SOLUTION

Signal Flow Graph

The signal flow graph for a system can be reduced to obtain the transfer function of the system using the following rules.

Rule 1: Incoming signal to a node through a branch is given by the product of a signal at the previous node and the gain of the branch.

Rule 2: Cascaded branches can be combined to give a single branch whose transmittance is equal to the product of individual branch transmittance.

Rule 3: Parallel branches may be represented by a single branch whose transmittance is the sum of individual branch transmittances.

Rule 4: A mixed node can be eliminated by multiplying the transmittance of the outgoing branch (from the mixed node) to the transmittance of all incoming branches to the mixed node.

Rule 5: the equations to find the ratio of output to input. This ratio gives the gain of the resultant branch. A loop may be eliminated by writing equations at the input and output node and rearranging.

The signal flow graph of a system can be reduced either by using the rules of a signal flow graph algebra (i.e.) by writing equations at every node and then rearranging these equations to get the ratio of output and input (Transfer function).

The signal flow graph reduction by the above method will be time-consuming and tedious S.J. Mason has developed a simple procedure to determine the transfer function of the system represented as a signal flow graph. He has developed a formula called by his name Mason’s gain formula which can be directly used to find the transfer function of the system.