Exercise 1-2. Velocity of Propagation EXERCISE OBJECTIVE

Size: px

Start display at page:

Download "Exercise 1-2. Velocity of Propagation EXERCISE OBJECTIVE"

Crystal Morton
6 years ago
Views:

1 Exercise 1-2 Velocity of Propagation EXERCISE OBJECTIVE Upon completion of this unit, you will know how to measure the velocity of propagation of a signal in a transmission line, using the step response method. Based on the measurements, you will know how to determine the relative permittivity of the dielectric material used to construct this line. DISCUSSION Velocity of Propagation A radio signal travels in free space at the velocity of light (approximately m/s, or ft/s). In a transmission line, a signal will travel at a relatively lower speed. This is due mainly to the presence of the dielectric material used to construct the line. In fact, the velocity of propagation of a signal in a transmission line, v P, is dependent upon the distributed inductance and capacitance of the line, L' and C' (see Figure 1-34). The equation for calculating v P is: where v P = Velocity of propagation (m/s or ft/s); L' = Distributed inductance, in henrys per unit length (H/m or H/ft); C' = Distributed capacitance, in farads per unit length (F/m or F/ft). 1-35

Figure 1-34. Equivalent circuit of a two-conductor transmission line.

2 Figure Equivalent circuit of a two-conductor transmission line. Step (Transient) Response Method The velocity of propagation of a signal in a transmission line can be measured by using the step response method. This methods requires that a step generator and a high-impedance oscilloscope probe be both connected to the sending end of the line, using a bridging connection, as Figure 1-35 shows. The receiving end of the line is left unconnected [impedance of the load in the open-circuit condition ( )]. 1-36

3 Figure Measuring the velocity of propagation of a signal by using the step response method. The signal propagation through the line is described below (refer to Figure 1-35). At time t = 0, the step generator launches a fast-rising, positive-going voltage, V I, into the line. The rising edge of V I is called a step, or transient. This step is incident because it comes from the generator and is going to travel down the line toward a possibly reflecting load. Incident step V I propagates at a certain velocity, v P, along the line. It arrives at the receiving end of the line after a certain transit time, T. There its level has decreased by a certain amount due to the resistance of the line. Since the impedance of the load at the receiving end of the line is in the opencircuit condition ( ), it does not match the characteristic impedance of the line. This impedance mismatch causes the incident step to be reflected back toward the generator. The reflected step, V R, gets back to the step generator after a time equal to twice the transit time, 2T. 2T is synonymous with round-trip time, or back-and-forth trip time. The signal at the sending end of the line, as a function of time, is the step response signal. As Figure 1-36 shows, this signal is the algebraic sum of the incident step V I and reflected step V R. Step V R is superimposed on step V I, and is separated by a time 2T from the rising edge of V I. 1-37

Figure 1-36. Voltage at the sending end of the open-circuit line (step response signal).

4 Figure Voltage at the sending end of the open-circuit line (step response signal). By measuring time 2T on the oscilloscope screen, the velocity of propagation of a signal in a transmission line, v P, can be determined, using the formula below. where v P = Velocity of propagation (m/s or ft/s); l = Length of the line (m or ft); 2T = Round-trip time, i.e. time taken for the launched step to travel from the generator to the line receiving end and back again to the generator (s). Transmission lines that are lossy, and whose series losses are predominant, will appear as a simple RC network (resistor-capacitor network) for a short time following the launching of a voltage step, as Figure 1-37 shows. This is due to the highfrequency components contained in the voltage step. 1-38

5 Figure Lossy line with predominant series losses. The time constant,, of the RC network (not to be confused with the transit time T) is determined by constants R s and C, which are themselves derived from the distributed series resistance, R' s, series inductance, L', and parallel capacitance, C', of the line. Consequently, the time constant of the RC network is independent of the length of the line. In that case, the incident and reflected steps observed at the sending end of the line will first rise to a certain level, and then increase exponentially at a rate determined by the time constant of the RC network, as Figure 1-38 shows. This does not prevent the measurement of time 2T on the oscilloscope screen for calculation of the velocity of propagation. However, it is clear that lossy lines cause a degradation in the rise time of voltage steps. 1-39

6 Figure Incident and reflected steps at the sending end of a lossy line with predominant series losses. Velocity Factor The velocity of propagation of a signal in a transmission line is usually expressed as a percentage of the velocity of light in free space. This percentage is called the velocity factor, v F. For example, a transmission line with a v F of 66% will transmit signals at about 66% of the velocity of light. where v F = Velocity factor (%); v P = Velocity of propagation in the transmission line (m/s or ft/s); c = Velocity of light in free space (about m/s, or ft/s). 1-40

7 In the case of coaxial cables, the velocity factor varies from about 66 to around 85%, as indicated in Table 1-1. TYPE OF COAXIAL CABLE VELOCITY FACTOR, v F (%) RG-8 66 RG RG RG RG RG LMR RG-8X 84 LMR Table 1-1. Velocity factor of various types of coaxial cables. TRANSMISSION LINES A and B of the circuit board are RG-174 coaxial cables. Consequently, they have a theoretical velocity factor, v F, of 66%. Relative Permittivity (Dielectric Constant) The velocity of propagation of a signal in a transmission line is determined mainly by the permittivity of the dielectric material used to construct the line. Permittivity is a measure of the ability of the dielectric material to maintain a difference in electrical charge over a given distance. The permittivity of a particular dielectric material is normally expressed in relation to that of vacuum. This ratio is called relative permittivity, or dielectric constant. When the velocity of propagation in a transmission line is known, the relative permittivity of the dielectric material used to construct that line, r, can be determined by using the equation below. where r = Relative permittivity (dielectric constant); c = Velocity of light in free space ( m/s, or ft/s); v P = Velocity of propagation (m/s or ft/m). The formula for calculating relative permittivity indicates that a higher velocity of propagation indicates a lower relative permittivity, since the velocity of light is a constant value. Table 1-2 lists the relative dielectric constants of various materials. 1-41

8 MATERIAL RELATIVE PERMITTIVITY, r VELOCITY FACTOR, v F (%) Vacuum Air Teflon Polyethylene Polystyrene Polyvinyl chloride (PVC) Nylon Table 1-2. Relative dielectric constant of various materials. Procedure Summary In this procedure section, you will measure the velocity of propagation of voltage steps in the transmission lines of the circuit board. Based on the measured velocity, you will determine the relative permittivity of the dielectric material used to construct these lines. PROCEDURE Measuring the Velocity of Propagation 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 1-39, connect the STEP GENERATOR 50- BNC output to the BNC connector at the sending end of TRANSMISSION LINE A. Leave the BNC connector at the receiving end of TRANSMISSION LINE A unconnected (open-circuit). Then, connect the STEP GENERATOR 100- BNC output to the trigger input of the oscilloscope, using a coaxial cable. Finally, using an oscilloscope probe, connect channel 1 of the oscilloscope to the 0-meter (0-foot) probe turret at the sending end of TRANSMISSION LINE A. Make sure to connect the ground conductor of the probe to the associated 0-meter (0-foot) shield turret. Note: When connecting an oscilloscope probe to one of the five probe turrets of a transmission line, always connect the ground conductor of the probe to the associated (nearest) coaxial-shield turret. This will minimize noise in the observed signal due to the parasitic inductance introduced by undesired ground return paths. 1-42

9 Figure Measuring the velocity of propagation of voltage steps through TRANSMISSION LINE A. G 3. Make the following settings on the oscilloscope: Channel 1 Mode Normal Sensitivity V/div Input Coupling DC Time Base s/div Trigger Source External Level V Input Impedance M or more Note: Throughout this course, the oscilloscope settings for the time base and channel sensitivity are given as a starting point for guidance and may be modified as necessary to obtain the maximum possible measurement accuracy. 1-43

G 4. On the oscilloscope screen, observe the step response signal at the sending end of TRANSMISSION LINE A. This signal corresponds to the step response of TRANSMISSION LINE A.

10 G 4. On the oscilloscope screen, observe the step response signal at the sending end of TRANSMISSION LINE A. This signal corresponds to the step response of TRANSMISSION LINE A. Does the reflected step appear superimposed on the incident step, a certain time interval separating these two steps, as Figure 1-40 shows? G Yes G No Figure Incident and reflected steps at the sending end of TRANSMISSION LINE A. G 5. Observe that the incident and reflected steps first rise to a certain level, and then increase exponentially, as the voltage across a capacitor charging through a series resistor. Does this indicate that TRANSMISSION LINE A have predominant series losses? G Yes G No 1-44

11 G 6. When the incident step arrives at the receiving end of TRANSMISSION LINE A, it is reflected back toward the sending end because a. TRANSMISSION LINE A is not terminated by a load impedance equal to the Thevenin impedance of the STEP GENERATOR. b. TRANSMISSION LINE A is not terminated by a load impedance equal to its characteristic impedance. c. the receiving end of TRANSMISSION LINE A is open-circuit, causing the characteristic impedance of the line to be infinite. d. the Thevenin impedance of the STEP GENERATOR is not equal to the characteristic impedance of TRANSMISSION LINE A. G 7. Decrease the oscilloscope time base to 0.05 s/div. On the oscilloscope, measure the round-trip time, 2T, separating the rising edge of the incident step from the rising edge of the reflected step, as Figure 1-41 shows. This is the time required for the step launched by the step generator to travel to the receiving end of TRANSMISSION LINE A and then back to the step generator. 2T = 10 9 s Figure Measuring time 2T. 1-45

12 G 8. Based on the round-trip time, 2T, measured in the previous step, and on a line length, l, of 24 meters (78.7 feet), calculate the velocity of propagation, v P, through the line. v P = 10 8 m/s or 10 8 ft/s G 9. Express the velocity of propagation, v P, obtained in the previous step as a percentage of the velocity of light, or velocity factor, v F, using the formula below. Your result should be near the theoretical value of 66% for a RG-174 coaxial cable (type of cable used for TRANSMISSION LINES A and B of your circuit board). where c = velocity of light in free space ( m/s, or ft/s) v F = % Determining the Relative Permittivity (Dielectric Constant) G 10. Based on the velocity of propagation v P obtained in step 8, determine the relative permittivity, r of the dielectric material used to construct the RG-174 coaxial cables used for TRANSMISSION LINES A and B. The result should be quite near the theoretical value of 2.25 for polyethylene (dielectric material used to construct the RG-174 coaxial cables used for TRANSMISSION LINES A and B). where c = velocity of light in free space ( m/s, or ft/s) r = Effects that a Change in Line Length Has on the Round-Trip Time (2T) G 11. As Figure 1-42 shows, increase the length of the line from 24 to 48 meters (78.7 to feet) through end-to-end connection of TRANSMISSION LINEs A and B. To do so, connect the BNC connector at the receiving end of TRANSMISSION LINE A to the BNC connector at the sending end of TRANSMISSION LINE B, using a short coaxial cable. Leave the 1-46

BNC connector at the receiving end of TRANSMISSION LINE B unconnected (open-circuit). Figure 1-42. Increasing the length of the line from 24 to 48 meters (78.7 to 157.4 feet). G 12.

13 BNC connector at the receiving end of TRANSMISSION LINE B unconnected (open-circuit). Figure Increasing the length of the line from 24 to 48 meters (78.7 to feet). G 12. Set the oscilloscope time base to 0.2 s/div. Observe that the round-trip time, 2T, separating the rising edge of the incident step from the rising edge of the reflected step has doubled, as Figure 1-43 shows. 1-47

Figure 1-43. The round-trip time, 2T, separating the rising edges of the incident and reflected steps has doubled. Time 2T has doubled because the a.

14 Figure The round-trip time, 2T, separating the rising edges of the incident and reflected steps has doubled. Time 2T has doubled because the a. velocity of propagation has decreased by a factor of two. b. length of the line has doubled. c. relative permittivity has doubled. d. characteristic impedance of the line has doubled. G 13. On the oscilloscope screen, observe that the incident and reflected steps first rise to a certain level, and then increase exponentially as they did with the shorter 24-meter (78.7-foot) long line. These steps increase at the same rate as they did with the shorter length. This occurs because the time constant of the series RC network temporarily presented by the line is determined by the a. characteristic impedance, which is a constant. b. total series resistance and parallel capacitance of the entire line. c. series resistance, parallel capacitance, and series inductance of the line per unit length. d. velocity factor, which is a constant. G 14. Turn off the Base Unit and remove all the connecting cables and probes. 1-48

15 CONCLUSION The velocity of propagation of a signal in a transmission line can be measured by using the step response method: a fast-rising (transient) step is launched into the line. The time required for this step to travel from the generator to the receiving end of the line and then back to the generator is measured. This time, 2T, permits calculation of the velocity of propagation. 2T is synonymous with round-trip time, or back-and-forth trip time. The velocity of propagation of a signal in a transmission line is only a percentage of the velocity of light in free space. The velocity of propagation in a transmission line, when expressed as a percentage of the velocity of light in free space, is called the velocity factor. The velocity of propagation in a transmission line is determined mainly by the relative permittivity (dielectric constant) of the dielectric material used to construct that line. The lower the relative permittivity is, the higher the velocity of propagation will be. REVIEW QUESTIONS 1. In a transmission line, a signal travels at a velocity a. that is null if the impedance of the load at the receiving end of the line is in the open-circuit condition ( ). b. that is directly proportional to the relative permittivity of the dielectric material used to construct the line. c. that usually increases as the diameter of the line conductors is decreased. d. relatively less than m/s, or ft/s. 2. The permittivity of the dielectric material used to construct a transmission line a. is a measure of the ability of the material to maintain a difference in propagation velocity over a given distance. b. is called dielectric constant, or relative permittivity, when expressed in relation to the permittivity of vacuum. c. is usually expressed as a percentage of the velocity of light in free space. d. does not determine the velocity factor of that line. 3. The velocity of propagation of a signal in a transmission line can be determined by using a. a high-impedance oscilloscope probe connected to the sending end of the line and a step generator connected to the receiving end of the line. b. a simple formula, if the time required for a voltage step to travel to the receiving end of the line and back to the generator is known. c. the step response method, provided that the load impedance perfectly matches the characteristic impedance of the line. d. a step generator and a high-impedance oscilloscope connected to the receiving end of the line. 1-49

16 4. When the step response method is used, the signal observed on the oscilloscope at the sending end of the line consists of a. a reflected step superimposed on an incident step, the rising edge of the incident step being of higher voltage than that of the reflected step due to attenuation. b. an incident step superimposed on a reflected step, the rising edge of the incident step being of higher voltage than that of the reflected step due to attenuation. c. a reflected step superimposed on an incident step, the time separating these steps being directly proportional to the velocity of propagation. d. several incident steps, the time separating two successive incident steps being determined by the length of the line. 5. When a voltage step is launched into a lossy line whose series losses are predominant, a. the high-frequency components contained in the voltage steps make the line temporarily appear as a simple RC network. b. the incident and reflected steps will first rise to a certain level and then decrease exponentially. c. it is not possible to measure the time separating the incident and reflected steps. d. the line will appear as a simple LC network from the perspective of the load. 1-50

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

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