 Hello everyone. Today we will see the topic introduction to S parameters, 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 coefficients for microwave junctions and will be able to describe the properties of scattering matrix. These are the contents for this session. So, microwave system consists of several microwave components including the source and load are connected to each other by waveguide or coaxial or transmission line. So, any microwave system which having this waveguide junction or it is also called as microwave junction, these are connected to the different ports. So, the figure 1 shows the microwave junction which having the 4 ports port 1, 2, 3 and 4. So, there is need of any microwave system to split the microwave energy into the different microwave components. So, this microwave junction or waveguide junction can be connected to the different source and the loads. So, this is the figure which having the microwave junction or it is also waveguide junction. So, it is analogous to the traffic junction where the number of vehicles are coming on a road and from that junction it will be leaving or coming to that junction. So, here it is connected the microwave source at port 1 and the different microwave loads are connected to the port 2, 3 and 4. That means, when you are applying the input to this microwave source at port 1, some part of this microwave energy is going towards this port 2, port 3 and port 4 and may be some part of the microwave energy is reflected back towards this microwave source that is port 1 because of the mismatch between the microwave source input and the microwave junction. So, before going to start the S parameters, you recall here what is 2 port network. The 2 port network is also known as a 4 terminal network or quadrupole is an electrical circuit or device with 2 pairs of terminals that is the circuit connects to 2 dipoles to connect the external circuits. So, this microwave junction also has the 4 ports similar to the low frequency 2 port networks. The parameters used to describe this 2 port network are Z, Y, H, G, T and ABCD parameters. They are usually expressed in matrix notation and they establish relation between the different variables. If the circuit consists of the linear elements, this can be represented by a set of linear equations relating to independent variables and the dependent variables like currents and voltages. But in case of the higher frequencies, the concept of voltages and currents becomes more difficult to relate the network performance especially in networks using transmission lines like waveguides. So, in this case we can use the scattering matrix is denoted with S matrix. So, what is scattering matrix? So, it is a matrix which gives all the combinations of power relationship between the various inputs and output ports of a microwave junction and the elements of this matrix are known as the scattering coefficients or scattering parameters. So, these S parameters are related to the power waves instead of voltages and currents. The S matrix for an n port contains the n square coefficients. It is also known as the S parameters each one representing a possible input output path. Suppose the 2 port network is there. So, in that case it becomes the 4 coefficients of the S matrix. S parameters are usually displayed in a matrix format. In matrix format, the number of rows and columns denotes number of ports. In general, the scattering coefficient is denoted with S ij is nothing but it is the scattering coefficient resulting due to the input at ith port and output taken out of the jth port. For example, S11 is the ratio of amplitude of the signal that reflects from port 1 to the amplitude of the signal incident on the same port that is port 1. So, parameters along the diagonal of this S matrix are referred to as the reflection coefficients and parameters along the off diagonal S parameters are referred to as the transmission coefficients. So, here are the example of the S matrices which having the 1 port, 2 port and 3 port matrix. So, here the 2 port matrix which having the scattering coefficients are S11, S12, S21 and S22 whereas S11 and S22 are the diagonal elements are referred to as the reflection coefficients whereas the off diagonal elements are referred to as the transmission coefficients. So, generally the input and output reflection coefficients of the network can be plotted on the Smith chart. Why the transmission coefficients are usually not plotted on the Smith chart? So, you can recall here what is Smith chart? Yes, the Smith chart is one of the most useful graphical tools for high frequency circuit applications and it is used to draw the characteristics impedances which having the complex values that is you can draw the real as well as the imaginary part on the Smith chart. The Smith chart can be used to display the multiple parameters including impedances, admittances, reflection coefficients, scattering parameters and noise figure circles etc. Now, consider the microwave junction with the 2 ports as shown in this figure that is this one is the port 1 and this is the port 2 which having the scattering coefficients are S11, S12, S21 and S22. So, if you observe at port 1, A1 and B1 is there whereas A1 is the input applied to port 1, B1 is the output taken out of the port 1. So, here you can say that As are the inputs to the particular ports while Bs are the output out of various ports and in general the scattering coefficients is denoted with Sij that is for S11, i equal to j equal to 1. Now, how to write the matrix for the 2 port network and how to calculate the output is denoted with this matrix notation as given below. So, this is the matrix B1, B2 is the used for the output that is corresponding to the reflected waves whereas this matrix is the scattering column matrix and A1, A2 are the column matrix A corresponding to incident wave or it is denoted as a inputs. So, from this matrix notation you can calculate the different outputs or the scattering coefficients by representing it with the equations. Thus, the output matrix B can be written as scattering matrix multiplied with the input. Now, expanding this above matrices into the equations we getting the equations for the output B1 and B2 whereas B1 is the output taken out of the port 1 and B2 is the output reflected at the port 2. So, B1 becomes equal to S11A1 plus S12A2, why B2 equal to S21A1 plus S22A2. So, this each equation gives the relation between the reflected and incident power waves at each one of the network ports 1 and 2 in terms of the networks individual S parameters. So, from this equation you can calculate the S parameter for each port. So, here S11, 12, 21 and 22 are the 4 scattering parameters are calculated by given B1 by A1, B1 by A2, B2 by A1 and B2 by A2 respectively. So, as you compare this equation this S11 is nothing but input is applied to the port 1 and you are taking the output at the same port. Thus, it becomes output at port 1 when the input is applied to the port 1. Similarly, S12 is denoted as B1 by A2. So, here the output is at port 1 when you are applying the input to port 2. So, these above equations for S11 and S21 are derived from network analysis of the measurement by setting the value of incident signal at the port 2 equal to 0 that is A2 equal to 0 and solving for the above S parameter ratios as the function of A1. Similarly, S12 and 22 are derived by setting the value of A1 equal to 0 and solving for the other ratios. These are the properties of the S matrix. S matrix is always a square matrix of order n by n. So, if you are considering the matrix is of 3 by 3 or 2 by 2 then you are getting the scattering coefficients or the parameters as n square. Second property is S matrix is a symmetric matrix that is the meaning of symmetric is scattering matrix of one of the Sij becomes equal to the scattering matrix of Sji that is here i denotes the number of rows equal to the number of columns for the second matrix and the number of columns for the first matrix becomes equal to the number of rows for the second matrix. Next is the scattering matrix is a unitary matrix that is you can write this property in equation form as scattering matrix multiplied with the complex conjugate of that matrix gives you the unit matrix or identity matrix of the same order. These are the references for this session. Thank you.