 Welcome to this session. I am Priyanka Bidla and today we are going to discuss about block diagram fundamentals and reduction rules. These are the learning outcomes of this video lecture. At the end of the session students will be able to represent the block diagram with transfer function and second outcome will be to apply the rules for reducing the block diagram. These are the contents of this video lecture. Basically block diagram is a pictorial representation of the cause and effect relationship between input and output of a system. It is very difficult to analyze the entire system as it is without breaking into small parts. So, to analyze such system block diagrams are used. This is the simplest form of block diagram. Here it is a single block with one input that is x and the one output that is y. A mathematical operation performed by that system and output is produced according to the input. The arrow indicates the direction of flow of signal from one block to another block. Now recall what is the transfer function of system. Now pause the video for a while and you have to recall the transfer function of a system transfer function. The system parameters are designed and their values are selected as per the requirement of input. Therefore, the performance of system parameters are expressed in terms of their output. So, we can write down system parameters are nothing but that is a gain. So, gain is equal to output upon input or Laplace transform of output to the Laplace transform of input. So, we can write down output is equal to gain into input. So, gain is nothing but performance of the system parameters. Then we consider here the components of block diagram. There are the four basic components. So, we see one by one the first is the signals. So, here this is the R of s and this is the C of s. R of s is nothing but reference signal and C of s is the controlled signal. So, R of s is the reference input that is a Laplace transform of the R of t and C of s is the controlled output. This is the Laplace transform of C of t. Second component transfer function of components shown inside the block diagram. Here G of s is nothing but transfer function of system input is nothing but R of s and this is the output is nothing but C of s. So, this is the representation of system. Then summing point here this is summing point one or more than one elements can added or subtracted at the summing point. Here R 1 of s and R 2 of s are added directly. So, here we get C of s is equal to R 1 of s plus R 2 of s. Suppose here the sign is negative then we can write here R 1 of s minus R 2 of s. By default if there is no sign then it will be considered as a plus. Then the next four component fourth component that is takeoff point this is a takeoff point. The point from which signal is taken for the feedback purpose R of s R of s R of s. So, these signals are taken from this point for the feedback purpose. Then we see the advantages of block diagram that the four advantages. So, first is the very simple to construct for complicated system. Then second we can write down the individual elements from the block diagram. Then third individual as well as the overall performance of the system can be studied using the transfer functions. And then the fourth is you can calculate the closed loop transfer function of a system directly by using the block diagram reduction rules. So, there are the total 10 rules. In this video lecture we will see the first three rules. Then these advantages of block diagram. Block diagram does not involve any information about the system. Then second energy source is not shown in the block diagram. And third one is nothing but it can be applied only for the linear time invariant systems. Then here we can calculate the simple form of closed loop system. For that we have to calculate the transfer function. All these signals are Laplace transform of the particular signals. So, R of s is nothing but reference signal Laplace of R of t. E of s is the Laplace of error signal that is E of t. Then B of s is here we can write minus and plus. It may be minus or it may be negative or it may be positive. The meaning of this is the feedback signal. So, feedback may be negative or positive. It is a Laplace transform of B of t. Then G of s is nothing but it is a forward path transfer function. And H of s is the feedback path transfer function. Feedback path means here the path indicates the flow of direction from output to input. So, this is feedback path and the forward means it is from input to output direction. Then C of s is the control output that is a Laplace transform of C of t. Here G of s and H of s can be obtained by reducing the complicated block diagram. So, from that diagram we can write these equations. So, E of s is equal to reference signal minus plus B of signal. Then B of s is equal to is the product of C of s and H of s. Then C of s is the product of E of s and G of s. From this you can write down E of s is equal to C of s divided by G of s. Then here that is B of s is equal to C of s H of s. So, put the value of equation 2 in the equation 1. So, we get E of s is equal to R of s minus plus C of s into H of s. Now, compare the equation 4 and 5. Here left hand side E of s and E of s are equal. So, right hand side are equal. So, we can write down this equation. After solving these equations we get the transfer function that is C of s divided by R of s is equal to G of s upon 1 plus or minus G of s into H of s. Transfer function is nothing, but output upon input. So, C of s divided by R of s we get this equation. So, this is the transfer function of closed loop system. And remember if there is a plus sign then this is a negative feedback. If there is negative sign then this is a positive feedback. Then we see the rules for the block diagram reduction. There are three rules for reduction of the block diagram. So, first is what any complicated system is brought into the simpler form. First we have to reduce the block diagram. Then second using proper logic of the system and value of the feedback should not get disturbed. And third we will now analyze some common topologies for interconnecting subsystem. Some common topologies are used for the reduction of the block diagram. So, we see the rules. So, rule 1 is associative law. Here this is the cascade connection of the summing parts. Here R 1 is given to this first summing point and R 2 and R 3 are given respectively. So, here this is a minus sign. So, R 1 minus R 2 and then this output is given to the next summing point. So, we get R 1 minus R 2 plus R 3. This is the final output of this summing points. Now, in this figure you have to consider exchange of summing points. So, here we have to exchange the summing points. So, instead of R 2 here R 3 is there and instead of R 3 here R 2 is there. So, here we get R 1 plus R 3 and then R 1 plus R 3 is given to second summing point. So, here we get R 1 plus R 3 minus R 2. So, after exchanging the position of the summing points then also we get the same output. This is associative law. Second blocks in series or cascade. The transfer function of the blocks which are connected in series get multiplied with each other. If the blocks are in series then we can get multiplied directly. If n blocks are there and the transfer function of respective blocks g 1 to g n then take the product of these blocks. So, it is represented by this equation. Consider example R of s is given to g 1 and here we get R of s into g 1 and this is applied to next block. So, here we get R of s into g 1 into g 2. This is the before reduction. After reduction in this case g 1 and g 2 are in series. So, directly take a product and we get the output R of s g 1, g 2 blocks in parallel. If the blocks are in parallel then directly added these blocks. Consider the sign. If there is negative sign then you have to consider the negative sign. Now consider an example here g 1, g 2 and g 3 these blocks are connected in parallel. So, directly added to at this summing point. So, we get g 1 of s plus minus g 2 of s plus minus g 3 of s and the R of s is common to all. So, we can multiply with R of s and how to reduce this block? If the blocks are in parallel then added directly. So, here this is the equivalent transfer function of the parallel subsystem. Remaining rules we will see in the next video lecture. These are the references. Thank you.