-
8/8/2019 Mason Gain Formula
1/15
Bloc k Diagram s, Signal
Flow Graphs , and Masons
Gain Form ula
EE-371 / EE-502 Control SystemsMilwaukee School of
Engineering
Fall Term 2005
Dr. Glenn Wrate, P.E.
-
8/8/2019 Mason Gain Formula
2/15
10/4/2005 2005, Milwaukee School of Engineering 2
Bloc k Diagram s
The transfer
function is giveninside the block
The input in this
case is E(s) The output in this
case is C(s)
C(s) = G(s) E(s)
G(s)
C(s)E(s)
-
8/8/2019 Mason Gain Formula
3/15
10/4/2005 2005, Milwaukee School of Engineering 3
Signal Flow Graphs
Each signal is a
node in the graph E(s) and C(s) in
the case shown
Each transferfunction is abranch
G(s) in this case C(s) = G(s) E(s)
C(s)E(s) G(s)
-
8/8/2019 Mason Gain Formula
4/15
10/4/2005 2005, Milwaukee School of Engineering 4
Sum m ing J unc t ion
Block format
Circle representsthe summing
junction
Plus or minussigns at eachsignal into the
junction
G1(s)
G3(s)
G2(s)
E1(s)
E2(s)
E3(s)
C(s)+
-
+
C(s) = G1(s) E1(s) + G2(s) E2(s) G3(s) E3(s)
-
8/8/2019 Mason Gain Formula
5/15
-
8/8/2019 Mason Gain Formula
6/15
-
8/8/2019 Mason Gain Formula
7/15
10/4/2005 2005, Milwaukee School of Engineering 7
Def in i t ions
Source Node
A node for which signals only flowaway from the node
Sink Node
A node for which signals only flowtowards the node
A sink node can be created byadding a branch of gain 1.0
-
8/8/2019 Mason Gain Formula
8/15
10/4/2005 2005, Milwaukee School of Engineering 8
Examples
Add a branch to make a sink
E(s) = R(s) H(s)C(s)
R(s) 1 C(s)G(s) C(s)1
- H(s)
E(s)
-
8/8/2019 Mason Gain Formula
9/15
10/4/2005 2005, Milwaukee School of Engineering 9
Examples
Add a branch to make a source?
E(s) = R(s) H(s)C(s) + E(s)?
R(s) 1 C(s)G(s) C(s)1
- H(s)1E(s)
E(s)
-
8/8/2019 Mason Gain Formula
10/15
10/4/2005 2005, Milwaukee School of Engineering 10
Terms
Path
A continuous connection of branchesin the same direction
Loop
A closed path in which no node istraveled through twice
E(s) G1(s) C(s)G2(s)
-
8/8/2019 Mason Gain Formula
11/15
10/4/2005 2005, Milwaukee School of Engineering 11
Term s Cont inued
Forward Path
A path that connects a source nodeto a sink node in which no
node istraversed more than once
Path Gain
The product of all the transferfunctions in the path, G
1G
2E(s) G1(s) C(s)G2(s)
-
8/8/2019 Mason Gain Formula
12/15
10/4/2005 2005, Milwaukee School of Engineering 12
Final Term s
Loop Gain
The product of all the transferfunctions in the loop
Nontouching
Two loops are nontouching if theyhave no nodes in common
A loop and a path are nontouching ifthey have no nodes in
common
-
8/8/2019 Mason Gain Formula
13/15
10/4/2005 2005, Milwaukee School of Engineering 13
Transfer Func t ion
The transfer function from a
source to a sink node
p is the number of forward paths
Mk is the path gain for the kth
forward path
1
1p
k k
k
T M=
=
-
8/8/2019 Mason Gain Formula
14/15
10/4/2005 2005, Milwaukee School of Engineering 14
The Del t a Func t ion
Usually only the first two terms
1 individual loops
product of nontouching loops
takentwo at a time
roduct of nontouching loopstakenthree at a time
=
+
+
-
8/8/2019 Mason Gain Formula
15/15
10/4/2005 2005, Milwaukee School of Engineering 15
Examples
Only one loop
Two loops, nottouching each
other Three loops,
two touching
