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., ai and bi).
 Signal-flow graphs are also used for a number of other engineering
applications (e.g., in control theory).
Note: In the signal-flow graph, ai(0) and bi(0) are denoted as ai and bi for simplicity.
2
Signal-Flow Graph Analysis (cont.)
Construction Rules for signal-flow graphs
bL
b2
aL
ad
a2
Lo
or
k
a1
b1
et
w
bg
ag
N
ur
ce
Each wave function (ai and bi) is a node.
S-parameters are represented by branches between nodes.
Branches are uni-directional.
A node value is equal to the sum of the branches entering it.
So
1)
2)
3)
4)
In this circuit there are eight nodes in the signal flow graph.
3
Example (Single Load)
Single load
Signal flow graph
1
aL
Z0
ZL
bL  aL  L
L 
Z L  Z0
Z L  Z0

aL
L
bL
1
bL   L aL
4
Example (Source)


Z0
b g  a g  s  VTh 

Z

Z
Th
0


ZTh
bg
+
s 
Z0
VTh
1
Z0
Z Th  Z 0
Z Th  Z 0
ag
-
Hence
bs
b g  a g  s  bs

1
1
where
bg
b s  VT h
s
b g  bs  a g  s
1

Z0 


Z Z 
0 
 Th
ag
5
Example (Two-Port Device)
Z0
a2
a1
b1
b2
a1
b1  S 11 a1  S 12 a 2
Z0
b2  S 21 a1  S 22 a 2
1

S21
S11
b1
1
1
b2
S22
S12
1
a2
6
Complete Signal-Flow Graph
bs
bg
1
a1
s
S 21
bL
b2
aL
b2
S11
S 22
1
ag
1
1
b1
S12
a2
ad
a2
Lo
or
k
a1
b1
et
w
bg
ag
N
So
ur
ce
A source is connected to a two-port device, which is terminated by a load.
aL
L
bL
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
a1
aL
a1
S 21
b2
S11
b1
b1
S 22
S12
ad
b2
or
k
bL
Lo
a1
a2
et
w
 in 
a1
b1
b1
N
A two-port device is connected to a load.
L
a2
9
Example: Direct Solution Technique (cont.)
a1
 in 
a1
b1
S 21
b2
S11
a1
b1
b1
L
S 22
S12
a2
b2  a1 S 21  S 22 a 2
a 2  b2  L
b1  S11 a1  S12 a 2
S o lv e :  in 
b1
a1
 S 11 
S 21 S 12  L
1   L S 22
For a given a1, there are three equations and three unknowns (b1, a2, b2).
10
Decomposition Techniques
1) Series paths
a 2  S 21 a1
a1
1
a2
S 21
a 3  S 32 a 2

a 3  S 21 S 32 a1
a1
1

S 21S32
1
a3
S32
1
a3
Note that we have removed the node a2.
11
Decomposition Techniques (cont.)
2) Parallel paths
Sa
a 2  S a a1  S b a 1
a1

a2 
a2

 S a  S b  a1
Sb
a1
a2
S a  Sb
Note that we have combined the two parallel paths.
12
Decomposition Techniques (cont.)
a1
3) Self-loop
a1
a1  a1  a 2 S b
a2
S 21
a2
Sb
a 2  a1S 21
a1
a1  a1  a1S 21 S b
a1
S 21Sb



1
a1  
 a1
 1  S 21 S b 

L
a1
S 21
a2
S 21
a2

a1
a2
a2
L
1
1  S 21Sb
Note that we have removed the self loop.
13
Decomposition Techniques (cont.)
4) Splitting
S 42
a 4  a 2 S 42
a1
a 3  a 2 S 32
a3
S 21
a 2  a1 S 2 1
a4
a2
S32

a4

S 21S 42
a3
a1
a 4  S 21 S 42 a1
S 21S32
a 3  S 21 S 32 a1
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 = b1 / a1
 in
Z Th
VT h
+-
a1
b1
Two-port device
Z0
S 
ZL
Z0
Note: The Z0 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:
Two-port device
a1
S 21
bs
s
S11
b1
b2
S 22
S12
L
a2
16
Example (cont.)
Consider the following decompositions:
a1
S 21
bs
s
b2
S 22
S11
b1
a2
S12

a1
S 21
bs
s
The self-loop at the end is rearranged
To put it on the outside (this is optional).
b2
L
S11
b1
L
S12
S 22
a2
17
Example (cont.)
Next, we apply the self-loop formula to remove it.
a1
S 21
bs
s
b2
bs
S 22
L
S11
a1

s
b2
S 21
S11
S 22  L
Rewrite self-loop
b1
S12
b1
a2
S12  L

a1
Remove self-loop
S 21 L1
b2
bs
s
S11
S12  L
b1
L1 
1
1   L S 22
18
Example (cont.)
a1
S 21 L1
bs
s
b2
L1 
1
1   L S 22
S11
S12  L
b1
b1  a1 S 1 1  a1  S 2 1 L1   S 1 2  L 
Hence:
 in 
b1
a1
 S 11   S 21 L1   S 12  L 
We then have
 in  S 11 
S 21 S 12  L
1   L S 22
19
Example
A source is connected to a two-port device, which is terminated by a load.
Solve for b2 / bs
 in
Z Th
VT h
a1
b1
+-
bs
a2
b2
Two-port device
Z0
ZL
Z0
S 
N ote :

VL  V2
 0  1   L   b2
Z 02 1   L 
(Hence, since we know bs, we could find the load voltage from b2/bs if we wish.)
20
Example (cont.)
Using the same steps as before, we have:
L1 
a1
S 21 L1
bs
s
1
1   L S 22
b2
S11
S12  L
b1
21
Example (cont.)
a1
S 21 L1
b2
bs
s
S11
a1  b s  a1  s S 1 1  a1  s  S 2 1 L1   S 1 2  L 

S12  L
b1
Rewrite self-loop on the left end
a1
S 21 L1
bs
L2 
b2
s
1
S11 S
1   S S 11
a1
bs
s

L2
Remove self-loop
S 21 L1
b2
S12  L
b2
S12  L
bs
  L 2 S 21 L1  L 3

Remove final self-loop
1
1  L 2 S 21 L1 S 12  L  S
bs
 L2 S21L1  L3
1  L 2 S 21 L1 S 12  L  S

L3 
L 2 S 21 L1
a1
b2
22
Example (cont.)
 in
Z Th
b2

bs

S 21 L1 L 2
1  S 21 S 12 L1 L 2  L  s
S 21
1
L1 L 2

+-
VT h
a1
b1
bs
a2
b2
Two-port device
Z0
 S 21 S 12  L  s
ZL
Z0
S 
S 21
1   S S 11  1   L S 22   S 21 S 12  L  s
Hence
b2
bs

S 21
1   L S 22  1   S S 11   S 21 S 12  s  L
23
Example (cont.)
a1
bg
bs
Alternatively, we can
write down a set of
linear equations:
s
b2
S 21
S11
b1
S 22
L
a2
S12
b g  b s   s b1
a1  b g
S o lv e to fin d
b1  S 11 a1  S 12 a 2
b 2  S 21 a1  S 22 a 2
b2
a 2   L b2
bs
There are 5 unknowns: bg, a1, b1, b2, a2.

S 21
1  S 11 S  1  S 22  L   S 21 S 12  s  L
24