Lecture: 23

[Music] [Music] namaskar so today in this lecture we will discuss about the singularity functions what are those functions and we will see how we can use those functions in our circuit analysis so let's start what is singularity functions singularity functions are those functions that are either discontinuous or have discontinuous derivatives now the basic understanding of singularity function will help you to understand the response of first order circuit to certain application of independent voltage or current source so in the circuit if you apply either voltage or current source suddenly to the circuit so that is the sudden injection of current into the circuit so that would be something which you can represent with the help of a mathematical function so we will see how will represent that particular phenomena with the help of a mathematical function another phenomena is that whenever you apply any voltage there might be some sudden surge coming up or maybe any event which is happening very for very small time period you will have one surge kind of voltage appearing in the circuit so that is also another practical event which happens in the circuit and we will see how we can equivalently represent that event also into the mathematical function so that we can use them in our circuit analysis so let's see that the singularity functions which we use in the circuit theory are nothing but the functions which are like unit impulse unit step and unit ram so when you suddenly apply the voltage source to the circuit it means that you are adding unit step to the circuit similarly when there is a sudden surge coming up into the circuit then you will say this is the unit impulse being applied to the circuit and then if the voltage is building up gradually to the in the circuit then you will say that is can be represented with the help of unit ramp function now these basically the unit step serves as a good approximation to the switching signals that arise in the circuit with the switching operations and it is very helpful in the neat and compact description of some circuit phenomena especially the step response of rcn rl circuit so in the later session of our discussion we will try to understand that what will happen to the circuit response when you apply step response to the circuit so this would be the important activity which we will try to understand before that let us try to understand what are the various functions which we just mentioned here like unit step unit impulse and unit ram functions so first let's see what is unit step function unit step function is 0 for negative value of time t and 1 for positive value of time t as shown in the figure so if you see the function value is 0 up to time just less than 0 and suddenly it becomes 1 at time 0 plus so this kind of function is called unit step function mathematically how we represent we represent ut that is unit step function equal to 0 for t less than 0 and 1 when t greater than 0 now important thing to understand is that unit step function is undefined at time t is equal to 0 because it is changing abruptly from 0 to 1 and that's why it is called as a singularity function it is dimension like less like any any other mathematical function such as sine or cosine now if the abrupt change occurs at any time ah t is equal to t naught where t naught is greater than 0 then instead of t is equal to 0 the unit step function will become 1 at time t naught plus so how the unit step function will look like in that case your value just up to t naught minus this just before t naught will be zero and just after t not you will get the value as 1 at t naught it is undefined so that is why it is shown as a dotted line so how you will represent mathematically you will say u t minus t naught is equal to 0 to 1 where t less than t naught would be the condition when the value is 0 t greater than t naught the value of unit step function would be 1 so in this case what we can say we can say that unit step is delayed by t naught seconds so it is delayed by t not second because it is starting not from 0 it is starting from at some time t naught now if you change t to minus t naught the unity step function will now become u t plus t naught and the value would be 0 and 1 it will be 0 when t less than minus t naught and it will be 1 when t greater than minus t naught so you can see from the figure also that just before minus t naught it is 0 and just after minus t naught it will become 1. so in this way you can say that the unit step function is advanced by d naught seconds now step function is used to represent an abrupt change in the voltage or current so whenever you apply voltage or current to the circuit through some switching arrangement it is nothing but you suddenly apply the source to the circuit and it is represented with the help of a function because the value of voltage will not be 1 it will be some value so you will simply say it is a step function now it will occur in the circuit generally you will see in the control system and digital computer computers also so in the same manner you will see in the electrical circuit also the application of step function now if you apply a step voltage to the circuit you can represent that phenomena with the help of this step function how you will represent you will say the value of voltage is 0 when time t less than t naught and then at t greater than t naught the value becomes v naught so you can simply represent in terms of unit step function also v naught is the constant value you can take it out and then multiplied by u t minus t naught so this is nothing but a unit step function then if t is equal to zero t naught is equal to zero then v t is simply the step voltage v naught u t so if you put t naught is equal to zero then it will simply become v naught multiplied by unit step function now the voltage source of v naught ut which is shown in the figure so if you see the between two terminals a b a voltage source v naught u t is applied that is nothing but the condition which you represent with the help of switch in the circuit that the circuit is represented like this and this is the switch which you can initially put at at this position circuit would be like this and when at time t is equal to 0 when you apply the switch from this terminal to this terminal then the updated circuit would be like this so this is the this is the switch so if you put this switch to this side then after that switching even the circuit will become this so the unit step response can be represented with the switching event which is represented with the help of switch in the circuit so we can say that when time t is equal to 0 the switch is thrown from this position to this position so it will at t 0 plus condition the voltage across terminal a b will become v naught so this is how you represent the voltage similarly for current the let us say that the current source is i naught multiplied by unit step function so i naught u t is the current source which is connected between terminal a and b now if you want to represent through the switch this would be the circuit now what we are doing initially it is like this the circuit is connected so your ah current between a and b is zero and when you at time t is equal to you open the switch and connect from here to here the circuit will become like this so that also says that in in the circuit the unit stop response can be represented for voltage as well as current source as shown in the figure now derivative of the unit step function is the unit impulse function you generally represent it as delta t so how you will represent you will say delta t is nothing but derivative of unit step function and it is represented as value 0 when t less than 0 and it is again 0 when t greater than 0 because the unit step function at time t greater than 0 is e 1 so derivative of that constant value will become zero so for t less than zero and t greater than zero it will be zero at time t is equal to zero it will be undefined so because of this property this will become a singularity function so generally the unit impulse function we also call as direct delta function and is shown in the figure so you will see that it's have only one impulse kind of activity in the time scale at t is equal to zero otherwise its value is zero so unit impulse function is 0 everywhere except at t is equal to 0 where it is undefined because it's it cannot be defined at time t is equal to 0. now impulse currents and voltage occur in electric circuit as a result of switching operation or impulse sources so when you apply any voltage source to the circuit there would be a sudden impulse which would be generated in the circuit so that particular phenomena can also be represented with the help of this unit impulse function now i will represent we will see later on now although the unit repair function is not physically realizable similar to what we have seen in case of ideal sources like ideal voltage source and ideal current source practically we do not have something called ideal voltage or ideal current source similarly ideal register does not exist but as these phenomena are very important and we used extensively in our circuit calculations and circuit analysis similar to that also the unit impulse function can also be utilized for our circuit analysis and it is very important mathematical tool now unit impulse can be regarded as an applied or resulting shock into the circuit it may be visualized as very short duration pulse of unit area so mathematically how you will represent you will represent that value from 0 minus 2 0 plus delta t dt equal to 1 so this is how you will represent the unit impulse or you will say direct delta function so t 0 minus denotes the time just before time t is equal to 0 and t 0 plus is the time just after t is equal to 0 now the unit area which we have just saw in case of this expression the value of unit area is called as a strength of the impulse function so when the impulse function has a strength other than the unity the area of the impulse is equal to its strength say for example if the impulse function is 10 delta t means it has an area equal to 10. so how you will represent them you will represent with some scale and say in this figure you have you can see easily that this is the value of 10 delta t and 5 delta t plus 2 is this value because it is if you compare with respect to unit function unit step we discuss that it is advanced by sometime minus t naught so similarly here also it is advanced by minus 2 second that is why the value of the unit the impulse function is 5 delta t plus 2 and this function is delayed by three second so that's why it is minus the value is also negative and the value of this function is minus of 4 delta t minus 3. now how this impulse function affects the other functions let's see when you club the impulse function with any other function what will happen so let us say that one function v f t we have and we are associating one direct delta function with it so if we want to integrate between a to b for f t delta t minus t naught dt where t naught ranges between a to b so if you see this expression this delta t minus c naught is 0 except at t is equal to t naught so it means that your integrand is 0 except at time t is equal to 0 so therefore you can simplify this function as a to b f of t naught delta t minus t naught dt because for other values this value is zero so you will left with only this at time t is equal to t naught now at a particular value this function value will be constant so you can take this value out from the integral what you left with this for integration a to b delta t minus t naught dt which is nothing but the unit area as we saw from the definition of unit impulse function so what you get finally is the value of f at time t naught so whenever you associate unit impulse function with some any other function the value of that particular function combined function will become the function value at time t is equal to t naught so this shows that when the function is integrated with the impulse function we obtain the value of function at the point where impulse occurs and this is highly useful property of the impulse function which is known as sampling or shifting property so when you want to sample a particular function you will associate the impulse function with that function at multiple intervals then you get the samples of that particular function at various time intervals so this is very important property and we use extensively in the signal processing now let us understand the third function which is unit ram function now if you integrate the unit step function so in case of unit impulse function we differentiated the unit step function here we are integrating the unit step function when you integrate you will get a unit ram function as shown in the figure now if you see the unit rep function you can define it like minus infinity to some time t the integration u t d t so you can simply simply say it as t into u t why because u t is 0 for time less than 0 and 1 for time t greater than 0 so you can define it like this the unit ramp function will be nothing but 0 at time t less than or equal to 0 and t equal to the value t when time t greater than or equal to 0 so because of this t you will say that you will see a ramp at time t is equal to 1 you will get the value of ramp as 1 so this function will be represented by this particular graph as you are seeing in the figure so unit ramp function is 0 for negative values of t and has a unit slope for positive value of t now in general a ramp is a function that changes at a constant rate the unit ramp function may be delayed or advanced as we saw in case of unit step and unit impulse also so here also you can advance or delay when you delay with some time t naught you get ramp function t minus t naught like as shown in the figure when you advance then your ramp will start from minus t naught and the ram function will be r of t plus t naught so important thing which you can see from here is that at time t is equal to minus 1 t naught is equal to minus 1 you will get the value of unit ramp now with the definitions of all those unit step unit impulse and unit ramp functions let us try to summarize what we have discussed till now for delayed ramp you will say that for time t less than or equal to t naught the value would be zero for time t greater than or equal to t naught it will be t minus t naught when the delay tram function is defined when you define the advanced unit time function r t plus t naught you can simply write value is 0 when time t is less than or equal to minus of t naught and it is t plus t naught when time t is greater than or equal to minus t naught so now you can summarize that the unit impulse you can simply write the delta t is nothing but derivative of unit step function and the unit step function is derivative of unit ramp function similarly unit ramp is integration of unit step function and unit step function is integration of unit impulse function so you can simply say that the these three functions are either the derivative or integrant integration of the functions which we discussed so now let's start the step response of a particular circuit because we have understood the basic functions which are called as singularity functions especially the unit step function is very important because when we talk about the step response of an rc circuit we will say that when we apply the voltage that means that we are talking about the step input to the circuit so is step response is the response of the circuit due to sudden application of dc voltage or current source now let's see the circuit here now if you see the circuit at the left you will simply switch on the circuit at time t is equal to 0 when you switch on the circuit it will be circuit like shown in the figure so you can simply say that vs is nothing but the unit step because this unit step is being defined by the switch so now here bs is constant because you are applying a constant volt dc voltage source now what we have to do we have to select the capacitor voltage as a circuit response which we have to determine so for determination let us assume that the capacitor is initially charged with some voltage v naught so when you will say that it is charged with some voltage v naught we can say that at time t 0 minus or time t 0 plus voltage will remain same as v naught so when 0 minus is the voltage across the capacitor just before the switching happens and 0 plus is the voltage immediately after the switching happened now if you apply kcl if in this circuit what you can do let us take this particular as a node a and you say that this is the current ir this is current ic so you can simply say ir plus ic is equal to 0 what is the value of ic ic is nothing but c dv by dt because this is the voltage across the node so this will become c dv by dt ic and r i r would be nothing but if this is voltage across capacitor v so this value will be v then v minus v s divided by r so we'll write v minus v s into the unit step function because we are considering the switching operation equal to 0 or we can simplify it as d by dt plus v by rc is equal to vs by rc ut now when v is the voltage across the capacitor for t greater than 0 you can simply say that equation 2 can be represented as d by dt plus v by rc equal to vs by rc because at time t greater than 0 this value of unit square function will become 1 so now you have the value dv by dt is nothing but minus of v minus vs by rc you can simplify with the help of this equation when you rearrange you will get dv upon v minus v s is equal to minus dt by rc now what you have to do you have to integrate at both sides and use the initial condition what was the initial condition the voltage across the capacitor was 0 at time 0 plus so we apply that we use that particular initial condition and take integration at both side we get ln v minus vs with the value varying from v naught to vt is equal to minus t by rc where time ranging between 0 to t so you can simply say that ln v minus vs upon when you use the initial condition it will become v naught minus v s is equal to minus t by rc and when you take exponential at both sides you will simply get v minus v s upon v naught minus v s is equal to e minus t by rc so when you discussing about the source free response of rc circuit we termed this tau as time constant which was nothing but equal to rc so we'll use the same time constant and we will replace rc as tau so this will be simplified as v minus v s is equal to v naught minus v s into e to the power minus t by tau or you can simply say that it is v t is equal to this v s will go here then it will become v s plus v naught minus v s bracket e to the power minus t by tau now this condition is valid only for time t greater than 0 for time t less than 0 the value is equal to the initial voltage across the capacitor that is v naught so what we can say that when we apply a step input to the rc circuit the capacitor which was initially charged with voltage v naught will remain same when time t less than 0 and when time t greater than 0 this value will be we can be calculated with the help of this expression that is v s plus v naught minus v s into e to the power minus t by tau and this is known as the complete response of the rc circuit to a certain application of dc voltage source with the assumption that capacitor is initially charged so with this we close our today's session and we continue this journey in the next lecture where we will discuss few more concepts related to rc circuit we will see some examples and then we will move on to rl circuit also thank you [Music] so [Music] you