Bode Plot Construction

Today we are going to learn about Bode plot and today I am going to teach you in Hindi how to draw Bode plot on semi log sheet for that we have taken an example of Bode plot that is G of s a given task of function r is equal to 80 divided by s into s plus 2 into s plus 20 and h is equal to 1 which means it is a unity feedback system so let's solve this so first of all we will see step by step how to solve this in 4 to 5 steps so first step first of all we will arrange given transfer functions in standard form standard form means in time constant form this given format like g into h is equal to 80 divided by s into s plus 2 to s plus 20 this is not standard format or time constant format how will we know that this is not standard format or time constant format so we will see the value of denominator and numerator which are the given values is it in the term of 1 plus ts or not that means here it is 2 plus s that is not 1 plus ts here also 20 plus s this is also not in the term of 1 plus ts So, if we take 20 and 2 as common from here, then it will be 1 plus and some constant into S and this will also be 1 plus in its form. So, we convert it. So, we take 2 and 20 as common from here. So, 80 divided by S into, if we take 2 as common, then it becomes 1 plus S by 2 and if we take 20 as common from here, then it becomes 1 plus S by 20. Now, this 1 plus is in the form of 81S. this is also a term of 1 plus t , where t1 is 1 by 2 and t2 is 1 by 20 so now 20 into 2 is 40 which will be cancelled out from the numerator and only 2 will be left that means 2 divided by our new dancer function in this standard form is 2 divided by s into 1 plus 1 by 2s into 1 plus 1 by 20s Let's consider this as equation number 1. Now let's move on to step 2. So the second step is to find the corner frequency. Corner frequency in simple words is where two slopes meet or where the slope changes. So we will see how the slope changes and how to find out. In other words, if we have a given task function, then we will see, in a numerator or denominator, if your function is in the term of 1 plus t s, then we can find out the corner frequency. And the one which is not in the term of 1 plus t s, will not have a corner frequency. For example, if we see in equation 1, we have three terms. One is a constant term, which we will not consider generally. 2 by s square a cut term consider Thank you. 2nd term is 1 plus 1 by 2s and 3rd term is 1 plus 1 by 20s total of 3 terms out of which 2 terms are 1 plus 2s and this single term 2 by s always with the consonant with the individual term we will consider 2 by s as a separate term so it means that we will have only 2 corner frequencies and the other one is the s-catorn. So, we remove the current frequency and denote it with omega1 or omega c1. And we can see that this is 1 plus 1 by 2s that means this 1 plus t1s where t1 is 1 by 2 so omega c1 will be equal to 1 by t1 and if we put 1 by t1 and 1 by 1 by 2 we will get 2 radians per second. we will take omega c2 from 1 plus 1 by 20s and here it is in the form of 1 plus t2s so value of t2 will be 1 by 20 and omega c2 value will be 1 divided by t2 because frequency of time periods is reversed so here we put t2 1 divided by 1 divided by 20 means 20 radians per second We will keep the frequency in radium per second so that we can plot on semi-log sheet easily There is an important note here that the number of frequencies that we get, here we have only 2 corner frequencies and we have 3, 4, 5, whatever number of corner frequencies we get we will arrange them in serial wise in the increasing order, in the smallest to the largest order and we will arrange it in ascending order the smallest frequency is 2 radians per second, then the first corner frequency will be there and the 20 radians per second will be the second corner frequency Now let's move on to step 3 In the third step, we will do Magnitude Plot First of all, because the semi log plot will be in two parts First will be Magnitude Plot and second will be Phase or Frequency Plot Magnitude Plot is also known as Gain Plot okay so I'm pilot In step 3, we will first make a table which is called magnitude and slope table and in this table we will have four terms we will write four main terms Term is the corner frequency in radium per second Slope is the dB per decade and change in slope is also dB decibel per decade dB means decibel per decade So first we will see the terms So I have told you that we have three terms How to decide the first term, second term, and third term? First of all, we will see what is the constant term in the numerator and denominator. For example, from equation 1, the constant term is 2. And with that, the terms of individual s are these. The terms of individual s are either s, s square, or s cube, in the numerator and denominator. For example, if 2s were here, then we would have taken the first term as 2s. but here 2 by s is there so our first term will be 2 by s which we can also call as 2 by j omega because we will put s value later in j omega so here we will take first term as 2 by s which has no corner frequency because it will not have corner frequency it is not in the term of 1 plus t s but the slope will be minus 20 dB per decade in my previous bote plot example I told you how it is I will explain it in a minute Whenever there is a term of S in the denominator like 2 by S, 2 by S square 2 by S contributes minus 20 dB per decade slope and if S is above, that is, if 2 S is above then it contributes plus 20 dB per decade slope okay so we can see that in all three terms we will not see any other terms other than s like s by 2, s by 5 or 5s we just have to see in denominator and numerator that how many terms of s we have if there is one term then it is in denominator then it is minus 20db and in numerator then it is plus 20db decade So, here we see that The first term is 2 by s which will contribute minus 20 dB per decade slope and since the slope is starting from here there is no change in slope so we have kept it as s The second term is 1 divided by 1 plus s by 2 Based on the corner frequency we have decided that this will be the second term because this is a small corner frequency So its corner frequency is 2 i.e. omega c1 and because in denominator it has S term maybe minus 20 dB per decade slope Now, how do we calculate the change in slope? We add the slope which was minus 20 and the slope which came later. So, we have minus 20 and minus 20. So, the slope will change to minus 40 dB per decade. Now, the third term is 1 divided by S into S by 20. Its corner frequency is 20 radium per second. And the slope also contributes minus 20 dB per decade. because in denominator also there is individual term of S so earlier slope was minus 40 and now it is minus 20 dB per decade so total change in slope is minus 60 dB per decade so we can say that in this magnitude and slope table we have taken out all the terms and change in slope now we will move ahead to the fourth step In the fourth step, we have to understand that we have only two corner frequencies omega c1 is equal to 2 and omega c2 is equal to 20 so what we do is we assume one of the smallest corner frequencies to be the smallest and we assume one of the largest to be the largest meaning that the smallest frequency can be anything here it can be 1, 0.1 0.2, 0.3 we can take anything whatever we take, our plot over it will be same and there will be no error so for our convenience we have taken the lowest frequency which is 1 and here we have taken more than 20 you can take 50,60,100,200,300 but we have taken 100 so that it is easy for us to calculate now we have 4 frequencies one is omega c1 and one is omega c2 and one is the smallest and one is the largest frequency Now, what we will do on these four frequencies? We will remove the magnitude or gain. So, for this we have to remember only two formulas. What is the first formula? On the two magnitudes of the beginning, if you have any problem, any question, any numerical, for the two frequencies of your beginning, a common formula will be used for omega L and omega C1. And after 2, that is, from omega C2, for all the number of corner frequencies or all the number of frequencies a common formula will be used for that so let's go to the first formula so, at omega L magnitude at omega L is equal to 1 20 log first term mode of first term is written because we don't want to take negative term and our at omega is equal to 1, so first term is 20 log here first term is 2 by s 20 log 2 by s and we will put s in j omega term where if we put j in mode then only omega will come so we can write this as 20 log 2 by omega and here we will put omega value as 1 so we got 20 log 2 divided by 1 means if we calculate this with scientific calculator then our value will be 6 dB means our first frequency the value of magnitude is 6db now we will write the same formula for second so, magnitude at omega c1 is equal to 2 20 log again first term 2 divided by omega and here we will write omega is equal to 2 because omega is equal to 2 so 20 log 2 divided by 2 is 20 log 1 and 20 log 1 is 0db Now, our formula changes from the third frequency i.e. from the equal to quantity of omega c2 And this formula will remain the same for all i.e. we will write this as change in slope between omega c1 to omega c2 because we have taken the formula till omega c1 what will be the slope till here? So, change in slope between omega c1 to omega c2 into log modulus of omega c2 by omega c1 2 or plus previous sum of magnitude plus omega c1 so the value of omega c1 is 0 now what we do is we put the value of b omega c2 is equal to 20 so change in slope how is this minus 40? let's see it once so here is our change in slope this is our slope table here we see the value of omega c1 and omega c2 So, the slope between omega c1 to omega c2 will be minus 40 and the slope of the frequency which is ahead of omega c2 will be minus 60 and the slope before omega c1 will be minus 20 So, this means that the slope between omega c1 to omega c2 will be minus 40 and from omega c2 to omega h, the change in slope will be minus 60 so we have put the change in slope as minus 40 log and this is the result omega c2 is 20 and omega c1 is 20 so 20 divided by 2 plus the magnitude at omega c1 we will put the first frequency measurement so 0 is here and 0 is here so, the value of log10 is equal to what will be the value of log10? it will be equal to the third frequency of omega C2 so, the magnitude of omega C2 is equal to the third frequency of omega C2 so, the third frequency same formula will be used change in slope between omega c2 to omega h into log omega h divided by omega c2 plus we will keep the matwood at omega c2 what will we get change in slope minus 60 log and omega h value is 100 omega c2 is 20 so we will divide by 20 plus the magnitude is minus 40 to omega ct so we will add minus 40 and the total value we will get is minus 81.95 db which is approximately minus 82 db minus 60 is wrong so don't consider it this is correct So, this is the result of the magnitude plot. data which we have calculated from a table here that omega is their frequency and corresponding to that is our magnitude that is omega L is equal to 1 and our magnitude is 60 omega C1 is equal to 2 and our magnitude is 0 omega C2 is equal to 20 and our magnitude is 40 and omega H is equal to 100 and our magnitude is 81.9 and our magnitude is 52 now in the last step now we will do phase plot analysis So in the equation 1, we will put s is equal to j omega So after putting s is equal to j omega, we will get g of j omega into h of j omega is equal to 2 divided by j omega into 1 plus j omega by 2 into 1 plus j omega by 20 Now we will find the angle of this equation and we have already told you to draw this angle in the previous videos of Bode plot we will find that the individual term i.e. 1 by s has a angle of minus 90 degree and 1 by s square has a angle of minus 180 degree if s is the numerator then it is plus 90 degree and if s is the numerator then it is plus 180 degree so here I have written my term in denominator so it will be minus 90 degree and here this term 1 plus we can write it as x plus jy and we know that the magnitude of x plus jy is tan inverse y by x that is y term is omega y2 divided by 1 so because all these are in denominator so all these will have negative sign and if it was in numerator that is in upper then all these will have positive sign So, if we write the angle of this equation, it will be Let's assume this as equation number 2. And let's make a table for this equation. We put the value of omega in equation number 2. And we calculate the corresponding degree of omega. So, many students ask this question. Sir, how will we decide the value of omega? We will randomly put the value of omega For example, since we started with 1, we will put 1 first and then we will see how much our value is coming in it When we put 1, our value is 119.4 degrees So, what we do is, before that, we have considered 0.1 as well because we will put the value of omega in this way so that the value of phase is from minimum minus 90 to maximum minus 180 plus i.e. minus 190, minus 200, minus 220 if after putting many values of omega the value of phi is not coming to minus 180 plus then we will stop but in this question we saw Omega value is 0.1, so it is minus 93.1 1.1 is 119.4 2.2 is minus 140.7 3.3 is minus 20 and we have found that our value is minus 219.3 Now we will draw body plot on the semi log graph plot In the last step, we have the table of border plots of face and dude. we will draw this on this semi log graph sheet we have a magnitude table we have found the magnitude at every frequency at omega L is equal to 1, we have found the magnitude at 6 dB at omega C is equal to 1, we have found the magnitude at 0 dB at omega C is equal to 20, we have found the magnitude at minus 40 dB and at omega H is equal to 100, we have found the magnitude at minus 81.9 that is 82 dB we will draw this on the semi log graph sheet In semi log graph sheet, x axis is logarithmic scale and y axis is normal scale Logarithmic scale means that the first values we assume Like minimum value of frequency is 1 and maximum value is 100 So, we have taken 0.1 here because the minimum values of phase So if we assume the first value of 0.1, then its value will be 10 times this value. That is, 0.1 will be 1. And after this, our main values will be 10. Then after this, 10 times 100, after this 1000 and 10,000. You can see that the variation is logarithmic. That is why we call it logarithmic scale. And this is a normal scale. its values constantly vary so because our maximum value is 6dB plus 6dB and minimum value is minus 82dB so we take zero at the top so that we have less space and at the bottom we have more space so we have taken zero this will be in dB so zero plus 20 minus 20 minus 40 minus 60 minus 80 minus 100dB So, as we know that on 1, the magnitude of 1 is 6 dB, so we have kept approximately 6 dB here. On 2, our value is 0 dB, so we have kept 0. Then, on our value 20, the magnitude of the frequency is 20, that is, we keep the frequency on this axis. The frequency of x axis is the same in radian per second. So, on 20, we have kept minus 40 dB, which is in the magnitude table. and then on 100, our approximate is minus 82 dB and all these points will be added through scale from here to here, then from here to here, then from here to here and the first one will be minus 20 dB per decade and the second slope will be minus 40 dB per decade and the third slope will be minus 60 dB per decade or isco we can do the magnitude plot or gain plot that means our magnitude plot is complete similarly, our phase table where we have taken the values of minimum 0.1 I have already told you how to take the values of 0.1 minimum value should be around and maximum value should be and the difference so that the value of the phase is not too different that's why we have taken the value like this so at minimum 0.1, we can see our table is it is minus 93, at 1 it is approximately minus 120 and we will put the same value here at 0.1, we can see that the value of the phase in the degree is kept aside, opposite to decibel magnitude and here we have started from minus 240 because the maximum value is around minus 220 and the minimum value is around minus 90 so we have kept the interval of 20 on this side so that it does not overlap with the magnitude now at 0.1 our value is approximately 93 so this is our value of 100 and this is our value of 97 let's see our value of 93.1 So this is our approximate 3-point. because 90 is the value of the pitch so 93 similarly, on 1, our values are minus 119 point something so we put this similarly, on 2, our values are 139 point something and on 3, our values are 157 so we put this and all this graph here we matched the magnitude plot with scale and here we will mix the values with free hand and this plot is called phase plot now what we do is after plotting we draw a line at 0 dp and a line at minus 1 degree we have to find this in every case and every question if we want to find out the gain margin and phase margin so we draw a line at 0 dp and a line at minus 1 degree The 0 dB key line we will cut the gain plot of phase magnitude where it will cut the frequency this is the vertical line frequency this frequency we will call as gain crossover frequency which is read as omega gc and we can see that the plot of gain is cut here omega is equal to 2 radium per second and we will bring this line downwards this omega gc and where it will cut the phase at the point where the curve is being cut and this point is of minus 140 degree so the difference between 180 degree and minus 140 degree we will call it as phase margin so we will write the formula as 180 degree plus the value of minus 140 degree you can see that we have not taken minus 180 degree So, you have to keep in mind that the phase margin will be 180 degree. we have to add the value of gain cross over frequency so 180 degree and this value is minus 140 degree so we add both and the difference is 40 degree so our phase margin is 40 degree now we will see the line of minus 180 degree where we have cut the phase plot you can see at this point the value of phase and minus 180 degree line is being cut and this frequency is approximately 6.3 so our phase cross-over frequency will be because this phase is cut by the line of minus 1 atteria so this is called phase cross-over and the line which is cutting the gain is called gain cross-over frequency so now we will take this point upwards and see where it is cutting the gain So where it will cut the gain plot, difference from 0 to minus 20 dB so the difference from 0 to minus 20 dB is the gain margin so the difference from 0 to minus 180 dB is the gain margin The difference is that we can subtract the value from 0 and we add the value from 180 degree because it is a negative value so we have to take it opposite so we hope that you have understood this plot well and again I want to explain in short that the line of 0 dB cross the gain and the line of minus 180 degree the phase where it crosses is called phase cross of frequency now we come to gain margin from 0 dB the phase cross of frequency where it cuts the gain that difference is called gain margin we subtract from 0 and the phase margin is from minus 180 degree and where the gain cross of frequency cuts the phase that difference means from 180 degree where it will cut, that difference is the phase margin so this is our complete plot and as we know on y-axis always we keep the value of decibel magnitude and on the opposite side of y-axis we keep the value of phase in degree and we plot the graph in such a way that both the graphs are separated so that it is easy to cut it So, I hope you have understood this well. If you have any doubt, then you can mail us at harimohandadaret.gmail.com or you can mail me on my website. Thanks for watching this video. I hope you have liked it. Thank you.

So, 80 divided by S into, if we take 2 as common, then it becomes 1 plus S by 2 and if we take 20 as common from here, then it becomes 1 plus S by 20. Now, this 1 plus is in the form of 81S. this is also a term of 1 plus t , where t1 is 1 by 2 and t2 is 1 by 20 so now 20 into 2 is 40 which will be cancelled out from the numerator and only 2 will be left that means 2 divided by our new dancer function in this standard form is 2 divided by s into 1 plus 1 by 2s into 1 plus 1 by 20s Let's consider this as equation number 1. Now let's move on to step 2. So the second step is to find the corner frequency. Corner frequency in simple words is where two slopes meet or where the slope changes. So we will see how the slope changes and how to find out.

In other words, if we have a given task function, then we will see, in a numerator or denominator, if your function is in the term of 1 plus t s, then we can find out the corner frequency. And the one which is not in the term of 1 plus t s, will not have a corner frequency. For example, if we see in equation 1, we have three terms. One is a constant term, which we will not consider generally. 2 by s square a cut term consider Thank you.

2nd term is 1 plus 1 by 2s and 3rd term is 1 plus 1 by 20s total of 3 terms out of which 2 terms are 1 plus 2s and this single term 2 by s always with the consonant with the individual term we will consider 2 by s as a separate term so it means that we will have only 2 corner frequencies and the other one is the s-catorn. So, we remove the current frequency and denote it with omega1 or omega c1. And we can see that this is 1 plus 1 by 2s that means this 1 plus t1s where t1 is 1 by 2 so omega c1 will be equal to 1 by t1 and if we put 1 by t1 and 1 by 1 by 2 we will get 2 radians per second. we will take omega c2 from 1 plus 1 by 20s and here it is in the form of 1 plus t2s so value of t2 will be 1 by 20 and omega c2 value will be 1 divided by t2 because frequency of time periods is reversed so here we put t2 1 divided by 1 divided by 20 means 20 radians per second We will keep the frequency in radium per second so that we can plot on semi-log sheet easily There is an important note here that the number of frequencies that we get, here we have only 2 corner frequencies and we have 3, 4, 5, whatever number of corner frequencies we get we will arrange them in serial wise in the increasing order, in the smallest to the largest order and we will arrange it in ascending order the smallest frequency is 2 radians per second, then the first corner frequency will be there and the 20 radians per second will be the second corner frequency Now let's move on to step 3 In the third step, we will do Magnitude Plot First of all, because the semi log plot will be in two parts First will be Magnitude Plot and second will be Phase or Frequency Plot Magnitude Plot is also known as Gain Plot okay so I'm pilot In step 3, we will first make a table which is called magnitude and slope table and in this table we will have four terms we will write four main terms Term is the corner frequency in radium per second Slope is the dB per decade and change in slope is also dB decibel per decade dB means decibel per decade So first we will see the terms So I have told you that we have three terms How to decide the first term, second term, and third term?

First of all, we will see what is the constant term in the numerator and denominator. For example, from equation 1, the constant term is 2. And with that, the terms of individual s are these. The terms of individual s are either s, s square, or s cube, in the numerator and denominator.

For example, if 2s were here, then we would have taken the first term as 2s. but here 2 by s is there so our first term will be 2 by s which we can also call as 2 by j omega because we will put s value later in j omega so here we will take first term as 2 by s which has no corner frequency because it will not have corner frequency it is not in the term of 1 plus t s but the slope will be minus 20 dB per decade in my previous bote plot example I told you how it is I will explain it in a minute Whenever there is a term of S in the denominator like 2 by S, 2 by S square 2 by S contributes minus 20 dB per decade slope and if S is above, that is, if 2 S is above then it contributes plus 20 dB per decade slope okay so we can see that in all three terms we will not see any other terms other than s like s by 2, s by 5 or 5s we just have to see in denominator and numerator that how many terms of s we have if there is one term then it is in denominator then it is minus 20db and in numerator then it is plus 20db decade So, here we see that The first term is 2 by s which will contribute minus 20 dB per decade slope and since the slope is starting from here there is no change in slope so we have kept it as s The second term is 1 divided by 1 plus s by 2 Based on the corner frequency we have decided that this will be the second term because this is a small corner frequency So its corner frequency is 2 i.e. omega c1 and because in denominator it has S term maybe minus 20 dB per decade slope Now, how do we calculate the change in slope? We add the slope which was minus 20 and the slope which came later. So, we have minus 20 and minus 20. So, the slope will change to minus 40 dB per decade.

Now, the third term is 1 divided by S into S by 20. Its corner frequency is 20 radium per second. And the slope also contributes minus 20 dB per decade. because in denominator also there is individual term of S so earlier slope was minus 40 and now it is minus 20 dB per decade so total change in slope is minus 60 dB per decade so we can say that in this magnitude and slope table we have taken out all the terms and change in slope now we will move ahead to the fourth step In the fourth step, we have to understand that we have only two corner frequencies omega c1 is equal to 2 and omega c2 is equal to 20 so what we do is we assume one of the smallest corner frequencies to be the smallest and we assume one of the largest to be the largest meaning that the smallest frequency can be anything here it can be 1, 0.1 0.2, 0.3 we can take anything whatever we take, our plot over it will be same and there will be no error so for our convenience we have taken the lowest frequency which is 1 and here we have taken more than 20 you can take 50,60,100,200,300 but we have taken 100 so that it is easy for us to calculate now we have 4 frequencies one is omega c1 and one is omega c2 and one is the smallest and one is the largest frequency Now, what we will do on these four frequencies? We will remove the magnitude or gain.

So, for this we have to remember only two formulas. What is the first formula? On the two magnitudes of the beginning, if you have any problem, any question, any numerical, for the two frequencies of your beginning, a common formula will be used for omega L and omega C1.

And after 2, that is, from omega C2, for all the number of corner frequencies or all the number of frequencies a common formula will be used for that so let's go to the first formula so, at omega L magnitude at omega L is equal to 1 20 log first term mode of first term is written because we don't want to take negative term and our at omega is equal to 1, so first term is 20 log here first term is 2 by s 20 log 2 by s and we will put s in j omega term where if we put j in mode then only omega will come so we can write this as 20 log 2 by omega and here we will put omega value as 1 so we got 20 log 2 divided by 1 means if we calculate this with scientific calculator then our value will be 6 dB means our first frequency the value of magnitude is 6db now we will write the same formula for second so, magnitude at omega c1 is equal to 2 20 log again first term 2 divided by omega and here we will write omega is equal to 2 because omega is equal to 2 so 20 log 2 divided by 2 is 20 log 1 and 20 log 1 is 0db Now, our formula changes from the third frequency i.e. from the equal to quantity of omega c2 And this formula will remain the same for all i.e. we will write this as change in slope between omega c1 to omega c2 because we have taken the formula till omega c1 what will be the slope till here? So, change in slope between omega c1 to omega c2 into log modulus of omega c2 by omega c1 2 or plus previous sum of magnitude plus omega c1 so the value of omega c1 is 0 now what we do is we put the value of b omega c2 is equal to 20 so change in slope how is this minus 40? let's see it once so here is our change in slope this is our slope table here we see the value of omega c1 and omega c2 So, the slope between omega c1 to omega c2 will be minus 40 and the slope of the frequency which is ahead of omega c2 will be minus 60 and the slope before omega c1 will be minus 20 So, this means that the slope between omega c1 to omega c2 will be minus 40 and from omega c2 to omega h, the change in slope will be minus 60 so we have put the change in slope as minus 40 log and this is the result omega c2 is 20 and omega c1 is 20 so 20 divided by 2 plus the magnitude at omega c1 we will put the first frequency measurement so 0 is here and 0 is here so, the value of log10 is equal to what will be the value of log10?

it will be equal to the third frequency of omega C2 so, the magnitude of omega C2 is equal to the third frequency of omega C2 so, the third frequency same formula will be used change in slope between omega c2 to omega h into log omega h divided by omega c2 plus we will keep the matwood at omega c2 what will we get change in slope minus 60 log and omega h value is 100 omega c2 is 20 so we will divide by 20 plus the magnitude is minus 40 to omega ct so we will add minus 40 and the total value we will get is minus 81.95 db which is approximately minus 82 db minus 60 is wrong so don't consider it this is correct So, this is the result of the magnitude plot. data which we have calculated from a table here that omega is their frequency and corresponding to that is our magnitude that is omega L is equal to 1 and our magnitude is 60 omega C1 is equal to 2 and our magnitude is 0 omega C2 is equal to 20 and our magnitude is 40 and omega H is equal to 100 and our magnitude is 81.9 and our magnitude is 52 now in the last step now we will do phase plot analysis So in the equation 1, we will put s is equal to j omega So after putting s is equal to j omega, we will get g of j omega into h of j omega is equal to 2 divided by j omega into 1 plus j omega by 2 into 1 plus j omega by 20 Now we will find the angle of this equation and we have already told you to draw this angle in the previous videos of Bode plot we will find that the individual term i.e. 1 by s has a angle of minus 90 degree and 1 by s square has a angle of minus 180 degree if s is the numerator then it is plus 90 degree and if s is the numerator then it is plus 180 degree so here I have written my term in denominator so it will be minus 90 degree and here this term 1 plus we can write it as x plus jy and we know that the magnitude of x plus jy is tan inverse y by x that is y term is omega y2 divided by 1 so because all these are in denominator so all these will have negative sign and if it was in numerator that is in upper then all these will have positive sign So, if we write the angle of this equation, it will be Let's assume this as equation number 2. And let's make a table for this equation. We put the value of omega in equation number 2. And we calculate the corresponding degree of omega. So, many students ask this question.

Sir, how will we decide the value of omega? We will randomly put the value of omega For example, since we started with 1, we will put 1 first and then we will see how much our value is coming in it When we put 1, our value is 119.4 degrees So, what we do is, before that, we have considered 0.1 as well because we will put the value of omega in this way so that the value of phase is from minimum minus 90 to maximum minus 180 plus i.e. minus 190, minus 200, minus 220 if after putting many values of omega the value of phi is not coming to minus 180 plus then we will stop but in this question we saw Omega value is 0.1, so it is minus 93.1 1.1 is 119.4 2.2 is minus 140.7 3.3 is minus 20 and we have found that our value is minus 219.3 Now we will draw body plot on the semi log graph plot In the last step, we have the table of border plots of face and dude. we will draw this on this semi log graph sheet we have a magnitude table we have found the magnitude at every frequency at omega L is equal to 1, we have found the magnitude at 6 dB at omega C is equal to 1, we have found the magnitude at 0 dB at omega C is equal to 20, we have found the magnitude at minus 40 dB and at omega H is equal to 100, we have found the magnitude at minus 81.9 that is 82 dB we will draw this on the semi log graph sheet In semi log graph sheet, x axis is logarithmic scale and y axis is normal scale Logarithmic scale means that the first values we assume Like minimum value of frequency is 1 and maximum value is 100 So, we have taken 0.1 here because the minimum values of phase So if we assume the first value of 0.1, then its value will be 10 times this value.

That is, 0.1 will be 1. And after this, our main values will be 10. Then after this, 10 times 100, after this 1000 and 10,000. You can see that the variation is logarithmic. That is why we call it logarithmic scale. And this is a normal scale. its values constantly vary so because our maximum value is 6dB plus 6dB and minimum value is minus 82dB so we take zero at the top so that we have less space and at the bottom we have more space so we have taken zero this will be in dB so zero plus 20 minus 20 minus 40 minus 60 minus 80 minus 100dB So, as we know that on 1, the magnitude of 1 is 6 dB, so we have kept approximately 6 dB here.

On 2, our value is 0 dB, so we have kept 0. Then, on our value 20, the magnitude of the frequency is 20, that is, we keep the frequency on this axis. The frequency of x axis is the same in radian per second. So, on 20, we have kept minus 40 dB, which is in the magnitude table.

and then on 100, our approximate is minus 82 dB and all these points will be added through scale from here to here, then from here to here, then from here to here and the first one will be minus 20 dB per decade and the second slope will be minus 40 dB per decade and the third slope will be minus 60 dB per decade or isco we can do the magnitude plot or gain plot that means our magnitude plot is complete similarly, our phase table where we have taken the values of minimum 0.1 I have already told you how to take the values of 0.1 minimum value should be around and maximum value should be and the difference so that the value of the phase is not too different that's why we have taken the value like this so at minimum 0.1, we can see our table is it is minus 93, at 1 it is approximately minus 120 and we will put the same value here at 0.1, we can see that the value of the phase in the degree is kept aside, opposite to decibel magnitude and here we have started from minus 240 because the maximum value is around minus 220 and the minimum value is around minus 90 so we have kept the interval of 20 on this side so that it does not overlap with the magnitude now at 0.1 our value is approximately 93 so this is our value of 100 and this is our value of 97 let's see our value of 93.1 So this is our approximate 3-point. because 90 is the value of the pitch so 93 similarly, on 1, our values are minus 119 point something so we put this similarly, on 2, our values are 139 point something and on 3, our values are 157 so we put this and all this graph here we matched the magnitude plot with scale and here we will mix the values with free hand and this plot is called phase plot now what we do is after plotting we draw a line at 0 dp and a line at minus 1 degree we have to find this in every case and every question if we want to find out the gain margin and phase margin so we draw a line at 0 dp and a line at minus 1 degree The 0 dB key line we will cut the gain plot of phase magnitude where it will cut the frequency this is the vertical line frequency this frequency we will call as gain crossover frequency which is read as omega gc and we can see that the plot of gain is cut here omega is equal to 2 radium per second and we will bring this line downwards this omega gc and where it will cut the phase at the point where the curve is being cut and this point is of minus 140 degree so the difference between 180 degree and minus 140 degree we will call it as phase margin so we will write the formula as 180 degree plus the value of minus 140 degree you can see that we have not taken minus 180 degree So, you have to keep in mind that the phase margin will be 180 degree. we have to add the value of gain cross over frequency so 180 degree and this value is minus 140 degree so we add both and the difference is 40 degree so our phase margin is 40 degree now we will see the line of minus 180 degree where we have cut the phase plot you can see at this point the value of phase and minus 180 degree line is being cut and this frequency is approximately 6.3 so our phase cross-over frequency will be because this phase is cut by the line of minus 1 atteria so this is called phase cross-over and the line which is cutting the gain is called gain cross-over frequency so now we will take this point upwards and see where it is cutting the gain So where it will cut the gain plot, difference from 0 to minus 20 dB so the difference from 0 to minus 20 dB is the gain margin so the difference from 0 to minus 180 dB is the gain margin The difference is that we can subtract the value from 0 and we add the value from 180 degree because it is a negative value so we have to take it opposite so we hope that you have understood this plot well and again I want to explain in short that the line of 0 dB cross the gain and the line of minus 180 degree the phase where it crosses is called phase cross of frequency now we come to gain margin from 0 dB the phase cross of frequency where it cuts the gain that difference is called gain margin we subtract from 0 and the phase margin is from minus 180 degree and where the gain cross of frequency cuts the phase that difference means from 180 degree where it will cut, that difference is the phase margin so this is our complete plot and as we know on y-axis always we keep the value of decibel magnitude and on the opposite side of y-axis we keep the value of phase in degree and we plot the graph in such a way that both the graphs are separated so that it is easy to cut it So, I hope you have understood this well. If you have any doubt, then you can mail us at harimohandadaret.gmail.com or you can mail me on my website.

Thanks for watching this video. I hope you have liked it. Thank you.

Transcript for:Bode Plot Construction

Transcript for:
Bode Plot Construction