A Friendly Approach to Transient Processes in Transmission Lines
Andrew Rusek, Subramaniam Ganesan, Daniel N. Aloi
Department of Electrical and Computer Engineering
Oakland University, Rochester, MI 48309
Email: rusek or ganesan or aloi@oakland.edu
Abstract
This paper discusses the Bergeron I-V (current-voltage) diagrams, applied to transmission lines,
supported by a very simple interpretation of the transmission line operating as a source of a
delayed voltage wave. The Bergeron method and the PSPICE simulations are used to support
three Oakland University courses: undergraduate course on electromagnetics, graduate course on
high frequency electronics, and graduate course on electromagnetic compatibility.
In this paper, a short description of the contents of the courses are provided and application of
the Bergeron method and simulation are included.
Introduction
For many years the transient processes in transmission lines have been treated formally by
complicated equations, bouncing wave diagrams, or using V-I (voltage-current) trajectories with
negative and positive moving observers1-6. The last method has been a modified graphical
approach based on the Bergeron diagrams, but still complicated and not easy to interpret.
The Bergeron diagram method is a method to value the reflection's effects on an electric
signal. This graphic method—based on the real line's characteristic—is valid both for linear and
non linear models and helps to calculate the delay of an electromagnetic signal on an electric
line.
Using the Bergeron method, on the I-V characteristic chart starts from the regime point before
transition, then move with a straight line with 1/Z0 slope (Z0 is the line's characteristic
impedance) to the new characteristic; then move with lines with −1/Z0 or +1/Z0 slope until
reaching the new regime situation. The “-“ value is considered always the same at every
reflection because the Bergeron method is used only for first reflections.
The paper discusses the Bergeron I-V (current-voltage) diagrams, applied to transmission lines,
supported by a very simple interpretation of the transmission line operating as a source of a
delayed voltage wave7. The introduction of this delayed equivalent circuit of the transmission
line, similar to the Thevenin’s equivalent circuit greatly simplifies the solutions, leads to
graphical solutions of consecutive voltage dividers, and helps in interpretations of the results.
This method, as well as the other mentioned methods cannot help in cases with inductive or
capacitive loads8.
Proceedings of the 2011 ASEE NC & IL/IN Section Conference
Copyright © 2011, American Society for Engineering Education
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The Bergeron method and the PSPICE simulations are used to support three Oakland University
courses: undergraduate course of electromagnetics, graduate course of high frequency
electronics, and graduate course of electromagnetic compatibility9-10. In this paper, a short
description of the contents of the courses and application of the Bergeron method and
simulations are included. Several examples are demonstrated for both linear and non-linear loads
of the transmission lines. Various steps of the solutions are also verified using PSPICE software.
Transient Voltages in Circuits with Transmission lines
Presented below are several basic steps leading to a multi-step process of plotting transient
voltages in circuits with transmission lines driven by a pulse source and terminated with a
resistor. The resistances, Rs and RL, and the transmission line characteristic impedances were
selected arbitrarily to satisfy the following: RL > Rs > Z0, in order to show reflections from the
load and from the source. The first step shown in Figure 1 demonstrates how the input incident
voltage wave is created for the front of the source pulse. The load is separated in time from the
input, so the equivalent circuit of the input circuit includes only the source and the transmission
line impedance, which is shown in Figure 2.
The incident voltage wave, vin1+, created at the input, moves towards the output of the
transmission line. For the time 0<t<TD, where TD is the delay time of the transmission line, the
line can be represented in terms of a voltage source 2vin1+ and an internal resistance Z0. The
doubled voltage vin1+ can be determined based on the transmission line incident and reflected
wave equations listed below.
v = (v+)
+
i = ( i+ )
+
( v-), where v+ is the voltage incident wave, v- is the voltage reflected wave.
( i-), where i+ is the current incident wave, i- is the current reflected wave.
The above currents are related to the voltages as follows
i+ = v+/Z0 and i- = v-/Z0
Eliminating v- from the first equation leads to the following
v = 2(v+) - i Z0, or i = (2v+ - v)/ Z0
In the discussed case, for the time t = TD, v+ = vin1+, and the voltage at the end of the line can
be calculated from
vout1+ = 2 (vin1+ ) RL/(RL+Z0)
Proceedings of the 2011 ASEE NC & IL/IN Section Conference
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Figure 3 illustrates the above equationwhile Figure 4 shows the corresponding equivalent circuit.
Themajor part of the equivalent circuit is the voltage source represented by the doubled incident
voltage and Z0 which is the output impedance of this voltage source.
The voltage vout1+, which is a part of 2 (vin1+) appears as the output voltage step, but the
remaining part of 2 (vin1+), across Z0 becomes a reflected, or returning voltage wave. This
returning voltage, doubled again, returns to the front of the transmission line, and it is divided
between Rs and Z0, and this part is added to vin1+, based on straightforward superposition of the
effects of actions of the original step source and the returning voltage wave. This process is
shown graphically in Figure 5.
Following this pattern, it is very straightforward to complete the construction of major transitions
from the end of transmission line and from the front of the transmission line.
The switching off process, although looking more complex, is the inverted image of the
previously described process. The initial conditions are described in terms of the motion of the
source line from the final steady state conditions of the previous process (point A of Figure 5) to
zero (point B), so the process can be treated in the same way as the previously described
switching on process with the new origin (point A) and the base line at the level of the new
origin. The procedure of obtaining the voltage waves for the input and output is shown in Figure
6. The time scale for both switching phases is related to transitions between the input and the
output. The graphical method presented here also allows for plotting the current waves, since the
current steps associated with the corresponding time sections could be plotted in the same
manner as the demonstrated voltage steps.
Figure 7 shows the PSPICE circuit and the voltage waves for this circuit similar to the waves
obtained graphically and demonstrated in Figure 1. Next figure (Figure 8) illustrates how the
same graphical method can be used to analyze transient processes in the transmission line
terminated with nonlinear loads. The PSPICE simulationsin Figure 9 are presented to show the
same characteristics of the voltage waves as the waves drawn with help of graphical analysis.
TL
RL 120
+
Rs 75
Vpulse
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Figure 1a) Circuit to analyze reflections in transmission line. TL = Transmission Line
i
V sm ax/R s
i = v/Z0
i = v/R L
i = -v/Rs
i = [Vsm ax - v]/R s
v
0
Figure 1b. For Circuit
shown in figure 1a, the
Voltage vs Current for
Rs, RL, and Z0 and Time
Delay. Vsmax is the
source step amplitude.
Vsm ax
vin1+
v
TD
2TD
vin1+ = Vsm ax Z0/[R s+Z0]
T D - Line Delay T im e
3TD
t
Vs
Rs
Z0
Figure 2.
The equivalent circuit for the first input step
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i
V sm a x/R s
i = v /Z 0
i = v /R L
i = [V sm a x - v ]/R s
v
0
2 v in 1+
v in1 +
V sm a x
v
TD
v in
2TD
vout
T D - L in e De la y T im e
3T D
v in 1 + = V sm a x Z 0 /[R s+Z 0 ]
v L 1 + = 2 v in 1 + R L /([R L + Z 0 ]
Figure 3 The voltage wave reaching the end of transmission line
Z0
2 v in 1 +
RL
Figure 4. The equivalent circuit for the first output voltage wave step
i
Vsmax/Rs
i = v/Z0
i = v/RL
A
i = [Vsmax - v]/Rs
v
0
2vin1+
vin1+
Figure 5. First two steps for input
and output voltage waves leading
to the steady state point A
Vsmax
v
TD
2TD
TD - Line Delay Time
3TD
vin1+ = Vsmax Z0/[Rs+Z0]
vout
Vin
vL1+ = 2vin1+ RL/([RL+Z0]
t
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i
Vsmax/Rs
Figure 6. Switching off
waves for input and
output voltages of the
transmission line
i = v/Z0
2vin1+off
i = v/RL
A
vin1+off
i = [Vsmax - v]/Rs
v
B
0
new starting time point
Vsmax
vin
v
TD
vL
2TD
TD - Line Delay Time
3TD
t
Figure 7a. Circuit to demonstrate voltage waves on the transmission lines
Proceedings of the 2011 ASEE NC & IL/IN Section Conference
Copyright © 2011, American Society for Engineering Education
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Figure 7b. The time-domain voltage Waves for the circuit shown in Figure 7a.
i
load line
source line
transmission line
v
0
vin1+
2vin1+
v
0
TD
vin
2TD
3TD
vout
t
Figure 8. Reflections in a transmission line terminated with a diode circuit as shown in Figure 9a.
Proceedings of the 2011 ASEE NC & IL/IN Section Conference
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Figure 9a Circuit Diagram used for PSpice simulation
Figure 9 b. PSPICE circuit to demonstrate voltage waves corresponding to time-domain
voltages shown in Figure 8. The time-domain voltage waves for the circuit of Figure 9a.
In addition to the described processes of plotting the voltage waves versus time, it is also
possible to plot input and output current waves as functions of time. Instead of vertical voltage
level projections along the vertically positioned time axis, the horizontal current level projections
along the horizontal time axis are shown.
Proceedings of the 2011 ASEE NC & IL/IN Section Conference
Copyright © 2011, American Society for Engineering Education
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OU Graduate Course Descriptions
High Frequency Electronics Course Contents: (ECE 525)
1.
2.
3.
4.
5.
6.
7.
8.
Passive lumped components at high frequencies
Transmission lines with sinusoidal and pulse excitation
Active devices operating at high frequencies
Pulse operation of active devices
High frequency amplifiers
High frequency oscillators
High frequency communication circuits
Introduction to high frequency measurements and instrumentation
Software Applied: PSPICE, Serenade, MATLAB, SMITH 191, APPCAD
Electromagnetic Compatibility High Frequency Electronics Course Contents: (ECE 546)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Overview of EMC
EMC Requirements
Review of Electromagnetic Principles
Distributed and Lumped Components
Signal Spectra and Spectrum Measurements
Introduction to EMC Pre-compliance and Compliance Tests, Component and System
Level Measurements
Radiated Emissions and Susceptibility
Conducted Emissions and Susceptibility
Crosstalk
Shielding and Guarding
Electrostatic Discharge
Comparison of our method with other methods
There are two major graphical methods related to transient processes in transmission lines. One
is the bouncing wave method, the other, graphical method based on the Bergeron method.
Bergeron method involves the formal process of memorization of the straight line slope
sequences for incident and reflected voltage waves. Our method could be named an enhanced
graphical method that is based on the interpretation of the equivalent circuits for the pulse
source, transmission line, and the load in terms of simple voltage dividers. For instance, at t= 0+
the pulse source voltage is divided between the transmission line input impedance at this
moment, which is Zo and the source resistance. This forms the first step of the incident wave.
The graphical solution represents this simple voltage divider. When this incident voltage wave
reaches the end of the transmission line, the other simple voltage divider is formed, which is
composed of a doubled incident voltage, transmission line characteristic impedance, Zo, and the
Proceedings of the 2011 ASEE NC & IL/IN Section Conference
Copyright © 2011, American Society for Engineering Education
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load resistance. Again, the graphical solution for this voltage divider yields the first step of the
output voltage. The part of the voltage that is not "absorbed" by the load comes back towards the
front of the transmission line and the graphical process is repeated. This time the source
resistance creates a new load, and this part of the voltage is superimposed on the first step of the
previously determined incident voltage wave.
The doubled incident or reflected voltage levels used for the voltage waves reaching the end or
the front of the transmission lines come from the fact that 'lossy" resistances are used to represent
the transmission line characteristic impedance, Accordingly, simple circuit equations can be
applied to determine voltage, and/or current levels. In addition, this doubled voltage can be
formally derived from the voltage wave equations, which is also shown in the paper. Our
explanation helps understand the Bergeron method, especially the starting steps, without
unnecessary memorization of the slopes for each step.
The graphical method, either the formal Bergeron, or our proposed "enhanced Bergeron
method”, helps in finding solutions for transient processes when nonlinear loads and/or source
resistances are applied. It has been observed over the years of teaching transient processes in
transmission lines in our High Frequency Electronics course that the students have learned and
followed this interpretation of the graphical method much faster than previously applied method
described in [7-10].
The bouncing wave method can be only used for linear resistive sources and loads, since it
applies reflection coefficients, which involve linear passive components. The Bergeron method
does not have this limitation. All methods discussed here do not cover capacitive or inductive
types of loads or sources.
Conclusion
The use of equivalent circuits of the transmission line simplifies the interprettation of the
Begeron I-V diagrams and provides a good understanding of the fundamentals of transient
processes in transmission lines discussed in undergraduate and graduate courses. Understanding
of such processes is also very important in the interpretation of the results obtained during
processes of verification of signal integrity in electronic circuits and systems operating at high
switching speeds. A couple of examples analyzed here are also verified using PSPICE software.
PSPICE and MATLAB are widely used in the courses listed in this paper, to provide practical
demonstrations of transmission lines.
Proceedings of the 2011 ASEE NC & IL/IN Section Conference
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References
1.
Stuart M. Wentworth, Applied Electromagnetics, Early Transmission Lines Approach, John Wiley & Sons,
2007
2.
William H. Hayt, Jr., John A. Buck, Engineering Electromagnetics, Mc Graw Hill, 2006
3.
Fawwaz T, Ulaby, Eric Michielssen, Umberto Ravaioli, Fundamentals of Applied Electromagnetics,
Prentice Hall, 2010
4.
Howard Johnson, Martin Graham, High-Speed Digital Design, A Handbook of Black Magic, Prentice Hall,
1993
5.
Howard Johnson, Martin Graham, High-Speed Signal Propagation, Advanced Black Magic, Prentice Hall,
2003
6.
Reinhold Ludwig, Gene Bogdanov, RF Circuit Design, Theory and Applications, Prentice Hall 2009
7.
The Bergeron Method, A Graphic Method for Determining Line Reflections in Transient Phenomena,
Texas Instruments, 1996
8.
C. Q. Lee, A Novel Approach to Transients in Transmission Lines, IEEE Transactions on Education, Vol.
E.30 No. 2, May 1987
9.
Rusek A. and Oakley B., AC 2007-246: Easy to Do Transmission Line Demonstrations of Sinusoidal
Standing Waves and Transient Pulse Reflections, ASEE 2007 Conference.
10. P. J. Longlois, “Graphical Analysis of Delay Line Waveforms: A Tutorial” IEEE Trans Education, Feb,
1995, pp 27-32.
Proceedings of the 2011 ASEE NC & IL/IN Section Conference
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