 Good afternoon everyone. Myself Piyusha Shedgarh. We will see the rectangular waveguide. These are the learning outcomes for this session. At the end of this session, students will be able to explain the characteristics of rectangular waveguide. They will be able to derive the expression for different field components of wave in rectangular waveguide. These are the contents for this session. Let us see the characteristics of waveguide, cut-off frequency. The lowest frequency at which the wave will propagate through the rectangular waveguide. So, the cut-off frequency decides some threshold value. Above this value, the wave will propagate through the rectangular waveguide. Critical wavelength, lambda c. It is the wavelength through which the wave is travelling through the rectangular waveguide. Phase velocity. It is the velocity at which the wave changes its phase. Generally, the phase velocity having the value is greater than the group velocity. Phase velocity is the ratio of omega by beta that is angular velocity by the phase shift. Group velocity. The velocity is considered for the group of the wave is given by the rate of change of angular velocity with respect to the phase shift. So, generally, Vg having the value is between the c and Vp where c is the velocity of light. Vp and Vg both are related by this equation. Vp into Vg is equal to c square where c is the velocity of light. Propagation wavelength in the waveguide that is lambda g. It is also known as guide wavelength which is considered with respect to the travelling wave travels in the waveguide. Waveguide characteristics impedance Z0. It is the impedance considered with respect to the cutoff frequency in turn which in turn determined by the guide dimensions that is breadth and width of the rectangular waveguide. Generally, the characteristics impedance is greater than 377 ohm. Now, before going to the further slide, you can pause video here for a second and tell that what are T and Tm modes in rectangular waveguide because you have to define the different field components for these modes. So, you must know what are T and Tm modes. Yes, as you know that in T mode electric field is always perpendicular to the propagation of the wave that is Z component of electric field is equal to 0 because you are considering that the propagation of the wave in Z direction and HZ is not equal to 0. Opposite of that in T mode, Z component of the magnetic field is equal to 0 while the electric field is existing. Now, you know the different Helmholtz equation. Helmholtz equation can be defined for T wave as well as the Tm wave. For T wave, EZ component equal to 0. Therefore, del square HZ is equal to minus omega square mu epsilon HZ and the second equation is del square EZ equal to minus omega square mu epsilon EZ. So, these are the Helmholtz equation. In Helmholtz equation, del operator is there. Del operator is nothing but the partial differentiation with respect to axis. If the wave propagation is considered in the rectangular coordinate axis, thus the partial differentiation is with respect to X, Y and Z and thus the del operator can be defined with respect to X, Y and Z axis. So, here dou by dou Z square can be replaced by the gamma square and this bracket which having the all constant values. So, all these constant values can be replaced by a single constant that is H square. So, if you know these two equations. So, by putting H square here, you can get this equation for T wave as well as for the Tm wave, where in this case you are putting dou by dou Z square is equal to gamma square and gamma is the propagation constant given by alpha plus j beta, where alpha is the attenuation constant and beta is the phase constant. Now, consider the general Maxwell's equations. So, here the two Maxwell's equations del cross E bar equal to minus j omega mu H bar and del cross H bar equal to j omega epsilon E bar. Now, consider the Maxwell's first equation here del cross E bar is nothing but the curl operator and this curl operator can be written with the determinant form like this. Again, the wave propagation is in rectangular coordinate axis and therefore, the differentiation is with respect to X, Y and Z, whereas the right hand side as it is. So, solve this determinant for this X, Y and Z component. So, this is the equation number 3 for the X component. Similarly, you can write the equation for Y component and the Z component thus you are getting the equation number 4 and 5. Now, consider the second Maxwell equation. Again, you have to solve this second Maxwell equation by curl operation of H bar. Again, you can write the equation for X bar, Y bar and Z bar component and give that equation number 6, 7 and 8. Now, from the equation number 4 which is written already in previous slide. So, here in equation number 4 the Y component of magnetic field is existing. So, from this equation you can find the equation for this Y component of magnetic field by transferring this j omega mu at the LHS side. So, here this is the equation and you can give that equation number as a 9. Now, you know the 9th equation which having the Y component of magnetic field and the 6th equation which having this H Y you can put this value of H Y from 9th equation in equation number 6. So, by putting this value getting the equation number 10. Now, in this equation number 10 you can rearrange this equation number 10 thus you getting again at the RHS side this bracket which having the constant. Again, this constant can be replaced by H square and rewrite that equation. Here, E X bar component can be find by transferring this H square at the LHS side thus you getting the equation for the X component of the electric field. Similar to this you can find the other components field components for magnetic field and the electric field. So, these 11 to 14 equations give the general relationship for the field components within the waveguide. Now, by putting the values of E Z and H Z we can find the old field components. So, by putting the values of E Z and H Z for the different mode of a propagation. So, first consider the more transverse electromagnetic wave where E Z as well as the H Z both components are equal to 0. So, by putting this E Z and H Z equal to 0 in above equation all field components becomes equal to 0 that is wave cannot propagate through the rectangular waveguide for T M wave. For T E wave that is transverse electric wave E Z component equal to 0 and H Z is existing that is the new electric line is in the direction of the propagation. While in case of the T M mode H Z equal to 0 that is no magnetic line is in the direction of the propagation of the wave. And the combination of both that is H E mode all components are existing. Now, represent the T E and T M mode consider the rectangular waveguide the width is placed along the x axis and the height is placed along the y axis with the dimension B and this is the dimension A. Now, the mode is considered with respect to this T E and T M mode can be defined at the subscript M N where M A is the number of half wave variation of the electric field intensity along the width while the N is the number of half wave variation of the electric field in the height that is B direction. What is dominant mode the wave which has the lowest cutoff frequency and having the greater value of the wavelength is known as the dominant mode. In rectangular waveguide generally T E 1 0 mode has the lowest cutoff frequency and therefore, T E 1 0 mode is called as the dominant mode of the waveguide. Dominant mode if the T E 1 0 that is one half cycle variation along the dimension A and no half variation along the short dimension that is at the B. Modes with the next higher cutoff frequency are T E 0 1 mode and T E 2 0 mode which having the cutoff frequency twice than that of the dominant mode that is T E 1 0 mode. So, this is the T different mode configuration for the T E wave for in figure A and B there is a variation with respect to the wider dimension that is A dimension and there is no any variation along the dimension B while in case of C and D there is the half wave variation along the A as well as in B direction. So, here this is a T E 1 0 mode and T E 2 0 mode. In T E 2 0 mode 2 half wave variation and therefore, it is known as a T E 2 0 mode. These are the references for this session. Thank you.