Control System – Mason’s Gain Formula

As we discussed earlier that the objective of  Block Reduction technique and Signal Flow graph is same i.e to find overall transfer function.So in SFG, we use Mason’s Gain Formula to compute the same. Before we start , you should go through my previous post(then only you can understand the terms used here).So let’s see the equation first:

Mason’s Gain Equation

Overall Transfer function = T.F =

Let’s see all the terms used here:

i = Number of forward paths

= Gain of  forward    path

= System determinant which is calculated as follows

= 1 – (Addition of each loop gains including self loop gains) +(Addition of all the gain products of two non-touching loops) – (addition of all gain products of the three non touching loops) + …………………….

 

 

= The value of  ‘‘  for the part of the graph not touching to the  forward path

 

i.e   = (1 – All the loops that do not touch the    forward path)

 

Initially, you may find all these terms and formulae a little complicated, but if you practice more problems, then eventually everything will become ample clear and easy.

 

So let’s have a look into the steps required to solve SFG using Mason’s Gain Formula

We need to find all possible forward paths i.e   (till n forward paths) and also their respective gains.

 First find out all single loops, i.e    and their gains.    stands   for first single loop,   stands for second single loop(Here m single loops are taken).

 1. Find out all the non touching loops out of step 2 i.e         and     their   gains(if any) . Here p two non touching loops are taken.    is the first product of 2 non   touching    loop,   is the second product of 2 non touching loops etc.

 2. Find all three nontouching loops out of step 2 (if any) i.e  and their respective gains. Similarly for other higher number of non touching loops if any present.

 3. The value of  ‘Δ’  is given by,

Take . Find    by equation given in step 5 , Repeat step 6 for all forward paths   to find 

4. Find Transfer Function ( i.e T ) =   

At first , you may get confused with the subscript notation used for ‘L’ ,so let’s make it simpler:

You should note that in two subscript notations used, first letter is serial number of loop and second number stands for loops non touching.

 

Eg    stands for 1^st , 2^nd , 3^rd …….of single loops

 

stands for 1^st , 2^nd , 3^rd ………of two non touching loops

 

stands for 1^st , 2^nd , 3^rd ………of three non touching loops

 

So without wasting much time, we should now look into the applications of Mason’s Gain Formula in block reduction technique in my next post. Before that please go through this post again to learn the terms that we will be using in the formula.