 Welcome to the 41st lecture in the course Engineering Electromagnetics. Today we have the last lecture of this course related to the last major topic of discussion that is radiation. The topics for discussion today are the two element arrays and the Yagi Uda array. Before we plan into discussing the behavior of two element arrays, it will be useful to consider first what are arrays and then why one may need to use arrays. And antenna array is a system of similar antennas which are similarly oriented that is very simply put an array. There could be minor differences in the elements of an array as we will see in the example of the Yagi Uda array, but otherwise by definition an antenna array is a system of similar antennas which are similarly oriented. Now why would we need a number of antennas which are similar in nature. The reason for this is as follows. We have seen the behavior particularly with respect to the radiation pattern of the simple antennas such as the electrically small antennas and the half wave dipole antenna and we have seen that the radiation pattern of these antennas is not very directional. In fact in the plane perpendicular to the antenna axis, the radiation is non-directional. It is completely uniform in all directions. In fact in that plane the plane normal to the antenna axis for all these linear antennas, the antenna can be considered to be radiating uniformly in all directions that is it can be called an isotropic radiator. Although in reality there is no radiator which radiates uniformly in the three dimensional space around it. For example even the Hertzian dipole the alternating current element which is perhaps the smallest radiating entity we could consider does not radiate uniformly in the three dimensional space around it, but if we consider the plane normal to the antenna axis the radiation is uniform there. For many purposes this may not be adequate. We may like to enhance the radiation in certain directions and by the same logic we may like to suppress the radiation in other directions for whatever purpose. This will work for transmission as well as for reception. Now we have seen that at low frequencies where also one needs to use antennas many times. It is not possible to increase the antenna size in terms of wave length because of mechanical constraints and the single antenna has a very radiation pattern which is not very directional. This aspect can be improved upon by putting a number of antennas together. That system where we put a number of antennas together becomes an array. So this is where the arrays come into picture. We improve the characteristics improve upon the characteristics of a single antenna by putting in similar antennas. This improvement could be of different types. Here what we are saying is that one improvement which is usually sought is the improvement in the radiation pattern. In particular to make the radiation pattern more directional we may put more than one antenna together. Obviously the simplest array or the simplest arrays would be two element arrays where we put two antennas which are otherwise identical together. The system of two antennas may appear like this. Now what we are considering here are two antennas labelled as antenna zero and antenna one. The separation between these antennas is D. Antenna one, antenna zero is carrying a current I0 and in general let antenna one carry a current I1 which is related to the current I0 in this manner that is I1 is K I0 at an angle alpha. That is notionally the amplitude is different by a factor of K and the phase of antenna one is ahead of the phase of antenna zero by an angle alpha. This is notional because depending on the value of alpha it may be ahead in phase actually or may be lagging in phase. To begin with to keep the consideration simple we consider that these antennas are isotropic radiators that is if you put any one of them alone it will radiate uniformly in the three dimensional space around it. This is a hypothesis it does not work in practice as we have just seen. But in a limited sense for example the plane perpendicular to the axis of the linear antennas the antenna can be considered an isotropic radiators. So we could perhaps say that these are linear antennas which are oriented normal to this plane of display. Then these could be dipole antennas also and then in this plane they are radiating individually they are radiating isotropically or uniform. One can see that if we consider some point P and we consider the total field because of the system due to the due to both the antennas then because of the path difference to the point P from these two antennas they will arise phase difference. Which phase difference will depend upon the orientation of this point P alright and therefore we can foresee a kind of an interference pattern. So individually these may radiate uniformly the radiation pattern may be a circle but put together we expect some different pattern more interesting patterns. How do we go about this one does make some approximations which are applicable as far as the far field is concerned. As long as the point P is at a large distance from the system we can make the following approximations. For example for a distant point P as far as the magnitude of the contribution of these two radiators at this point P is concerned it will be different by the factors r1 and r0. The far field varies as 1 by r strictly speaking the amplitudes are different but for a distant point P the difference in these distances and therefore the difference in the amplitudes is going to be really of no consequence. And therefore we say that as far as the magnitude factor is concerned we can say that r1 and r0 are virtually the same. However the phase cannot be ignored the phase differences cannot be ignored and we say that for the phase factor we consider the path difference in these two paths such that r1 is equal to r0 minus d cos phi where phi angle has been shown in the diagram. So corresponding to this path difference there will be a phase difference which will arise which can be put down as beta d cos phi the phase difference the path difference is simply d cos phi r0 minus r1 is simply d cos phi corresponding to this the phase difference is going to be beta times d cos phi beta being the phase shift constant the phase shift per unit length. This is the phase difference simply because of the path difference. In addition we are saying that the current in antenna 1 notionally leads the current in antenna 0 by an angle alpha which also should be taken into account therefore the overall phase difference becomes beta d cos phi plus alpha which we represent as psi. In most array theory considerations you will find that the phase difference is represented by this symbol psi. So with this kind of phase difference and the amplitudes being more or less similar we want to consider the total field at the point p actually it should be a vector addition and a phasor addition. However since the point p is assumed to be located far away the vectors the electric field vectors due to these two antennas will be more or less co-directional. So vector addition will be of really no significance but the phasor addition taking the phase difference into account will certainly have to be done and therefore we can proceed further and say that the total field is given by E equal to E naught times 1 plus k times E to the power j psi. The reason for these factors should be obvious k is the factor by which the amplitudes are different and psi is the phase difference that we have just evolved and therefore this is the total electric field and as I said we do not really need to consider the vector addition because they are both in the same direction and the magnitude of the total field can be found out by considering the magnitude of the expression on the right hand side. Conceptually that is how it works. Now this can be developed further and we can find out the magnitude corresponding to this factor what we have is Et which is E naught and considering that E naught is real we have the magnitude of 1 plus k times cos psi plus jk sin psi alright which is equal to E naught and then we have the square root of 1 plus k cos psi whole squared plus k squared sin squared psi which simplifies to E naught into 1 plus k squared and then twice k cosine psi. Now the system of two elements the two element array for this a very important case special case is that where k is equal to 1 where the currents are they have the same amplitude in the two antennas but the phases could be different for which case it is simply E naught into square root of 2 plus 2 cosine psi alright which can be simplified by considering the by expanding cosine of psi in terms of half its angle and then one gets a very simple expression namely Et becomes twice E naught cosine of psi by 2 for course k equal to 1 where what is psi? Psi is the phase difference that we had identified beta d cos phi plus alpha and therefore the argument of the cosine function becomes pi d cos phi by lambda recognizing that beta is 2 pi by lambda plus alpha by 2. Now this process that we have followed can be represented by this kind of phasor addition of the field due to the zeroth antenna and the field due to the first antenna in general it is E1 but for the kind of current we put down k times I naught at an angle alpha it becomes like this and psi is the overall phase difference that we have taken into account and then you see that Et is the whole square root of the whole square of this term plus square of this term that is precisely the mathematical procedure that was coming up. Now this is the final expression that one can work with and for the sake of illustration what is the kind of effect of putting these two antenna elements together with similar excitation but with different phases one can see by considering different simple cases involving two element arrays. For example we could have a situation where d is lambda by 2 separation is half a wavelength and let us say alpha is 0 alright and therefore this factor is what is going to govern the radiation pattern becomes simply pi cos phi by 2 for alpha equal to 0 one can put it down here for a two element array with d equal to lambda by 2 and alpha equal to 0 Et is going to be let us say proportional to cosine of pi by 2 cos of phi so phi is the angle that we had shown the radius vector to the point p makes with the line joining the two antenna elements alright and therefore one can give different values of phi and find out what is this factor and then plot this as a function of phi. This has been done here in this diagram and let me put it at the center so that perhaps it can be enlarged okay we are right now looking at this pattern only alright for d equal to lambda by 2 and alpha equal to 0 and we see a pattern which is roughly a figure of 8 as long as we continue to assume that the antennas are isotropic radiators this is the kind of pattern we will get in any plane containing this line joining the two antennas right but if in a certain plane the antennas themselves are directional that directional radiation pattern will become in some manner superimposed over this pattern alright so that consideration we are not going into at the moment. So let us say that in any plane passing through containing the line joining these two antennas if we assume that these antennas are isotropic radiators this is the kind of pattern one will get okay individually each one had a radiation pattern which was a circle but when we put these two together like this with this kind of phasing the pattern becomes more directional this we said was one of the objectives of arraying the antenna elements so that is being achieved one can see by a simple logic that yes this is how the pattern should be for example in this direction either way the antennas have the same phase of excitation but the spacing is lambda by 2 and therefore the radiation becomes out of phase along this line okay so in this direction phi equal to 0 or phi equal to 180 degrees there is no radiation which is how the result is coming out in the plane of symmetry since the excitations are in phase the radiations add together the interfere constructively so to say and therefore we get a maximum okay so even without such an expression one can figure out these simple things that where the radiation will be minimum where it will be maximum and particularly for the case of K equal to 1 where there will be complete cancellation if it occurs in certain directions the patterns can be made out fairly simply now of course the requirement may be different we may require the maximum radiation in a in some other direction that there is a very simple solution to that let us say we change the phasing keep the same separation and now that is D equal to lambda by 2 and alpha becomes pi or 180 degrees in which case you will see that for D equal to lambda by 2 and alpha equal to 180 degrees we will have a total electric field magnitude which is proportional to sin of pi by 2 cos of phi and the pattern when plotted still looks like a figure of 8 but with respect to this it is rotated by 90 degrees but it is not exactly 90 degree rotation one can make out that if you change phi by 90 degrees you do not get this expression so that difference is there but qualitatively loosely speaking it remains a figure of 8 but the maximum direction direction of maximum radiation has now become different and once again it can be explained or understood in similar terms in this direction the spacing is lambda by 2 so that leads to a phase difference of pi radians in addition there is a phase difference of pi radiation therefore in either direction the radiation interferes constructively whereas in the plane of symmetry since the excitations are out of phase they interfere destructive alright therefore we get this kind of a pattern of course these are two very simple situations other situations are equally well possible for example one can have a situation where d is lambda by 4 the spacing between the elements and alpha the phase by which the antenna one excitation leads antenna zero excitation and that is minus 90 degrees therefore actually the antenna one excitation is lagging in phase by 90 degrees then one gets a pattern which is typically that of a cardio when the excitations are of equal amplitude and with this kind of phase here again one can consider now this kind of patterns are unidirectional in the sense that there is radiation in this direction say forward direction but no radiation in the backward direction which becomes a case of practical importance how is radiation there considerable radiation there in this direction in the 5 equal to 0 degree direction and no radiation in this direction let us see this element when we consider this direction is ahead by a distance lambda by 4 which corresponds to a phase difference of pi by 2 radiance or 90 degrees so just because of the spacing the radiation from this is ahead in phase in this direction now the phase of this is minus 90 degrees and therefore that phase difference is completely nullified and in this direction the fields or the radiation interferes constructively that does not happen when we reverse direction okay there the phase difference is accumulate to give us an overall phase difference of pi radiance as you can see very simply pi radiance of 180 degrees therefore in this direction there is no radiation and there is there are intermediate values of the total field in other directions which one can make out by using the mathematical expression we have developed so this way you can play around with the separation and the phasing and you can generate patterns with different behavior or suiting different requirements if we go to a larger separation we increase the separation between the elements we make it a one wave length and keep the phasing the same that is they are excited in phase with the equal amplitude but the separation is one way then we get a more interesting pattern which is shown here so here what one notices is that there are directions of maximum radiation which are not just two as we saw previously or not just one there are four directions in which the radiation is maximum and there is a very simple explanation to that using the arguments we have been providing earlier in the plane of symmetry the radiation is interfering constructively and in this plane also in this direction also because of the separation the radiation is not experiencing any phase difference and therefore once again it is constructive interference that does not happen in some other directions where the interference is distracted alright so this was the behavior of some very simple two element arrays now let us keep in mind for a while this particular system where the phase difference was 180 degrees it can be made out that the there will be changes in the pattern but the basic nature will not change even if the spacing is changed alright so this is the point that I like you to keep in mind before we proceed to the next topic of discussion today which is the Yagi Uda array the Yagi Uda array was considered to be a very important development when it was proposed around 1930s it was proposed by Japanese Uda and it was made known to the scientific community by an American Yagi and therefore the name Yagi Uda array the Yagi Uda array is a parasitic array so this is an important difference in the Yagi Uda array and the arrays that we have just considered the arrays that we just considered the simple two element arrays each antenna had an individual separate excitation of its own which means you have to provide a feed network a feed for each antenna and then phase the feeder currents appropriately to get the desired results so in those two element arrays each element was driven alright but in the Yagi Uda array there is only one element which is driven and the other elements derive their excitation through mutual coupling or through induced fields or through induced currents alright so therefore there is one driven element as we shall just see in the examples and other elements are parasitic they are not driven separately but it does mean that if they are not connected to a driver or to a generator they do not have any effect they have an important effect because they do have induced currents which will depend upon their separation and their size alright and therefore by controlling the separation and the size of the parasitic elements one can once again so to say engineer the radiation pattern to a desired degree because only one element is driven it becomes a very simple array and therefore it is a popular low cost antenna okay and it can provide a more directional radiation pattern compared to a single radiating element alright now as an example let us consider the following system what is the system it consists of two elements one is the driven element as indicated by provision for connecting a field and the other is a parasitic element in this in this illustrative example the separation between these two elements is kept very small alright and what we are considering is the pattern in a plane which is normal to the axis of these two elements okay that is in the terminology of the dipole antennas we are considering the H plane pattern in which plane individually these will be isotropic radiators that is their radiation pattern will be that of a circle okay H plane pattern a plane which is normal to the antenna axis where individually these are uniformly radiative we make out the effect or the current that is induced on the parasitic element in the following manner we say that since the antennas are since the elements are placed very close together therefore the field which is incident on the parasitic element is the field of the driving element or the driven element which we call in this case the driver if we consider that these are made of very high conductivity materials then the tangential field over this perfect conductor should be zero total tangential field and therefore there will be an electric field which will be generated by this parasitic element which should nullify the field which is incident on it that is e parasitic should be equal to the negative of the incident field which is the field due to the driver so the parasitic element has a field which is exactly the negative of that of the driver the driven element and therefore it becomes a system of two elements like those two dots we showed earlier with a phase difference which is 180 degrees and one can calculate the overall radiation pattern using the expression we had put down earlier using this separation between the elements and it comes out to be of this form okay in the plane of symmetry since they are these are out of phase excited elements there is no radiation otherwise the pattern varies in this manner so straight away from a uniform radiation we have achieved radiation in restricted directions alright but this is not the complete story if we change the dimensions of the parasitic element is the same system alright one driven element one parasitic element but we are changing the length of the driven element as a result of this the impedance that this element presents for this driven element is now different and therefore the phase of the current that is induced on it or the field that it generates is now different from earlier since the length is a little more than this and if you consider that this is the resonant length one can work out from simple principles that the phase that will be phase of the current and the field that we generated because of this will be leading the phase of this element and therefore one will get the effect of the cardioid pattern that we got where antenna one was lagging in phase or effectively antenna zero was leading in phase that kind of effect is coming in but the separation is not exactly time by four nor is the phase difference exactly 90 degrees because that depends upon the separation and the dimensions the effect is somewhat like this essentially the radiation in the forward direction if we call this the forward direction towards the driver is enhanced okay and this enhancement will depend upon the separation and the size typically if this is a lambda by two dipole or a resonant length dipole which resonant length is in practice slightly less than lambda by two because of the field extension beyond the open circuit end and we keep this length somewhat greater than the resonant length then we get this kind of pattern and a parasitic element which is longer slightly longer than the resonant length is called a reflector in some manner the field is being reflected in the forward direction away from this element and the field towards the reflector is getting reduced alright so this is the kind of effect that a longer so called reflector element has on the radiation back we proceed further from here and instead of placing the parasitic element behind the driven element we place it in front but we still look for the same effect that is the radiation is enhanced in the forward direction and suppressed or reduced in the backward direction and then if we keep the dimensions of this parasitic element such that it is slightly less than the resonant length then the current that is induced in this is going to be lagging in phase compared to the driver element and the overall radiation pattern in the plane normal to the antenna axis that is the H plane pattern is going to be like this so once again you see that we may place a parasitic element in front or at the back but if we choose its dimension suitably the radiation in the forward direction in our notation 5 equal to 0 degree direction can be enhanced by choosing the dimensions judiciously and therefore to continue this enhancement of radiation in the forward direction we consider this kind of an array which is a yagi Uda array there is one driven element the center element and there are two parasitic elements one in front the other at the back and their dimensions have been suitably chosen so that each one enhances radiation in the forward direction this is slightly less than the resonant length this is slightly more than the resonant length to give you an idea of the kind of lengths that we have in mind here the resonant length L driver which we want to be exactly half wavelength or to resonate around half a wavelength kind of frequency is typically 0.478 lambda close to lambda by 2 but slightly less of course this particular resonant length depends also upon the thickness of the say conducting rod which is used to make the antenna that effect is small but it can change this value slightly on the other hand the length of the reflector which is expected required to be slightly greater than this in this particular example that we are considering which is fairly typical is 0.49 lambda and the length of the parasitic element that is in front this element is called a director because it in some sense directs the radiation in the forward direction is 0.45 okay the example that I am showing in that example the spacing between these elements is still small but in practice the spacing is of the order of 0.15.2 lambda and as you can see all dimensions are related to wavelength their absolute value is not so important it is the relative value with respect to the wavelength that matters for this kind of dimensions and with the spacing equal to in this case 0.04 lambda the pattern that we get is the following okay the plane in which we are showing it is again the H plane pattern the plane normal to the antenna axis so that in that plane these three elements look like these three dots okay this is the director element this is the driven element this is the reflector element and the pattern that we get is like this the pattern in the other plane that is the plane in which we are displaying this these elements the plane containing these antennas will be somewhat different because in that plane these individual elements have a directional somewhat directional pattern okay so which will multiply the field values that we get on the basis of isotropic radiation patterns and that can be done since the individual radiation pattern is not very complicated we get a pattern in the plane containing the antennas like this this is called the H plane pattern and this is the E plane pattern this plane is parallel to the orientation of the electric field and the other plane is normal to it and now you cannot mistake this this is the element that we use so frequently as the antenna for receiving the television signals there is one difference the antenna that we use in practice has a center element which is the driven element which is a folded dipole from the point of view of the considerations we have already discussed otherwise this is the television receiving antenna and only one antenna needs to be connected to the transmission line only one element the other two elements do not have any physical connection with that feeder transmission line or the transmission line that connects this receiving antenna to the receiver therefore it is so simple the exact analysis is not so simple because the amplitude of the current that is induced and phase of the current that is induced depends upon the separation and the dimensions okay but some attempts have been made at that and fairly complete design information is available on the design of this antenna and even when you buy this kind of antennas from the market you would notice that some antennas perform well some others do not so antennas which conform to these guidelines they will perform better and those that are connected to good quality transmission lines with a proper connection keeping the impedance levels in mind they will perform better others will not the guidelines are evolved in this manner the general Yagi Uda array antenna configuration is like this while we may have a number of director elements the addition of more reflector elements does not help okay so there is usually one reflector and depending on how much enhancement of the radiation pattern you require in the forward direction you can add more directors and one can make out that as the number of director elements increases although the pattern will become more and more directional but they will arise a point of diminishing returns okay there will be saturation in the directionality of the radiation pattern that one can achieve through a given driven element okay so initially as you add the director elements the gain or the directionality of the radiation pattern will increase rather rapidly and then this rate of increase will drop and finally it will saturate to some value complete optimization analysis of such a system exists okay and when you want to optimize this you will like to have the separations optimized and the dimensions optimized so that with a given number of elements you maximize the directionality of the overall radiation pattern alright so that kind of theory exists the results based on such a theory are like this okay here the directionality of the overall radiation pattern is expressed in terms of an antenna parameter which is called gain which we have not defined here but has a very simple definition right now it is enough to say that as the number of elements increases and it is the number of director elements that is increasing the gain increases and then it the rate of increase of gain tapers tapers down and one may have up to 8, 9, 10 director elements and one can have a radiation pattern which is roughly 11, 12 times more directional than the radiation pattern of a single antenna say the driven element okay so now we have reached the end of the course you will recall that in the beginning of the course we said that electromagnetic deals with the study of electric and magnetic fields the behavior of these fields how these are generated or what is the effect of these fields we identified certain very important phenomena which can be described in terms of fields very conveniently these phenomena were the phenomena of wave propagation and radiation so that is been the main theme of this course we have studied extensively the wave propagation taking the vehicle of transmission lines first then the parallel plane waveguide and finally the rectangular waveguide we also saw how we may build resonators using sections of certain lengths from transmission lines or the rectangular waveguides and finally the second important phenomenon together with the propagation that was mentioned that is radiation we made some very simple considerations regarding how radiation may arise in different structures and then we considered the mathematical framework with the help of which we can calculate the radiation fields from a given structure and we applied this framework to some very simple antennas the electrically short antennas and the half wave dipole antenna fine finally as we have done today we considered the effect of putting this kind of antennas together that is forming arrays and we have tried to explain the evolution the design evolution of the commonly used television receiving antenna thank you very much.