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Exercise
3-1
Standing Waves and Voltage Standing Wave Ratio (VSWR)
EXERCISE OBJECTIVES
Upon completion of this exercise, you will know how standing waves are created on
transmission lines. You will be able to describe the characteristics of a standing wave
based on the nature of the impedance mismatch at the origin of this wave. You will
know what is the standing-wave ratio and how to measure it.
DISCUSSION
Standing Waves on an Open-Ended Line
Figure 3-4 shows a sinusoidal-voltage generator connected to a transmission line.
The receiving end of the line is in the open-circuit condition (ZL = ȍ).
•
At time t = 0, the generator produces a sinusoidal voltage that is launched into the
line.
•
After a certain transit time, T, the voltage reaches the receiving end of the line,
where it is reflected back to the generator due to the impedance mismatch. At
that very moment, the incident and reflected voltages have the same phase,
because the incident voltage is reflected with the same phase as if it would have
continued if the line had not ended.
Figure 3-4. Creation of a standing wave on an open-ended line (ZL = ȍ).
•
After time T, the reflected and incident voltages travel through each other, but in
opposite directions, along the line, thereby combining vectorially. This results in
3-7
Standing Waves and Voltage Standing Wave Ratio (VSWR)
the creation of a standing wave of voltage along the line (the boldest line in
Figure 3-4). The standing wave is the algebraic sum of the instantaneous values
of the incident and reflected voltages at each point all along the line. This wave
does not move or travel along the line, hence the term "standing".
Figure 3-5 shows standing waves on an open-ended line versus distance (D) at
different instants of time. The amplitude of the standing wave is different in each
case, due to the fact that the incident voltage has a different phase when it reaches
the receiving end.
•
At the receiving end, the incident and reflected voltages are always in phase,
whatever the phase of the incident voltage may be.
•
Along the line, the voltage of the standing wave can vary between twice the
positive maximum and twice the negative maximum of the incident voltage
(assuming a lossless line), depending on the phase of the incident voltage (or
time of observation) of the incident voltage.
•
Along the line, the voltage of the standing wave invariably reaches a minimum or
maximum at multiples of Ȝ/4 from the receiving end of the line, as Figure 3-5
shows.
– At odd multiples of Ȝ/4 from the receiving end, the incident and reflected
voltages are 180( out of phase. Consequently, the voltage of the standing
wave is at a minimum.
– At even multiples of Ȝ/4 from the receiving end, the incident and reflected
voltages have the same phase. Consequently, the voltage of the standing
wave is at a maximum (positive or negative).
3-8
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-5. Standing waves on an open-ended line versus distance (D) at three different
instants, tn.
Even if the voltage of standing waves continually changes polarity with time, the
conventional way of representing these waves is with their negative and positive halfcycles pointing upward.
Figure 3-6 shows the conventional representation of a standing wave on an openended line. This representation actually corresponds to the result obtained when
measuring the amplitude of a standing wave after rectification and filtering as with
a peak detector.
•
The points where the voltage is minimum are called nodes. At a node, the
voltage is null if the line is lossless. A node occurs at every odd multiple of Ȝ/4
from the receiving end. Since Z = V/I, the input impedance of an open-ended
lossless line that is exactly Ȝ/4 long (or 3Ȝ/4, 5Ȝ/4, 7Ȝ/4 long, ...) is null (0 ȍ).
•
The points where the voltage is maximum are called loops, or antinodes. A loop
occurs at every even multiple of Ȝ/4 from the receiving end. At a loop, the current
is null if the line is lossless. Consequently, the input impedance line of an openended lossless line that is exactly Ȝ/2 long (or Ȝ, 3Ȝ/2, 5Ȝ/2 long, ...) is infinite
( ȍ).
3-9
Standing Waves and Voltage Standing Wave Ratio (VSWR)
The higher the frequency of the sinusoidal voltage launched into a line, the longer
the electrical length of the line and, therefore, the greater the number of loops and
nodes along the line.
Figure 3-6. Conventional representation of a standing wave of voltage on an open-ended lossless
line.
Standing Waves on a Short-Ended Line
Figure 3-7 shows a sinusoidal-voltage generator connected to a transmission line.
The receiving end of the line is in the short-circuit condition (ZL = 0 ȍ).
3-10
•
At time t = 0, the generator produces a sinusoidal voltage that is launched into the
line.
•
After a certain transit time, T, the voltage reaches the receiving end of the line,
where it is reflected back to the generator due to the impedance mismatch. At
that very moment, the incident and reflected voltages are 180( out of phase,
because the incident voltage is reversed (shifted in phase) as it is reflected.
•
After time T, the reflected and incident voltages travel through each other, but in
opposite directions, thereby combining vectorially. This results in the creation of
a standing wave of voltage along the line. The standing wave is the algebraic sum
of the instantaneous values of the incident and reflected voltages at each point
all along the line.
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-7. Creation of a standing wave on a short-ended line (ZL = 0 ȍ).
Figure 3-8 shows standing waves on a short-ended line versus distance (D) at
different instants of time. The amplitude of the standing wave is different in each
case, due to the fact that the incident voltage has a different phase when it reaches
the receiving end of the line at time T.
•
At the receiving end, the incident and reflected voltages are always 180( out of
phase, whatever the phase of the incident voltage may be. Consequently, the
voltage of the standing wave is always 0 V (assuming a lossless line).
•
Along the line, the voltage of the standing wave invariably reaches a minimum or
maximum at multiples of quarter-wavelengths (Ȝ/4) from the receiving end of
the line, as Figure 3-8 shows.
– At odd multiples of Ȝ/4 from the receiving end (Ȝ/4, 3Ȝ/4, 5Ȝ/4, etc.), the
incident and reflected voltages have the same phase. Consequently, the peak
voltage of the standing wave is at a maximum.
– At even multiples of Ȝ/4 (2Ȝ/4, 4Ȝ/4, 6Ȝ/4, etc.) from the receiving end, the
incident and reflected voltages are 180( out of phase. Consequently, the peak
voltage of the standing wave is at a minimum.
3-11
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-8. Standing waves on a short-ended line versus distance (D) at different instants of time.
Figure 3-9 shows the conventional representation of a standing wave of voltage on
a short-ended line.
3-12
•
On a short-ended line, a node invariably occurs at the receiving end of the line.
•
Nodes also occur at every even multiple of Ȝ/4 from the receiving end. At a node,
the voltage is null if the line is lossless. Consequently, the input impedance of a
short-ended lossless line whose electrical length is an even multiple of Ȝ/4 is null
(0 ȍ).
•
Loops occur at every odd multiple of Ȝ/4 from the receiving end. At a loop, the
voltage is maximum, and the current is null if the line is lossless. Consequently,
the input impedance of a short-ended lossless line whose electrical length is an
odd multiple of Ȝ/4 is infinite ( ȍ).
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-9. Conventional representation of a standing wave of voltage on a short-ended lossless
line.
Voltage Standing-Wave Ratio (VSWR)
The ratio of the loop voltage to the node voltage of a standing wave is called the
voltage standing-wave ratio (VSWR). In equation form:
where VSWR = Voltage standing-wave ratio (dimensionless number) ;
V LOOP = Voltage of the standing wave at a loop, = VMAX. (V);
V NODE = Voltage of the standing wave at a node, = V MIN. (V).
The VSWR indicates the degree of mismatch between the load impedance and the
characteristic impedance of the line. The higher the VSWR, the more severe the
impedance mismatch. Conversely, the lower the VSWR, the better the impedance
match and, therefore, the better the efficiency of power transfer.
•
When a line is properly matched, all the transmitted energy is transferred to the
load. There is no reflection and, therefore, no standing waves. The voltage
remains constant along the line. Consequently, the ratio VLOOP/VNODE is equal to
unity if the line is lossless. Consequently, the VSWR is 1, and the efficiency is
optimum.
•
In the worst situation, the load is in the open- or short-circuit condition, so that the
node voltage is equal to 0 V (if the line is lossless). Consequently, the VSWR is
infinite ().
•
In any other situation, the VSWR is comprised between 1 and .
3-13
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Note: So far, we have been talking in terms of energy transfer, and
energy transfer efficiency through transmission lines that carry transient
(short-duration) signals. From now on, however, we will use the terms
power transfer and power transfer efficiency instead, because power,
which is energy per unit of time, is more relevant to the transfer of
signals that repeat periodically and that show negligible change over a
relatively long period of time.
Line Terminated by ZL > Z0
Figure 3-10 shows a standing wave on a lossless line terminated by a purely
resistive load having an impedance ZL > Z0, in comparison to that obtained when the
line is in the open-circuit condition (ZL = ȍ).
The wave obtained with ZL < Z0 is weaker than for the open-ended line, because part
of the received voltage is absorbed by the load. Thus, the maximum voltage of the
wave is not as high as when the line is open-ended. Moreover, the minimum voltage
of the wave is not as low as when the line is open-ended. Consequently, the VSWR
is lower than for an open-ended line (ZL = ȍ).
In fact, when ZL > Z0,
In Figure 3-10, for example, ZL = 2Z0, so that the VSWR = 2. If ZL is increased to 4Z0,
then the VSWR will be 4, and so on.
Figure 3-10. Standing waves on a line when ZL > Z0, as compared to when ZL = ȍ.
3-14
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Line Terminated by ZL < Z0
Figure 3-11 shows a standing wave on a lossless line terminated by a purely
resistive load having an impedance ZL < Z0, in comparison to that obtained when the
line is in the short-circuit condition (ZL = 0 ȍ).
The wave obtained with ZL < Z0 is weaker than for the short-ended line, because part
of the received voltage is absorbed by the load. Thus, the maximum voltage of the
wave is not as high as when the line is short-ended. Moreover, the minimum voltage
of the wave is not as low as when the line is short-ended. Consequently, the VSWR
is lower than for a short-ended line ().
In fact, when ZL < Z0,
In Figure 3-11, for example, ZL = Z0/2, so that VSWR = 2. If ZL is decreased to Z0/4,
then the VSWR will be 4, and so on.
Figure 3-11. Standing waves on a line when ZL < Z0, as compared to when ZL = 0 ȍ.
The VSWR is a scalar quantity: it consists of a real number that does not take
account of the phase of the incident voltage as it reaches the load. This implies that
a same VSWR can be caused by different load impedances.
For example, as you have seen, a VSWR of 2 can be caused by a load impedance
of 2Z0 (i.e. higher than Z0) or of Z0/2 (i.e. lower than Z0). However, the standing
waves produced in each case differ in the location of their loops and nodes:
•
With a load impedance higher than Z0, the situation is similar to that of an openended line: the nodes occur at odd multiples of Ȝ/4, while the loops occur at
even multiples of Ȝ/4.
3-15
Standing Waves and Voltage Standing Wave Ratio (VSWR)
•
With a load impedance lower than Z0, the situation is similar to that of a shortended line: the loops occur at odd multiples of Ȝ/4, while the nodes occur at
even multiples of Ȝ/4.
Procedure Summary
In this procedure section, you will determine the effect that a change in electrical
length has on the characteristics of the standing wave created on a short-ended line.
You will then observe the differences between the standing waves and the VSWR
that occur on a line of given electrical length when ZL < Z0 and when ZL > Z0.
PROCEDURE
Standing Waves on a Short-Ended Line for Different Electrical Lengths
G
1. Make sure the TRANSMISSION LINES circuit board is properly installed
into the Base Unit. Turn on the Base Unit and verify that the LED's next to
each control knob on this unit are both on, confirming that the circuit board
is properly powered.
G
2. Referring to Figure 3-12, connect the SIGNAL GENERATOR 50-ȍ output
to the sending end of TRANSMISSION LINE A, using a short coaxial cable.
Connect the receiving end of TRANSMISSION LINE A to the sending end
of TRANSMISSION LINE B, using a short coaxial cable. Connect the
receiving end of TRANSMISSION LINE B to the input of the LOAD
SECTION, using a short coaxial cable.
Using an oscilloscope probe, connect channel 1 of the oscilloscope to the
0-meter (0-foot) probe turret of TRANSMISSION LINE A. Connect the
SIGNAL GENERATOR 100-ȍ output to the trigger input of the oscilloscope,
using a coaxial cable.
In the LOAD section, set the toggle switches in such a way as to connect
the input of this section directly to the common (i.e. via no load). This places
the impedance of the load at the receiving end of the line made by
TRANSMISSION LINEs A and B connected end-to-end in the short-circuit
condition (0 ȍ).
The connections should now be as shown in Figure 3-12. (The toggle-switch
setting is not shown). In the exercise, the oscilloscope probe will be
successively connected to each probe turret along the transmission line, as
indicated by the dashed lines in Figure 3-12.
3-16
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-12. Measuring the standing-wave voltage along a short-ended line for different electrical
lengths.
G
3. Make the following settings on the oscilloscope:
Channel 1
Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal
Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 V/div
Input Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AC
Time Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.5 ȝs/div
Trigger
Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External
Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.5 V
Input Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Mȍ or more
G
4. In the SIGNAL GENERATOR section, set the FREQUENCY knob to the
fully clockwise (MAX.) position. This sets the frequency of the sinusoidal
voltage produced by this generator to a maximum (about 5 MHz).
The voltage present at the sending end of the line (as displayed on the
oscilloscope) should be as shown in Figure 3-13.
3-17
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-13. The frequency of the sinusoidal voltage observed at the sending end of the line is
maximum (5 MHz approximately).
G
5. Decrease the frequency of the SIGNAL GENERATOR voltage from
maximum to minimum (i.e. from 5 MHz to 5 kHz approximately). To do this,
very slowly turn the FREQUENCY knob of this generator fully
counterclockwise. As you do this, observe the voltage present at the
sending end of the line on the oscilloscope.
You should observe that, as the frequency is decreased, the amplitude of
the displayed voltage continually varies, alternating between some
maximum and some minimum values.
This occurs because, as the frequency of the generator voltage is
decreased, the wavelength of this voltage increases, causing the electrical
length of the line, lȜ, to decrease, as Figure 3-14 shows. This decrease in
electrical length causes the frequency of the standing wave present along
the line to decrease, causing the voltage at any given point, z, along the line
to vary in consequence.
3-18
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-14. The voltage at point z continually varies as the frequency of the standing wave
decreases.
G
6. Make sure the FREQUENCY knob of the SIGNAL GENERATOR is set to
the fully counterclockwise (MIN.) position.
While observing the voltage on the oscilloscope, slowly turn the
FREQUENCY knob clockwise and stop turning it as soon as the amplitude
of the voltage reaches a first maximum.
The frequency of the voltage should now be nearly 1 MHz approximately
(T x 1 ȝs), as Figure 3-15 shows.
Since the frequency of the voltage is around 1 MHz, and the theoretical
velocity of propagation is 1.96 # 108 m/s (6.43 # 108 ft/s), the wavelength of
the voltage, Ȝ, is 196 m (643 ft) approximately. Consequently, the electrical
length of the 48-m (157.4-ft) line made by TRANSMISSION LINEs A and B
connected end-to-end is about
a.
b.
c.
d.
2Ȝ
Ȝ/2
Ȝ/4
Ȝ
3-19
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-15. The amplitude of the voltage reaches a maximum at around 1 MHz.
G
7. Measure the peak (positive) amplitude of the sinusoidal voltage along the
entire length of the line. To do so, connect the oscilloscope probe to each
of the probe turrets listed below, and record the amplitude at each distance
from the sending end in the graph of Figure 3-16.
•
•
•
•
•
•
•
•
•
3-20
0-m (0-ft) turret of line A (sending end);
6-m (19.7-ft) turret of line A;
12-m (39.4-ft) turret of line A;
18-m (59.0-ft) turret of line A;
24-m (78.7-ft) turret of line A;
6-m (19.7-ft) turret of line B [30 m (98.4 ft) from the sending end)];
12-m (39.4 -ft) turret of line B;
18-m (59.0-ft) turret of line B;
24-m (78.7-ft) turret of line B (receiving end);
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-16. Standing waves on a short-ended line for different electrical lengths.
G
8. Connect the dots your recorded in the graph of Figure 3-16. The obtained
curve corresponds to the standing wave on the short-ended line when the
electrical length is Ȝ/4. This should resemble that shown in Figure 3-17.
3-21
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-17. Standing wave on a short-ended Ȝ/4 line.
G
9. Connect channel 1 of the oscilloscope to the 0-meter (0-foot) probe turret
of TRANSMISSION LINE A. Set the oscilloscope time base to 0.2 ȝs/div.
Increase the frequency of the generator voltage. To do this, slowly turn the
FREQUENCY knob clockwise and stop turning it as soon as the amplitude
of the displayed voltage reaches a minimum. The frequency of this voltage
should be around 2 MHz approximately (T x 0.5 ȝs), as Figure 3-18 shows.
3-22
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-18. The amplitude of the voltage reaches a minimum at around 2 MHz.
G 10. Since the frequency of the voltage is 2 MHz approximately, the wavelength
of the voltage is around 98 m (321 ft). Consequently, the line is nearly
Ȝ/2 long.
Measure the peak (positive) amplitude of the voltage along the Ȝ/2 line.
Record your results in the graph of Figure 3-16. Then, connect the dots to
obtain the standing wave for the Ȝ/2 line.
G 11. Connect channel 1 of the oscilloscope to the 0-meter (0-foot) probe turret
of TRANSMISSION LINE A.
Further increase the generator frequency until the amplitude of the
displayed voltage reaches a new maximum. The frequency of this voltage
should now be around 3 MHz (T x 0.33 ȝs), as Figure 3-19 shows.
3-23
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-19. The amplitude of the voltage reaches another maximum at around 3 MHz.
G 12. Since the frequency of the voltage is 3 MHz approximately, the wavelength
of the voltage is 65 m (213 ft) approximately. Consequently, the line is
nearly 3Ȝ/4 long.
Measure the peak (positive) amplitude of the voltage along the 3Ȝ/4 line.
Record your results in the graph of Figure 3-16. Then, connect the dots to
obtain the standing wave for the 3Ȝ/4 line.
G 13. Connect channel 1 of the oscilloscope to the 0-meter (0-foot) probe turret
of TRANSMISSION LINE A.
Further increase the generator frequency until the amplitude of the
displayed voltage reaches a new minimum. The frequency of this voltage
should now be around 4 MHz (T x 0.25 ȝs), as Figure 3-20 shows.
3-24
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-20. The amplitude of the voltage reaches another minimum at around 4 MHz.
G 14. Since the frequency of the voltage is 4 MHz approximately, the wavelength
of the voltage is 49 m (161 ft) approximately. Consequently, the line is
nearly 4Ȝ/4 long, i.e. Ȝ long.
Measure the peak (positive) amplitude of the voltage along the entire length
of the Ȝ-long line. Record your results in the graph of Figure 3-16. Connect
the dots to obtain the standing wave of the Ȝ-long line.
G 15. The four standing waves you plotted in the graph of Figure 3-16 should be
similar to those shown in Figure 3-21.
As the frequency of the generator voltage is increased,
a.
b.
c.
d.
the wavelength of the generator voltage decreases.
the electrical length of the line increases.
the frequency of the standing wave increases.
All of the above.
3-25
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-21. Standing waves on a short-ended line for different electrical lengths.
G 16. On your graph of Figure 3-16, locate the points of minimum voltage (nodes)
in the standing waves plotted for the Ȝ/2, 3Ȝ/4, and Ȝ lines. Observe that the
voltage at the nodes is not null (0 V), due to the resistance of lossy
TRANSMISSION LINEs A and B.
According to the plotted waves, a node occurs at
a.
b.
c.
d.
Ȝ/4 from the receiving end of the line, for any electrical length.
Ȝ/2 from the receiving end of the line, for any electrical length.
Ȝ/2 from the receiving end of the line for the Ȝ/2 line only.
Ȝ/4 from the receiving end of the line for the 3Ȝ/4 line only.
G 17. On your graph of Figure 3-16, locate the points of maximum voltage (loops)
in each standing wave. Note that as the electrical length increases, the
voltage at the loop(s) of a standing wave decreases, due to the fact that
attenuation through lossy TRANSMISSION LINEs A and B increases with
frequency.
3-26
Standing Waves and Voltage Standing Wave Ratio (VSWR)
According to the plotted waves, a loop occurs at
a.
b.
c.
d.
Ȝ/2 from the receiving end of the line, for any electrical length.
Ȝ/4 and Ȝ/2 from the receiving end of the line for the 3Ȝ/4 line.
Ȝ/4 from the receiving end of the line, for any electrical length.
Ȝ/2 from the receiving end of the line for the Ȝ/4 line.
G 18. Based on your graph of Figure 3-16, the distance between two successive
loops, or between two successive nodes in any standing wave is invariably
equal to
a.
b.
c.
d.
2Ȝ.
Ȝ/2.
Ȝ/4.
Ȝ.
G 19. Based on your graph of Figure 3-16, for which electrical length is the voltage
at the sending end of a short-ended line maximum and, therefore, the input
impedance of the line maximum?
a.
b.
c.
d.
Ȝ/4
Odd multiples of Ȝ/2, such as Ȝ/2, 3Ȝ/2, 5Ȝ/2, etc.
Ȝ
Odd multiples of Ȝ/4, such as Ȝ/4, 3Ȝ/4, 5Ȝ4, etc.
G 20. Leave the connections set as they are. Proceed with the exercise.
Standing Waves on a Line for ZL < Z0 and ZL > Z0
G 21. Connect channel 1 of the oscilloscope to the 0-meter (0-foot) probe turret
of TRANSMISSION LINE A.
Make sure the frequency of the displayed voltage is 4 MHz approximately
(T x 0.25 ȝs), in order for the line to be Ȝ long. Make sure the impedance of
the line load is in the short-circuit condition (ZL = 0 ȍ).
Measure the peak (positive) amplitude of the sinusoidal voltage along the
line. In the graph of Figure 3-22, record the measured amplitude at each
distance from the sending end. Then, connect the dots to obtain the
standing wave for ZL = 0 ȍ.
3-27
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Figure 3-22. Standing waves on a Ȝ-long line for ZL < Z0 and ZL > Z0.
G 22. In the LOAD section, modify the setting of the toggle switches in order for
the LOAD-section input to be connected to the common through resistor R2
(25 ȍ).
Measure the peak (positive) amplitude of the voltage along the line. Record
your results in the graph of Figure 3-22. Then, connect the dots to obtain the
standing wave for ZL = 25 ȍ.
G 23. In the LOAD section, modify the setting of the toggle switches in order for
the LOAD-section input to be connected to the common through resistor R4
(100 ȍ).
Measure the peak (positive) amplitude of the sinusoidal voltage along the
line. Record your results in the graph of Figure 3-22. Then, connect the dots
to obtain the standing wave for ZL = 100 ȍ.
G 24. In the LOAD section, set all the toggle switches to the O (OFF) position.
This places the impedance of the load in the open-circuit condition ( ȍ).
Measure the peak (positive) amplitude of the sinusoidal voltage along the
line. Record your results in the graph of Figure 3-22. Then, connect the dots
to obtain the standing wave for ZL = ȍ.
3-28
Standing Waves and Voltage Standing Wave Ratio (VSWR)
G 25. The standing waves you plotted in the graph of Figure 3-22 should resemble
those shown in Figure 3-23:
•
The wave obtained for ZL = 100 ȍ is weaker than for the open-ended
line (ZL = ȍ), because part of the voltage is absorbed by the 100-ȍ
load;
•
Similarly, the wave obtained for ZL = 25 ȍ is weaker than for the shortended line (ZL = 0 ȍ), because part of the voltage is absorbed by the
25-ȍ load.
Figure 3-23. Standing waves on a Ȝ-long line for ZL < Z0 and ZL > Z0.
G 26. Locate the nodes and loops in each of the standing waves you plotted in the
graph of Figure 3-22. Observe that for any given wave, the voltage at the
loops and nodes of the wave decreases over distance from the sending end
of the line, due to attenuation through the line.
3-29
Standing Waves and Voltage Standing Wave Ratio (VSWR)
Also, observe that the waves differ in the location of their loops and nodes,
depending on whether ZL is lower or greater than Z0. In fact,
a. loops occur at even multiples of Ȝ/4 when ZL < Z0.
b. nodes occur at odd multiples of Ȝ/4 when ZL < Z0, whereas they occur
at even multiples of Ȝ/4 when ZL > Z0.
c. loops occur at odd multiples of Ȝ/4 when ZL > Z0.
d. loops occur at odd multiples of Ȝ/4 when ZL < Z0, whereas they occur at
even multiples of Ȝ/4 when ZL > Z0.
G 27. Based on the standing wave you plotted in the graph of Figure 3-22 for
ZL = 0 ȍ, calculate the VSWR at this impedance. To do so, use the loop
voltage measured at 3Ȝ/4 from the receiving end and the node voltage
measured at the sending end.
Note: The voltage at the loops and nodes of the standing wave
decreases over distance from the sending end, due to attenuation
through lossy TRANSMISSION LINEs A and B. Consequently,
the VSWR varies over distance, depending on the loop and
adjacent node used for its calculation. The loop and node used
here have been chosen arbitrarily, just for the sake of showing
how the VSWR is calculated.
VSWR (0 ȍ) =
According to your result, the VSWR for ZL = 0 ȍ is
a.
b.
c.
d.
not equal to the theoretical value of 1 because the line is lossy.
much lower than the theoretical value of because the line is lossy.
nearly .
lower than the theoretical value of 1 for a lossless line.
G 28. Based on the standing wave you plotted in the graph of Figure 3-22 for
ZL = ȍ, calculate the VSWR at this impedance. To do so, use the loop
voltage measured at the sending end and the node voltage measured at
3Ȝ/4 from the receiving end.
VSWR ( ȍ) =
According to your result, the VSWR for ZL = ȍ is
a.
b.
c.
d.
nearly .
not equal to the theoretical value of 1 because the line is lossy.
lower than the theoretical value of 1 for a lossless line.
lower than the theoretical value of because the line is lossy.
G 29. Based on the standing waves you plotted in the graph of Figure 3-22 for
ZL = 25 ȍ and ZL = 100 ȍ, calculate the VSWR for each of these
3-30
Standing Waves and Voltage Standing Wave Ratio (VSWR)
impedances. To do so, use the loop and node voltages measured at the
sending end and at 3Ȝ/4 from the receiving end.
VSWR (25 ȍ) =
VSWR (100 ȍ) =
G 30. Compare the VSWR's obtained in the previous steps for ZL = 25 ȍ and
ZL = 100 ȍ to those obtained for ZL = 0 ȍ and ZL = ȍ.
The VSWR's for ZL = 25 ȍ and ZL = 100 ȍ are both
a.
b.
c.
d.
lower than those for ZL = 0 ȍ and ZL = ȍ.
greater than 1.
lower than .
All of the above.
G 31. Turn off the Base Unit and remove all the connecting cables and probes.
CONCLUSION
•
When a line is mismatched at its load, standing waves are created along the
line. The points of maximum voltage in a standing wave are called loops. Those
of minimum voltage are called nodes.
•
When ZL is higher than Z0, loops occur at even multiples of Ȝ/4 from the receiving
end, and nodes at odd multiples of Ȝ/4 from the receiving end. This implies that
the input impedance of an open-ended lossless line is null when the line is Ȝ/4
long, and infinite when the line is Ȝ/2 long.
•
When ZL is lower than Z0, nodes occur at even multiples of Ȝ/4 from the receiving
end, and loops at odd multiples of Ȝ/4 from the receiving end. This implies that
the input impedance of a short-ended lossless line is infinite when the line is Ȝ/4
long, and null when the line is Ȝ/2 long.
•
The ratio of the loop voltage to node voltage is called the voltage standing-wave
ratio (VSWR). The VSWR is comprised between 1 (no standing wave) and (short- or open-circuit load). The closer the VSWR is to 1, the better the
impedance match between the line and load and, therefore, the better the
efficiency of power transfer on the line.
REVIEW QUESTIONS
1. The electrical length of a line is determined by the actual (physical) length of the
line and on the
a.
b.
c.
d.
amplitude of the standing wave present on the line.
severity of the impedance mismatch at the load.
frequency of the sinusoidal voltage it carries.
amplitude of the sinusoidal voltage it carries.
3-31
Standing Waves and Voltage Standing Wave Ratio (VSWR)
2. If the electrical length of a lossless line with standing waves is an exact odd
multiple of quarter wavelengths (Ȝ/4) and is open-ended, its input impedance
looks like
a.
b.
c.
d.
a partial short circuit.
a partial open circuit.
a short circuit.
an open circuit.
3. If the electrical length of a lossless line with standing waves is an odd multiple
of quarter wavelengths (Ȝ/4) long and is short-ended, its input impedance looks
like
a.
b.
c.
d.
a partial short circuit.
a partial open circuit.
a short circuit.
an open circuit.
4. What voltage standing-wave ratio (VSWR) would a lossy line have if it were
partially short- or open-ended?
a.
b.
c.
d.
A value that is higher than 1 but lower than .
A value lower than 1.
1
5. A line can have the same VSWR for both ZL < Z0 and ZL > Z0—the difference
lying in the location of the loops and nodes in the standing wave created. For
example, a lossless line will have the same VSWR if
a.
b.
c.
d.
3-32
ZL = 3Z0 or ZL = Z0/3.
ZL = 0.25 # Z0 or ZL = 2Z0.
ZL = 0.25 # Z0 or ZL = 4Z0.
Both (a) and (c)