Exercise 2-1 Determining Characteristic Impedance and Velocity EXERCISE OBJECTIVES Upon completion of this exercise, you will know how to measure the distributed capacitance and distributed inductance of a line. You will use the measured values to determine the characteristic impedance of the line and the velocity of propagation of the signals along the line. DISCUSSION Relationship Between the Characteristic Impedance of a Line and its Distributed Parameters The characteristic impedance, Z 0, of a line is an intrinsic property of the line. Because of this, Z 0 is determined by the geometrical and physical characteristics of the line not by the length of the line. Z 0, therefore, is of constant value, whether the line is short, long, or infinite. The physical characteristics that determine Z 0, which are the diameter of the conductors, the relative spacing between the conductors, and the insulating material used, also determine the value of the distributed parameters of the line. Consequently, a relationship exists between the value of Z 0 and the value of the distributed parameters. Figure 2-6. Distributed parameters of a line. 2-7
When R' S is negligible and R' P is relatively high (G' is relatively low), and the frequency of the carried signals, f, is relatively high (such that 2 fl' R' S and 2 fc' R' P ), the reactive component of Z 0 can be disregarded and Z 0 be considered as purely resistive. In this condition, it can be demonstrated that where Z 0 = Characteristic impedance ( ); 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). Measuring the Distributed Capacitance and Distributed Inductance of a Line The theoretical values of C' and L' are normally specified by the line s manufacturer. However, these values can also be measured by using the step response method. To do so, a step generator and a high-impedance oscilloscope probe are both connected to the sending end of the line, using a bridging connection. Distributed Capacitance To measure the distributed capacitance, the resistive component of the step generator impedance, R TH, is set to a value that is much greater than the characteristic impedance of the line, while the receiving end of the line is left unconnected (open-circuit condition), as Figure 2-7 shows. This creates an impedance mismatch at both the sending and receiving ends of the line, and permits a step response signal with measurable time constant. Figure 2-7. Measuring the distributed capacitance by using the step response method. 2-8
At time t = 0, the step generator launches an incident voltage step, V I, into the line. This step arrives at the receiving end of the line after a certain transit time, T. There, it is reflected back toward the generator due to the impedance mismatch between the line and the load. The reflected step, V R, which is of positive polarity, gets back to the generator at time t = 2T (twice the transit time, equal to the round-trip time). There, it is re-reflected down the line due to the impedance mismatch between the line and the generator. This phenomenon continues for a certain time, the reflected step bouncing back and forth on the line, and becoming lower and lower in level. Consequently, the step response signal is the algebraic sum of the incident and successive reflected steps. This causes the rising edge of each pulse in this signal to look like the step response of an RC series circuit, as Figure 2-8 shows. Figure 2-8. Step response of the RC circuit made by R TH and C T. In fact, the overall shape of the successive steps that form the rising edge of each pulse form an exponential transition having a time constant,. The time constant,, is determined by the resistive component of the generator impedance, R TH (neglecting the resistance of the line, which is low by comparison), and the total line capacitance, C T. Consequently, the time constant is where = Time constant of the RC circuit (s); R TH = Resistive component of the step generator Thevenin impedance ( ); C T = Total line capacitance, equal to the sum of the distributed capacitances along the entire line length (F, i.e. s/ ). 2-9
The difference between the final and initial voltages of the exponentially-rising edge of the pulse is called voltage excursion, V, as Figure 2-9 shows: At time t = 5, the voltage approximately reaches its final value. At time t = 0.69, the voltage reaches 50% of the voltage excursion, as the figure shows. By measuring the time required for the voltage to increase by 50%, that is, from the initial level to V/2 on the oscilloscope screen, and then dividing this time by 0.69, the time constant,, can be determined and C T be calculated: Figure 2-9. At time t = 0.69, the voltage reaches 50% of its excursion ( V/2). Once C T is known, the distributed capacitance of the line is determined simply by dividing C T by the length of the line: where C' = Distributed capacitance of the line (F/m or F/ft); C T = Total line capacitance (F); l = Length of the line (m or ft). 2-10
Distributed Inductance A similar method is used to measure the distributed inductance of the line, except that this time, the resistive component of the step generator impedance, R TH, is set to a value that is much lower than the characteristic impedance of the line, while the impedance of the load at the receiving end of the line is placed in the short-circuit condition (0 ), as Figure 2-10 shows. This creates an impedance mismatch at both the sending and receiving ends of the line, and provides a step response signal with measurable time constant. Figure 2-10. Measuring the distributed inductance by using the step response method. At time t = 0, the step generator launches an incident step, V I, into the line. This step arrives at the receiving end of the line after a certain transit time, T. There it is reflected back toward the generator due to the impedance mismatch. The reflected step, V R, which is of negative polarity, gets back to the generator at time t = 2T (twice the transit time, i.e., equal to the round-trip time). There, it is re-reflected down the line due to the impedance mismatch. This phenomenon continues for a certain time, the reflected step bouncing back and forth on the line, and becoming lower and lower in level. Consequently, the step response signal is the algebraic sum of the incident and successive reflected steps. This causes each pulse in this signal to look like the step response of an RL circuit, as Figure 2-11 shows. 2-11
Figure 2-11. Step response of the RL circuit made by R TH and L T. In fact, the overall shape of the successive steps that form the falling part of each pulse form an exponential transition having a time constant,. The time constant is determined by the resistive component of the generator impedance, R TH, and the total line inductance, L T. Consequently, the time constant is where = Time constant of the RL circuit (s); L T = Total line inductance, equal to the sum of the distributed inductances along the entire line length (H, i.e. s); R TH = Resistive component of the step generator Thevenin impedance ( ). 2-12
The difference between the initial and final voltages of the exponentially-decreasing section of the pulse is the voltage excursion, V, as Figure 2-12 shows: At time t = 5, the voltage approximately reaches its final value. At time t = 0.69, the voltage reaches 50% of the voltage excursion, V. At time t = 1.4, the voltage reaches 75% of the voltage excursion, V. By measuring the time required for the initial voltage to decrease by, for example, 50% of V and then dividing this time by 0.69, or to decrease by 75% of V and then dividing this time by 1.4, the time constant,, can be determined and L T be calculated: Figure 2-12. At time t = 0.69, the voltage reaches 50% of its excursion (0.5 V). At time t = 1.4, the voltage reaches 75% of its excursion (0.75 V). 2-13
Once L T is known, the distributed inductance of the line is determined simply by dividing L T by the length of the line: where L' = Distributed inductance of the line (H/m or H/ft); L T = Total line inductance (H); l = Length of the line (m or ft). Calculating Characteristic Impedance and Velocity of Propagation from the Values Measured for C' and L' Once the distributed capacitance, C', and distributed inductance, L', of a line have been measured, the characteristic impedance, Z 0 of this line can be calculated by using the formula earlier stated: Moreover, the velocity of propagation of the signals, v P, in this line can also be calculated, using the equation below: where v P = Velocity of propagation (m/s or ft/s); L' = Distributed inductance of the line (H/m or H/ft); C' = Distributed capacitance of the line (F/m or F/ft). To summarize, the characteristic impedance and velocity of propagation can be calculated by using the values measured for L and C when 2 fl' R' S and 2 fc' R' P. Procedure Summary In this procedure section, you will measure the distributed capacitance and distributed inductance of a line. You will then use the measured values to calculate the characteristic impedance of the line and the velocity of propagation of the signals along the line. PROCEDURE 2-14
Measuring the Distributed Capacitance 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 2-13, connect the STEP GENERATOR 500- output to the sending end of TRANSMISSION LINE A, using a coaxial cable. Leave the BNC connector at the receiving end of TRANSMISSION LINE A unconnected. Figure 2-13. Measuring the distributed capacitance. Connect the STEP GENERATOR 100- output to the trigger input of the oscilloscope, using a coaxial cable. 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 shield turret. 2-15
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. The connections should now be as shown in Figure 2-13. G 3. Make the following settings on the oscilloscope: Channel 1 Mode........................................ Normal Sensitivity.................................... 0.2 V/div Input Coupling.................................... DC Time Base....................................... 5 s/div Trigger Source...................................... External Level........................................... 0.3 V Input Impedance........................... 1 M or more G 4. In the step response signal, note that the rising edge of the pulses increases exponentially, going from an initial voltage of about 0 V to a final maximum voltage, like the voltage across a capacitor charging through a series resistor (see Figure 2-14). The signal on the oscilloscope is the step response of the RC circuit made by a. the resistive component of the STEP GENERATOR impedance and the total line inductance. b. the reactive component of the STEP GENERATOR impedance and the total line capacitance. c. the resistive component of the STEP GENERATOR impedance and the total line capacitance. d. the resistive component of the STEP GENERATOR impedance and the distributed line capacitance. 2-16
Figure 2-14. Step response of the series RC circuit made by R TH and C T. G 5. Decrease the oscilloscope time base to 2 s/div to better see the exponentially-rising edge of a pulse in the step response signal, as Figure 2-15 shows. Measure the voltage excursion, V, of the rising edge of the pulse: V = V 2-17
Figure 2-15. Measuring the voltage excursion, V. G 6. Divide the measured voltage excursion, V, by 2. Record below this voltage, V/2. V/2 = V G 7. Decrease the oscilloscope time base to 0.5 s/div. Observe that the first reflected steps that form the exponentially-rising edge of the pulse are distinguishable, as Figure 2-16 shows. Note that the reflected steps are somewhat rounded, due to a low rise time caused by attenuation and dispersion of their high-frequency components. Measure the time required for the exponentially-rising edge of the pulse to rise from 0 V to V/2 as accurately as possible. This time, t 50%, corresponds to 0.69. t 50% = 10-9 s 2-18
Figure 2-16. Measuring t 50% = 0.69. G 8. Using t 50% obtained in the previous step, calculate the time constant,, of the RC circuit: = 10-9 s G 9. Using the time constant,, obtained in the previous step, calculate the total line capacitance, C T. (Consider R TH to be equal to 500 ): C T = 10-9 F 2-19
G 10. Using the total line capacitance, C T, obtained in the previous step, calculate the distributed capacitance of the line, C'. Consider the length of the line, l, to be 24 m (78.7 ft). C = 10-12 F/m Is C near the manufacturer s specified value of 101 10-12 (30.8 10-12 F/ft)? F/m G Yes G No Measuring the Distributed Inductance G 11. Modify the connections as indicated below in order to be able to measure the distributed inductance (refer to Figure 2-17). Remove the coaxial cable between the STEP GENERATOR 500- output and the sending end of TRANSMISSION LINE A. Connect the STEP GENERATOR 5- output to the sending end of TRANSMISSION LINE A, using a coaxial cable. Connect the BNC connector at the receiving end of TRANSMISSION LINE A to the BNC connector at the LOAD-section input. 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 will place the impedance of the load at the receiving end of TRANSMISSION LINE A in the short-circuit condition (0 ). Leave the STEP GENERATOR 100- output connected to the trigger input of the oscilloscope. Leave the 0-meter (0-foot) probe turret at the sending end of TRANSMISSION LINE A connected to channel 1 of the oscilloscope via the oscilloscope probe. The connections should now be as shown in Figure 2-17. 2-20
Figure 2-17. Measuring the distributed inductance. G 12. Make the following settings on the oscilloscope: Channel 1 Mode......................................... Normal Sensitivity.................................... 0.2 V/div Input Coupling..................................... DC Time Base....................................... 5 s/div Trigger Source...................................... External Level.......................................... 0.5 V Input Impedance........................... 1 M or more G 13. On the oscilloscope screen, observe that the pulses first peak very quickly to an initial voltage, and then decrease exponentially to a final voltage, which corresponds to the step response of an RL circuit (see Figure 2-18). 2-21
Figure 2-18. Step response of the RL circuit made by R TH and L T. The signal on the oscilloscope is the step response of the RL circuit made by a. the resistive component of the STEP GENERATOR impedance and the total line inductance. b. the reactive component of the STEP GENERATOR impedance and the distributed line inductance. c. the resistive component of the STEP GENERATOR impedance and the total line capacitance. d. the reactive component of the STEP GENERATOR impedance and the total line inductance. 2-22
G 14. Decrease the oscilloscope time base to 0.5 s/div to better see the section of the pulse where the voltage is decreasing exponentially, as Figure 2-19 shows. Note that the first reflected steps that form this section are distinguishable. These steps are somewhat rounded, due to some attenuation and dispersion of their high-frequency components. Figure 2-19. First reflected steps in the pulse section where the voltage is exponentially decreasing. Further decrease the oscilloscope time base to 0.1 s/div. As Figure 2-20 shows, measure the voltage (height) of the rising edge of the pulse at midpoint between the beginning and the end of the rounded part (first reflected step). This approximately corresponds to the initial level of the exponentially-decreasing voltage, V INIT. V INIT. = V G 15. Set the oscilloscope time base to 2 s/div. As Figure 2-21 shows, measure the final level of the exponentially-decreasing voltage, V FIN.. V FIN. = V 2-23
Figure 2-20. Measuring the initial level of the exponentially-decreasing voltage. Figure 2-21. Measuring the final level of the exponentially-decreasing voltage. 2-24
G 16. Using the initial and final levels of the exponentially-decreasing voltage measured in the previous steps, calculate the excursion of this voltage, V. V = V G 17. Multiply the voltage excursion, V, by 0.75. Note: Due to the shape and peaks of the step response signal here, the time constant,, will be determined through measurement of the time required for the initial voltage to decrease by 75% (rather than by 50%, the usual method). This will provide a better accuracy of measurement. 0.75 V = V G 18. Set the oscilloscope time base to 0.2 s/div. Referring to Figure 2-22, measure the time required for the exponentially-decreasing voltage to decrease from the initial level (V INIT. ) to V INIT. 0.75 V as accurately as possible. This time, t 75%, corresponds to 1.4. t 75% = 10-6 s 2-25
Figure 2-22. Measuring t 75% = 1.4. G 19. Using the value measured for t 75% in the previous step, calculate the time constant,, of the RL circuit. = 10-6 s G 20. Using the time constant,, obtained in the previous step, calculate the total line inductance, L T. (Consider R TH to be equal to 5 ). L T = 10-6 H (or s) 2-26
G 21. Using the total line inductance, L T, obtained in the previous step, calculate the distributed inductance of the line, L. Consider the length of the line, l, to be 24 m (78.7 ft). L = 10-7 H/m The obtained value should theoretically be near of the manufacturer s specified value of 2.52 10-7 H/m (7.7 10-8 H/ft). However, since the obtained value is dependent upon the accuracy of the measurements made on the oscilloscope (i.e. the voltage excursion and the time constant) and on the rounding accuracy used for the calculations, this value may differ fairly from the manufacturer's value. Calculating Characteristic Impedance and Velocity of Propagation G 22. Using the distributed inductance, L, and distributed capacitance, C, obtained in the previous section, calculate the characteristic impedance, Z 0, of the line by using the formula below. Z 0 = The obtained value for Z 0 should theoretically be quite near the characteristic impedance of 50 specified for the RG-174 coaxial cable used as TRANSMISSION LINE A. However, since the obtained value is dependent upon the accuracy of the measurements and calculations used to find C' and L', Z 0 may differ fairly from the manufacturer's value of 50. G 23. Using the distributed inductance, L, and distributed capacitance, C, obtained in the previous section, calculate the velocity of propagation of the signals, v P, using the formula below. v P = m/s (or ft/s) The obtained value for v P should theoretically be near the theoretical velocity of propagation of 1.96 10 8 m/s (6.43 10 8 ft/s) for the RG-174 coaxial cable used as TRANSMISSION LINE A. However, since the obtained value is dependent upon the accuracy of the measurements and calculations used to find C' and L', v p may differ fairly from the manufacturer's value. 2-27
G 24. Turn off the Base Unit and remove all the connecting cables and probes. CONCLUSION The characteristic impedance and the distributed parameters of a line are both related to geometrical and physical properties of the line. Consequently, the values of Z 0 and of each distributed parameter are constant, regardless of how long the line may be. When R' S is negligible and R' P is very high (G' is very low), and the frequency of the carried signals is relatively high, Z 0 can be considered as purely resistive. In that case, the step response method can be used to measure the distributed capacitance, C', and distributed inductance, L', of the line. To do so, an impedance mismatch is created at both the sending and receiving ends of the line, in order to create multiple reflections of the incident step and obtain a step response signal with measurable time constant. The measured time constant permits calculation of C' or L'. Once C' and L' are known, Z 0 and v P can be calculated by using simple formulas. This occurs because Z 0 and v P are related to the distributed C' and L' of the line. REVIEW QUESTIONS 1. When measuring the distributed capacitance of a line by using the step response method, a. the resistive component of the step generator is set to a value that is much greater than Z 0, while the impedance of the load at the receiving end of the line is placed in the short-circuit condition (0 ). b. the resistive component of the step generator is set to a value that is much lower than Z 0, while the receiving end of the line is in the open-circuit condition. c. an impedance mismatch is created at both the sending and receiving ends of the line. d. the impedance of the step generator and that of the load must both be equal to Z 0. 2. When R' S is negligible and R' P is very high (G' is very low), and the frequency of the carried signals, f, is relatively high (such that 2 fl' R' S and 2 fc' R' P ), which is a common condition, a. the resistive and reactive components of Z 0 are perfectly equal. b. the resistive component of Z 0 can be disregarded. c. Z 0 can be considered as purely reactive. d. Z 0 can be considered as purely resistive. 2-28
3. When measuring the distributed inductance of a line by using the step response method, a. the resistive component of the step generator is set to a value that is much lower than Z 0, while the impedance of the load at the receiving end of the line is placed in the short-circuit condition (0 ). b. the resistive component of the step generator is set to a value that is much greater than Z 0, while the impedance of the load at the receiving end of the line is placed in the short-circuit condition (0 ). c. the resistive component of the step generator is set to a value that is much lower than Z 0, while the impedance of the load at the receiving end of the line is placed in the open-circuit condition ( ). d. the impedance of the step generator and that of the load must both be equal to Z 0. 4. Once C' and L' have been determined by using the step response method, it is possible to calculate a. Z 0 by extracting the square root of L' divided by C'. b. v P by extracting the square root of L' divided by C'. c. Z 0 by multiplying C' by L'. d. v P by dividing L' by C'. 5. When 2 fl' R' S and 2 fc' R' P, the velocity of propagation is approximately equal to a. the reciprocal of the square root of R' S times R' P. b. the reciprocal of the square root of L' times C'. c. the square root of R' S divided by R' P. d. the reciprocal of C' divided by L'. 2-29
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