 Welcome to the 30th lecture in the course engineering electromagnetics continuing with our discussion of rectangular wave kites today we go on to consider the characteristics of the dominant mode and then consider the excitation of various modes in a waveguide you would recall that considering the rectangular waveguide shown here we have worked out the fields for the transverse magnetic modes and the transverse electric modes for example the fields for the TE MN modes were worked out as shown here. And then we were able to identify the lowest order mode as the TE10 mode by following the convention that the x axis is oriented along the longer dimension of the cross section of the waveguide. So out of the two possibilities TE10 or TE01 we identified TE10 as the lowest order mode that is this mode has the lowest cutoff frequency and if you are interested in having a single mode of propagation in the rectangular waveguide then TE10 mode is the candidate and therefore we should consider the characteristics of this lowest order mode the TE10 mode or the dominant mode in some more detail. We consider these field components for the TE MN mode and for the dominant mode that is the TE10 mode now what are the various parameters simplifying to for M equal to 1 and N equal to 0 that is what we identified first we have B equal to M pi by A in general and for M equal to 1 it goes to pi by A on the other hand A which is N pi by B simply goes to 0. So as the value 0 of the index N would indicate along the y direction there is not going to be any variation of the fields whatsoever when N becomes 0 A is going to become 0 and therefore we will find that the parameter A which signified the variation along the y direction that will completely drop out fields will be constant for various values of y that is what is going to happen for whenever the index N becomes 0. So we make these substitutions in these field expressions and then we get the following simplified expressions for the fields we will also require the value of H squared which is A squared plus B squared and therefore in this case it simply becomes pi by A whole square. So we substitute these values of A B and H squared in the expressions just shown and we obtain the following set of field expressions we have H z equal to C times cosine of pi by A x and then let us write down these field expressions in the frequency range where propagation without attenuation takes place in this frequency range that is where omega is greater than the corresponding cutoff frequency we can have gamma bar going to J beta bar. So this substitution can also be made and therefore one gets the simplified field expressions as H z equal to C times cosine of pi by A x e to the power minus J beta bar z. Similarly the other field expressions which are non-zero are E y equal to minus J omega mu A C by pi and then sine of pi by A x and then e to the power minus J beta bar z which factor will be there in the same manner with all non-zero field expressions and finally H x comes out to be J beta bar A times C by pi sine of pi by A x e to the power minus J beta bar z. are non-zero what about the other field expressions that is E x and H y examination of the complete set shows that E x and H y have A as a multiplier and therefore once A goes to zero these field expressions will become zero for the TE modes E z is as it is zero that is a defining condition for the transverse electric modes and therefore these are the three field expressions which are non-zero and the other three that is E z, E x and H y are all zero. So it is a relatively simpler set of equations and these are the field expressions for the TE 1 0 mode which we say is the dominant mode as we have already identified. So what would be the properties for this mode for this case we will have the phase shift constant beta bar which in general has the expression omega squared mu of silent minus m pi by A whole squared minus n pi by B whole squared whole square root it will simplify to omega squared mu of silent minus pi by A whole squared whole square root and of course it is a function of the frequency the constitutive parameters of the medium filling the waveguide and the dimension the longer dimension A and as we have already commented upon the fields for this particular mode are independent of this shorter dimension B. The cutoff frequency for this mode following the general expression 1 by 2 pi square root of mu of silent and then m pi by A whole squared plus n pi by B whole squared whole square root comes out to be simply see that the velocity of light in the medium divided by twice the longer dimension A. So for this mode the longer dimension plays a very important role if we consider the corresponding cutoff wavelength lambda C which can be obtained simply by dividing the velocity of light by the cutoff frequency we get particularly simple expression that is twice A and therefore one could interpret this result by saying that if the wavelength in free space or in the medium filling the waveguide of the signal is greater than twice A or the half wavelength is greater than the width of the waveguide A then this signal even in the dominant mode will not be propagated in the waveguide one could look at it as if the half wavelength should be accommodated within the width of the waveguide and then propagation is possible that will be a very crude way of looking at it but it works apart from the properties the phase shift constant the cutoff frequency in the cutoff wavelength we will also like to see visualize the field lines okay the field lines play a very important role in the utilization of the waveguide we use the waveguide for transmission of signals and we also try to condition the signal as it propagates to the waveguide in a number of ways for example we will like to change the level of the signal we like to measure the frequency we like to measure the wavelength we may like to change the phase so on we like to excite this mode so lots of things will need to be done and all these things will depend upon the kind of field configuration that the waveguide supports for this particular mode so it is very important to see what kind of field configuration this mode has corresponding to itself and the procedure which should by now be fairly standard is to express these field expressions which are now in phasor notation incorporating the time variation explicitly and we multiply these field expressions by e to the power j omega t and take out the real part the procedure is fairly straightforward and we say that we are incorporating the time variation for which mode for the TE 1 0 mode and the expressions that we get are as follows we change the order of the field components slightly we write E y first E y is omega mu A C by pi sin of pi by A x and then sin of omega t minus beta bar z and we find that once this is done once the time variation is incorporated explicitly there is nothing complex about the field expressions this is what actually exists phasor notation is only a mathematical means to express this in a compact notation the other field components the magnetic field components will become H x equal to minus beta bar A C by pi sin of pi by A x and sin omega t minus beta bar z one can compare this with the phasor expression for H x and H z turns out to be C times cosine of pi by A x and then this one is cosine of omega t minus beta bar z so these are the field expressions incorporating the time variation explicitly we have one electric field component which is oriented along the y direction and you would be able to recall this is the shorter dimension of the wave chi along the y direction we have the shorter dimension of the cross section as shown here and as I mentioned this is the convention the magnetic field has two components one along the x direction the other along the z direction so that is entirely in the x z plane and we can use the method that we used earlier for the parallel plane guide for displaying these field components let's consider as far as display of the y component of the electric field is concerned the cross section of the wave guide that is this is the x axis and this is the y axis and these are the points x equal to 0 x equal to A and y equal to 0 y equal to B we can choose some time instant and value of z so that sin omega t minus beta bar z has some value let's say positive value and then depending on that value the amplitude can be calculated so with that certain amplitude the y component of the electric field is going to have a sinusoidal variation from x equal to 0 to x equal to A this clearly shows the effect of the index m we said that index m represents the order of variation along the x direction since m is one there is one maximum in the field variation between the plates obviously if m were equal to 2 if you have to consider t20 mode there will be two maxima like this and the field will change sin at the midpoint and therefore clearly m is the order of variation along the x direction and this needs to be indicated as y direction because this could be any field component this is the magnitude representation of the electric field. Now since this does not show very clearly the orientation of this field component we can go to the alternative description of the field component okay and we can use the directed line segment kind of display for the same parameters and that will be strongest field in the middle so that closely spaced y directed arrows are shown in the middle and then as we move out towards the walls at x equal to 0 and at x equal to A the spacing between the arrows keeps on reducing and of course this y component is a tangential field component at the x equal to 0 and x equal to A walls and is it must vanish at the walls okay so and there is no variation of the field along the y direction consistent with the value of the index n which represents the order of variation along the y direction taking up the magnetic field components next okay here we have two field components the x and the z magnetic field components and we choose to display this in the x z plane so that we mark this as z this as x and these are the walls at x equal to 0 and x equal to A you would recall that we need to mark various values along the z direction for the parameter omega t minus beta bar z and let us say that this is 0 this is minus pi by 2 this is minus pi by 2 etc when this argument is 0 Hx is going to be completely 0 and Hz at least right at this location is going to be maximum alright and then it is going to vary as cosine of pi x by A it will be maximum at x equal to 0 and at x equal to A but the sign will be different between 0 and A so with this in mind and keeping the sign convention making some sign convention we can plot this in the following manner let us say that the positive values are indicated towards the right of this reference plane where omega t minus beta bar z is 0 and negative values are displayed to the left and then this is the cosine pi x by A variation for Hz and as has been mentioned earlier this should be visualized in the third dimension perpendicular to the plane of the display because as we shift from this plane in the z direction the value is going to be different depending on the value of this function alright when the value of this argument is minus pi by 2 Hz is 0 and Hx is maximum and taking care of this sign and this sign Hx will have a maximum positive value and being consistent in our notation it can be displayed by a curve which looks like this okay this is a sinusoidal variation the earlier one was a cosine sinusoidal variation. The other points can be filled up just by inspection and we just need to change the sign and therefore Hz at this plane is going to have this kind of variation and similarly Hx at this plane will have the variation just completed so this is the magnitude representation for the magnetic field components but it shows the it gives the information but in a fragmented manner the complete information is obtained by combining these in the form of continuous field lines and that can be done by adopting the directed line segment representation this is the xz plane and the various z equal to constant values are marked in symmetry with the upper figure and therefore we can now try and complete this here we will have Hz like this and in the positive z direction recalling our convention and as we move towards the middle of the wave kite this z component should become smaller and smaller and therefore as we move in we draw it we show it by a increased spacing here and we also shorten the length anticipating the other field component appearing as we move away from this plane inside and as we move away from this plane in the z direction similarly here we have completely symmetric directed line segments with a direction which is opposite to the one drawn around x equal to 0 the same thing can be completed here around omega t minus beta bar z equal to minus pi and all we need to do is to change the direction in this manner. Next we consider the x directed field component and how that may be represented in this notation and then we find that it is going to have the strongest value around this plane in this manner and it is it has a positive sign so that we can draw an arrow which is looking in the positive x direction and similarly here we are going to have a field line which looks like this and as we move away from this plane because of the sign function according to which there is variation the magnitude is going to reduce as we move inwards and therefore this is the way perhaps we can represented at other locations and at omega t minus beta bar z equal to minus 3 pi by 2 also we can complete the picture in a similar manner and we can draw the field lines with the same magnitude but the arrow direction is now reversed in this manner. So, these are just the skeleton lines for the actual magnetic field lines which must form closed loop kind of figure and therefore now we can complete this and make out that actually the field lines magnetic field lines have this kind of a shape qualitatively this is quite fine but there is some idea of the quantitative value if it is done properly and in a more detailed fashion. So, that by showing the density of lines we can give an idea about the magnitude of the various field components and along the y direction the same field configuration persists since there is no variation along the y direction for any field component. We can also look at the y component of the electric field and try and indicate it in the same figure the y axis is oriented so that it is coming upwards from this plane of display and the expression for the y component of the electric field is written here and wherever this argument is plus minus pi by 2 or multiple of pi by 2 this is going to have a non zero value maximum value and then it will smoothly decrease around those planes. So, therefore, at omega t minus beta bar z equal to minus pi by 2 sin function is going to have the maximum value but of negative sin and therefore, it can be shown here maybe we can indicate it on this figure itself it is going down and therefore, it should be shown in the form of crosses and as we move on either side the number of crosses should decrease to indicate the reduction in this field value according to the sin function. Similarly, at this location where this argument is minus 3 pi by 2 we are going to have this electric field component directed upwards in the positive y direction and therefore, can be indicated in this manner and then decreasing on either side of this plate and the blue represents the electric field line the red represents the magnetic field line and if you consider e cross h that will always be in the positive z direction showing that there is power flow taking place in the positive z direction along the direction of propagation of the wave. The magnitude is different at different locations it is maximum here then it drops to 0 and so on but the direction is always the z direction there is one more thing that we will like to discuss while we are at this point and that is the kind of current flow that takes place on the walls of the wave kind for this mode. Let me see I have not thought about this so perhaps we can discuss this later what you think seems reasonable but let us see as far as the current direction is concerned we will use the relation that the linear current density J s is equal to n cap cross h tan and let us consider the upper wall of the wave kind the one at y equal to b so that n cap is pointing downwards as far as that upper wall is concerned and for this upper wall the current that is going to be flowing will be indicated by the kind of lines that we are drawing in green and they are going to be in this form alright these are the current lines on the upper wall of the wave kind and these are going to be in this direction in a similar manner here also we will have similar magnitudes and similarly oriented current lines and since the direction of the magnetic field is different the current direction will be like this and as we approach the mid point x equal to a by 2 point the current is more or less horizontal I mean in the z direction this is the way it is going to look like and the arrow directions are going to be consistent in this form if one were to consider the current in the lower broad wall the opposite wall wall opposite to what we have considered just now the current direction will be opposite to these and then one can see that there is going to be continuity of this conduction current that is flowing on the walls walls of the waveguide in this direction here then down the narrow wall and then in the other direction in the lower broad wall on one side and similarly on the other side one can visualize the current direction as far as the narrow walls are concerned they are going to have current only which is y directed as indicated by this kind of current flow on the broad walls and of course each lambda g by 2 or lambda bar by 2 the current direction is going to get reversed consistent with the reversal of all the other field directions alright. Now what is the significance of this observation the value here it should be less than the value here it will depend upon the relative value of the magnitudes of the x and the z components so straight away I do not think one can say that just a moment so is this point over it will depend upon the magnitudes straight away one cannot say which one will be greater or it will depend upon the value of beta a x and so on see of course is there in both field components okay yes the next question was are these current lines perpendicular to the magnetic field lines yes as given by this expression the current the unit normal and the magnetic field lines they all form a mutually orthogonal system the utility of this consideration that is what is the current flow on the walls of the waveguide can be in many situations the utility or the application we have in mind right now is supposing we have the requirement that we want to probe the fields in the wave guide for the purpose of making out the standing wave ratio for example right then we have to have a probe which is able to move along the z direction the direction along which the wave propagation takes place fine so we have to have a slot in the waveguide walls to provide access to some sort of a probe which can pick up some field value and then accordingly one can generate some voltage or current in an indicator so we have to have a z oriented slot somewhere in the waveguide allowing this access of the probe where to make that slot we want to probe the field but we do not want to disturb the signal flow we do not want to disturb the wave that is propagating unduly we want to minimize this disturbance to the extent possible where can that slot be such that this requirement is fulfilled supposing we make the slot in the narrow wall of the waveguide okay the slot visualizes slot along the z direction the currents are all flowing in the y direction positive y or negative y depending on the location and the time instant the slot is going to completely disrupt the current lines okay and when such a thing happens the slot is going to radiate and a lot of power from the signal flowing into the wave flowing in the waveguide is going to be leaked out okay so that is not a very good location for the slot similarly if you make the slot somewhere on the broad wall but such that the current lines are disturbed similar thing is going to happen perhaps the extent may be somewhat less depending on its location but if we make the slot right in the middle where the current lines are entirely z directed they have no x component then the current lines will be least disturbed okay and even though we make a slot in the waveguide for probing the fields the signal is not going to leak out as long as the slot is fairly nap in any case the leakage whatever leakage there may be in practice is going to be extremely small compared to any other location and that is why in slotted waveguides which are used for the purpose of measuring the voltage standing wave ratio for waveguide signals the slot is made at this location right in the middle of since both broad walls are completely symmetric in behavior it can be any broad wall that is hardly a point to be stress if there are any questions here we can try those otherwise we proceed further one consequence if a is equal to b will be that you can have a T10 mode propagating with the electric field either in the y direction or in the x direction okay and there is going to be degeneracy as far as the cutoff frequency and the field configuration is concerned and that is not a very good situation so normally unless some application particularly requires this condition normally such as thing is avoided and as I mentioned by convention a is greater than b and we took up the example of an X band waveguide last time and with this choice a much greater than b various cutoff frequencies for various modes come out fairly well separated I mentioned in the beginning that the field configuration for the various modes of the waveguide is very important and we have seen that for the simple question of where to put a slot for probing the fields the field configuration gives us a very clear answer similarly the field configuration helps us in making out how to launch power into the waveguide in a particular mode and whatever mechanism we choose to launch or excite a particular mode will be the mechanism which when operated in reverse will be used for collecting power for that particular mode from that particular signal this excitation of various types of modes is also a very important point so keep in mind as far as the T10 mode is concerned that the electric field is y directed along the short dimension of the waveguide and the magnetic field lines form closed loops in the xz plate and therefore if we go back to our template of the waveguide cross section and say that this is the x direction and this is the y direction the electric field is y oriented magnetic field is in the xz plate what is the thumb rule for arranging excitation the thumb rule is that the exciting agency should have a field configuration should have an associated field structure which is as close to the field configuration of the desired mode as possible this is the thumb rule the closer this compatibility the correspondence the better will be the efficiency of excitation. So with this thumb rule in mind and keeping the field configuration for the T10 mode in mind the scheme that may work well for the T10 mode the objective is to launch this into the waveguide could be the following we have a coaxial cable it comes to the let us say lower broad wall of the waveguide and it has a centre conductor which continues like this inside the outer conductor and then we make the centre conductor protrude a little inwards in this manner and the outer conductor is soldered to the broad wall of the waveguide now within the coaxial cable this is the coaxial cable the kind of field configuration that exists is fairly straight forward the electric field is radially directed between the two conductors and the magnetic field forms closed loops concentric with the axis of the coaxial cable when the probe is taken up somewhat the radial field lines acquire some why orientation some vertical orientation the extent of which will depend upon how much this central conductor is taken upwards the magnetic field lines continue to have the azimuthal variation and therefore they are in the extended plane and we have provided some component of the electric field in the y direction and therefore this can excite the T10 mode and we will have fields which are going to be as required by the T10 mode more or less of this form and the magnetic field is in the XZ plane in this cross section they may perhaps be shown in this manner the field configuration once any power is launched into the T10 mode will become that of the T10 mode because that is the field the waveguide can support the boundary condition will enforce the field configuration to be exactly that of the T10 mode but how much power is transferred to the T10 mode from the exciting probe will depend upon how much match we are able to provide between the fields of the probe and of the mode that we want to excite if we take up another example the thing should become somewhat clearer instead of T10 mode let us consider the T20 mode and see what kind of changes we will require in the excitation scheme of course the starting point will be the identification of the field configuration that we want to launch that is the starting point and we have written the field expressions for the TEMN mode earlier we can obtain the fields for this mode by substituting M equal to 2 and N equal to 0 there and we will have the EY0 dropping the amplitude part and the Z part EY0 will be minus j omega mu AC by 2 pi this time and then sin of 2 pi by A x and then they will be an x component of the magnetic field and a Z component of the magnetic field which can also be written in a similar manner. Now at x equal to A by 2 the field is 0 it is not maximum and therefore a probe put there is not going to be able to accomplish much so the probe location has to be changed that is one thing the second thing is the field is maximum at A by 4 and 3A by 4 and it has opposite polarity at these two locations keeping these things in mind if we have two probes located at A by 4 3A by 4 in this manner and otherwise the center conductor is protruding up and these probes are provided signals so that they are out of phase out of phase inputs we can expect to launch TE20 mode provided the waveguide dimensions are such that the frequency of the signal is above the cutoff frequency for this particular that is a waveguide constraint but if the waveguide does support this mode and if you want to excited then this kind of an arrangement can excite this particular mode similarly one can consider the fields for any other mode that we desire to excite make sure that the waveguide will support this mode and then think of a suitable scheme for excitation and whatever is true of excitation as I said earlier will hold good for collecting power efficiently from that particular mode and therefore it is quite clear that in practice we should have signal propagation in a single mode because one single mode can be excited efficiently we cannot excite multiple modes efficiently and then when we want to collect power then also the power collection is more efficient for a particular okay so this is where we stop today in today's lecture we have considered the characteristics of the dominant mode in a rectangular waveguide and we have also considered the excitation schemes for various types of modes thank you.