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Unformatted text preview: Block Diagrams Signal Flow Graphs
The block diagram is useful for graphical representation of a control system. However, for a very complicated system, the block diagram reduction method to find the overall transfer function becomes too time consuming. We shall use an alternative method involving signal flow signal flow graph. A signal flow graph is a pictorial representation of a set of simultaneous linear algebraic equations describing a system. A signal flow graph is a frequency domain manipulation. A signal flow graph consists of a network in which nodes are connected by directed branches. Each node represents a system variable or signal, and each branch connected between 2 nodes acts as a signal multiplier.
1 / 34 C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 2 / 34 A block diagram of a system is a graphical representation of a physical system, illustrating the functional relationship among its components Each functional block is connected by arrows which show the direction of signal flow Each block contains information concerning dynamic behavior, but it does not contain any information about the physical structure of the system A block diagram is NOT unique. C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 Signal Flow Graphs  Definitions
Node a node is a point representing a system variable or signal Transmittance is a gain between 2 nodes. Branch a branch is a directed line segment joining 2 nodes. The gain of a branch is a transmittance. Input node is a node that has only outgoing branches. This is independent variable. Output node is a node that has only incoming branches. Path a path is a traversal of connected branches in the direction of the branch arrows.
If no node is crossed more than once, the path is open. If the path ends at the same node from which it began and does not cross any other node more than once, it is closed.
C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 3 / 34 Signal Flow Graphs  Definitions
Loop a loop is a closed path which originates and terminates on the same node, and along the path no node is met twice. Selfloop is a closed loop consisting of a single branch and single node. Loop gain is the product of the branch transmittances of a loop Forward path is a path from an input node to an output node which does not cross any nodes more than once. Forward path gain is the product of the branch transmittance of a forward path. Nontouching loops loops that do not possess any common nodes.
C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 4 / 34 Converting Block Diagram to Signal Flow Graphs Converting Block Diagram to Signal Flow Graphs C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 5 / 34 C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 6 / 34 Mason's Gain Formula
The Mason's gain formula is given as: Tij (s) = where Ci (s) = Rj (s) k The Determinant The determinant can be interpreted as = 1  (sum of all individual loop gains) + (sum of gain products of all possible combination of 2 nontouching loops)  (sum of gain products Pk k Pk : is the path gain of k th forward path. k : is the cofactor of the k th forward path determinant of the graph with the loops touching the k th forward path removed. : is the determinant of the graph. + (1)k (sum of gain products of all possible combination of 3 nontouching loops) + + of all possible combination of k nontouching loops). That is, =1 i Li + 2 j Lj  3 j Lj + + (1) k k j Lj
8 / 34 C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 7 / 34 C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 Summary on Mason Gain Formula
Let k be the number of feedforward paths. Let n be the number of closed loops. 2 Find the number of feedforward paths. (i.e., Determine the value of k .) 3 Determine all the k th forward path gain, Pk . 4 Find the number of closed loops. (i.e., Determine the value of n.) 5 Determine each loop gain, Li . 6 Determine the loop gains of 2 nontouching loops, 3 nontouching loops, etc. (Note that all the possible combinations of j nontouching loops n out of n loops are .) j 7 Determine the determinant of the graph, . C. S. George Lee Block January 22, 8 Determine ()the cofactor Diagrams &kSignalFlow Graphspath determinant2010 the/ 34 of the th forward of 9 graph with the loops touching the k th forward path removed. (i.e., Determine P k for all feedforward paths.) P Ci (s) 9 Tij (s) = Rj (s) = k k k Mason's Gain Formula  Example 2
1 Mason's Gain Formula  Example 1 G3 R(s) 1 E1 1 E2 G 1G 4 H1 H 2 G2 1 C(s) C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 10 / 34 Mason's Gain Formula  Example 3 G3 R(s) + + + C(s)  G1 + H1 G4 G2 R(s) +  G1 +  H3 G2 + G3 G4 C(s)  H1
H2 H2 C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 11 / 34 C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 12 / 34 Mason's Gain Formula  Example 4 Mason's Gain Formula  Example 5
r1 1 a11 x1 G4 H2 R(s) + + G1 + G2 G3 + + C(s) a21 1 r2 a22 x2 a12 H1 C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 13 / 34 C. S. George Lee () Block Diagrams & SignalFlow Graphs January 22, 2010 14 / 34 ...
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This note was uploaded on 02/04/2012 for the course ECE 382 taught by Professor Staff during the Spring '08 term at Purdue University.
 Spring '08
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