 Good afternoon everyone. Today we will see the topic H-plane T-junction, myself Piyusha Shedgarh from WIT, Solapur. These are the learning outcomes for this session. At the end of this session, students will be able to derive the scattering matrix for H-plane T-junction and they will be able to apply the properties of S-matrix to H-plane T-junction. These are the contents. So, what is Microwave T-junction? So, Microwave T-junction is an interconnection of three waveguides in the form of English alphabet T. So, there are the several types of the T-junctions, H-plane T-junction, E-plane T-junction and the combination of both these H-plane T and E-plane T-junction. It is also called as Magic T-junction. So, E-plane T-junction in E-plane T-junction, the E-arm that is the side arm is connected to the main waveguide at the wider dimension. Whereas in H-plane T-junction the side arm that is it is also known as the H-arm is connected to the broader dimension of the main waveguide. And the combination of both these H-plane T and E-plane T forms the hybrid plane T or it is also known as the Magic T-junction. So, we will see the details of H-plane T-junction. H-plane T-junction is formed by cutting the rectangular slot along the width of the main waveguide. And attaching another waveguide it is also known as the side arm or the H-arm as shown in this figure. So, here the port 1, port 2 and port 3. Thus, H-plane T-junction which having the three ports. H-plane T-junction is so called because the axis of the arm parallel to the plane of main transmission line. As all the arms of H-plane T lie in the plane of magnetic field the magnetic field divide itself into the arms and therefore this is also called as the current junction. So, here the three ports are there. If the input is applied to port 3 that is to H-arm then whatever is the microwave energy equally divides into port 1 and port 2. So, here microwave T-junction H-plane T-junction having the three ports that is three inputs and three possible outputs and therefore the for defining the scattering matrix for H-plane T we can define it with the order 3 by 3. So, how to calculate the scattering parameters for H-plane T-junction? So, here you can use the scattering matrix properties. So, the properties of H-plane T can be defined by its scattering matrix. So, here the order of the matrix is 3 by 3 and therefore it can be defined with the given equation A. So, it having the scattering parameters S11, S12, S13 like that it defines with the row and column number. Before going to further slide you can pause video here and recall that what are the properties of scattering matrix. So, you can write these properties of the scattering matrix such as symmetric property, unitary property and the other properties. Now, how to calculate the S parameters? We will see it stepwise. Step 1, because of the plane of the symmetry of the junction scattering coefficients S13 and S23 must be equal in phase that is if the input is applied to port 3 you are getting the output at port 2 and port 1 is equal. Second property from the symmetry property that is number of column for the first matrix is equal to the number of rows for the second matrix thus you can define this property as Sij equal to Sji. So, by using this property you are getting these equations S12 becomes equal to S21, S13 becomes S31 and S23 equal to S32 and from equation 1 S23 is also equal to S13 and therefore this can be written as S23 is also equal to S13. Now, since the port 3 is perfectly matched to the junction that is it is there is no any reflection to the port 3 because there is the perfect matching between the junction and the source and therefore you can represent that parameter or coefficient equal to 0. Now, using all these values in the S matrix from the equation 1, 2 and 3 thus you are getting the matrix as shown in equation number 4 thus we are having the 4 unknown values S11, S12, S13 and S22. So, you have to find the values of these all scattering parameters and by putting these values in this scattering matrix you are getting the equations for the scattering coefficients for H-plane t-junctions. Next type is from the unitary property that is the scattering matrix multiplied with the complex conjugate of that scattering matrix getting that equation is equal to the identity matrix. Therefore, define this scattering matrix as shown in this figure. Next is you are taking the each element complex conjugate of that is multiplied with this matrix equal to identity matrix. Now, by multiplying we are getting R1 is multiplied with the C1 that is rho 1 is multiplied with the column 1 of the second matrix. These are the equations S11 square plus S12 square plus S13 square equal to 1. So, as similarly you can calculate the equation for the R2 C2. So, if you are comparing the equations 5 and 6 S11, S12 and S13 and in equation 6 it is S12, S22 and S13. So, if you are comparing these two equations S12 term is there, S13 is also term is there and therefore, you can write S11 square equal to S22 square. Similarly, R3 C3 equal to S13 square plus S13 square equal to 1 and taking the another combination R3 C1 you can write the equation as shown in equation number 8. Now, equating 5 and 6 you can write S11 square equal to S22 square and therefore, S11 becomes equal to S22. From that equation number 7 you can calculate the value of the scattering parameter S13. So, from this equation you are getting the value of the S13 as 1 by root 2 and from equation 8 S13 taken as a common in bracket complex conjugate of S11 plus S12 conjugate equal to 0. So, from equation 10 as you know the value of S13 should not be equal to 0 because you already get the value is equal to 1 by root 2 and therefore, making this bracket is equal to 0 again you are getting S11 equal to minus of S12 or you can write S12 equal to S11 minus of S11. So, using all these equations and putting all these values in equation number 5 you can calculate the value for S11. So, again you are getting the scattering coefficient S11 equal to half. Now, one more parameter is remaining as a S12. So, from previous equation you are getting the value of S12 is the negative of S11. Thus you can write the value for S12 equal to negative of 1 by 2 and S22 is the positive of 1 by 2. Now, you know all these value for unknown values by putting all these values in the scattering matrix you are getting this scattering matrix for the H plane T junction. So, as you know the scattering matrix you can find out the output for this scattering matrix. So, output matrix B is defined with scattering matrix is multiplied with the input matrix. So, the output matrix which having this outputs B1, B2, B3 that is B1, B2, B3 are the outputs for the port 1, 2 and 3 respectively. Similarly, for input we are using the notation as A, A1, A2, A3 are the inputs applied to port 1, port 2 and port 3. So, the output matrix is defined with S into A. So, from this matrix you can define the equations for the output B1, B2, B3 as shown in equations 15, 16 and 17. So, you can use the number of cases to consider the case 1. Now, case 1 is input is applied to port 3 that is no input is for the port 1 and port 2 and therefore, you can equate A1 equal to A2 equal to 0 and A3 is not equal to 0. So, by putting these conditions in above equations you getting the output for port 1 and port 2 as 1 by root 2 times of the A3 and the same output can be observed at the port 2. Thus, if the P3 is the power input at the port 3, then this power divides equally between the port 1 and port 2 which is in phase. And you know that the power at the port 3 is the addition of the power at the port 1 and port 2. So, if you are defining this value in decibel you are getting the value for this power as a minus 3 decibel. Hence, the power coming out of port 1 or port 2 is 3 decibel down with respect to input power at port 3. Hence, H plate E is also called as a 3 dB splitter. Now, consider the case 2. A1, A2 equal to A that is the same input is applied to port 1 and port 2 and the input is not applied to port 3 and therefore, A3 is equal to 0. So, again by putting these conditions in above equations for the calculation of output, thus we are getting the outputs at port 1 equal to 0, output at port 2 is also equal to 0 and output at port 3 is equal to 1 by root 2 A plus 1 by root 2 A that is the addition of these two inputs. Thus, the output at port 3 is the addition of the two equal inputs at port 1 and port 2 in phase. Thus, you can calculate the output for this by considering the different cases. These are the references for this session. Thank you.