 Welcome to this session. I am Priyanka Bidla and today we will see about signal flow graph and MESSON's GAN formula. These are the learning outcomes of this video lecture. At the end of this session, students will be able to represent the block diagram by its equivalent signal flow graph and second outcome will be use MESSON's rule for reducing a signal flow graph to a system transfer function. These are the contents of this video lecture. First, we see the procedure of converting block diagram into the signal flow graph. There are total four steps. So, step one, label all the summing points and takeoff points on a block diagram and labeling the summing points by S1, S2 and so on and labeling the takeoff points by T1, T2 and so on. Step two, assign node to every summing point and takeoff points on a block diagram. Then step three, add dummy node at input and output side. Then step four, connect nodes as per block diagram and write gains associated with each branch. So, we see the example. So, this is consider one example. Here first, we identify the summing points and takeoff points and represented by S1, S2 and T1, T2 respectively. So, step one, there are three summing points which can be labeled as S1, S2 and S3 and there are two takeoff points that can be labeled as T1 and T2. This is T1 and T2. Then step three, add dummy nodes as R of S and C of S. So, R of S is the input side and C of S at the output side. Then this is the representation of summing point and takeoff points from the block diagram. So, there are total three summing points S1, S2 and S3 and two takeoff points T1 and T2. And R of S is the input node and C of S is the output node. Then connect nodes as per block diagram. So, according to block diagram you have to connect the nodes. Then next step, that is the last step, write the gains associated with each branch and we get the final signal flow graph. So, this is a representation of block diagram into the SFG with gains G1, G2, G3, G4 and the feedback loop also there. So, in this way you have to represent the block diagram into the signal flow graph. Now pause the video for a minute and you have to recall the terminologies that are used in signal flow graph. So, here consider this example. For that example you have to identify the forward path loops and write down the loop gain of signal flow graph. Now pause the video. So, here there is only one forward path that is represented by P1. So, P1 is equal to G1, G2, G3, G4. Now we see the loops. So, there are total 3 loops. First loop that is represented by L1. So, write down the loop gain L1 is equal to minus G2, G3, H2. Then L2 is, L2 will be G1, G2, G3, G4 minus H2 minus H3. Then this is a loop 3 and the loop gain for L3 will be L3 is equal to minus G3, G4, H1. So, these are the loops and these are the loop gains respectively. Now we see the MESSONS GAN formula. MESSONS formula is related with the signal flow graph and simultaneous equations that can be written from the graph. In order to arrive the system transfer function, block diagram reduction technique requires successive application of fundamental relationships. Whereas, MESSONS rule for reducing a signal flow graph to a single transfer function requires the application of one formula. So, MESSONS GAN formula is calculated by the relationship between the input and output variable of a signal flow graph is given by net gain and the formula is given by transfer function is equal to summation of pk into delta k divided by delta where k is nothing but number of forward path. Then pk means gain of kth forward path. Then delta is equal to 1 minus sum of all individual loop gains plus sum of product of loop gains of all possible combinations of two non-touching loops minus sum of product of loop gains of all possible combinations of three non-touching loops plus and so on. So, each time we have to consider first individual loop gains, then two non-touching loops, then loop gains of three non-touching loops, then loop gains of four non-touching loops with alternate sign. Then delta k is nothing but value of delta for the part of block diagram that does not touch the kth forward path. If there are no non-touching loops to the kth path, then delta k is equal to 1. So, consider this example and apply MESSONS rule to calculate the transfer function of the system represented by signal flow graph in figure two. So, first we have to calculate the forward paths. There are two forward paths. So, first forward path will be G1, G4, G2. P1 is equal to G1, G4, G2. Then second forward path will be G1, G4, G3. Now we consider the loops. So, there are three feedback loops. So, first feedback loop will be this, then loop gain for this, L1 is equal to G1, G4, H1. Then second feedback loop will be G1, G4, G2, minus H2, then L3 will be this. So, loop gain for L3 will be minus G1, G4, G3, H2. There are three feedback loops. Then here we see the non-touching loops, but here there are no non-touching loops. So, calculate the delta is equal to 1 minus sum of all individual loop gains. Therefore, delta is equal to 1 minus in bracket L1 plus L2 plus L3. So, L1, L2, L3 are nothing but these are the loop gains. So, write down the loop gains respectively L1, L2, L3. This is nothing but this is a delta. Then we have to calculate delta k. Delta k is nothing but value of delta for the part of the block diagram that does not touch the kth forward path. So, here first forward path, consider forward path 1, G1, G4, G2. To this forward path, all the loops are touching. So, neglect all the loop gains. So, delta 1 is equal to 1. Again consider forward path 2 that is G1, G4, G3. Here also there are all the loops are touching. So, that all the loop gains are neglected. So, delta 2 is equal to 1. If there is third forward path, at that time again you have to calculate the delta 3. So, apply the formula that is C by R is equal to P1 delta 1 plus P2 delta 2 divided by delta. If there is third forward path, then it must be plus P3 delta 3. But here in this example only two forward paths. So, you have to write down P1 delta 1 plus P2 delta 2 divided by delta. Then put the values respectively. After simplification we get this output. So, this is nothing but this is the transfer function of signal flow graph. In this way we have to calculate the transfer function of signal flow graph. These are the references of this video lecture. Thank you.