3. 3
INTRODUCTION
ā¢ A signal flow graph consists of a network in which nodes are connected by
directed branches.
ā¢ It depicts the flow of signal from one point of a system to another and gives
the relationship among the signal.
4. FUNDAMENTALS OF SIGNAL FLOW GRAPHS
ā¢ Consider a simple equation below and draw its signal flow
y ļ½ ax
graph:
ā¢ The signal flow graph of the equation is shown below;
a
yx
ā¢
ā¢
Every variable in a signal flow graph is designed by a Node.
Every transmission function in a signal flow graph is designed
Branch.
Branches are always unidirectional.
by a
ā¢
ā¢ The arrow in the branch denotes the direction of the signal flow.
5. An input node or source contain only the outgoing branches. i.e., X1
An output node or sink contain only the incoming branches. i.e., X4
A path is a continuous, unidirectional succession of branches along
node is passed more than ones. i.e.,
ā¢
ā¢
ā¢ which no
X2 to X3 to X4X1 to X2 to X3 to X4 X1 to X2 to X4
ā¢ A forward path is a path from the input node to the output node. i.e.,
X1 to X2 to X3 to X4 , and X1 to X2 to X4 , are forward paths.
A feedback path or feedback loop is a path which originates and terminatesā¢ on
the same node. i.e.; X2 to X3 and back to X2 is a feedback path.
TERMINOLOGY
6. MASONāS RULE (MASON,
1953)
ā¢ The block diagram reduction technique requires successive
application of fundamental relationships in order to arrive at the
system transfer function.
On the other hand, Masonās rule for reducing a signal-flow graphā¢
to a single transfer function requires the application of one
formula.
The formula was derived by S. J. Mason when he related theā¢
signal-flow graph to the simultaneous equations that can be
written from the graph.
7. Where,
n = number of forward paths.
i thPi = the forward-path gain.
ithāi = Determinant of the forward path
ā is called the signal flow graph determinant or characteristic function.
Since, ā=0 is the system characteristic equation.
ļ„Pi ļi
C(s)
T ļ½ iļ½1
ļ½
ļR(s)
n
MASONāS RULE
8. ā = 1- (sum of all individual loop transmittance) + (sumof the products of
loop transmittance of all possible pairs of Non Touching loops) ā (sum of
the products of loop transmittance of Triple of Non Touching loop) + ā¦
āi = Calculate ā for ith path
=1- [loop transmittance of single Non Touching loops with forward path]
+ [loop transmittance of pair of Non Touching loops withforward path]
MASONāS RULE
9. ā¢ Calculate forward path gain Pi for each forward path i.
ā¢ Calculate all loops transfer function.
ā¢ Consider non touching loops 2 at a time
ā¢ Consider non touching loops 3 at a time
ā¢ Calculate Ī from steps 2,3,4 and 5
ā¢ Calculate Īi as portion of Ī not touching forward path i.
SYSTEMATIC APPROACH
12. ā¢ Obtain total number of single loop:
L11= -H1,
L21 = H2,
L31 = G3H4,
L41= G2G3H3
ā¢ Obtain total number of 2 non-touching loops:
L12= L11 and L21 = H1H2
L22 = L11 and L31 = G3H4H1
ā¢ There are no 3 non touching loops.
ā¢ Find out the value of
L11 + L21 + L31 + L41 + L12 + L22
-H1 + H2 + G3H4 + G2G3H3 + H1H2 + G3H4H1
13. ā¢ Find out value of ā1 and ā2
Take F1 and Find ā1
Since all loops touch the F1 forward path, ā1=1
Take F2 and ā2
L 11 and L21 do not touch the forward path
ā2= 1 - (L11 + L21)
ā2= 1 + H1+ H2
ā¢ Obtain the transfer function:
š¶(š )
š (š )
=
šŗ1 šŗ2 šŗ3 + šŗ4(1 + š»1 + š»2)
1 + H1+ H2 + G3H4 + G2G3H3 + H1H2 + G3H4H1