 Good afternoon everyone. Myself Piyusha Shedgarh. Today we will see the topic microwave component magic tea. These are the learning outcomes. At the end of this session, students will be able to derive the scattering matrix for magic tea junction. They will be able to apply the properties of s matrix to magic tea junction. These are the contents. So what is microwave tea junction? Microwave tea junction is an interconnection of three waveguides in the form of English alphabet T. So there are the several types of the tea junctions. These are the microwave components. H-plane tea junction, E-plane tea junction and the combination of E-plane tea and H-plane tea junction that is it is also called as magic tea junction. So before going to start the microwave component magic tea junction, you can pause video here for a second and recall that what are the properties of s matrix. Now what is magic tea junction? Magic tea junction it is also called as E-H-plane tea junction. It is formed by attaching two simple waveguides one parallel and the other is in series to a rectangular waveguide which already have the two ports. This is called as a magic tea junction. It is also known as hybrid or 3 dB coupler. The arms of rectangular waveguide mix the two ports are called collinear ports that is if the rectangular waveguide having the already two ports port 1 and port 2 are collinear while port 3 is called as a H-arm or it is connected in parallel therefore it is also called as a parallel port. Port 4 is connected in series so it is called as E-arm or difference port or series port. So this is the magic tea junction. So here the rectangular waveguide which having the ports 1 and port 2 already are collinear with each other if the H-arm is connected here at the side arm of the rectangular waveguide and E-arm is connected here. So you can say that it is the combination of E and H-plane tea therefore it is known as a magic tea junction. In practical session you can use this magic tea junction which having the ports 1 and port 2 and the port 3 and port 4 are nothing but the E-arm and H-arm. So what are the characteristics of the magic tea? So if the signal is applied to port 1 and port 2 which having the equal phase and the magnitude then the output at port 4 is equal to 0 whereas output at port 3 is nothing but the addition of the input at port 1 and port 2 it is known as the additive property of the magic tea. So if the signal is fed at the port 3 then the power is divided between the port 1 and port 2 equally while the output at port 4 is equal to 0. That is if the input is applied to port 3 the output at port 4 is equal to 0 it is denoted with S4 3 equal to 0. S4 3 is one of the scattering coefficient of the magic tea junction. So if the signal is fed at one of the collinear port that is at port 1 or port 2 then there appears no output at the another collinear port as the E-arm produces a phase delay and the H-arm produces a phase advanced. Therefore S12 equal to S21 equal to 0. Now let us discuss the properties of the magic tea. So as you know that the port 1 and port 2 are collinear port but if you are applying the input to port 1 you are not getting the output at the another collinear arm. This is the important property of the magic tea. It is somewhat like a magic and therefore it is known as the magic tea junction. So it having the 4 ports port 1, 2, 3 and 4 therefore the S matrix for the magic tea is defined by 4 by 4 order. Thus it having the 4 possible inputs and the 4 possible outputs. Now how to calculate the S matrix for the magic tea? Let us calculate the each parameter defined by this S matrix. Since it is a 4 port junction the scattering matrix is of order 4 by 4. So it having the scattering coefficient or it is also known as a scattering parameters. So total 16 parameters are there S11 to S4, 4. The properties of the magic tea can be defined by this matrix. So let us applying the properties of the S matrix to define the S matrix for the magic tea junction. Now because of the plane of the symmetry in H plane tea junction S23 equal to S13. That is if the input is applied to port 3 you are getting the same output at port 1 and port 2. So meaning of this is nothing but the plane of symmetry. Therefore the S23 becomes equal to S13. Now port 1 and port 2 are out of phase for the E plane tea junction and therefore S24 is equal to minus S14. That is if the input is applied to E arm you are getting the output at port 2 and port 1 are same but out of phase and therefore it can be written as S24 equal to minus of S14. Now because of the geometry of the junction at port 3 and port 4 S34 equal to S43 equal to 0. That means if the input is applied to port 4 you are getting the output at port 3 equal to 0 and if the input is applied to port 3 output at port 4 is equal to 0. Now consider the next property as a symmetry property. Here symmetry property is nothing but Sij equal to Sji. That is the number of column for this matrix is equal to the number of column for the second matrix. So by using this property you can write these coefficients are equal that is S12 equal to S21, 13 equal to 31, 23 equal to 32 and from the figure equation number 1 S23 is nothing but S13. Therefore S23 equal to S32 equal to S13. Similarly S34 equal to S43, S24 equal to 42, S41 equal to S14 which is nothing but equal to minus S24 from equation number 2. Now the next property as the port 3 and port 4 are perfectly matched to the junction S33 equal to S44 equal to 0. That is if the input is applied to the port 3 the same port which having the output is equal to 0. Now using all these values for the scattering coefficients you can rewrite the matrix defined in above slide. So you are getting the matrix for this magic T is nothing but this one while in this case the lower rectangular box becomes is equal to 0. Thus you can observe this matrix which having the 4 unknown value which you have to define the values for this S parameters. Now from the unitary property unitary property is nothing but the scattering matrix multiplied with the complex conjugate of that scattering matrix it becomes equal to the unitary matrix. So here the scattering matrix is considered each parameter is complex conjugated which becomes equal to the unitary matrix. Now by solving this equations taking the combination of the row 1 and the column 1 for the second matrix R1 C1 R2 C2 R3 C3 and R4 C4 you can write the equations 6 7 8 9 for this. Now from the equation 8 and 9 you are getting the equation twice of S13 square is equal to 1. Here by solving this equation getting the equation for the scattering coefficient S13 is equal to 1 by root 2. Similarly S14 is equal to 1 by root 2. Now from equation 6 and 7 you know the equations 6 and 7 these two are the equations. So by comparing these equations all parameters are equal except this S11 and S22. Therefore you can write S11 and S22 from equation 6 and 7 putting these values from equation 10 into the equation number 6 you getting this equation. Now from this equation you can write S11 square plus S12 square is equal to 0. So by solving this equation both parameters S11 and S12 becomes equal to 0. From this equation you know that S11 equal to S22 and therefore in above slide you are getting the value of S11 as equal to 0 and therefore S22 is also equal to 0. Now you know all these four parameters you can put all these parameters into the S matrix. So this is the S matrix for the magic t which having the values is 1 by root 2 1 by root 2 and minus 1 by root 2 because of the E plane t junction. Now you know that how to find out the output for these all ports. So this is the equation output matrix B equal to S into A. Why this output matrix is the B and the input matrix is defined with this column matrix as a A and this is the scattering matrix. So by using this you can write the equation for the four outputs B1, B2, B3 and B4. Thus you are getting this equation 15, 16, 17 and 18. Now consider the different cases. The case 1 input is applied to port 3 that is no input is at port 1 and 2 and 4. Thus by putting all these values in the above equation you are getting B1 equal to and B2 equal to 1 by root 2 A3. This is the property of H plane t junction which have discussed in the previous slides. Now the case 2 consider that now the input is applied to port 4 and the other ports are equal to 0. Put all these values in the above equations as you are getting B1 and B2 having the same values but these are out of phase to each other and this is the property of E plane t junction. While the output at port 3 and port 4 is equal to 0. Consider the case 3 if the input is applied to port 1 and the other inputs are at other ports equal to 0. Now by putting all these values you are getting B3 equal to B4 the same values 1 by root 2 of A1. That is when the power fed to port 1 there is no any output at port 2 even though they are collinear ports thus it is like magic and therefore these are called as isolated ports. Similarly if the input is applied to port 3 you are not getting any output at port 4 thus E and H ports are also isolated ports. These are the references for this session. Thank you.