 Today we will see the one of the Microwave T-junction, E-plane T-junction, Myself, Piyusha, Shedgar. These are the learning outcomes for this session. At the end of this session students will be able to derive the scattering matrix for E-plane T-junction and they will be able to apply the properties of S-matrix to A-plane T-junction. These are the contents. First we will see what is the Microwave T-junction. Microwave T-junction is an interconnection of the three waveguides in the form of the English alphabet T. Therefore, it is known as the Microwave T-junction. So, there are several types of the T-junctions. H-plane T-junction, E-plane T-junction and the combination of E-plane T-junction and H-plane T-junction is also known as EH-plane T-junction or magic T-junction. So, the first figure shows E-plane T-junction where the side arm is connected to the wider portion of the main waveguide. Whereas in H-plane T-junction the side arm is connected to the narrower portion of the main waveguide. It is also known as the H-arm and the combination of both E-plane T-junction and the H-plane T-junction is nothing but the hybrid plane T-junction or magic T-junction. Now we will see the details of this E-plane T-junction. As you know that the side arm is connected to this slot is connected, cut along the broader dimension of a long waveguide and a side arm is attached as shown in this figure. These are having the three ports, port 1, 2 and port 3. Port 1 and port 2 are the collinear arms and port 3 is also known as the E-arm. In E-plane T-junction these are the inputs are applied to port 3. You can apply the input to port 3, port 1 or port 2. Here when the input is applied to port 3 the outputs at port 1 and port 2 will have the phase shift of 180 degree. Since the electric field line change their direction when they come out of the port 1 and 2 it is called as E-plane T-junction. E-plane T is also known as the voltage or series junction symmetrical about the central arm. As shown in this figure you can observe that if the input is applied to port 3 whatever are the direction of the output at port 1 and the direction of the output at port 2 are out of phase with each other. Now we will see the calculation of the S-parameters. As this matrix which having this three port junction the scattering matrix is consider is of the order of 3 by 3. The properties of E-plane T can be defined by its S-matrix. Determination of S-parameters by applying properties of the S-matrix. Now consider the S-matrix which having these coefficients and the S-matrix is the order of 3 by 3 since it having the three ports. Before going to start the calculation you can pause video here and you can write down the properties of the S-matrix. So you can write the properties such as symmetrical property, unitary property and the other properties. Now we will see the calculation of S-parameters stepwise. Consider the step 1 since the outputs at port 1 and port 2 are out of phase by 180 degree with an input at port 3. The scattering coefficients S-1-3 and S-2-3 must be out of phase. Meaning of this S-1-3 as input is applied to port 3 and you are taking the output at port 1 whereas S-2-3 means input is applied to port 3 and you are taking the output at port 2. Since these are out of phase with each other S-2-3 can be written as negative of S-1-3. Now use the second property of the S-matrix, symmetry property as Sij equal to Sji. Since the number of columns equal to the number of rows you can write these three conditions for this S-1-2 equal to S-2-1, S-1-3 equal to S-3-1, S-2-3 equal to S-3-2 and from equation 1 S-2-3 equal to minus of S-1-3. Therefore you can add this minus of S-1-3 for the equation number 2. Next is since port 3 is perfectly matched to the junction that is there is no any mismatching between the junction and the source that is there is no any reflection to the port 3. Thus you can write S-3-3 equal to 0. Using these values equation 1, 2 and 3 in the above matrix you can rewrite this matrix as shown in this equation number 4. Thus we are getting the 4 unknown values S-1-1, S-1-2, S-1-3 and S-2-2. So you can find these all values and you can put all these values in this equation to calculate the S-matrix for the E-plate t-junction. Now for this calculation consider the next property unitary property that is S-matrix is multiplied with the complex conjugate of the S-matrix equating to the identity matrix. Thus this is the S-matrix S-1-1, 1, 2, 1, 3 and take the complex conjugate of this matrix multiplied with this matrix. Thus you are getting the identity matrix or it is the unit matrix. Now multiplying this the combination R-1 is multiplying with the C-1. Thus you are getting these equations 5, 6, 7, 8 from the first equation S-1-1 square plus S-1-2 square plus S-1-3 square is equal to 1 and R-2 C-2 combination is S-1-2 square plus S-2-2 square plus S-1-3 square equal to 1. Now from equation 5 and 6 if you are comparing these two equations these S-1-1 and S-2-2 are also equal as S-1-3 and S-1-2 is equal. And therefore from equation number 5 and 6 you can write the equation number 9 as S-1-1 equal to S-2-2. From equation 7 you can make it twice of S-1-3 square is equal to 1. Thus you are getting the value of S-1-3 parameter that is input is applied to port 3 and you are taking the output out of the port 1 equal to 1 by root 2. Similarly from equation 8 take S-1-3 as a common and the complex conjugate of S-1-1 minus complex conjugate of S-1-2 equal to 0. From equation 10 you can observe that S-1-3 is never equal to 0 because the calculated value for S-1-3 equal to 1 by root 2 and therefore S-1-3 is not equal to 0 thus you can equate this bracket is equal to 0. So from this you can getting the equation S-1-1 equal to S-1-2 or S-1-2 equal to S-2-2 equal to S-1-1. Now using all these values in equation number 5 to calculate the value of S-1-1 parameter thus by putting all these values in this equation by solving this equation you are getting the value of S-1-1 as 1 by 2. Now from equation 11 S-1-2 equal to minus of S-1-1 as you know S-1-2 equal to 1 half thus S-2-2 is also becomes equal to 1 by 2. Now substitute all these four unknown values in the above matrix equation number 4 thus you are getting this matrix S matrix it is also known as the scattering matrix for E plane T junction. Now to calculate the output values consider the output matrix B this matrix is the column matrix B it is also known as the output matrix is equal to scattering matrix multiplied with the column matrix which having these inputs. Now B-1, B-2, B-3 are the outputs taken out of the port 1, 2 and 3 equal to whatever is the scattering matrix you are getting then it is multiplied with the number of inputs A-1, A-2, A-3 are applied to port 1, 2 and 3. Now by solving this matrix you are getting these equations as shown by equation number 15, 16 and 17. So B-1 equal to 1 by 2 A-1 plus 1 by 2 A-2 plus 1 by root 2 A-3. B-2 equal to 1 by 2 A-1 plus 1 by 2 A-2 minus 1 by root 2 A-3 and B-3 equal to 1 by root 2 A-1 minus 1 by root 2 A-2. Now to find out the outputs at different ports consider the different cases. So consider the case 1 in case 1 input is given at port 3 that is input is applied to the port 3 and no input is at port 1 and port 2 that is A-1 and A-2 equal to 0 and A-3 is not equal to 0. By putting these conditions in the above equation 15, 16 and 17 you are getting the value of the output B-1, B-2 and B-3. And input at port 3 equally divides between the port 1 and port 2 from this B-1 and B-2 but these two are 180 degree out of phase with each other. Hence E plane T acts as a 3 dB splitter. Consider the case 2 if A-1, A-2 equal to A that is the same input is applied to port 1 and port 2 and A-3 is equal to 0. Thus getting the values of the output B-1, B-2, B-3 as shown in this equation. Thus if the equal input is applied to port 1 and port 2 there is no any output at port 3 that is B-3 is equal to 0. Case 3 if A-1 is not equal to 0 and A-2 and A-3 are equal to 0 they are getting this equation B-1 and B-2 which having the same value and B-3 is negative of A-1 by root 2 value. Thus you can find out the output for the different ports by applying the different cases. These are the references for this session. Thank you.