SE201_071_Signal Flow graphs

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    Signal Flow Graphs

    Dr. Samir Al-Amer Term 071

    SE201: Introduction to Systems Engineering

    Term 071

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    Out lines

    Basic Elements of Signal Flow GraphBasic PropertiesDefinitions of Signal Flow Graph TermsMason TheoryExamples

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    B asic Elemen ts of Signal Flow Graph

    A Signal flow graph is a diagram consistingof nodes that are connected by severaldirected branches .

    node node

    branch branch

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    B asic Elemen ts of Signal Flow Graph

    A node is used to represent a variable (inputs,outputs, other signals)

    A branch shows the dependence of one variable

    ( node ) on another variable ( node )Each branch has GAIN and DIRECTION

    A signal can transmit through a branch only in thedirection of the arrow

    If gain is not specified gain =1

    A B

    GB = G A

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    N odes

    A node is used to represent a variableSource node (input node)

    All braches connected to the node are leaving the nodeInput signal is not affected by other signals

    Sink node (output node) All braches connected to the node are entering the nodeoutput signal is not affecting other signals

    A B

    VU

    C

    X Y Z

    DSource node

    Sink node

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    R ela t ionship B etween Variables

    U (input)X=A U+Y

    Y=BXZ=CY+DXV=Z (output)

    A B

    VU

    C

    X Y Z

    DSource node

    Sink node

    Gain is notshown means

    gain=1

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    A no ther Example

    X=A U+Y Y=BX+KZ

    Z=CY+DX+HWW=3U

    V=Z

    A B

    VU

    C

    X

    YZ

    DSource node

    Sink node

    3W H

    K

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    B asic Proper t ies

    Signal flow graphs applies to linear systems onlyNodes are used to represent variables

    A branch from node X to node Y means that Y dependson XValue of the variable (node) is the sum of gain of branch * value of nodeNon-input node cannot be converted to an input nodeWe can create an output node by connecting unit branch

    to any node

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    T erminology: Pa thsA path: is a branch or a continuous sequence of branches that can be traversed from one node toanother node

    A BU CX Y Z

    A B

    VC

    X

    YZ

    3

    W H

    K

    U

    Z

    3W HPaths from U to Z

    U

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    T erminology: Pa thsA path: is a branch or a continuous sequence of branches that can be traversed from one node toanother nodeF orward path : path from a source to a sink

    Path gain : product of gains of the braches that make thepath

    A B

    VU

    C

    X

    Y

    Z3

    W H

    K

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    T erminology: loop A loop: is a closed path that originates andterminates on the same node, and along the

    path no node is met twice.N

    ontouching loops:

    two loops are said to benontouching if they do not have a commonnode.

    A B

    VU

    C

    X

    YZ

    3W H

    KC Y

    K

    B

    X

    Y

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    A n example

    a 11 x1 + a 12 x2 + r 1 = x 1a 21 x1 + a 22 x2 + r 2 = x 2

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    A n example

    (1- a 11 )x1 + (- a 12 ) x2 = r 1( - a 21 ) x1 + (1- a 22 )x2 = r 2

    This have the solutionx1= (1- a 22 )/( r 1 + a 12 /( r 2

    x2= (1- a 11 )/( r 2 + a 21 /( r 1

    ( = 1 - a 11 - a 22 + a 22 a 11 - a 12 a 21

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    A n example

    ( = 1 - a 11 - a 22 + a 22 a 11 - a 12 a 21

    Self loops a 11 , a 22 , a 12 a 21Product of nontouching loops a 22 a 11

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    SGF : in general

    The linear dependence (T ij) between theindependent variable x i (input) and thedependent variable (output) x j is given byMasons SF gain formula

    ijk ijk

    ijk

    k ijk ijk

    ij

    P

    k P

    P T

    pa ththeof cofac tor

    graphtheof tde terninanxtoxfrom pa th jith

    !(

    !(!

    (

    (!

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    T he de terminan t (

    Or ( =1 (sum of all different loop gains) +(sum of the gain products

    of all combinations of 2 nontouching loops)-(sum of the gain products of all combinations of 3 nontouching

    loops)

    The cofactor is the determinant with loopstouching the k th path removed

    gainlooptheis

    ...11

    ,

    1,1

    q

    N

    n

    Q M

    qmt sr qmn

    L

    L L L L L L ! !!

    !(

    ijk (

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    Example

    The number of forward paths from U to V = ?Path Gains ?Loops ?Determinant ?Cofactors ?Transfer function ?

    A B

    VU

    C

    X

    YZ

    3W H

    K

    Determine the transfer function between V and U

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    Example

    The number of forward paths from U to V = 2Path Gains ABC, 3HLoop Gains B, CKTransfer function (ABC+3H-3HB)/(1-B-CK)

    A B

    VU

    C

    X

    YZ

    3W H

    K

    Determine the transfer function between V and U

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    A n example

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    A n example:

    Two paths : P1, P2Four loopsP

    1= G

    1G

    2G

    3G

    4, P

    2= G

    5G

    6G

    7G

    8L1=G 2H2 L2=G 3H3 L3=G 6H6 L4=G 7H7(! - (L1+L2+L3+L4)+(L1L3+L1L4+L2L3+L2L4)Cofactor for path 1 : ( 1= 1- (L 3+L4)Cofactor for path 2 : ( 2= 1-(L 1+L2)T(s) = (P 1( 1 + P 2( 2)/(

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    A no ther example

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    3 Paths8 loops

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    Su mmary

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    K eywords

    NodeBranchPathLoopNon-touching loopsLoop gain

    Sink nodeSource node

    Forward pathcofactor