Transcript Notes 15 - Signal flow graph analysis
ECE 5317-6351 Microwave Engineering
Fall 2011
Prof. David R. Jackson Dept. of ECE
Notes 15
Signal-Flow Graph Analysis 1
Signal-Flow Graph Analysis This is a convenient technique to represent and analyze circuits characterized by
S
-parameters.
• It allows one to “see” the “flow” of signals throughout a circuit.
• Signals are represented by wavefunctions (i.e.,
a i
and
b i
).
Signal-flow graphs are also used for a number of other engineering applications (e.g., in control theory).
Note: In the signal-flow graph,
a i
(0) and
b i
(0) are denoted as
a i
and
b i
for simplicity .
2
Signal-Flow Graph Analysis (cont.)
Construction Rules for signal-flow graphs
1) Each wave function (
a i
and
b i
) is a node.
2) S
-parameters are represented by branches between nodes.
3) Branches are uni-directional.
4) A node value is equal to the sum of the branches entering it.
S ou rc e
b g a g a b
1 1 N et w or k
a
2
b
2
b L a L
In this circuit there are eight nodes in the signal flow graph.
Lo ad 3
Example (Single Load) Single load
a L Z
0
b L
L L
L Z L Z L
Z
0
Z
0
Z L
Signal flow graph 1
a L
L b L
1
b L
L a L
4
V Th
+ -
b s Z Th
1 Example (Source)
Z
0
b g b g a g s V Th
Z Th Z
0
Z
0 1
Z
0
s Z Th Z Th
Z
0
Z
0
a g
s
1 1
b g a g b g b g
Hence
a s b s b s a g s
where
b s
V Th
Z Th Z
0
Z
0 5
Z
0
a
1
b
1 Example (Two-Port Device)
a
2
b
2
Z
0
b b
2 1
S a
11 1
S a
21 1
S a
12 2
S a
22 2
a
1 1
b
1 1
S
21 1
S
11
S
12
S
22 1
b
2
a
2 6
Complete Signal-Flow Graph A source is connected to a two-port device, which is terminated by a load.
S ou rc e
b g a g a
1
b
1 N et w or k
a
2
b
2
b L a L
Lo ad
b s b g
1
a g
s
1
a
1
S
21
b
2 1
a L b
1
S
11
S
12
a
2
S
22 1
b L
L
When cascading devices, we simply connect the signal-flow graphs together. 7
Solving Signal-Flow Graphs a) Mason’s non-touching loop rule: Too difficult, easy to make errors, lose physical understanding.
b) Direct solution: Straightforward, must solve linear system of equations, lose physical understanding.
c) Decomposition: Straightforward graphical technique, requires experience, retains physical understanding.
8
Example: Direct Solution Technique A two-port device is connected to a load.
b
1
a
1
a
1
b
1 N et w or k
a
2
b
2
b L a L
Lo ad
a
1
b
1
b a
1
S
11 1
S S
21
b
2 12
a
2
S
22
L
9
Example: Direct Solution Technique (cont.)
b
1
a
1
a
1
a
1
S
11
S
21
b
2
S
22
L b
1
b
1
S
12
a
2
b a
2
b
2 1
a S
1 21
b
2
S a L
11 1
S a
22 2
S a
12 2 Solve :
b
1
a
1
S
11
L
1
L S
22 For a given
a
1 , there are three equations and three unknowns (
b
1 ,
a
2 ,
b
2 ).
10
Decomposition Techniques 1) Series paths
a
2
a
3
S a
21 1
S a
32 2
a
3
S S a
21 32 1
a
1 1
a
1 1
a
2
S
21
S S
21 32
S
32 1
a
3 1
a
3 Note that we have removed the node
a
2 .
11
Decomposition Techniques (cont.) 2) Parallel paths
a
2
S a a
1
S a b
1
a
2
S a
b
1
a
1
a
1
S a
S b S a
S b a
2
a
2 Note that we have combined the two parallel paths.
12
Decomposition Techniques (cont.) 3) Self-loop
a
1
a
1
S
21
a
2
a
1
a
2 1 1 21 2
b a
1
a
1
S b
S
21
a
2
a
1 1 1 21
b S S
21
b
a
1 1 1
S S
21
b
a
1
L a a
1
L
1 1
S S
21
b
1
S
21
a
2
a
2
a
2
a
2 Note that we have removed the self loop.
13
Decomposition Techniques (cont.) 4) Splitting
a
4
a a
2 3
a S
2 42
a
2
S
32
a
1
S
21
a
1
S
42
S
21
a
2
S
32
a
4
a
3
S S a
21 42 1
S S a
21 32 1
a
1
S S
21 42
S S
21 32
a
3
a
4
a
4
a
3 Note that we have shifted the splitting point.
14
Example A source is connected to a two-port device, which is terminated by a load.
Solve for
in
=
b
1 /
a
1
in Z Th V Th
+-
a
1
b
1
Z
0 Two-port device
Z
0
Z L
Note: The
Z
0 lines are assumed to be very short, so they do not affect the calculation (other than providing a reference impedance for the
S
parameters).
15
Example The signal flow graph is constructed:
b s
s
Two-port device
a
1
S
21
b
2
b
1
S
11
S
12
a
2
S
22
L
16
Example (cont.) Consider the following decompositions:
b s b s
s a
1
S
11
S
21
b
2
S
22
L
s b
1
a
1
S
12 ß
a
2 The self-loop at the end is rearranged To put it on the outside (this is optional).
S
21
b
2
S
11
L S
22
b
1
S
12
a
2 17
b s
Example (cont.) Next, we apply the self-loop formula to remove it.
s a
1
S
11
b
1
S
12
b s S
21
a
2
b
2
L a
1
S
22
b s
Rewrite self-loop
S L
21 1
s b
2
a
1
S
21
b
2
b
1
S
11
S
12
L S
22
L
Remove self-loop
s S
11
S
12
L b
1
L
1 1 1
L S
22 18
Example (cont.)
a
1
b s
s b
1
b
1
a S
1 11 1 21 1
S
12
L
S
11 Hence:
S L
21 1
S
12
L b
1
a
1
S
11
S L
21 1
S
12
L
We then have
in S
11 1
S S
21 12
L
L S
22
b
2
L
1 1 1
L S
22 19
Example A source is connected to a two-port device, which is terminated by a load.
Solve for
b
2 /
b s
in Z Th V Th
+-
b s a
1
b
1
Z
0 Two-port device
Z
0
a
2
b
2
Z L
Note :
V L
V
2
L
b
2
Z
02 1
L
(Hence, since we know
b s
, we could find the load voltage from
b
2 /
b s
if we wish.) 20
Example (cont.) Using the same steps as before, we have:
b s a
1
s b
1
S
11
S L
21 1
S
12
L b
2
L
1 1 1
L S
22 21
b s b s L
2 1 1
S S
11
s
s L
2
a
1
b
1
L
3 1 1
L S L S
2 21 1 12
L S b s a
1
S
11 Example (cont.)
S L
21 1
b
2
a
1
s a
1
s S
11
S
12
L
a
1
s
S L
21 1
S
12
L
Rewrite self-loop on the left end
a
1
b s b
2
s S
11
S S
12
L
Remove self-loop
S L
21 1
b
2
S
12
L b
2
b s
21 1 3 1
L S L
2 21 1
L S L S
2 21 1 12
L S
Remove final self-loop 21 1 3
a
1
b
2 22
Example (cont.)
in b
2
b s
1 2
S S L L
21 12 1 2
L s
S
21 1
L L
1 2
S S
21 12
s
1
S S
11 1
S
21
L S
22
S S
21 12
V Th s
+-
Z Th a
1
b
1
b s Z
0 Two-port device
Z
0
a
2
b
2 Hence
b
2
b s
1
L S
22 1
S
21
S S
11
S S
21 12
s L Z L
23
Example (cont.)
b g a
1
S
21
b
2
b s
Alternatively, we can write down a set of linear equations:
s S
11
S
22
L b
1
S
12
a
2
b a
1
g
s s b
1
b g b b
1
a
2 2
S a
11 1
S a
21 1
S a
12 2
S a
22 2
L b
2 There are 5 unknowns:
b g
,
a
1 ,
b
1 ,
b
2 ,
a
2 .
Solve to find
b
2
b s
1 11
S
1
S S
22 21
L
S S
21 12
s L
24