitted on telegraph lines, these are all signals and the question of processing a signal arises when the signal that is available to you, when the signal that is available to you is not the desirable, is not the desired signal, you wish to convert it to some other form, some other form for example if you are given simply the current in the telephone receiver but that is not your desired signal, you want to process the signal so that it produces a sound, in other words you wish to transuse the electrical signal into a sound signal that is the form of that is the desired signal, so this the telephone receiver or the micro or the microphone that acts as a signal processing element, the signal may be very weak, all right for example an old record for example which has gone very weak and you wish to amplify this, all right. Amplification is a processing or you might have a slowly varying signal like this and you wish to find out where it is varying at the most rapid rate, then what do you do, you differentiate the signal, all right, so differentiation is processing or you might wish to have the integrated effect of a signal, in other words you might want to integrate a signal that integration is also signal processing, so anything that you do to the signal another example could be that the signal is mixed with noise which is almost always the case, a signal is the desired information, it is always mixed with some undesired information or noise, if you wish to separate the signal from the noise then you have to subject the signal to a signal processing circuit which gets rid of the noise and retains the signal only, so these are the various examples of signal processing. We shall come to signal processing circuits later but let us study some of the some of the common signal waveforms and a signal waveform is said to be continuous or so called BC, the ordinate here could be either a voltage or current, all right because signal is a voltage or current, time varying voltage or current, this could be V or I, we shall not write this again and again, excuse me and the independent variable is the time, so a signal which remains constant, a voltage or current which remains constant for all values of time, this is called a DC waveform or also called a continuous waveform, continuous signal waveform but as I said a non-time varying voltage or current does not carry any information so in that sense this is not a signal, it is simply a waveform, a DC or continuous waveform, consider now a battery, let us say a voltage V in series with a switch and a resistance R, all right and suppose the switch is put on at time t equal to 0, then if you measure the voltage across this V of t and you plot the voltage V of t versus t, this is t equal to 0, then for t less than 0 it is 0 and for t greater than 0 it is equal to capital V, is not it right, so t greater than 0 it is equal to capital V, in other words the signal now the voltage rises from 0 to a value capital V abruptly at t equal to 0, such a waveform is called a step waveform, step, it just it does look like a step in a staircase, all right and as I said it could be the voltage or the current, if you measure this current I obviously the current I shall also be of the form of a step, all right, I would be equal to simply V by R, V divided by R, so the current as well as the voltage in this circuit shall be a step, if the amplitude or the height of the step is unity, if this is 1 volt or 1 ampere then you call it a unit step, all right, a unit step therefore is a function which is 0 for t less than 0 and which rises to 1 at t equal to 0 and this function is given a special name of unit step function and is denoted by U of t, this symbol U is reserved for a unit step and therefore the voltage in the previous case, in the previous circuit which had a step of height capital V, if you have a voltage waveform of height capital V voltage step V of t versus t then obviously V of t can be written as capital V times U of t, all right, U of t is the unit step and this symbol is more or less universal, we shall reserve the symbol small U for unit step, it could be it could be a voltage, it could be a current, it could be a force, it could be an acceleration, it could be anything but U simply stands for unity height and when you write U of t it means that the abrupt rise or the step occurs at t equal to 0, t equal to 0 minus it is 0, at t equal to 0 plus it is equal to unity, this is the symbol that is reserved for this. Now suppose we have a battery, consider another circuit, a battery in series with a switch and then an R, a resistance R, you measure the voltage V t across the resistance R or the current that is flowing through R they are simply related by Ohm's law and what you do is you operate this circuit on and off, let us say that it is on for 0 to let us say t1, all right, that is it is put on at t equal to 0, then it is kept on up to time t1, then it is put off that is t1 to let us say t2, then again maybe it is put on let us say t2 to t2 plus t1 and off from t2 plus t1 to some value you calculate, all right. Suppose this goes on that is this switch is put on for a time t1, then put off for a certain period of time t2 minus t1, then again put on for t1, again put off for t2 minus t1 and so on, suppose it goes on doing this, then what would be the waveform, what would be the shape of the signal, if I plot V of t versus t then we shall have, we shall have for the on period the voltage shall be V, for the off period the voltage shall be 0, again for the on period voltage is V, for the off period the voltage is 0 and so on, such waveforms they look like rectangles, they are called pulses, pulses, so this is a pulse waveform and you have seen how it can be generated, pulse waveform, this is also a special kind of signal which is, it is a signal because it is time varying and if you call the on pulse, the on voltage is 1 and the off voltage is 0, then this forms the basic waveform of a digital computer, a digital computer after all works on the basis of just two levels of voltage or current, one level is called 1 and the other level is called 0, 1 sell 0, all right. So a pulse is in the heart of operation of a digital computer, there are other kinds of waveforms which are also of interest, for example you have a waveform once again let us say this is a voltage waveform, you have a waveform which rises linearly up to a certain instant of time and then it falls abruptly, okay, then again it rises linearly falls abruptly and so on. This waveform looks like the teeth of a saw and therefore it is called a sawtooth waveform, sawtooth waveform, it is just that similarity, there is no other, no other interpretation associated with this. Now a sawtooth waveform is extremely useful in a cathodic oscilloscope, in CRO, the electron beam that you see on the face of the CRO, have you seen a cathodic oscilloscope? Yes sir. The electron beam that you see, well it starts from the left end, it starts from the left end, goes linearly across the screen and then in almost zero time it comes back, so it is coming back you cannot see, all right, then again it goes back like this, then again comes back like this, this is the waveform to which it is subjected, that is the electron beam is subjected to a force which varies like this and therefore it goes linearly up to a certain point then comes back immediately, again goes, again comes back. The TV scanning is also done exactly by a similar waveform, a sawtooth waveform. The other kinds of waveforms which are useful in electrical engineering and electronics are the exponential and the other one is sinusoidal. An exponential waveform simply varies exponentially with time, for example you could have a V of t which is V e to the power At, all right, it is an exponential that is it varies as a power of E, well if A is 0, A is 0 then you see it is a DC or a continuous waveform, if A is greater than 0 then it is a growing waveform, if A is positive then with time t the amplitude or the value of the voltage continues to increase, all right, if this is let us say t equal to 0 this is t then when A is greater than 0 it is a waveform like this at t equal to minus infinity it goes to 0 as t goes what is the value here, this is obviously capital B, all right, when t equal to 0 it is capital V and then it continues to rise, so this is called a building exponential, an exponential which builds or which grows building or growing exponential, all right, that is a better term I support, growing exponential. On the other hand if A is negative, if A is negative then obviously the exponential will be a decaying exponential, if A is negative then you see what happens is it decays like this, again let us say V of t equal to V2, V to the at then this voltage, this level shall be equal to V2, all right, this level is V2 and then as t increases it decays, so this is called a decaying exponential, well these are not hypothetical quantities, a growing exponential and decaying exponential occur very frequently, very frequently in circuits, in electronic circuits and also in passive circuits, for example, for example let us consider a capacitor which is charged to a voltage let us say V, all right, a capacitor C which is charged to a voltage V so the charge in the capacitor shall be equal to C times V, let us have a switch here and the resistance here R, the switch is closed at t equal to 0, at t equal to 0 minus the charge is Q, let us call this Q0, Q0 is the charge on the capacitor at t equal to 0 minus, all right, so Q0, all right, let us use a different colour, at t equal to 0 minus the charge is Q0 and when the switch is put on the charge on the capacitor now finds a path for flowing, R this resistance allows a path to flow, if the voltage is V then obviously the current through the capacitor shall be V by R and you know when current flows through a resistor heat is generated, so the resistor consumes energy and therefore the charge in the capacitor decreases, so does its voltage and this decrease as you shall see later is exponential, the decay is exponential, all right or let us say let us say we have a battery V, we have a switch and then we have a resistance and then we have a capacitor C, suppose we measure the voltage across this V of t across the capacitor C, this is resistance R and this switch is put on at t equal to 0, all right, at t equal to 0 minus that is when the switch is off the capacitor voltage is 0, capacitor has no charge in it, so the capacitor voltage is 0, so if I plot the capacitor voltage V of t, t equal to 0 minus it is 0 then at t equal to 0, at t equal to 0 the switch is on, the capacitor cannot change its energy instantaneously and therefore t equal to 0 plus the voltage across this will still be 0 and therefore a current V by R passes in this circuit and tends to charge the capacitor, as soon as it accumulates a little charge, a voltage appears across it and as it continues to accumulate charge the voltage across the capacitor rises as you shall see exponentially and if sufficient time is allowed for the capacitor to charge, which capacitor will ultimately charge to a voltage equal to that of the, that of the battery V, that is in the equilibrium condition when the capacitor has been fully charged the current in the circuit shall be equal to 0 or the capacitor voltage shall be rising to a value of capital V, all right, so this is a case of a growing exponential and this is a case, the previous one was the case of a decaying exponential, in exponentials as I said if I consider a decaying exponential that is a V t equal to V e to the minus at, let us make it specific where a is greater than 0, all right, suppose we wish to find out the value of time at which V of t, you see V of t diminishes with time, it starts at capital V and it diminishes with time, suppose we want to find out the current at which the voltage reduces to V by e, that is the e-th fraction of the original voltage, then let us call this time as t1, so t1 would be given by V by e equal to capital V e to the power minus at1 and therefore we shall have e to the minus 1 as equal to e to the minus at1, right, V and V cancels, 1 by e is e to the minus 1 and therefore 1 should be equal to at1 or t1 should be equal to 1 over a, all right, this time t1 is given a special name, it is called the time constant of the exponential wave form and is denoted usually by capital t, capital t the time constant and therefore in terms of the time constant of the exponential wave form we could rewrite our exponential wave form as V e to the power minus t by capital t which makes it obvious that when small t is equal to capital t then the value of the wave form shall be equal to V by e, all right, it is a matter of simple calculation to show that a wave form like an exponential wave form like this e to the minus t by t let us consider the normalized wave form that is the normalized with respect to the value at t equal to 0, all right, in other words V t by V is the normalized and it starts from 1 then it decays like this, all right, it is very simple to show that if t is equal to 5 capital t then the value V t by V decays from 1 to e to the power minus 5 that is 1 by e to the power 5 which is approximately equal to 0.0067 less than 0.7 percent, all right, less than 0.7 percent which is approximately which is smaller than the accuracy of the ordinary instruments available in the laboratory, if this is a voltage you cannot measure a voltage to an accuracy less than 0.7 percent and therefore 0.0067 is for all practical purposes can be taken to be 0 and this is the reason why electrical engineers say that an exponential wave form has a lifetime of 5 times the time constant, is this, is the phrase understood? It is very loosely said it is not exactly 0 but since the value is less than the accuracy of an instrument available in the laboratory we say that for all practical purposes an exponentially decaying wave form has decayed to 0, it means it is dead after 5 times the time constant and that is why we say the lifetime of an exponentially decaying wave form is approximately 5 times the time constant. Exponential wave forms are extremely interesting not only they are important they are also very interesting because as you know d dt of an exponential wave form e to the at is also exponential, is it not right? Integral of e to the at dt is also exponential it is e to the at divided by a all right indefinite integral, so this makes life very simple to work with exponential wave forms as you shall see later and to illustrate the importance of exponential wave forms let us consider an example, we have a current generator which generates a 2 times e to the minus 250 t milli amperes okay current generator which generates not a constant value but it varies with time therefore it is a current signal 2 e to the minus 250 t milli amperes what is the time constant? 1 by 250 that is point double not 4 okay and this is passed through a series combination of a resistance R which is 3k and an inductor L which is 400 what you are required to find out is V R that is the voltage of across R V L voltage of across L and the total voltage V across the combination all right as you can see V R is simply I R is simply the current multiplied by R and this is milli amperes this is kilo so it would be 6 times e to the minus 250 t unit volts do you understand why it is 6? 3k is 3000 and 2 milli ampere is 2 times 10 to the minus 3 and therefore 3 times 2 now V L similarly would be L that is 4 d i d t that is times 2 times 10 to the minus 3 milli ampere mind you we want to convert it into volt times d d t of minus e to the minus 250 t that will be minus 250 e to the power minus 250 t is that okay can you tell me how much is this? Minus 2 e to the power minus 250 t volts all right so the polarity that is shown here is not the correct polarity V L is negative means that this and this positive and this and this negative all right this is because the current is a decaying exponential so d d t is negative the slope is negative and therefore V would be the simply the sum of the 2 sum of the 2 and therefore this would be 4 times e to the minus 350 t volts is that okay similarly we could have taken a capacitor and we could have found out the voltage across the capacitor by taking 1 by c and then integral of i d t that would also have been an exponential this is the beauty of an exponential waveform that an exponential signal generates in linear circuits if it is non-linear then of course you have complications but in a linear circuit an exponential signal input produces exponential signals as output because integral and differential of an exponential is again an exponential finally we consider the most important waveform of all that is the sinusoidal waveform a sinusoidal waveform is very much a practical proposition because if you take a coil if you take a coil and rotate it in a constant magnetic field then you know that a voltage an emf is generated in the coil which is sinusoidal so electricity generation at generating stations the generated waveform is sinusoidal and a sinusoidal waveform in general can you give me another example of sinusoidal sinusoidal quantity in any any other field it does not matter whether it is electrical or non-electrical simple harmonic motion simple harmonic motion who executes a simple harmonic motion a pendulum a damped pendulum or undamped pendulum undamped obviously a damped pendulum all that you have to do is to is to add an exponential term you a damped pendulum will execute oscillations of this form sin omega t plus theta the exponential term in if it is included then this takes care of the decaying amplitude that is a pendulum gradually executes smaller and smaller arcs all right therefore a pure sinusoid let us to bring variety into experience let us say we now consider a current I of t will be of the form capital I sometimes we call it I m sin of let us say omega t plus alpha let us say it could be sin or it could be cosine all right or it could be let us say V of t is equal to some V m cosine of let us say omega t plus alpha sin or cosine it does not matter both are sinusoidal there is a favoritism in favor of sin we do not say cos sinusoidal because it uses more letters C O is to be added that is all economic so we say sinusoidal both are sinusoidal now the various parameters of a sinusoidal waveform are firstly this quantity which is called the peak value obviously this is the maximum of the right hand side because sin can either be the maximum value is 1 and therefore I m is the maximum value so I m is called the maximum value or peak value P E A K or simply called amplitude amplitude these are the three days by which it can be called omega is called the frequency in radiance per second in radiance per second in a particle executing simple harmonic motion can be described by another particle revolving round a circle so the number of complete circles it executes in 1 second is the frequency in radiance okay omega is the frequency in radiance per second T is the time in second and alpha is called the phase or sometimes well the actual actual nomenclature should be initial phase that is when t equal to 0 t equal to 0 mark these words carefully when t equal to 0 the angle is simply alpha that is why alpha is called the initial phase or sometimes loosely simply the phase actually phase stands for whatever the argument of sin is that is the total total quantity omega t plus alpha is the phase at time t alpha is the initial phase but loosely it is often called the phase the sketch of a sinusoidal waveform let us say V m cosine of omega t plus alpha is very familiar it is something like this where this amplitude is V m and this is minus V m the maximum in the positive direction is plus V m the maximum in the negative direction is also the same it is minus V m and this value would be V m cosine alpha this is t equal to 0 what is the value of time corresponding to this obviously okay if this is omega t then this would be minus alpha agree what is the difference between between this time and this time in terms of omega t what is the difference of omega t between this point and this point pi by 2 all right so what is this value then I mean minus alpha plus pi by 2 wonderful and what is the difference between these 2 points that is the point at which 2 pi omega t equal to 2 pi so if I denote the corresponding time by capital t then omega capital t is equal to 2 pi and what is capital t the time period and what is the relation between time period and frequency inverse and therefore if we say that this sinusoid has a frequency of f in cycles per second capital t is the time for one cycle so the number of cycles per second is 1 by t so if we say f is the frequency 1 by t then the unit of f is cycles per second or hertz all right and so omega is 2 pi by capital t because omega capital t is 2 pi and therefore this is equal to 2 pi f omega is in radiance per second and f is in hertz after the name of Hendrick hertz to illustrate this sinusoidal sinusoidal will be your bread and butter in this course on introduction to electrical sir electronic circuit it would also be extremely useful in the course 120 which is electromechanical energy conversion which in which you will see that all generators generate sinusoidal waveform all right then what you do with it is somebody else's business but originally you always generate a sinusoidal waveform let us take an example a non-trivial example I am warning the example is on sinusoidal waveform express the following sinusoidal signal as a function of time the description of a sinusoidal given signal is given you have to express it as an equation or as an expression and the information that is given is the frequency is 100 hertz okay amplitude is let us say it is a current amplitude is 10 milliampere and this is I of t and it passes through these are the information that is supplied passes through 0 value with positive slope with positive slope at t equal to 1 millisecond this is the information that is supplied about the waveform you have to write an analytical expression for this what are the things given the frequency is given the amplitude is given and about phase obviously what remains is the phase isn't it about phase of the initial phase alpha the information that is given is that it passes through 0 the value of the waveform passes through 0 with positive slope at t equal to 1 millisecond all right so I can write my I of t as 10 times 10 to the minus 3 always normalized always in terms of SI 10 milliampere then would you use a sign or a cosine we have not been asked we have not been told so we can use anything we like let us say we use a cosine cosine of omega t omega is now 2 pi times 100 which is 200 pi t all that remains to be found out is alpha all right and it has been told to us that the value is 0 the value is 0 it is 0 at t equal to 1 millisecond with positive slope now the importance of positive slope and negative slope you can see that a cosine waveform is like this here at this point it is positive slope it passes to 0 with positive slope at this point it passes to 0 with negative slope so it is this point that is given all right now what is the condition for positive slope let us take a cosine of omega t plus alpha d dt of cosine omega t plus alpha is minus omega sin of omega t plus alpha so if the slope is to be positive it is not a trivial question you follow this carefully if the slope is to be if d dt is to be positive then obviously sin must be negative because there is a negative sign here how when is sin negative when omega t plus alpha itself is negative and therefore the condition that it equal to 0 at t equal to 1 millisecond with positive slope implies that this quantity should be equal to minus this quantity should be negative minus pi by 2 that is which will now give away alpha that is our condition will be 200 pi 1 millisecond so 10 to the minus 3 plus alpha should be equal to minus pi by 2 is that clear how we found this information out we are good that the argument omega t plus alpha must be negative and since cosine is 0 the value must be minus pi by 2 and therefore alpha is equal to this is minus 0.5 pi and this is this is 0.2 pi so minus 0.7 pi is that okay and therefore my final answer is 10 to the minus 2 10 into 10 to the minus 3 cosine of 200 pi t minus 0.7 pi so many amperes all right this is the final answer be careful about handling sinusoidal quantities they depending on the slope and other information it is very easy it is very easy to make a mistake. Now sinusoidal waveforms are a special kind of waveforms which go by the general name of periodic waveforms periodic periodic waveform a function f of t we will not use f of t because small f we have used for frequency a function x of t is said to be periodic is said to be periodic if x of t is equal to x of t plus n t where capital t is a constant and n is an integer positive or negative including 0. X of t is said to be periodic the quality of meaning is that x of t values repeat after every regular interval and this interval of repetition is said to be the time period so capital t here is the time period and n is an integer n is an integer obviously n could also be 0 x of t is obviously identically equal to x of t and n could be any other integer positive or negative for example a cosine omega t plus alpha is periodic because it is equal to a cosine omega t plus capital t plus alpha where capital t is equal to 2 pi divided by omega it is therefore a periodic waveform now a periodic waveform because it repeats after regular intervals how do you measure a periodic waveform how do you device a meter to measure a periodic waveform there are various kinds of measurements for example you wish to see the periodic waveform then you have to display it on a cathodic oscilloscope that displays the total waveform that is a cosine omega t plus alpha all right now if you want a parameter for the periodic waveform there are various parameters possible for example one could think of the average value the average value is always spoken of with regard to one period all right average value is over one period what is the average value of a sinusoidal waveform then average value of a sinusoidal waveform 0 if you take a sine omega t for example the average value equal positive equal negative and the average value is 0 however if I take let us say a triangular waveform like this a symmetrical triangular waveform like this let us say this is 0 t and let us say I of t all right what is the average value suppose this is I m yes I m by 2 this is the time period and therefore the average value is I m by 2 half of the peak value in general the average value of a periodic waveform let us say I average is equal to integral you integrate over one period 0 to t I t dt and divide by capital T this is the average value definition now does it matter whether the lower limit is 0 or some other value can we have t1 yes then the upper limit should be t1 plus t you can start anywhere but you have to end after an interval of capital T so this average value is independent of small t1 all right now since in a sinusoidal waveform the average value is 0 counted over one period in a sinusoidal waveform the average value is usually calculated over half a period for a sinusoidal waveform I average is calculated over half a waveform that is you take you take half of the waveform the positive cycle find out the area and then divide by divide by t by 2 and you can easily show that this is given by twice I m divided by pi you can easily show you take a sin omega t for example integrate from 0 to t by 2 then divide by 2 which is equal to 0.637 I m you must remember the definition of the average value for a sinusoidal waveform if you take over one period it is identically equal to 0 we always peak of half a period next time next time would be Monday again isn t it this Friday is a holiday oh I see a smile on the face so Monday we shall talk of other kinds of parameters for periodism that is all for today.