Short Notes: Block Diagrams & Signal Flow Graphs | Control Systems - Electrical Engineering (EE) PDF Download

 Page 1


Block Diagr ams & Signal Flow Gr aphs Notes
Introduction
Block diagr ams and signal flow gr aphs are gr aphical representations used in
control systems and signal processing to model the flow of signals through a sys-
tem. They simplify the analysis and design of complex systems b y illustr ating
functional relationships and signal paths.
K ey Concepts
• Block Diagr am : A pictorial representation of a system using blocks (func-
tional units) and arrows (signal paths).
• Signal Flow Gr aph (SFG) : A directed gr aph with nodes (signals) and br anches
(gains), repres enting system equations.
• Applications : Used to analyze system behavior , derive tr ansfer functions,
and simplify c ontrol system design.
Block Diagr ams
• Components :
– Blocks : Represent system elements (e.g., tr ansfer functionsG(s) ).
– Arr ows : Indicate signal flow direction.
– Summing Points : Combine signals (e.g.,e(t)=r(t)-y(t) ).
– T ak e-off Points : Distribute a signal to multiple paths.
• Tr ansfer Function : Ratio of output to input in the Laplace domain,T(s)=
Y(s)
R(s)
.
• Configur ations :
– Series : T(s)=G
1
(s)·G
2
(s) .
– Par allel : T(s)=G
1
(s)+G
2
(s) .
– F eedback : F or negative feedback, T(s) =
G(s)
1+G(s)H(s)
, where G(s) is the
forward path andH(s) is the feedback path.
Signal Flow Gr aphs
• Components :
– Nodes : Represent signals (e.g., inputR(s) , outputY(s) ).
– Br anches : Directed edges with gains (e.g., tr ansfer functionsG(s) ).
• K ey Definitions :
1
Page 2


Block Diagr ams & Signal Flow Gr aphs Notes
Introduction
Block diagr ams and signal flow gr aphs are gr aphical representations used in
control systems and signal processing to model the flow of signals through a sys-
tem. They simplify the analysis and design of complex systems b y illustr ating
functional relationships and signal paths.
K ey Concepts
• Block Diagr am : A pictorial representation of a system using blocks (func-
tional units) and arrows (signal paths).
• Signal Flow Gr aph (SFG) : A directed gr aph with nodes (signals) and br anches
(gains), repres enting system equations.
• Applications : Used to analyze system behavior , derive tr ansfer functions,
and simplify c ontrol system design.
Block Diagr ams
• Components :
– Blocks : Represent system elements (e.g., tr ansfer functionsG(s) ).
– Arr ows : Indicate signal flow direction.
– Summing Points : Combine signals (e.g.,e(t)=r(t)-y(t) ).
– T ak e-off Points : Distribute a signal to multiple paths.
• Tr ansfer Function : Ratio of output to input in the Laplace domain,T(s)=
Y(s)
R(s)
.
• Configur ations :
– Series : T(s)=G
1
(s)·G
2
(s) .
– Par allel : T(s)=G
1
(s)+G
2
(s) .
– F eedback : F or negative feedback, T(s) =
G(s)
1+G(s)H(s)
, where G(s) is the
forward path andH(s) is the feedback path.
Signal Flow Gr aphs
• Components :
– Nodes : Represent signals (e.g., inputR(s) , outputY(s) ).
– Br anches : Directed edges with gains (e.g., tr ansfer functionsG(s) ).
• K ey Definitions :
1
– Path : Sequence of br anches from input to output.
– F orwar d Path : Path from input to output without touching an y node
more than once.
– Loop : Path that starts and ends at the same node.
– Non-touching Loops : Loops with no common nodes or br anches.
• Mason’ s Ga in F ormula : Used to find the over all tr ansfer function:
T(s)=
?
P
k
?
k
?
where:
– P
k
: Gain of thek -th forward path.
– ? : Determinant,?=1-
?
L
i
+
?
L
i
L
j
-
?
L
i
L
j
L
k
+··· , whereL
i
are
loop gains and higher terms involve non-touching loops.
– ?
k
: Cofactor , determinant excluding loops touching the k -th forward
path.
Conversion Between Block Diagr ams and SFGs
• From Bloc k Diagr am to SFG :
– Represent each summing point and tak e-off point as a node.
– Replace blocks with br anches having tr ansfer function gains.
• From SFG t o Block Diagr am :
– Convert nodes to signals or summing points.
– Represent br anch gains as blocks with tr ansfer functions.
Simplification T echniques
• Block Diag r am Reduction :
– Combine series blocks: G
1
(s)·G
2
(s) .
– Combine par allel blocks: G
1
(s)+G
2
(s) .
– Move summing/tak e-off points to simplify feedback loops.
• SFG Simplification : Use Mason’ s Gain F ormula or eliminate nodes b y com-
bining paths and l oops.
Pr actical Consider ations
• Stability : F eedback loops must be analyzed for stability using1+G(s)H(s)=
0 .
2
Page 3


Block Diagr ams & Signal Flow Gr aphs Notes
Introduction
Block diagr ams and signal flow gr aphs are gr aphical representations used in
control systems and signal processing to model the flow of signals through a sys-
tem. They simplify the analysis and design of complex systems b y illustr ating
functional relationships and signal paths.
K ey Concepts
• Block Diagr am : A pictorial representation of a system using blocks (func-
tional units) and arrows (signal paths).
• Signal Flow Gr aph (SFG) : A directed gr aph with nodes (signals) and br anches
(gains), repres enting system equations.
• Applications : Used to analyze system behavior , derive tr ansfer functions,
and simplify c ontrol system design.
Block Diagr ams
• Components :
– Blocks : Represent system elements (e.g., tr ansfer functionsG(s) ).
– Arr ows : Indicate signal flow direction.
– Summing Points : Combine signals (e.g.,e(t)=r(t)-y(t) ).
– T ak e-off Points : Distribute a signal to multiple paths.
• Tr ansfer Function : Ratio of output to input in the Laplace domain,T(s)=
Y(s)
R(s)
.
• Configur ations :
– Series : T(s)=G
1
(s)·G
2
(s) .
– Par allel : T(s)=G
1
(s)+G
2
(s) .
– F eedback : F or negative feedback, T(s) =
G(s)
1+G(s)H(s)
, where G(s) is the
forward path andH(s) is the feedback path.
Signal Flow Gr aphs
• Components :
– Nodes : Represent signals (e.g., inputR(s) , outputY(s) ).
– Br anches : Directed edges with gains (e.g., tr ansfer functionsG(s) ).
• K ey Definitions :
1
– Path : Sequence of br anches from input to output.
– F orwar d Path : Path from input to output without touching an y node
more than once.
– Loop : Path that starts and ends at the same node.
– Non-touching Loops : Loops with no common nodes or br anches.
• Mason’ s Ga in F ormula : Used to find the over all tr ansfer function:
T(s)=
?
P
k
?
k
?
where:
– P
k
: Gain of thek -th forward path.
– ? : Determinant,?=1-
?
L
i
+
?
L
i
L
j
-
?
L
i
L
j
L
k
+··· , whereL
i
are
loop gains and higher terms involve non-touching loops.
– ?
k
: Cofactor , determinant excluding loops touching the k -th forward
path.
Conversion Between Block Diagr ams and SFGs
• From Bloc k Diagr am to SFG :
– Represent each summing point and tak e-off point as a node.
– Replace blocks with br anches having tr ansfer function gains.
• From SFG t o Block Diagr am :
– Convert nodes to signals or summing points.
– Represent br anch gains as blocks with tr ansfer functions.
Simplification T echniques
• Block Diag r am Reduction :
– Combine series blocks: G
1
(s)·G
2
(s) .
– Combine par allel blocks: G
1
(s)+G
2
(s) .
– Move summing/tak e-off points to simplify feedback loops.
• SFG Simplification : Use Mason’ s Gain F ormula or eliminate nodes b y com-
bining paths and l oops.
Pr actical Consider ations
• Stability : F eedback loops must be analyzed for stability using1+G(s)H(s)=
0 .
2
• Complexity : Simplification reduces computational effort but must pre-
serve system beh avior .
• Software T ools : MA TLAB, Simulink, or Python can automate block dia-
gr am and SFG anal ysis.
3