Exercise 3-2. Effects of Attenuation on the VSWR EXERCISE OBJECTIVES

Exercise 3-2 Effects of Attenuation on the VSWR EXERCISE OBJECTIVES Upon completion of this exercise, you will know what the attenuation constant is and how to measure it. You will be able to define important terms related to the transfer and loss of power in mismatched transmission lines: insertion loss, return loss, and mismatch loss. You will know how to calculate the VSWR in a lossless line in terms of the reflection coefficient at the load. Finally, you will know how attenuation modifies the VSWR in lines that are lossy. DISCUSSION Attenuation of Sinusoidal Signals As for transient (pulsed) signals, sinusoidal signals always lose some power as they travel down a line. The power losses cause the transmitted signal to become more and 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 occurs because the shunt losses in the dielectric between the conductors are low as compared to the I 2 R losses of the conductors. R' S decreases as the diameter of the conductors is increased, and therefore so does attenuation. R' S increases as the frequency of the carried signal is increased, and therefore so does attenuation. In the theoretical example of an infinite line, the transmitted signal would gradually lose all of its power. Consequently, there would be no power reflection toward the generator, as if a perfectly matched load were continually absorbing all the received power. 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 (V a ) is higher than the amplitude of the voltage at point b (V b ). 3-33

Figure 3-24. Attenuation of the voltage over distance. The attenuation of the voltage between points a and b is determined by the quantity 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 being used. This constant is determined by the geometrical and physical characteristics of the line. It is therefore related to the distributed parameters of the line at the frequency 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). 3-34

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 nepers by 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 attenuation constant as a function of frequency. Figure 3-35, for example, shows the attenuation constant of two typical coaxial cables as a function of frequency. For both cables, the attenuation constant increases as the frequency of the carried signal increases. The attenuation constant of the RG-58 cable is lower than that of the RG-174, for any given frequency. This occurs because the conductors of the RG-58 cable have a larger diameter than those of the RG-174. Note that, in this example, the American Wire Gauge (AWG) standard is used to specify the conductor diameters. The lower the AWG of a conductor, the greater the diameter of the conductor. Figure 3-25. Attenuation constant-versus-frequency for two typical coaxial cables. 3-35

Insertion Loss The insertion loss is measured in decibels (db). It corresponds to the total loss that occurs along the entire length of a line. The insertion loss can be determined by measuring the power or voltage of the signal at the sending and receiving ends of the line: where log = Base-10 logarithm; P R = Power of the signal at the receiving end (V); P S = Power of the signal at the sending end (V). V R = Amplitude of the voltage at the receiving end (V); V S = Amplitude of the voltage at the sending end (V). Since the power or voltage ratio is always lower than 1, the insertion loss always has a negative value. Dividing the insertion loss by the length of the line gives the distributed attenuation constant of the line, '. For example, the insertion loss in the 24-m coaxial cable used as TRANSMISSION LINE A of your circuit board, will be 2.4 db if ' at the frequency of the carried signal is 0.1 db/m. Return Loss and Mismatch Loss Part of the power transmitted on a line, in addition to being lost through the distributed series and parallel resistances of the line, is also lost by reflection whenever a discontinuity, or impedance change, occurs along the line. If, for example, the impedance of the load does not perfectly match the characteristic impedance 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 reflection coefficient L : where L = Reflection coefficient at the load (dimensionless number, comprised between +1 and -1); Z L = Impedance of the load ( ); Z 0 = Characteristic impedance of the line ( ). Note: Z L is a complex quantity composed of a real, resistive part R, and an imaginary, reactive part X. Consequently, when Z L is not purely resistive, L is a vectorial quantity having both magnitude and phase information. Important terms relating to the loss of power must be known when studying the behavior of lines with mismatched load impedances. These terms include the return loss and the mismatch loss. 3-36

Return Loss The return loss is the ratio of the power or voltage incident at the load to the power or voltage reflected at the load: Since the power or voltage ratio is always lower than 1 (except when the impedance of the load is 0 or ), the return loss always has a negative value. The greater the absolute value of the return loss, the lower the power or voltage lost by reflection at the load. If, for example, the return loss is -10 db, then about 30% of the voltage incident at the load is reflected, as shown below: Since then When the reflection coefficient at the load, L, is known, the return loss can also be calculated in terms of this coefficient: Mismatch Loss The mismatch loss is the difference between the power or voltage incident at the load and the power or voltage reflected at the load. When there is no impedance mismatch, there is no reflection, so that all the power received at the load is absorbed 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 of 50, the return loss and mismatch loss will be -9.5 db and -0.51 db, respectively. 3-37

Relationship Between L and the VSWR You will recall that when an impedance mismatch occurs at the load, a standing wave is created on the line. The voltage standing-wave ratio (VSWR) is the ratio of the loop voltage to the node voltage of the standing wave: When the line is lossless, the VSWR, which stays constant along the line, can be calculated in terms of the reflection coefficient at the load L, using a simple formula: Note: Z L is a complex quantity composed of a real, resistive part R, and an imaginary, reactive part X. Consequently, when Z L is not purely resistive, L is a 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 and load and, therefore, the better the efficiency of power transfer from the generator to the 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 power is transferred from the generator to the load. In a lossy line, however, attenuation makes it difficult to use VSWR measurements as a direct indication of the efficiency of power transfer. This occurs because attenuation causes the incident power to become weaker as it travels toward the load, as Figure 3-26 shows. Consequently, the reflected power becomes weaker as it travels back toward the generator. The lost power is by heating of the line, which can result in the need for special cooling mechanisms such as copper tubing soldered along the sides of the guide, and carrying a liquid such as water or ethylene glycol. 3-38

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 VSWR decreases (become better and better) as we approach the sending end of the line. Consequently, a VSWR measurement made at the sending end can give an illusion of having a good VSWR and, therefore, an efficiency that is much better than reality. Figure 3-27 shows how attenuation improves the VSWR. For example, assume an insertion loss of 12 db. The line has a VSWR of 1.09 (good) if measured at the sending 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 than at the sending end. 3-39

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 the insertion loss. In the second part of this procedure section, you will measure the VSWR at the sending end and receiving end of a lossy line terminated by a mismatched load. This will 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 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. 3-40 G 2. Referring to Figure 3-28, connect the SIGNAL GENERATOR 50- output to 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 the sending end of TRANSMISSION LINE A [0-meter (0-foot) probe turret].

Using another probe, connect channel 2 of the oscilloscope to the receiving end of TRANSMISSION LINE A [24-meter (78.7-foot) probe turret]. Connect the SIGNAL GENERATOR 100- 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 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. 3-41

G 3. Make the following settings on the oscilloscope: Channel 1 Mode........................................ Normal Sensitivity..................................... 1 V/div Input Coupling.................................... AC Channel 2 Mode........................................ Normal Sensitivity..................................... 1 V/div Input Coupling.................................... AC Time Base...................................... 0.1 s/div Trigger Source...................................... External Level........................................... 0.5 V Input Impedance........................... 1 M or more G 4. Adjust the FREQUENCY knob of the SIGNAL GENERATOR until the frequency 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), as Figure 3-29 shows. Figure 3-29. The frequency of the voltage at the sending end of the line is set to 4.0 MHz approximately. 3-42

G 5. On the oscilloscope, measure the peak (positive) amplitude of the sinusoidal voltage at the sending end (V S ) and receiving end (V R ) of TRANSMISSION LINE A. Record your measurements below. V S = V V R = V G 6. Using the voltages measured in the previous step, calculate the insertion loss 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 using the following steps: Remove the coaxial cable between the receiving end of TRANSMISSION LINE A and the LOAD-section input. 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 LOAD-section input, using a short coaxial cable. Leave channel 1 of the oscilloscope connected to the sending end of the 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. 3-43

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 the line should still be set to 4.0 MHz (T 0.25 s). Measure the peak amplitude of the sinusoidal voltage at the sending end (V S ) and receiving end (V R ) of the line. Record your measurements below. V S = V V R = V G 9. Using the voltages measured in the previous step, calculate the insertion loss of TRANSMISSION LINEs A and B connected end-to-end. Insertion loss (48 m/157.4 ft) = db 3-44

G 10. Compare the insertion loss obtained for a single transmission line (as recorded in step 6) to that obtained for both transmission lines connected end-to-end (as recorded in step 9). Does the insertion loss double approximately when the length of the line is doubled? G Yes G No G 11. Calculate the distributed attenuation constant, ', of the 48-m (157.4-ft) line by 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 the procedure. 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 reflection coefficient at the load to 1 and, therefore, the theoretical return loss at the load 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 the frequency 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 Z L is higher than Z 0, nodes occur at odd multiples of /4 from the receiving end of the line. Consequently, a node occurs at the sending end of 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. V NODE (3 /4) = V 3-45

G 15. Adjust the FREQUENCY knob of the SIGNAL GENERATOR in order for the frequency 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 Z L is higher than Z 0, loops occur at even multiples of /4 from the receiving end of the line. Consequently, a loop occurs at the sending end of the 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. V LOOP (4 /4) = V G 16. Calculate the VSWR at the sending end of the line, based on the loop voltage previously measured at 4 /4 and on the node voltage previously measured at 3 /4. Note: Assume the attenuation constant of the line to remain approximately constant when the signal frequency is increased from 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 of TRANSMISSION LINE B. Since the line is 4 /4 long approximately, this turret is located at /4 from the receiving end. Since Z L is higher than Z 0, a node occurs at /4 from the receiving end of the line. Very slightly readjust, if necessary, the FREQUENCY knob of the SIGNAL GENERATOR in order for the voltage at that node to be minimum. Then, measure the peak (positive) amplitude of this voltage, and record your measurement. V NODE ( /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 Z L is higher than Z 0, a loop occurs at the receiving end of the line. Measure the peak (positive) amplitude of this voltage, and record your measurement. V LOOP (RECEIVING END) = V 3-46

G 19. Calculate the VSWR at the receiving end of the line, based on the loop voltage measured at that end, and on the node voltage measured at /4. Note: For this calculation, we will use the node voltage measured at /4, and therefore neglect the attenuation undergone by the incident voltage in transit between /4 and the receiving end of the line (around 1dB), as well as the attenuation undergone by the reflected voltage in transit between the receiving 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 that measured 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 at the receiving end. b. The improvement in VSWR caused by the insertion loss is more apparent 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 the insertion loss causes the difference in voltage at a loop and adjacent node 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 down a line. Usually, the distributed series resistance of the line, R' s, is responsible for most 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 length of the line. The attenuation constant increases with frequency. Consequently, manufacturers provide graphs or tables indicating the attenuation constant of a line as a function of frequency. Important terms relating to the loss of power are the insertion loss, the return loss and the mismatch loss, all expressed in decibels (db). The insertion loss is the total loss occurring along the line. The return loss is the ratio of the voltage incident at the load to the voltage reflected at the load. The mismatch loss is the difference between the voltage incident at the load and the voltage reflected at the load. 3-47

When a standing wave is present on a line, the VSWR can be calculated in terms of the reflection coefficient at the load, L, if the line is lossless or the losses can be neglected. In lines that are lossy, attenuation improves the VSWR. The improvement in VSWR by attenuation is greater at the sending end of the line than at the receiving end. Consequently, it is preferable to measure the VSWR at the receiving end of the line, or to measure the insertion loss of the line rather than the 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 the skin 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 reflected toward 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 whose reflection 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 can have high losses. If the line is lossless, the VSWR can be calculated, using a simple 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 an efficiency of power transfer that is much better than if the VSWR is measured at the sending end. 3-48

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. 3-49

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