 So, in today's lecture, we will cover a couple of review topics as they relate to acoustics engineering. So, essentially what I will be covering is a bunch of subjects and concepts which you may have gotten exposed to during your high school years or even during your first and second or third year while you have been doing your undergraduate program in engineering. Some of these system concepts we will be mapping directly later in the course in the acoustics course. So, it will be fruitful to review some of these and also practice a little bit more before we actually start walking deeply into the area of acoustics. So, essentially I will be covering four key areas. One is that I will in a very brief try to explain what is a linear system because we will be using a lot of linear principles in the area of acoustics. The second area which I will be covering again very briefly today is the area of complex variables and then associated with that would be complex time signals. And then that from there we will move on to RLC circuits as we use them in electrical engineering because later in the course we will find some of the tools and techniques we use in RLC circuits to be very fruitful and useful in area of acoustics. So, that is in a brief the overview of what we will try to accomplish today. So, we will start with linear systems and we have often heard that this system is linear. System A is linear system B is not linear, but specifically whenever we use a term linear there is a context behind it. So, one suppose one says system A is linear then that person should be unless it is really clear should be explicit in saying that system this particular system is linear in variables A and B because you can have a complex system and it could be linear in some variables and it could be non-linear in some other variables. So, we will take an example. Suppose I have a relationship a y equals a x plus b or I have another relationship d y over d t equals c d x over d t plus k. So, in this case the system is linear between y and x in this case the system is linear between d y over d t and d x over d t and of course, if I integrate this then I will find that this the same equation is also linear in y and x. So, here we can say y and x are linearly related and we can make a similar statement in context of this particular relation also. So, I will I will give you another example. So, I have a second derivative of y so second derivative of y in time let us say it equals a constant c 1 times second derivative of x in time plus another constant c 2 del x over del t plus c 3 x. So, again by inspecting this whole relation because the power of y is 1 in this whole relation power of x in the entire relation is 1. So, you can say that this is a linear relation between x and y because a x and y are on the opposite sides of the equivalence sign and the second thing is that both the power of x is 1 and power of y is 1. So, even though you are having higher order differentials the relationship can still be linear if the power of the variables are 1 and they are on opposite sides of the equivalent sign. So, typically if you have a linear relationship between two variables and when you try to plot variable a versus variable b you will get a straight line. The straight line can pass through the origin or it can cut the x axis or it can get the y axis at a nonzero intercept. So, that is another feature for a linear system that if I have two variables and I am plotting these two variables against each other everything else being same then I will get a linear response. The third feature is that I can use superposition principle position principle. So, in a linear system I am able to use a superposition principle what does that mean suppose I have a system and I have an input x 1 and it yields an output y 1 and then in another case I provide it an input x 2 and it yields an output y 2 then if I provide an input x 1 plus x 2 then if it is a linear system the output will be y 1 plus y 2. So, this is again an important feature of a linear system which later as we go deeper into the area of acoustics we will be using to understand the response of systems when they are receiving complex time signals. Suppose a system which is linear in nature it is receiving a frequency input of x hertz x 1 hertz at a certain amplitude level and it gives me a certain output response and then it is receiving another input at x 2 hertz which is different in frequency and it gives me another response then if I try to sum up x 1 plus x 2 and the response on the response that is the output will be summation of the individual outputs. So, this comes in very handy and we also use this idea in the context of transfer functions which we will learn later today itself. The third feature of a linear system is that if it is having a particular input shape or a particular type of a signal then the output signal in the steady state situation will be of a similar shape whatever is going in the output is going to be of a similar shape whatever is coming out and this is very important to understand because if we are sure that there is a linear system and it does not matter whether it is acoustics related or any other thing and if we know that if it is a linear system in nature and if it is receiving only sinusoidal inputs and in the output if I am seeing a stepped function or a non sinusoidal output then either my system is not linear or my experiment is not correct or there is something wrong in my understanding or analysis of the whole problem. So, consider a spring mass system which also has a damper and then the spring and the damper are now connected to a mass m the stiffness of the spring is k the damping constant is c and I am exciting this entire system by F naught E j omega t. So, my input in this case equals F naught E j omega t then my output response. So, suppose I am interested in knowing the displacement at this point and suppose I call it x. So, then the output response has so the shape of the input function is defined by this term E j omega t. So, output response also has to be such that it is some constant times E j omega t plus a phase because the phase this component is not going to alter the shape all it will do is it will induce a difference in the timing at which the peaks of input and the peaks of output happen. So, if phase is 0 then the peaks of input function and peaks of output will coincide in the phase is not 0 then they may not necessarily coincide, but the shape of the function for input and output will remain same. Another feature of a linear system is that if I have small changes in the input then the changes in the output response will be proportional to the magnitude of changes which are coming in. So, if I have small changes coming in the changes in the output will be small in a proportional sense because there will be a proportionality factor. If I have large changes coming in the input function then the changes in the output also will be accordingly larger. So, we will do another example. So, I have a beam equation E i del 4 w over del x 4 and where w is the displacement of a beam. So, in a beam equation I have this term coming up and this is related to the moment exerted in a beam. Now, here m and w are linearly related. Now, we know that I equals the thickness of the beam time cube of it times the width. So, b t cube over 12 if it is a rectangular cross section. So, this is i. So, if I put this in then I get E b t cube over 12 times 4 derivative of the deflection equals moment. So, again m and w this may appear very trivial, but it is important to note because as we do more detailed analysis m and w are linearly related, but m and t are not linearly related. So, that is all I wanted to capture about what is a linear system. So, that once we start solving complex problem in the ream of acoustics and also start doing experiments, we at the back of our minds be sure that the system which we are dealing with is it really linear or is it not so linear. So, that we are cognizant of whether our answers or experimental observations are consistent with our assumptions or not. So, the second thing I wanted to capture was complex variables. So, we know from our high school mathematics and also first and second year of engineering mathematics that exponent of E j theta is cosine of theta plus j sin theta where j in this course is minus 1. Typically we use i as square root of minus 1, but in this course we are using j as square root of minus 1 because we are using i for other variables such as correct. So, if I represent this graphically, I have a real axis, I have an imaginary axis, I have a vector as radius r equals 1 and this angle is theta. So, if I draw normal here, this value which is along the real axis is cosine theta and the vertical component is sin theta. So, this is my function and this is its graphical representation. So, understanding this picture, we can then generalize that if I have any complex variable of the form a plus b j, I am able to represent it in such a form. How do we do that? So, again let us say there is a complex number a plus b j and if I have to represent it graphically then the length of the radius will be the magnitude of this variable. So, magnitude is a square plus b square the square root of this term. Similarly, the angle or the value of theta will be such that it is tan inverse of b over a. So, this is my theta. So, using such an approach, again I construct a circle, this is my real axis, imaginary axis, that is my radius and the magnitude is a square plus b square and that is the value of theta and that is tan inverse b over a and this is in radians. So, if I have any number 3 plus 4 j or whatever number I have, I should be able to fairly easily represent it in this graphical format. So, we can use these complex numbers in a variety of ways, we can also use them to solve equations. So, let us say that I have an equation x cube plus 1 equals 0 and if I want to solve for x, so my x cube equals minus 1. Now, let us see how we can represent minus 1 in a graphical format. So, minus 1 could be written as e j pi, we can also write minus 1 as e 3 j pi, we can also write the same thing as e pi j pi and so on and so forth. So, my x is cube root of this thing, so is either it could be e j pi over 3, that could be one solution or it could be e j pi which corresponds to this value of minus 1 or it could be e pi j pi over 3 and so on and so forth. So, now let us start them on the complex plane. So, I draw a circle that is my real axis, this is my imaginary axis. So, my first solution was e j pi over 3, so pi over 3 is 60 degrees. So, that is my first solution x 1, x 1 equals e j pi over 3 and this angle is pi over 3. My second solution x 2 equals e j pi, so this is my x 2. My third solution is x 3 equals e pi pi j over 3. So, pi pi j over 3 is here, this whole angle, this angle is pi pi over 3, this is my x 3 and all other solutions x 4 because this will have infinite solutions will be basically repetitions. So, my x 4 will be same as x 1, x 5 will be same as x 2, x 6 will be same as x 3 and so on and so forth. So, I have a graphical representation of three unique solutions for the equation x cube plus 1 equals 0. We will very quickly cover some very fundamental basic identities now about complex variables. So, let us say z equals a plus b j. So, I can express it as a real number d times e j theta, where theta as we saw earlier is tan inverse of b over a and d is the magnitude a square plus b square the whole thing square root and then I have another number z star equals a minus b j. So, I can express the same thing as e minus j theta. So, z star is complex conjugate of z, this is a definition. So, complex conjugate of z which is a plus b j is a minus e j and a property of this is that z times z star equals d square which is the magnitude square. Couple of other identities, if z equals 1 then in complex plane I can write z equals e j theta, where theta could be 0, 2 pi, 4 pi and so on and so forth or it could be minus pi, minus 3 pi and so on and so forth. Similarly, if z equals j which is a unit length along the imaginary axis, then in complex plane z could be represented as e j pi over 2, actually you are right. So, actually it should have been yes it should have been 0, 2 pi 2 or an alternative solution could be z equals minus e j theta, where theta is this thing. So, I have two sets of solutions, one is e equals plus e j theta, where theta equals 0, 2 pi 4 pi and so on and so forth. And the other one is z equals minus j theta, where the associated theta values are minus pi 3 pi and so on and so forth. So, going back if z equals j then z could be e j pi over 2 and then we can again or it could be minus e j 3 pi over 2. So, similarly if z equals 1 over j then I can call it as e minus j pi over 2 equals minus j. So, these complex variables can be used in a variety of ways. If I have additions of different phases and if I use complex variables I can very easily add the exponential term and then express the exponential terms in terms of cosines and sines and I am able to manipulate complex additions and subtractions of vectors by using the idea of complex variables. So, for instance I know that cosine plus pi over 2 is minus sin theta, but if I have to use complex variables it is basically real component of e j theta plus pi over 2 equals real component of e j theta times e j pi over 2 equals real component of cosine theta plus j sin theta and then I now expand on e j pi over 2. So, cosine of pi over 2 is 1 plus now cosine of pi over 2 is 0 plus j times sin of pi over 2 is 1. So, I get real cosine theta times j minus sin theta equals minus sin theta. So, I can prove some complex identities also using such an approach. This is again a simple illustration, but we will use some of these principles and notations for calculation and that will make our life significantly easy. So, we have talked about what is a linear system. We have very briefly captured about some basic things about complex variables. So, now we will talk about complex time signals complex time signals. So, it happens that quite often we develop differential equations in time and also our space. For instance you have a spring mass damping system such as f equals m x double dot plus c x dot plus k x and we have to solve it and in such situations this notion of complex time signals comes in very handy. So, we will do that suppose you have f equals m x mass times acceleration plus damping times velocity plus stiffness times displacement. So, solution for this kind of an equation in a very general sense could be a form x bar E s t where x bar and s both can be complex numbers. So, again a very general solution could be of this form x bar E s t where x bar is a number and s is a number, but they need not be real numbers and x bar is called complex amplitude. Also s is called complex frequency and this can be decomposed further as sigma plus j omega where sigma and omega are real. So, sigma and omega are real numbers, but s itself could be complex thing. So, if I have this kind of a representation my velocity will be real component of s x bar E s t basically what I have done is differentiated the displacement and then I take its real component that is my velocity which I can physically measure. Also note that if I have to integrate this thing. So, that gives me x bar over s E s t. So, again differentiation and integration becomes fairly simple process here assuming that x bar does not contain time x bar is a constant. It can be imaginary, it can be purely real, but it does not have time embedded in it. Yes. So, the time is in t. So, basically this and that is that is because it is a linear system. We have assumed the shape of well I mean I am not using features of linear system to talk about this, but I have assumed that x bar E s t is a complex time signal it could be anything. I am not even saying right now it is a solution of this thing. If there is a complex time signal of this shape then this is by definition called complex amplitude x bar s by definition we are calling complex frequency which is equal to sigma plus j w omega and x bar is a number. It is not having time embedded in it. So, and then if I have to find velocity I know complex displacement. So, I just differentiate it and take its real component this is my velocity. If I have to integrate x all I have to do is divide that by s. So, that is all I am saying. So, we will do an example. So, let us say my x bar equals 3 E j pi over 4 and s equals minus 2 plus 4 j this is to black color. So, my x t equals the way we have defined is real component of x bar E s t equals real component of 3 E j pi over 4. So, again there is no time in it in x bar itself times E minus 2 plus 4 j times t. So, the time is right now only in the exponential part just to reiterate x bar and s are not functions of time themselves. So, now I rearrange. So, I get real component of 3 E minus 2 t. So, I am basically breaking up into real and imaginary component times E j pi over 4 plus 4 t. Did I do the math right minus 2 t j pi plus pi over 4 plus 4 t. So, which tells me. So, this is real I can take it out times and then the real portion of this guy is cosine pi over 4 plus 4 t this is my x. So, we have broken it x into 2 components 1 and s is an exponential component and then other one is an oscillatory component and we are multiplying both of them based on mathematical manipulations which we did here. So, now we will plot x as a function of time. So, we will plot this function over time. So, I have I am plotting x t here this is my time axis and. So, I get my plot something like this and so on and so forth and the envelope is defined something like this. So, this is the envelope which is a plot of 3 E minus 2 t which we got from here. So, this is defining the envelope and the oscillating signal is the actual value of the function over a period time. It just happens that because the phase difference here there is a non-zero phase value which is pi over 4. So, at t equals 0 this point is not touching the envelope. So, based on what we did earlier x bar equals 3 E j pi over 4 and this has 2 components 1 component is 3 the other component is pi over 4. So, this is like your envelopes upper limit that is the maximum value the function can get. This is your phase and then the complex frequency is minus 2 plus 4 j again it has a real component minus 2 and this is rate of decay and it is symptomatic of the damping characteristics of a system and then you have 4 and that is essentially your angular velocity. So, you have a complex amplitude which tells you the magnitude which defines the magnitude also and also the phase part of it and then you have complex frequency which helps you understand the damping characteristic of the oscillation or the signal and also the angular velocity or the frequency. So, that is all I wanted to talk about complex time signals at least today and we will talk more about complex time signals as we move forward. So, now we move on to the third big category what we will be talking about today and that is about RLC circuits. So, resistance inductance and capacitance circuits. So, we all know at the first electrical element very commonly used is a resistance if I have current I and R then voltage across the resistance is I times R. Similarly, if I have an inductor of value L and there is a current going through it I then voltage is rate of change of current d I over d t times L and then the finally, if I have a capacitance I is the current going through the capacitance of value C then the current and voltage are related such that current equals capacitance times rate of change of voltage. So, this is these are the three basic elements in electrical circuits which we will be using a lot in acoustics. Then we have Kirchhoff's current law which essentially says that if I have a node through which this flow of current I 1 I 2 I 3 then Kirchhoff's current law says summation of I 1 plus I 2 plus I 3 equals 0. Another law developed by Kirchhoff's is called KVL or Kirchhoff's voltage law. So, I have a loop a closed loop of electrical circuit let us say around this element I have a voltage of V 1 here I have voltage of V 2 and here I have these are three elements a, b, c and here I have a voltage difference of V 3 then across voltage law says that if I go around a loop then it is V 1 plus V 2 plus V 3 equals 0. So, we will be using these notions also in acoustic circuits so that is why good to recapitulate those. So, the other thing I wanted to talk about so let us look at a circuit. So, we will do a very simple example and use some of these electrical engineering elements and try to understand them a little better and also recapitulate some of the concepts which you have learnt earlier. So, let us say I have a circuit with the voltage source V i which generates current I flowing through a resistance R and also a capacitor of value C. So, the question is that if I know V i which could be pulsating then what is the voltage difference across the capacitor as a function of time. So, we know that V R plus V naught equals V i from Kerkhoff's voltage law and we also know that V R equals I R and we also know that I times I equals d V naught over d t plus no times capacitance. So, current and voltage across the capacitor are related by this thing. So, if I synthesize all these relationships what essentially I get is V i equals R C d V naught over d t plus V naught and what you have here is first order linear differential equation. So, this equation when we try to solve it a very general solution will have two parts one will be what we call a homogeneous function and the other will be a particular response. So, we will have a homogeneous response and we will have a particular response and the homogeneous response will be highly transient in nature. So, after a certain period of time it will decay. So, for instance if I have a system if I have a glass and a moving mass hits it initially the glass will shake, but after a certain time it will start moving steadily. So, the initial response is called the transient the transient part that dies over a period of time. In a lot of acoustic systems we are more interested in steady state response. So, we will focus a little bit more in this course on the steady state response of the system and we will focus not that much on the transient response of the system, because a lot of physical devices suppose you have an engine which is running and once I start an engine there is a perturbation in the system and you have here one particular type of noise coming out from the system, and once the engine has started running the sustained noise level is of a different it may be of a different type and in general people are bothered that over a period of time how does the engine sound rather than what happens right at the point when you are starting an engine. So, we will be focusing a little bit more on the steady state response of our acoustic systems and that we can do in a fairly straight forward way if we use complex variables and also Laplace approaches. So, for this particular function let us say that V i is real component of V i E s t this is known input I know how a power generator is working. So, I know this. So, if I know this and if the system is linear then my output also has to have a similar shape. So, I can assume that V naught is real component of a complex time signal V naught E s t. So, now I take this relation and I plug it. So, I put this relation and the relation for V naught in this entire equation. So, what I get is V i E s t equals r c times d over d t of V naught E s t plus V naught E s t and now I differentiate and do the math. So, what I get is r c s V naught plus V naught E s t. So, I get V naught equals V i over 1 plus r c s or V naught over V i equals 1 over 1 plus s c r. So, this will help me determine the steady state response of this. The time component is embedded in this exponential term I know V i. So, I can calculate V naught and then I take the real component of V naught E s t and I figure out what is the actual physical voltage across the capacitance. So, again we are seeing that the complex notation comes in very handy to solve some of these problems especially if we are interested in solving steady state response of systems. So, we had drawn this circuit V i and V naught. So, input was V i and output which we were trying to find out is V naught and we developed a ratio V naught over V i is this thing. This ratio is called transfer function in a strict sense here it just happens that V naught and V i are the complex magnitudes of the actual voltages. In a very strict sense the transfer function which can be designated as H s is basically response over input, but it just happens that E s t in this way thing goes away because it is a linear system. So, transfer function this is what we call a transfer function and linear systems the time wave component of it will always go away because it is a linear system. So, they will be they will cancel out and the idea of transfer function works only in a linear framework. We do not use a whole lot of transfer functions in non linear systems because they depend on time and then for each time instance you could have a different ratio. So, then it becomes very complicated rather there you solve equations in time in real time itself. So, we will write couple of transfer functions. So, H s equals if my H s or transfer function V over i then for a resistance my transfer function is R for an inductor my transfer function is. So, this is again current going through the resistance and the voltage across the resistance the ratio is R for an inductor with value l this number would be s times l for a capacitor of value c the transfer function would be 1 over s c. So, again it is handy to remember these ratios these values and what we will do in another example is that we will construct a circuit. So, those transfer functions for what? So, the transfer function is in these three elements the transfer function is basically the ratio of voltage across the element and the current going through the element not through the whole circuit current going through the element and voltage across the element. If I take the ratio of those then I get these values. So, there are the ratio of the complex voltage means complex voltage. So, transfer function can always be complex because they are ratios of complex entities remember when we did this V naught is complex and so is V i is complex. Similarly, I can be complex when I want to measure the actual current then I take its real component the phase is embedded in the imaginary part. But the ratios of volt V naught and V i. So, this is complex this is lower case that is real this is a real number. Similarly, V naught is complex, but lower case is real when I take real component of V naught E s t. And what about s? s is again complex it can be complex as we saw in an earlier case if you have vibrations and if they are damping damping over a period time it can have a real component and an imaginary component. And if it is a pure vibration it is a if it is a pure non-dampening vibration then it will be purely imaginary. So, we will do an example and see how we use some of these transfer functions to simplify circuit analysis. So, let us say I have a voltage source V i and I have a resistance R and inductor L and a capacitor C. And I am interested in finding what is the actual voltage across capacitor. So, one approach which we saw earlier was add these up develop a differential equation and solve it is a faster approach. So, the question is that what is what is V i. So, to do this maybe I can be well placed if I can figure out what is V naught over V i which is complex representation of V naught and V i. So, how do we do that? If I know this ratio that is the transfer function then I can very easily multiply this times this find V naught on the complex plane take its real value and that gives me V naught. So, a relatively straight forward way is that I transform this circuit in frequency space which means that I replace R by its equivalent transfer function which happens to be R I replace S L by its I replace L by S L and C by S C. So, I am transforming this in frequency space of frequency domain. So, V i transforms into a complex entity upper case R remains R inductor transforms to S L and there is a mathematical proof, but we are just capturing whatever is the learning from electrical engineering here directly and this capacitor gets transformed into let us see. So, now I compute the impedance of this circuit and I do not have to worry about phase and everything because the phases are embedded in these things themselves. So, let us just first compute the impedance of this block. So, impedance let us call it is Z. So, in that case because S L and S C these are in parallel it is 1 over S L we just assume that they behave like resistors even though because there is S embedded here. So, 1 over S L plus impedance of a capacitor is this and then I take inverse because they are in parallel and what I get is 1 plus S square C L over S L which is S L over I have to take inverse of this 1 plus S square C L. So, I make another circuit I have a resistance R, I have this impedance Z where Z equals we calculated S L over 1 plus S square C L. So, what we have done is just replaced and we are in frequency domain and we are trying to find the voltage across this impedance Z. So, now we can calculate V naught. So, V naught is what if my complex current is I then it is basically V I over R plus Z right that is my current complex current I and then I multiply that by Z that will be the voltage complex voltage across Z. So, I do the math. So, I have V I over R plus S L over 1 plus S square C L that is my complex current times Z S L over 1 plus S square C L. So, if I do this entire math carefully what I get my final relation as V naught equals V naught over V I is S L over S square C L R plus S L plus R. So, I know now I can take I can multiply this by V I and take its real component and I can figure out what is my actual voltage going to be. So, again you are using complex variables and also the idea of complex time signals and what you are doing is you are converting circuits which are in time domain into frequency domain by making these transformations you are replacing S I mean L by S L and capacitance by C and then you are adding them up appropriately and then calculating complex current complex voltage and then mapping it back by taking their real components. Sir, even in the frequency domain the Kirchhoff laws holds. It holds. I mean is that any special property or it can be taken just. See the current will still has to hold something conservation laws have to work similarly energy laws have to obey the same principles. So, they still hold if they do not hold then you cannot do a circuit analysis then you need another system. So, they have to hold. So, we will do one final example. So, suppose I have a some black box of some impedance. So, the complex current is let us say I and complex voltage is V and let us say that the transfer function which is a function of S which is complex frequency like in this case complex function is a function of S it depends on frequency. So, this is equivalent to S square plus 2 S plus 2 over S square plus S plus 1 and the question is that if my current signal if my current real current I is something like this it is a flat line. So, it is like a step function at t equals 0 before t equals 0 it is 0 at t equals 0 it goes up and it becomes 1 and this is my time axis. So, before t equals 0 it is 0 at t equals 0 it steps up becomes 1 and remains constant throughout the. So, if this is my current what will my voltage look like given that this is the transfer function. So, in this context we should remember that this is a linear system what does that mean that if my steady state input let us say I is constant then my steady state output if output the unknown is V that will also remain constant because it is a linear system we are using that feature bring it. So, I go to the complex domain and I say V over I equals S square plus 2 S plus 2 over S square plus S plus 1 now I know. So, I expand this I get S square V plus 2 S V plus 2 V equals S square I plus S I plus I now these terms relate to derivatives of current and voltage second derivatives and first derivatives. So, when everything is steady state these terms can be in a steady state situation they will be 0 is that right. So, 2 V equals I so if my current is 1 then voltage is 2. So, transfer functions tell us a lot and they are basically properties of the system they do not change with excitation signals input signals they are properties of the system for instance in an electrical circuit we saw that they are dependent on resistance inductance and capacitance and of course, there is a dependency on frequency in that sense they are dependent on the input, but they are a property of the system. So, if I know my input if I know my steady state input I should be able to figure out my steady state. So, that is all I wanted to cover for today and we will meet again and we will again review of a few more concepts in next one or two more lectures and then after that we will move into area of acoustics thank you very much.