Exercise 3-2 Effects of Attenuation on the · PDF file3-33 Exercise 3-2 Effects of Attenuation...

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Exercise 3-2

Effects of Attenuation on the VSWR

EXERCISE OBJECTIVES

Upon completion of this exercise, you will know what the attenuation constant is andhow to measure it. You will be able to define important terms related to the transferand loss of power in mismatched transmission lines: insertion loss, return loss, andmismatch loss. You will know how to calculate the VSWR in a lossless line in termsof the reflection coefficient at the load. Finally, you will know how attenuationmodifies the VSWR in lines that are lossy.

DISCUSSION

Attenuation of Sinusoidal Signals

As for transient (pulsed) signals, sinusoidal signals always lose some power as theytravel down a line. The power losses cause the transmitted signal to become moreand more attenuated over distance from the generator.

The power is lost in the distributed series resistance, R'S, and parallel resistance, R'p,of the line conductors. Usually, R'S is responsible for most of the losses. This occursbecause the shunt losses in the dielectric between the conductors are low ascompared to the I2R losses of the conductors.

• R'S decreases as the diameter of the conductors is increased, and therefore sodoes attenuation.

• R'S increases as the frequency of the carried signal is increased, and thereforeso does attenuation.

In the theoretical example of an infinite line, the transmitted signal would graduallylose all of its power. Consequently, there would be no power reflection toward thegenerator, as if a perfectly matched load were continually absorbing all the receivedpower.

Attenuation Constant

Figure 3-24 shows a sinusoidal signal propagating down a line between two points,a and b. Due to attenuation over distance, the amplitude of the voltage at point a (Va)is higher than the amplitude of the voltage at point b (Vb).


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Figure 3-24. Attenuation of the voltage over distance.

The attenuation of the voltage between points a and b is determined by thequantity e- D:

where e = Base of the Napierian logarithm (2.71);= Attenuation constant, in nepers (Np);

D = Distance between the two points.

The attenuation constant, , in nepers (Np), is specific to the particular line beingused. This constant is determined by the geometrical and physical characteristics ofthe line. It is therefore related to the distributed parameters of the line at thefrequency of the carried signals.

Rearranging the attenuation equation just stated in order to isolate gives:

where = Attenuation constant, in nepers (Np);ln = Napierian (base-2.71) logarithm.

Manufacturers often specify the attenuation constant of a line per unit length.Consequently, if D is a unit length in the attenuation-constant equation just stated,then

where ’ = Distributed attenuation constant per unit length (Np/m, or Np/ft).


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The distributed attenuation constant of a line can also be expressed as "decibels(dB) per unit length". 1 neper equals 8.686 decibels. Consequently, multiply nepersby 8.686 to obtain decibels.

Due to the skin effect, the attenuation constant of a line increases with frequency.For this reason, manufacturers provide graphs or tables indicating the attenuationconstant as a function of frequency.

Figure 3-35, for example, shows the attenuation constant ’ of two typical coaxialcables as a function of frequency.

• For both cables, the attenuation constant increases as the frequency of thecarried signal increases.

• The attenuation constant of the RG-58 cable is lower than that of the RG-174, forany given frequency. This occurs because the conductors of the RG-58 cablehave a larger diameter than those of the RG-174.

Note that, in this example, the American Wire Gauge (AWG) standard is used tospecify the conductor diameters. The lower the AWG of a conductor, the greater thediameter of the conductor.

Figure 3-25. Attenuation constant-versus-frequency for two typical coaxial cables.


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Insertion Loss

The insertion loss is measured in decibels (dB). It corresponds to the total loss thatoccurs along the entire length of a line. The insertion loss can be determined bymeasuring the power or voltage of the signal at the sending and receiving ends ofthe line:

where log = Base-10 logarithm;PR = Power of the signal at the receiving end (V);PS = Power of the signal at the sending end (V).VR = Amplitude of the voltage at the receiving end (V);VS = Amplitude of the voltage at the sending end (V).

Since the power or voltage ratio is always lower than 1, the insertion loss always hasa negative value. Dividing the insertion loss by the length of the line gives thedistributed attenuation constant of the line, '.

For example, the insertion loss in the 24-m coaxial cable used as TRANSMISSIONLINE A of your circuit board, will be 2.4 dB if ' at the frequency of the carried signalis 0.1 dB/m.

Return Loss and Mismatch Loss

Part of the power transmitted on a line, in addition to being lost through thedistributed series and parallel resistances of the line, is also lost by reflectionwhenever a discontinuity, or impedance change, occurs along the line.

If, for example, the impedance of the load does not perfectly match the characteristicimpedance of the line, not all the voltage incident at the load is absorbed by the load.Instead, part of this voltage is reflected back toward the generator by a reflectioncoefficient L:

where L = Reflection coefficient at the load (dimensionless number,comprised between +1 and -1);

ZL = Impedance of the load ( );Z0 = Characteristic impedance of the line ( ).

Note: ZL is a complex quantity composed of a real, resistive part R, and animaginary, reactive part X. Consequently, when ZL is not purely resistive, L isa vectorial quantity having both magnitude and phase information.

Important terms relating to the loss of power must be known when studying thebehavior of lines with mismatched load impedances. These terms include the return

loss and the mismatch loss.


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Return Loss

The return loss is the ratio of the power or voltage incident at the load to the poweror voltage reflected at the load:

Since the power or voltage ratio is always lower than 1 (except when the impedanceof the load is 0 or ), the return loss always has a negative value. The greater theabsolute value of the return loss, the lower the power or voltage lost by reflection atthe load.

If, for example, the return loss is -10 dB, then about 30% of the voltage incident atthe load is reflected, as shown below:

Since

then

When the reflection coefficient at the load, L, is known, the return loss can also becalculated in terms of this coefficient:

Mismatch Loss

The mismatch loss is the difference between the power or voltage incident at theload and the power or voltage reflected at the load. When there is no impedancemismatch, there is no reflection, so that all the power received at the load isabsorbed by the load.

A formula for calculating the mismatch loss, in terms of L, is

For example, given a load impedance of 25 and a characteristic impedance of50 , the return loss and mismatch loss will be -9.5 dB and -0.51 dB, respectively.


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Relationship Between L and the VSWR

You will recall that when an impedance mismatch occurs at the load, a standingwave is created on the line. The voltage standing-wave ratio (VSWR) is the ratio ofthe loop voltage to the node voltage of the standing wave:

When the line is lossless, the VSWR, which stays constant along the line, can becalculated in terms of the reflection coefficient at the load L, using a simple formula:

Note: ZL is a complex quantity composed of a real, resistive part R, and animaginary, reactive part X. Consequently, when ZL is not purely resistive, L isa vectorial quantity having both magnitude and phase information.

Effect of Attenuation on the VSWR

The closer to 1 the VSWR is, the better the impedance match between the line andload and, therefore, the better the efficiency of power transfer from the generator tothe load.

• In a lossless line, the VSWR remains constant over distance from the generator,so that VSWR measurements are useful to determine how efficiently the poweris transferred from the generator to the load.

• In a lossy line, however, attenuation makes it difficult to use VSWRmeasurements as a direct indication of the efficiency of power transfer. Thisoccurs because attenuation causes the incident power to become weaker as ittravels toward the load, as Figure 3-26 shows. Consequently, the reflected powerbecomes weaker as it travels back toward the generator.

The lost power is by heating of the line, which can result in the need for specialcooling mechanisms such as copper tubing soldered along the sides of the guide,and carrying a liquid such as water or ethylene glycol.


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Figure 3-26. Attenuation causes the incident and reflected power to become weaker over distance.

The result of the power gradually becoming weaker and weaker is that the VSWRdecreases (become better and better) as we approach the sending end of theline. Consequently, a VSWR measurement made at the sending end can give anillusion of having a good VSWR and, therefore, an efficiency that is much betterthan reality.

Figure 3-27 shows how attenuation improves the VSWR. For example, assumean insertion loss of 12 dB. The line has a VSWR of 1.09 (good) if measured at thesending end. However, the line will have a VSWR of 5 (poor) at the load.Because of this, the VSWR should be measured at the receiving end rather thanat the sending end.


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Figure 3-27. Attenuation improves the VSWR.

Procedure Summary

In the first part of this procedure section, you will measure the insertion loss in a line.You will then double the length of the line and see the effect that this has on theinsertion loss.

In the second part of this procedure section, you will measure the VSWR at thesending end and receiving end of a lossy line terminated by a mismatched load. Thiswill allow you to see the effect that attenuation has on the VSWR.

PROCEDURE

Measuring the Insertion Loss

G 1. Make sure the TRANSMISSION LINES circuit board is properly installedinto the Base Unit. Turn on the Base Unit and verify that the LED's next toeach control knob on this unit are both on, confirming that the circuit boardis properly powered.

G 2. Referring to Figure 3-28, connect the SIGNAL GENERATOR 50- outputto the sending end of TRANSMISSION LINE A, using a short coaxial cable.Connect the receiving end of this line to the input of the LOAD SECTION,using a short coaxial cable.

Using an oscilloscope probe, connect channel 1 of the oscilloscope to thesending end of TRANSMISSION LINE A [0-meter (0-foot) probe turret].


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Using another probe, connect channel 2 of the oscilloscope to the receivingend of TRANSMISSION LINE A [24-meter (78.7-foot) probe turret]. Connectthe SIGNAL GENERATOR 100- output to the trigger input of theoscilloscope, using a coaxial cable.

In the LOAD section, set the toggle switches in such a way as to connectthe input of this section to the common through resistor R3 (50 ).

The connections should now be as shown in Figure 3-28.

Figure 3-28. Measuring the insertion loss of a transmission line.


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G 3. Make the following settings on the oscilloscope:

Channel 1Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NormalSensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 V/divInput Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AC

Channel 2Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NormalSensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 V/divInput Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AC

Time Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 s/divTrigger

Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ExternalLevel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.5 VInput Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 M or more

G 4. Adjust the FREQUENCY knob of the SIGNAL GENERATOR until thefrequency of the voltage at the sending end of the line [0-meter (0-foot)probe turret of TRANSMISSION LINE A] is 4.0 MHz (T 0.25 s), asFigure 3-29 shows.

Figure 3-29. The frequency of the voltage at the sending end of the line is set to 4.0 MHzapproximately.


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G 5. On the oscilloscope, measure the peak (positive) amplitude of the sinusoidalvoltage at the sending end (VS) and receiving end (VR) of TRANSMISSIONLINE A. Record your measurements below.

VS = V

VR = V

G 6. Using the voltages measured in the previous step, calculate the insertionloss of TRANSMISSION LINE A.

Insertion loss (24 m/78.7 ft) = dB

G 7. Referring to Figure 3-30, double the length of the transmission line by usingthe following steps:

– Remove the coaxial cable between the receiving end ofTRANSMISSION LINE A and the LOAD-section input.

– Connect the receiving end of TRANSMISSION LINE A to the sendingend of TRANSMISSION LINE B, using a short coaxial cable. Connectthe receiving end of TRANSMISSION LINE B to the LOAD-sectioninput, using a short coaxial cable.

– Leave channel 1 of the oscilloscope connected to the sending end ofthe line [0-meter (0-foot) probe turret of TRANSMISSION LINE A].Connect channel 2 of the oscilloscope to the receiving end of the line[24-meter (78.7-foot) probe turret of TRANSMISSION LINE B].

The connections should now be as shown in Figure 3-30.


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Figure 3-30. Measuring the insertion loss of both transmission lines connected end-to-end.

G 8. On the oscilloscope, the frequency of the voltage at the sending end of theline should still be set to 4.0 MHz (T 0.25 s).

Measure the peak amplitude of the sinusoidal voltage at the sending end(VS) and receiving end (VR) of the line. Record your measurements below.

VS = V

VR = V

G 9. Using the voltages measured in the previous step, calculate the insertionloss of TRANSMISSION LINEs A and B connected end-to-end.

Insertion loss (48 m/157.4 ft) = dB


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G 10. Compare the insertion loss obtained for a single transmission line (asrecorded in step 6) to that obtained for both transmission lines connectedend-to-end (as recorded in step 9).

Does the insertion loss double approximately when the length of the line isdoubled?

G Yes G No

G 11. Calculate the distributed attenuation constant, ', of the 48-m (157.4-ft) lineby dividing the insertion loss of this line by the length of the line.

where ’ = Distributed attenuation constant per unit length (dB/m).l = Length of the line (m).

’ = dB/m

G 12. Leave all the connections as they are and proceed to next section of theprocedure.

Effect of Attenuation on the VSWR

Measuring the VSWR at the Sending End

G 13. In the LOAD section, set all the toggle switches to the O (OFF) position.This places the impedance of the load at the receiving end of the 48-m(157.4-ft) in the open-circuit condition ( ). This also sets the reflectioncoefficient at the load to 1 and, therefore, the theoretical return loss at theload to 0 dB.

G 14. With channel 1 of the oscilloscope connected to the sending end of the line,adjust the FREQUENCY knob of the SIGNAL GENERATOR in order for thefrequency of the voltage at that point to be 3.0 MHz (T = 0.33 s)approximately. This makes the line 3 /4 long approximately.

Since ZL is higher than Z0, nodes occur at odd multiples of /4 from thereceiving end of the line. Consequently, a node occurs at the sending endof the line (i.e., at 3 /4 from the receiving end).

Measure the peak (positive) amplitude of the voltage at the sending end.Record your measurement below.

VNODE (3 /4) = V


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G 15. Adjust the FREQUENCY knob of the SIGNAL GENERATOR in order for thefrequency of the voltage at the sending end of the line to be 4.0 MHz(T = 0.25 s) approximately. This makes the line 4 /4 long approximately.

Since ZL is higher than Z0, loops occur at even multiples of /4 from thereceiving end of the line. Consequently, a loop occurs at the sending end ofthe line (i.e., at 4 /4 from the receiving end).

Measure the peak (positive) amplitude of the voltage at the sending end.Record your measurement below.

VLOOP (4 /4) = V

G 16. Calculate the VSWR at the sending end of the line, based on the loopvoltage previously measured at 4 /4 and on the node voltage previouslymeasured at 3 /4.

Note: Assume the attenuation constant of the line to remainapproximately constant when the signal frequency is increasedfrom 3 MHz to 4 MHz.

VSWR (SENDING END) =

Measuring the VSWR at the Receiving End

G 17. Connect channel 2 of the oscilloscope to the 12-m (39.4-ft) probe turret ofTRANSMISSION LINE B. Since the line is 4 /4 long approximately, thisturret is located at /4 from the receiving end.

Since ZL is higher than Z0, a node occurs at /4 from the receiving end of theline. Very slightly readjust, if necessary, the FREQUENCY knob of theSIGNAL GENERATOR in order for the voltage at that node to be minimum.Then, measure the peak (positive) amplitude of this voltage, and record yourmeasurement.

VNODE ( /4) = V

G 18. Connect channel 2 of the oscilloscope to the receiving end of the line [24-m(78.7-ft) probe turret of TRANSMISSION LINE B].

Since ZL is higher than Z0, a loop occurs at the receiving end of the line.Measure the peak (positive) amplitude of this voltage, and record yourmeasurement.

VLOOP (RECEIVING END) = V


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G 19. Calculate the VSWR at the receiving end of the line, based on the loopvoltage measured at that end, and on the node voltage measured at /4.

Note: For this calculation, we will use the node voltagemeasured at /4, and therefore neglect the attenuation undergoneby the incident voltage in transit between /4 and the receivingend of the line (around 1dB), as well as the attenuationundergone by the reflected voltage in transit between thereceiving end of the line and /4 (around 1 dB).

VSWR (RECEIVING END) =

VSWRs Comparison

G 20. Compare the VSWR measured at the sending end of the line to thatmeasured at the receiving end of the line, and observe they are different.

Which of the following best describes your observation?

a. The VSWR at the sending end is closer to reality than that measured atthe receiving end.

b. The improvement in VSWR caused by the insertion loss is moreapparent at the receiving end of the line.

c. The VSWR at the receiving end is higher than that at the sending end,due to attenuation.

d. A poorer VSWR occurs at the receiving end of the line, because theinsertion loss causes the difference in voltage at a loop and adjacentnode to be lower at the receiving end of the line.

G 21. Turn off the Base Unit and remove all the connecting cables and probes.

CONCLUSION

• As for pulsed signals, sinusoidal signals undergo attenuation as they travel downa line. Usually, the distributed series resistance of the line, R's, is responsible formost of the attenuation. Power is lost by heating of the line.

• The attenuation constant of a line is a measure of the attenuation per unit lengthof the line. The attenuation constant increases with frequency. Consequently,manufacturers provide graphs or tables indicating the attenuation constant of aline as a function of frequency.

• Important terms relating to the loss of power are the insertion loss, the returnloss and the mismatch loss, all expressed in decibels (dB). The insertion loss isthe total loss occurring along the line. The return loss is the ratio of the voltageincident at the load to the voltage reflected at the load. The mismatch loss is thedifference between the voltage incident at the load and the voltage reflected atthe load.


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• When a standing wave is present on a line, the VSWR can be calculated interms of the reflection coefficient at the load, L, if the line is lossless or thelosses can be neglected.

• In lines that are lossy, attenuation improves the VSWR. The improvement inVSWR by attenuation is greater at the sending end of the line than at thereceiving end. Consequently, it is preferable to measure the VSWR at thereceiving end of the line, or to measure the insertion loss of the line rather thanthe VSWR per se.

REVIEW QUESTIONS

1. The attenuation constant of a line

a. decreases as the frequency of the carried signal is increased, due to theskin effect.

b. increases as the AWG of the line conductors is increased.c. corresponds to the quantity e- D.d. usually specified per unit length.

2. How much of the voltage incident at the load of a mismatched line is reflectedtoward the generator, if the return loss is -6 dB?

a. 25%b. 50%c. 75%d. 100%

3. What are the VSWR, return loss, and mismatch loss of a lossless line whosereflection coefficient at the load, L, is 0.333?

a. 2, -0.51 dB, and -9.55 dB, respectively.b. 0.5, -11.3 dB, and -0.336 dB, respectively.c. 2, -9.55 dB, and -0.51 dB, respectively.d. 0.5, -14 dB, and -0.177 dB, respectively.

4. When a line is improperly terminated, standing waves will result and the line canhave high losses. If the line is lossless, the VSWR can be calculated, using asimple equation, in terms of the

a. loop and maximum voltages measured at the receiving end of the line.b. distributed attenuation constant of the line.c. reflection coefficient at the load.d. insertion loss along the line.

5. In a lossy line with standing waves, the

a. VSWR measured at the receiving end can give an illusion of having anefficiency of power transfer that is much better than if the VSWR ismeasured at the sending end.


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b. incident voltage becomes weaker as it travels back toward the generator,and the reflected voltage decreases as it travels back toward the generator.

c. VSWR decreases as we approach the receiving end of the line.d. VSWR is better at the receiving end of the line.


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