 Welcome to the fourth section in the second module where we continued to discuss those two rotating complex numbers that we had rotating in opposite directions. As I was saying it seems like a silly thing to do to describe a sinusoid as a combination of two rotating complex numbers. As it is complex numbers are difficult to deal with as compared to real numbers and there you have not one but two of them. Why are we doing this? Well let us look at that inductor once again and even though this seems rather a figment of our imagination let us assume that the current were described by such a rotating complex number. Suppose you had an inductor there again with ELT and IL as before an inductor value L and let us assume that ILT were not I0 cos omega t plus phi of I0 but one of those rotating complex numbers. Let us take the count the clockwise number. So I0 by 2 e raised to the power j omega t plus phi t. And let us find out VLT. Now we have a very beautiful property VLT by ILT is equal to simply j omega L beautiful. So the ratio of the voltage to the current here is a constant independent of time and we can do the same for the other rotating sinusoid or other rotating complex number. So again this is a constant independent of time. The only change that you notice is that in the first case you had L times j omega. In the second case you have L times minus j omega. So in some sense here the actual angular frequency whether positive or negative is reflected if minus j omega here and v plus j omega in the previous. So you notice that the direction of rotation is also captured in this constant which comes independent of time a plus or a minus sign. So we have the notion of a so called positive angular frequency and a negative angular frequency. Now this is distinct from a sinusoid. In a sinusoid there is really no notion of a positive angular frequency or a negative angular frequency. You see except for this positive and negative distinction by and large the beauty is that the voltage and the current have a constant or proportional relationship with the proportion at the constant being complex. And in fact that makes it so much easier to analyze circuits which have resistances, inductances together and in fact you can also do the same kind of analysis for capacitances. The only difference in a capacitance and I will leave this to you as an exercise. Let me write down that exercise. I am going to leave an exercise for you to do. The exercise is repeat this analysis for a capacitance. We call that for a capacitance of value c with a voltage of vct across it and a current of Ict we have Ict is c times dvct dt and one could repeat the same analysis by using rotating complex numbers instead of the original sinusoids. And here one would get the relationship vct by Ict is again a constant independent of time find this constant and explain is the exercise that you need to do. Now what we have established then assuming that you have done this exercise is that when you have a circuit comprising of resistances, inductances and capacitances they can all be treated as constant multipliers to the current and therefore you have this notion of impedance as a generalization of resistance. The resistor has an impedance of R so let us write all of them down. So for sinusoidal inputs for sinusoidal voltages and currents the resistance of value R has an impedance of R and inductance of L operating an angular frequency of omega would have an impedance of g omega L and the capacitance of value c operating at angular velocity omega again angular frequency omega has an impedance of 1 by j omega c and now you could use the same principle as you do for resistances to analyze circuits comprising of resistances, inductances and capacitances. This is the beauty of sinusoids and in fact this is even more than sinusoids the beauty of rotating complex numbers. Now the only change that you have to make when going from the positively rotating to the negatively rotating complex number is to change the sign of omega plus omega for counter clockwise minus omega for clockwise that is all. So in fact whatever you are doing to the positively rotating complex number is mirrored by the negatively rotating complex number. You do not need to do both of them separately if you do one you have done the other. So with the same effort you can deal with one of those rotating complex numbers work the whole circuit out and then based on the principle of linearity of RLC circuits conclude that the overall response is the response that comes out of the positively rotating complex number and the negatively rotating complex number. This is the whole principle of what is called phasor analysis. In fact this is a term that we shall now introduce phasor. Phasors are essentially rotating complex numbers. Complex numbers that change their phase but not their amplitude. You know now you understand why I have been saying all to the first module that we would like to allow for complex signals as well. All to the first module we were saying well if I want to test additivity or homogeneity I must allow for a complex constant of multiplication in homogeneity or in additivity I need to allow for complex signals too to be added. And I should be worried about complex signals because here I have a concrete reason why I need to bring in complex signals even if we are dealing with a physically real signal the sinusoid. Though we are dealing with sinusoids the convenience of the sinusoid the mathematical convenience of the sinusoid is not completely exploited until we go to the corresponding rotating complex number. In a nutshell what is the reason for this? Dealing with amplitudes is easy but dealing with phases is a problem in sinusoids and that problem is overcome when you go to the phasor instead of the corresponding sinusoid. So, it is simple why do we go to complex rotating number you know complex signals in general because there is a certain convenience associated in dealing with changes of phase at least where sinusoids are concerned. And now I am going to say something which you understand better only over several sessions from now. I am saying that this sinusoid is also a very generic kind of signal what I mean by that is you could actually build up a very wide class of signals by taking linear combinations of sinusoids of different frequencies that is the catch. So, all this while we have been talking about sinusoids of the same angular frequency and there are some beautiful properties but when I bring together sinusoids of different angular frequencies with appropriately chosen amplitudes and phases I have the power to build a very wide class of signals. So, now in this session we have understood what happens when we deal with complex numbers rotating instead of sinusoids and in the next session we are going to ask the question what happens in general when this complex rotating number goes into a linear shift invariant system of which RLC circuits are one example. Thank you.