Welcome. In this web lecture, we're going to look into a special form of wire antennas. Namely the loop antenna. And we're going to introduce the magnetic dipole as well. Our particular objectives of this lecture are first to apply of course, the general formulation. That we introduced to analyze these loop antennas. We will show the similarity between a very small loop antenna and an electric dipole. And as a result, we're going to introduce the magnetic dipole. And we will show that if you combine the magnetic dipole and electric dipole into one combined antenna. You can create a perfect circle polarized antenna. Well, let's have a closer look at the configuration at as shown here. So we have a loop antenna, which is affect a circle of wire thin wire in the XY plane. And the radius of the circular loop is given by the Perimeter 8 and along the loop we have. Constant current flowing, so mathematically we can express it as shown here. So we have a constant current along the loop, so that is a valid approximation when the loop is relatively small. So if A is much smaller than the wavelength. But we will also use it for larger loops and you could create such configuration as well with larger groups. If you would use electronics in the loop to remain a kind of constant currency. But remember, it's it's in this lecture we will use uniform current distribution. So that would be an approximation for practical situations. Now the current is flowing in the file direction. And in particular in the not direction. So the source points are indicated by fine, not so keep that in mind and I will come back to that later. So and find not the direction in the final direction can be expressed, of course in X&Y coordinates as well as shown here. Now, because the wire is very thin and oriented in the XY plane, we can use again Delta functions to describe that mathematically. So this first Delta function in the XY plane indicates where the loop is located. And we use also a Delta function in these at direction. So apply the recipe so to determine the far field expression. So in this case we're going to start with the age field, which is given by this expression. Where we recognize again the vector product that we need to workout, and the integration of the current distribution. If we know the magnetic fields, we can calculate the electric field. Because there's a TM relation between the electric and magnetic fields in the far field region. An the current was already given, so we can substitute that in the integral. And the other thing we need to do in the integral is to workout the inner product. Between the unit vector in the radial direction an are not. Use the appendix in the book to perform this inner product yourself, and then you will find this expression as given here. An special attention should be taken to this cosine. Because we have cosine of 5- 5, not an fi is the observation point indicating the observation point P. And find not indicates the source point on the loop. So that's an important thing to mention here. Substituting all these relations in the integral, we find that the 8 field can be written in this form. Where we still have to workout the integral over fi, not an integral. Looks like like this, so we indicate over Finot. From zero to 2Ï€ along loop. But we recognize that the integrant the exponential has an argument where both fiifi not are present. So looking at the symmetry of the configuration at hand. We know that in the far field, the expressions for Ian H should be fi independence. So we can choose any file in the integration that we would like to use. So most simple choice would be to say okay, let's assume 5 zero and that's done and. In this expression here. So, finally, we have to workout this integral and this is a very standard integral which can be found in mathematical books. And the result of this, you can already guess it, because we have a circular problem will be a Bessel function. And indeed, the result of this integral is 2 pi j times the Bessel function of the first kind with arguments k(knot a) sine theta and directed in the u(file) direction. More details can be found in the book. The Bessel function can be calculated very easily, for example, with MATLAB or other mathematical tools. So let's take a closer look at the radiation pattern of a couple of Loop antennas. So radiation pattern can be calculated by using our well known expression. So we normalize the power density, of course, in the maximum direction. And as a result we get this expression of Bessel functions. Now if we apply this recipe for a couple of dimensions of the loop, we get this kind of radiation patterns. And looking a little bit more carefully, the full line shown here is the magnetic dipole, where we have a radius which is much smaller than the wavelength. And you might already recognized that the radiation pattern is very similar to the one of an electric dipole, so that's interesting. And we're going to use that later on in this lecture as well. If we increase the size of the loop to, for example, lambda knot over 2, then we see already that we have multiple maxima in the radiation pattern. And this changes even further, if we made the loop even more, even larger. And these kind of loops with the uniform current distribution might not be very useful, as you can imagine. Now, the magnetic dipole, so a loop which is very small as compared to the wavelength is interesting to investigate in a little bit more detail. Also, mathematically, so if a is very small as compared to the wavelength, then the argument of the Bessel function, so that's k(knot a) times sine theta will become also very small. And we can use an approximation that the Bessel function can also be written as half times the arguments as shown here. If we substitute that in the expressions for the electric and magnetic far field, then we get this as a result. So we have a theta component for the h field and a file component for the electric field, with a spherical expansion of course or constant and the sine theta relation. And you might recognize that hey, this looks very similar to the electric dipole. Well, this is the magnetic dipole. And the electric dipole expressions, we have introduced a couple of web plexus ago. And they look very indeed very similar, where in the electric dipole case we have file component of the h field and a theta component of the e field. But for the rest, they look very similar. And if we look a little bit more carefully at the constants in both expressions, for example, for the electric field. Then of course, we can still make a choice for the length of the electric dipole and for the size of the magnetic dipole. And if we make particular choices, for example, if we say okay, the current i(knot l) is equal to -k(knot pi a squared) times i(knot). So we make this particular choice, then they are really identical apart from affected j. So the theta component if we would combine both antennas, so we combine and we make a new antenna where we have a combination of a magnetic dipole and a electric dipole and we apply superposition, then we find that the total field, electric field will have two components. So e file component and an e theta component with this relation between them. And that is very special. Now let's look at it again. A little bit more careful just to make sure that you have the same picture as I have. So we have the electric dipole oriented along the z axis in the origin of a coordinate system. And we have the magnetic dipole indicated by the small circular loop. And they form both one antenna. Well, how to do that in practice is of course another discussion, the feeling and those kind of things. But let's assume we have this configuration. So by making the particular choice as we just discussed where we we introduced P and P is the dipole moment of the electric dipole and we made this particular choice. Then we have shown that we get as a superposition of both radiated fields. We get this property of the electric field that E phi is equal to J times E theta and that means that a s a 90 degree phase difference between the phi the TT component. And they are perpendicular to each other, so in the far field would look like one component in this direction and one component in that direction. And because they have a phase difference of 90 degrees, they will form what's called a left hand circular polarized wave. And that's very interesting because this circle polarization is perfect in all directions. So not only in one direction, but it's valid for all theta and phi directions. Now in the same way, of course we can make another choice for the relation between the dipole moment and the characteristics of the magnetic dipole. So if we make this particular choice, then there will also be a 90 degree phase shift between both components. But then w with the minus sign and that will result as you can guess already in a right hand circular polarized wave. So this is a very interesting feature of the combination of a magnetic and electric dipole. Well, to summarize this web lecture we have analyzed loop antennas with the uniform current distribution. We have introduced in magnetic dipole, which is effective small loop antenna. And we have shown you very interesting combination of an electric and magnetic dipole. That provides us with a perfect circle E polarized antenna. And hope to see you back next time.
