 Welcome to the third session in the second module and where we continue from the second session in which we identified that an amplitude change in a sine wave is very simply described by a multiplying factor. However, we said a phase change is a little more troublesome. Let us see what happens when we change the phase of a sine wave. Let us assume that we begin with a sine wave described by A cos omega t as usual plus 5. In fact, here we have an original phase of phi 1. So, I write phi 1 here and the changed phase is phi 2. So, I have A cos omega t plus phi 2 after changing the phase. Now, the changed wave form divided by the original wave form is not a constant independent of time unlike the change of amplitude. So, when we change the phase, there is a problem. It is not so easy to describe. Let us take a physical example where this becomes necessary. Let us take a simple inductor across which we know the current to be sinusoidal and let us find out the voltage. Let the inductor have value L in henry's. Let the current in the inductor be I t or I l t if you please and let the voltage across the inductor be V l t, whereupon V l t is l d I l t d t. Let I l t be sinusoidal. Whereupon V l t is going to be equal to minus l I o omega sin omega t plus phi naught. Now, of course, you can see that there is essentially a 90 degrees phase difference between the voltage and the current. We know this for a sinusoid. If you have a sinusoidal excitation in an inductor, there is a 90 degrees difference between the voltage and the current. The same can be worked out for a capacitance. But unfortunately, V l t, Y I l t is not a constant independent of time and that is where the problem is. So, you know if you talk about sinusoid as they are, you cannot quite think of the inductor like you think of a resistor where there is a proportional relation between voltage and current. On the other hand, let us adopt a slightly different paradigm of view of a sinusoid. Let us go back to the original sinusoid that we had. The original sinusoid that we had, the current sinusoid was I 0 cos omega t plus phi t. And we could think of this as a combination of two rotating complex numbers. So, visualize this to be the complex plane here. And let us show two rotating complex numbers in this plane. Let 1 begin at time t 0 with an angle of phi 0. Let it have a magnitude of I 0 by 2. And let the other begin with the same magnitude, I 0 by 2, but the opposite angle of starting. So, phi 0 minus phi 0 are angles of start. And let them both rotate with an angular velocity of capital omega. The only thing is this rotates with a counterclockwise angular velocity. And this one rotates with a clockwise velocity which you might call minus omega t like. Let us describe these complex numbers. Now, you know, let us recall how we describe a complex number. A complex number can be described either in what is called its rectangular form where we say real part plus j times the imaginary part. In fact, this quantity j is the square root of minus 1. So, it is the primordial complex number if you might. And what we are writing here is in what is called the polar form. We are writing the polar form. And remember this complex number is a function of time. You see this omega t plus phi 0 is the instantaneous angle. Similarly, the clockwise complex number would be essentially the same as the anticlockwise or the counterclockwise complex number, but with a negative sign for the angle. Simple. Now, when we add these two complex numbers, notice that they are complex conjugates. And lo and behold, when you add these two rotating complex numbers, it gives you back the original sinusoid all on the real line. So, essentially you could think of a sinusoid as a combination of two rotating complex numbers, rotating in opposite directions with the same magnitude and opposite starting angles. Now, why is this, you see, it looks like we are doing something silly. We had nice real functions to begin with and we are going to two complex functions. As it is complex numbers are more difficult to deal with than real functions or real numbers. And here we have not one but two of them. Why are we doing all this? We will see in the next discussion that this is the answer to how we can make dealing with phase more convenient. Thank you.