Signal Flow Diagram
The signal flow graph has long been used in many different applications (i.e. Feedback Control Systems).
It is a way of describing transitions with block diagrams. In fact, the transmission line is a perfect
feedback system with the signal flow point of view. The following examples illustrate the usage of the
signal flow graph.
(a) Single section
As shown in Figure 1, the TDR tester sends out a voltage signal down the transmission line. In addition
to the signal propagates down the line, there is a reflection in the junction of the test and the
transmission line. This reflection can be ignored if the tester’s characteristic impedance matches that of
the transmission line (Z0=Z1). The magnitude of the reflected signal can be calculated as
0 
Z1  Z 0
while the magnitude of the transmitted signal is T01  1  0 .
Z1  Z 0
Tester
T01  1  10
Transmission Line (Z1)
(TDR)
Z0
0 
Z1  Z 0
Z1  Z 0
Figure 1 – Initial Reflection and Transmission Coefficients
Most of the reflectometers have the detector/sampler installed inside the tester. The signal flow graph
of the system can be represented as shown in Figure 2.
Inpu
t
T01
Output
0
Figure 2 – Initial Response
1
With some time delay that is proportional to the wire length, the wave propagates further to the end of
the wire and reflects back to its original junction at the tester with another time delay that is equivalent
to the forward delay. Figure 3 shows the time delay behavior and the signal flow graph of the system is
shown in Figure 4, where 12 represents the reflection coefficient at the end of wire. 12 is 1 for an
open and is -1 for a short.
Tester
T01  1  10
Delay ( forward )
(TDR)
Delay (reflected )
Z0
12 
Z L  Z1
Z L  Z1
Figure 3 – Delay behavior and reflection at the end of wire
Input
T01
12
Delay
Delay
Output
0
Figure 4 – Response with delays
A portion of the reflected wave transmits into the tester while the remaining energy reflects back onto
the transmission line as shown in Figure 5. This process can be modeled in the signal flow graph as
shown in Figure 6.
Tester
11 
(TDR)
Z0
Z 0  Z1
Z 0  Z1
Transmission Line (Z1)
T10  1  10
Figure 5 – Distribution of the reflected wave at the tester-wire junction.
2
T01
Input
12
Delay
T10
Delay
Output
11
0
Figure 6 – Complete signal flow graph for a single section transmission line.
For simplicity, the responses have been pretty ideal so far. The basic model shown in Figure 6 can be
further developed in order to simulate behaviors in the real world situation. Attenuation for example,
can be added within or after the delay box. Filtering effect, for another example, can be included in the
block diagram as well. For some application specific configurations (i.e. aircraft environment), the noise
or vibration effects may be critical, thus, can be added to the basic block diagram shown above.
(b) Multiple sections
With a two-section transmission line configuration shown in Figure 7, the signal flow becomes more
difficult to keep in track. This is where the easiness of the bounce diagram method starts to fade.
Tester
11
(TDR)
T10
T01
Z0
21
Z1
Z2
T12
12
22
Figure 7 – Two-Section Configuration
With the basic signal flow graph of a single section transmission line shown in Figure 6, we can easily
expand the model to simulate multiple sections. Let’s assume the impedance of the tester matches the
first section of the transmission line. We can remove the initial reflection  0 from the graph. The
resulting signal flow graph is shown in Figure 8, where L1 block represents the first section while L2
represents the second section. Comparing the blocks L1 and L2 we noticed that they are functionally
identical. By representing them as big boxes with two pairs of IOs, we can greatly simplify the signal
flow diagram as shown in Figure 9.
3
L2
T12
 22
Delay2
T21
Delay2
21
L1
T01
Input
11
Delay1
T10
Delay1
Output
10
Figure 8 – A two-section signal flow graph
out 2
L2
out 1
in 1
out 2
input
in 1
in 2
in 2
L1
out 1
output
Figure 9 – Simplified block diagram for a 2-section configuration
With the same token, we can further cascade the configuration to N sections ash shown in Figure 10.
The ability of cascading block diagrams as connecting sections of transmission lines makes this method
easy to use and more flexible. The properties of the transmission lines can be built-in each block and
the users only need to connect them together. This is especially useful for concurrent GUI programming
languages.
out 1
in 2
out 1
L1
in 1
out 2
in 2
out 1
L2
in 1
in 2
LN
out 2
in 1
out 2
Figure 10 – N-section block diagram
(c) Branched networks
The signal flow graph also works for branched networks. Figure 11 shows a T-Junction (or T-Junction)
configuration.
4
12
11
Tester
(TDR) 0
T10
Z1
T12
T21
T01
22
Z2
21
T32
T23
T31
T13
31
Z0
Z3
32
Figure 11 – A Y-Junction branched network
The reflection and transmission coefficients of this Y-Junction network can be calculated as
0 
Z1  Z 0
Z1  Z 0
T01  1  0
11 
Z 0  Z1
Z 0  Z1
T10  1  11
Z 2 Z3
Z  ( Z 2 // Z 3 )
Z 2  Z 3 Z1Z 2  Z1Z 3  Z 2 Z 3
12  1


Z1  ( Z 2 // Z 3 ) Z  Z 2 Z 3
Z1Z 2  Z1Z 3  Z 2 Z 3
1
Z 2  Z3
Z1 
T12  T13  1  12
Z1Z 3
Z  ( Z1 // Z 3 )
Z1  Z 3 Z 2 Z1  Z 2 Z 3  Z1Z 3
 21  2


Z 2  ( Z1 // Z 3 ) Z  Z1Z 3
Z 2 Z1  Z 2 Z 3  Z1Z 3
2
Z1  Z 3
Z2 
T21  T23  1   21
5
Z 2 Z1
Z  ( Z 2 // Z1 )
Z 2  Z1 Z 3 Z 2  Z 3 Z1  Z 2 Z1
31  3


Z 3  ( Z 2 // Z1 ) Z  Z 2 Z1
Z 3 Z 2  Z 3 Z1  Z 2 Z1
3
Z 2  Z1
Z3 
T31  T32  1  31
Figure 12 shows the signal flow graph of the Y-Junction configuration stated above. With the same
concept described in the previous section, the Y-junction signal flow graph can be simplified as shown in
Figure 13.
A
in1
T12
in 3
 22
Delay2
Delay2
T21
in 2
out 2
to B
out 1
21
Input
in1
T01
in 3
out 2
Delay1
11
Delay1
T10
in 2
out 1
Output
10
B
in1
T13
in 3
32
Delay3
Delay3
in 2
out 2
31
Figure 12 – A Y-Junction Signal Flow Graph
6
T31
to A
out 1
out 1
in 2
L2
in 1
out 1
in 3
out 2
in 2
L1
in 1
out 2
out 1 in 3 in 2
L3
out 2
in 1
Figure 13 – The Simplified Y-Junction Block Diagram
By using the concepts described above, the signal flow graph and block diagram method can be easily
expanded into more complex configurations without much of programming efforts. A possible
application can be shown in Figure 15, which consists multiple sections of transmission lines after the Yjunction.
out 1
in 2
out 1
L2
out 2
in 1
out 1 in 3 in 2
out 1
in 1
out 1
in 3
in 2
out 1
out 2
in 1
in 2
out 1
in 2
LM
L4
out 2
in 2
L1
in 1
out 2
L5
L3
in 1
out 2
in 1
out 2
in 1
Figure 14 – A possible expansion of the Y-Junction configuration
7
in 2
LN
out 2
IV. Results
A Campbell Scientific TDR 100 (impedance = 50 ohms) was used for the measurement of the following
results. The simulation was done in Matlab Simulink. Several types coaxial cables were used as wires
under test and the specifications are listed as following:
RG58: 50 ohms, VOP=0.66
RG59: 75 ohms, VOP=0.66
RG62: 93 ohms, VOP=0.84
The effects of signal attenuation, tester/cable bandwidths, filtering and noise have not yet taken into
account for the simulation results in this report. With that in mind, the results are very close to the
measured data already.
RG58 (5m)
TDR
Figure 15 – Single Section RG-58
TDR1
1.5
Measured
Simulated
Magnitude
1
0.5
0
-0.5
0
5
10
15
Distance (m)
Figure 16 – Result of the single-section with RG-58 Coaxial Cable
8
RG62 (2.2m)
RG58 (5m)
TDR
Figure 17 – Two-Section Configuration with RG58 and RG62 Coaxial Cables
TDR2
1.5
Measured
Simulated
Magnitude
1
0.5
0
-0.5
0
5
10
15
Distance (m)
Figure 18 – Result of the two-section configuration with RG58 and RG62 Coaxial Cables
RG62 (2.2m)
RG58 (5m)
RG59 (2.74m)
TDR
Figure 19 – Three-section configuration with RG58, RG62 and RG59 coaxial cables.
9
TDR3
1.5
Measured
Simulated
Magnitude
1
0.5
0
-0.5
0
5
10
15
20
Distance (m)
25
30
Figure 20 – Result of three-section configuration with RG58, RG62 and RG59 coaxial cables
RG58 (9.55m)
RG58 (3.4m)
RG58 (3.66m)
TDR8
Figure 21 – A Y-Branched Network with 3 RG58 coaxial cables.
TDR8 Sampling-0.1
1.5
Measured
Simulated
Magnitude
1
0.5
0
-0.5
0
10
20
30
Distance (m)
40
50
Figure 22 – Result of a Y-Branched Network with 3 RG58 coaxial cables.
10
RG58 (9.7m)
RG58 (3.4m)
RG58 (5.03m)
TDR9
RG58 (3.66m)
RG58 (3.35m)
Figure 23 – A complex branched network with RG58 coaxial cables
TDR9 Sampling=0.1
1.5
Magnitude
1
0.5
0
Measured
Simulated
-0.5
0
20
40
Distance (m)
60
Figure 24 – Result of a complex branched network with RG58 coaxial cables
RG58 (3.35m)
RG58 (9.7m)
RG58 (3.4m)
TDR10
RG58 (5.03m)
RG58 (3.66m)
Figure 25 - Another complex branched network with RG58 coaxial cables
11
TDR10 Sampling=0.2
1.5
Magnitude
1
0.5
0
Measured
Simulated
-0.5
0
20
40
Distance (m)
60
Figure 26 – Result of another complex branched network with RG58 coaxial cables
VI. References
[1] F. T. Ulaby, “Fundamentals of Applied Electromagnetics”, Prentice Hall, 1999
[2] Chet’s forward solution paper
12
User’s Manual for Simulink Signal Flow Graph
Figure 27 shows the Simulink Block Diagram that simulates the setup described in Figure 17. The
reflection coefficients R0, R12 & R2 can be calculated as 0, 0.3, and -0.3 respectively. Figure 7 shows
these reflection coefficients, and equations are given on pages 5-6. The values of transmission
coefficients T01, T10, T12, T21 are 1, 1, 1.3 and 0.7 based on T  1   . Delay1 & Delay2 represent the
first section 5m RG58 coaxial cable. Delay 1 represents the forward wave, and Delay 2 represents the
reflected delay. The delays are divided in two, to account for the round trip data. This way the final
result can be shown in meters along the line, not meters traveled (there and back). The Delay is
normalized by the velocity of propagation for an RG58 cable (0.66 times the speed of light). Delay is
calculated as ½ the length of the cable (m) x 0.66 / VOP of the cable (normalized to the speed of light).
Set both of the Delays to ½ x 5 m x 0.66 / 0.66 = 2.5. Similarly, Delay3 and Delay4 represent the length
of the 2.2m RG62 Cable, which has a velocity of 0.84. Since the VOP is normalized to that of RG58,
Delay3 and Delay4 should be normalized as 1.1
0.66
 0.86 . The TDR1 block represents a variable
0.84
that stores the numerical result in Matlab. The Input and Output icons show the graphical
representation of the input step function (TDR source) and the output waveform as shown in Figure 28.
1.3
1
T 12
Delay3
0.7
R22
Delay4
T 21
-KR21
T DR1
T o Workspace
Input
1
Step
T 01
0.3
Delay1
1
R12
Delay2
T 10
0
R11
0
R0
Figure 27 – Signal Flow Graph of 2-Section Setup in Simulink
13
Output
Figure 28 – Graphical Representation of the Input and Output result. Give units
14