SFG and Mason s Rule : A revision Andersen Ang 2016-Nov-29 SFG and Mason s Rule Vu Pham
Review SFG: Signal-Flow Graph SFG is a directed graph SFG is used to model signal flow in a system SFG can be used to derive the transfer function of the system by Mason s Rule / Mason s Gain Formula. 2
SFG terminologies Node Edge Gain Input / Sources Output / Sinks Path Path gain Forward path Forward path gain Loop Self loop Loop Gain Non-touching loop 3
SFG terminologies Node: variables. e. g. X %, X ', X (, X ) Edge: directed branches. e. g. X % X ' Gain: transmission of that branch. e. g. A '% Input/Sources: nodes with out-going branches only e. g. X % Output/Sinks: nodes with incoming branches only e. g. X ) 4
SFG terminologies Path: successive branches without repeated nodes, e. g. X % X ' X ) Path gain: the gain of the path Forward path: path from input to output, e. g. X % X ' X ( X ) or X % X ' X ) Forward path gain: gain of forward path 5
SFG terminologies Loop / Feedback path: a closed path which originates and terminates on the same node, e. g. X ' X ( X ' Self loop: loop with only one branch, e.g. X ( X ( Loop gain: gain along the loop Non-touching loop: two loops are non-touching if they do not share any nodes nor branches 6
Some SFG simplifications Branch in parallel: Branch in series: Isolated loop: 7
SFG Example 1 Two forward path: G 1 G 2 G 3 G 4 G 5 G 7 & G 1 G 2 G 3 G 4 G 6 G 7 Four loops: G 2 H 1, G 4 H 2, G 4 G 5 H 3, G 4 G 6 H 3 Non-touching loops: G 2 H 1 & G 4 H 2, G 2 H 1 & G 4 G 5 H 3, G 2 H 1 & G 4 G 6 H 3 8
SFG Example 2 Three forward path: A 32 A 43 A 54 A 65 A 76 A 72 A 42 A 54 A 65 A 74 9
SFG Example 2 Loops A 32 A 33, A 43 A 34, A 54 A 45, A 65 A 56, A 76 A 67, A 77 A 42 A 34 A 23, A 65 A 76 A 57 * A 65 A 76 A 67 A 56 is not a loop since X 6 is repeated on the path! A 72 A 57 A 45 A 34 A 23 A 72 A 67 A 56 A 45 A 34 A 23 10
Mason s Rule How to derive transfer function: block diagram (BD) reduction or signal flow graph reduction. BD approach requires successive application of fundamental relationships in order to derive transfer function. SFG just applies one formula Mason s Rule 11
Mason s Rule Equation TF(s) = 1 Δ #789:;9< >;?@ 3 P 5 Δ 5 5A% P i = the i th forward-path gain. = Determinant of the system i = Determinant of the i th forward path = 1 - (sum of all individual loop gains) + (sum of products of gains of all 2 loops that do not touch each other) (sum of products of gains of all possible three loops that do not touch each other) + i = Δ for part of SFG that does not touch i-th forward path Δ i = 1 if no non-touching loops to the i-th path, or if taking out i-th path breaks all the loops 12
Example 1 X 2 X 3 X 4 X 5 X 6 Input: Output: X 6 Two Forward paths: paths along X 2 X 3 X 4 X 5 X 6 Two Forward path gains: P % = G % G ' G ) and P ' = G % G ( G ) Loops: X 2 X 3 X 2 and two loops along X 2 X 3 X 4 X 5 X 2 Loops gains: L % = G % G ) H %, L ' = G % G ' G ) H ' and L ( = G % G ( G ) H ' 13
Example 1 X 2 X 3 X 4 X 5 X 6 Δ = 1 - (sum of all individual loop gains) + (sum of products of gains of all 2 loops that do not touch each other) (sum of products of gains of all possible three loops that donot touch each other) Since no non-touching loop, thus Δ = 1 - (sum of all individual loop gains) Δ = 1 (L % + L ' + L ( ) 14
Example 1 X 2 X 3 X 4 X 5 X 6 i = Δ for part of SFG that does not touch i-th forward path (Δ i = 1 if no non-touching loops to the i-th path) Since no non-touching loop, thus Δ 1 = Δ 2 = 1 15
Example 1 X 2 X 3 X 4 X 5 X 6 #789:;9< >;?@ TF = 1 Δ 3 P 5 Δ 5 5A% = 1 Δ P %Δ % + P ' Δ % 1 = 1 (L % + L ' + L ( ) P % + P ' G % G ' G ) + G % G ( G ) = 1 G % G ) H % + G % G ' G ) H ' +G % G ( G ) H ' 16
Example 2 Input, Output X 8 Two Forward paths: X 5 X 6 X 7 X 2 X 3 X 4 X 8 and X 5 X 6 X 7 X 8 X 8 Two Forward paths gain: X 2 X 3 X 4 P % = G % G I G J G K P ' = G % G ' G ( G ) 17
Example 2 Loops: X 2 X 3 X 2, X 3 X 4 X 3, X 5 X 6 X 5, X 6 X 7 X 6 X 5 X 6 X 7 Individual Loop gains: X 8 L %% = G I H I L %' = G J H J, X 2 X 3 X 4 L %( = G ' H ' L %) = G ( H ( Two non-touching loops: L '% = L %% L %( L '' = L %% L %) L '( = L %' L %( L ') = L %' L %) Three non-touching loops: no Δ = 1 L %% + L %' + L %( + L %) + (L '% + L '' + L '( + L ') ) 18
Example 2 i = Δ for part of SFG that does not touch i-th forward path (Δ i = 1 if no nontouching loops to the i-th path.) X 5 X 6 X 7 X 2 X 3 X 4 X 8 Δ ' = Δ for part of SFG that does not touch 2nd forward path Δ % = Δ for part of SFG that does not touch 1 st forward path 19
Example 2 SFGs without the i-th forward path: X 8 X 8 For these SFGs, compute their Δ as: Δ = 1 - (sum of all individual loop gains) + (sum of products of gains of all 2 loops that do not touch each other) (sum of products of gains of all possible three loops that do not touch each other) 20
Example 2 Both SFGs do not havetwo/morenon-touching loops X 8 X 5 X 7 X 2 X 4 X 8 Δ ' = 1 L %% + L %' Δ% = 1 (L %( + L %) ) = 1 G I H I G J H J = 1 G ' H ' G ( H ( 21
Example 2 #789:;9< >;?@ X 5 X 6 X 7 TF = 1 Δ 3 P 5 Δ 5 X 8 5A% X 2 X 3 X 4 = P % Δ % + P ' Δ' 1 L %% + L %' + L %( + L %) + (L '% + L '' + L '( + L ') ) = G % G I G J G K 1 G ' H ' G ( H ( + G % G ' G ( G ) (1 G I H I G J H J ) 1 G ' H ' + G ( H ( + G I H I + G J H J + (G ' H ' G I H I + G ' H ' G J H J + G ( H ( G I H I + G ( H ( G J H J ) END 22