Resonant and Nonresonant Lines. Input Impedance of a Line as a Function of Electrical Length

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Dominick Banks
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1 Exercise 3-3 The Smith Chart, Resonant Lines, EXERCISE OBJECTIVES Upon completion of this exercise, you will know how the input impedance of a mismatched line varies as a function of the electrical length of the line. You will know what a Smith Chart is, and how it is used to determine the input impedance of a line that is not terminated by its characteristic impedance. You will know how quarterwavelength ( /4) line sections can be used for impedance transformation and matching. DISCUSSION Resonant and Nonresonant Lines When the input impedance of a line, Z IN, is equal to the characteristic impedance Z 0 and the load impedance Z L, there are no standing waves on the line. Changing the frequency of the generator will not change the input impedance of the line. In this case, the maximum possible power is transmitted to the load, and the line is said to be flat, or nonresonant. When the impedance of the load Z L is not perfectly equal to the characteristic impedance of the line, the line may appear like a parallel or resonant circuit, or as an off-resonance or reactive circuit to the generator input, depending on the electrical length of the line. Consequently, the input impedance of the line will vary as a function of the electrical length and, therefore, of the frequency of the carried signal. In this case, the line is said to be resonant. Input Impedance of a Line as a Function of Electrical Length It is easy to determine the input impedance of the line for specific load impedances, if the electrical length is an odd multiple of /4: When the line is open-ended, the input impedance is minimum for lengths that are odd multiples of /4; it is maximum for lengths that are even multiples of /4. When the line is short-ended, the input impedance is maximum for lengths that are even multiples of /4; it is minimum for lengths that are odd multiples of /4. Very often, however, it is necessary to know the value of the input impedance for line lengths that are not exact multiples of /4. Figure 3-31 shows how the input impedance, Z IN, of an open-ended line varies as a function of electrical length. Z IN varies according to a specific pattern that repeats at every half of wavelength. The figure shows that 3-51

2 Z IN is purely resistive at resonance, that is, for lengths that are either odd or even multiples of /4. Thus, Z IN is minimum for odd multiples of /4; conversely, Z IN is maximum for even multiples of /4. Z IN is capacitive (R - jx C ) for lengths that are between 0 and /4, or between an even /4 and the next longer odd /4; Z IN is inductive (R + jx L ) for lengths that are between an odd /4 and the next longer even /4. Figure Input impedance as a function of electrical length for open-ended lines. Figure 3-32 shows how the input impedance, Z IN, of a short-ended line varies as a function of electrical length. Z IN varies according to a specific pattern that repeats at every half of wavelength. The figure shows that Z IN is purely resistive at resonance, that is, for lengths that are either odd or even multiples of /4. Thus, Z IN is minimum for even multiples of /4. Conversely, Z IN is maximum for odd multiples of /4. Z IN is inductive (R + jx L ) for lengths that are between 0 and /4, or between an even /4 and the next longer odd /

3 Z IN is capacitive (R - jx C ) for lengths that are between an odd /4 and the next longer even /4. Figure Input impedance as a function of electrical length for short-ended lines. If, for example, the electrical length of an open-ended line is increased from /2 to nearly, but less that 3 /4, the input impedance of the line will change from purely resistive to capacitive. The Smith Chart The Smith Chart is a graphical computation tool developed by Dr. P.H. Smith in This chart greatly simplifies evaluation of transmission line parameters, such as the VSWR caused by a given load and the impedance at any point along a line for various line lengths and various load impedances. 3-53

4 Figure The Smith Chart. 3-54

5 Figure 3-33 shows a Smith Chart. It consists of a set of impedance coordinates used to represent impedance at any point along a line in rectangular form: R ± jx: R is the purely resistive component of the impedance; ± jx is the reactive component (reactance) of the impedance. All resistance and reactance values on the chart are normalized to the characteristic impedance of the line, Z 0. Resistance values correspond to R/Z 0. Reactance values correspond to ± jx/z 0. This allows the chart to be used with transmission lines of any characteristic impedance. As Figure 3-34 shows, the "R" coordinates are a set of circles tangent at the right end of the horizontal centerline of the chart. The point of tangency is called the common point, or infinity point. Each circle represents a constant resistance (R) value: the largest circle, which outlines the chart, corresponds to a constant R value of 0 ; the smaller circles correspond to higher, constant R values. Figure Circles of constant resistance. 3-55

6 Figure Circles of constant R/Z 0 values. 3-56

7 As Figure 3-35 shows, the horizontal centerline of the chart represents pure resistance, or zero reactance. The normalized values for R/Z 0 are marked all along this line. These values range between 0 and 50. The figure shows the circles for constant R/Z 0 values of 0, 0.3, 1.0, 2.0, and 5.0 emphasized. As Figure 3-36 shows, the "± jx" coordinates are a set of arcs starting from the common, or infinity point. Each arc represents a constant reactance value. The upper half of the chart contains coordinates for inductive reactance (+ jx L ). Thus, each arc curving upward represents a constant inductive reactance. The lower half of the chart contains coordinates for capacitive reactance (- jx L ). Thus, each arc curving backward represents a constant capacitive reactance. Figure Arcs of constant reactance. 3-57

8 Figure Arcs of constant jx/z 0 values. 3-58

9 As Figure 3-37 shows, the normalized values for ± jx/z 0 are marked on the inner scale just beneath the 0- R circle of the chart. These values range between 0 and 50. The figure shows the arcs for constant jx/z 0 values of +0.4, +1.2, +3.0, 0.6, 1.0, and 2.0 emphasized. 3-59

10 The Smith Chart and Its Applications This section consists of examples indicating how to perform various measurements with the Smith Chart. The given examples are for a lossless line (and, therefore, a constant VSWR throughout the line). Reading Off an Impedance Value Referring to Figure 3-38, find the normalized impedance corresponding to the point marked "A" on the Smith Chart. Then, convert this impedance to actual impedance, given a characteristic impedance of Point A is at the intersection of the 1.2 resistance circle and the 0.8 capacitive reactance arc. The normalized impedance for this point is, therefore, j Calculate the actual impedance as follows: 3-60

11 Figure Reading off a normalized impedance value. 3-61

12 Plotting a Normalized Impedance The following impedance is measured at a particular point along a 50- line: 30 + j10. Plot this impedance on the Smith Chart of Figure First, normalize the impedance as follows: 2. From the left-hand extremity [zero (0) point] of the horizontal centerline, move to the right to find the 0.6 resistance circle, as shown on the chart. 3. Move up around the 0.6 resistance circle to the point intersecting the 0.2 inductive reactance arc. This point, marked "A" on the chart, represents the normalized impedance j

13 Figure Plotting a normalized impedance. 3-63

14 Drawing the VSWR Circle of a Line A lossless line can be represented by a circle having its origin at the center point of the Smith Chart, when the load impedance Z L is known. This circle is called the VSWR circle. For example, draw the VSWR circle of a 50- line terminated by a load impedance Z L = 65 + j20 on the Smith Chart of Figure Normalize the load impedance and then plot this impedance on the Smith Chart (point marked "A" on the chart). 2. Using a compass, draw a circle having its origin at the center point of the chart [the mark "1.0" (i.e., 1 + j0)] on the horizontal centerline], and with a radius such that the circle crosses the load impedance (point A). This circle is a constant VSWR circle: all impedances on this circle would produce the same VSWR. 3. Read the VSWR from this circle at the point where it cuts the horizontal centerline on the right side of this line (VSWR = 1.55). Another way of determining the VSWR and the corresponding db value is by using the SWR scale in the RADIALLY SCALED PARAMETERS section below the Smith Chart, as Figure 3-40 shows. To do so, first set the compass for the distance from the center point of the Smith Chart to point A. Then, place one leg of the compass on the CENTER line (VSWR value of 1) of the RADIALLY SCALED PARAMETERS and determine where the other leg cuts the SWR scale. This scale gives the ratio and the db value. (VSWR = 1.55 or 3.75 db). 3-64

15 Figure Drawing the VSWR circle of a line. 3-65

16 Determining Impedance at Any Point Along a Line Referring to Figure 3-41, locate the dual-scale wavelength circle just beneath the outer rim of the Smith Chart. This circle has an outer and inner scales that permit measurement of the distance between any two points of a line, in wavelength ( ) units. Both scales start at the same point: the zero (0) point of the horizontal centerline. They are graduated in hundredths of a wavelength, going from 0 to The outer scale is marked "WAVELENGTHS TOWARD GENERATOR" (that is, from the load). The values on this scale increase when moving around the circle in a clockwise (CW) direction. The inner scale is marked "WAVELENGTHS TOWARD LOAD" (that is, from the generator). The values on this scale increase when moving around the circle in a counterclockwise (CCW) direction. The reason the maximum value of each scale is 0.5 is that the impedance variation along a line invariably follows a precise pattern that repeats cyclically at every halfwavelength. Consequently, distances greater than 0.5 are measured by turning around the circle as many times as necessary. 3-66

17 Figure The wavelength circle. 3-67

18 Once a lossless line has been represented by its VSWR circle, the impedance at any point along the line can be determined. This occurs because moving around the VSWR circle corresponds to traveling down the lossless line. For example, the Smith Chart of Figure 3-42 shows the VSWR circle of a line, based upon a normalized load impedance of j0.45 (point A). Since Z L > Z 0, the VSWR is 1.6 (point B). Note that a same VSWR of 1.6 would be obtained for Z L < Z 0, i.e. Z L = 30. Determine the input impedance of the line if it is 0.45 long. 1. Draw a line from the center point of the chart through point A to the wavelength circle. This line intersects the wavelength circle at on the CW (clockwise) scale, which is the scale to use in order to measure a distance toward the generator (from the load) is the load end of the line (point C). 2. The input of the line is at 0.45 from the load end, that is, at ( ) or clockwise from the zero (0) point of the horizontal centerline. Find this point on the CW scale (point D) and draw a line to the center point of the chart. 3. The point where this line crosses the VSWR circle (point E) corresponds to the normalized input impedance. Read off this impedance: j0.47. A similar procedure can be used to find the impedance at a distance (n ) from the load. 3-68

19 Figure Determining the input impedance of a 0.45 line. 3-69

20 Impedance Transformation and Matching Using Quarter-Wavelength ( /4) Line Sections Line sections that are exactly a quarter-wavelength ( /4) long can be used to achieve impedance transformation and matching. For example, consider an open-ended lossless line whose length is an odd multiple of /4: the input impedance of the line is null. Adding a /4 open-ended line section to this line causes the line length to become an even multiple of /4. As a result, the input impedance of the line, which was initially null, is now infinite. The computation of this example is shown on the Smith Chart of Figure Point "A" corresponds to the load impedance ( ) of the open-ended line in the initial condition (that is, before the addition of the /4 line section). The VSWR circle of this line therefore corresponds to the largest constant resistance circle (0- R circle). 2. Since the line length is an odd multiple of /4, the input impedance is 0.25 clockwise from the load impedance, that is, at the zero (0) point of the horizontal centerline (point B). 3. The addition of a /4 open-ended line section moves the input impedance of the line 0.25 clockwise from the zero (0) point of the horizontal centerline, that is, to 0.25 (point C). Consequently, the input impedance of the line, which was initially null, is now infinite. 3-70

Figure 3-43. Transformation of the line input impedance, using a /4 open line.

21 Figure Transformation of the line input impedance, using a /4 open line. /4 line sections can also be used for impedance matching. For example, Figure 3-44 shows a 100- transmission line that must transmit power to a 25- resistive load at around 2.0 MHz. 3-71

22 Figure Using a /4 line section to match a 100- transmission line to a 25- resistive load. In spite of the impedance mismatch between the line and load in Figure 3-44, an effective transmission can be obtained if the transmission line is connected to the load through a /4 line section having an impedance Z /4 and length l /4 of where Z /4 = Characteristic impedance of the /4 line section. Z 0 = Characteristic impedance of the transmission line. Z L = Impedance of the load. l /4 = Length of the /4 line section (m or ft); v P = Velocity of propagation in the /4 line section (m/s or ft/s); f = Frequency of the carried signal (Hz). Therefore, the line will see an impedance of 100 if the impedance of the /4 section is and if the length of the /4 section is Now let's look at the Smith Chart of Figure 3-45 to see the computation of this example. 1. The load impedance is normalized with respect to the characteristic impedance of the /4 line section, and then plotted on the chart (point A). 3-72

23 2. The VSWR circle corresponding to point A is drawn on the chart. 3. The addition of a /4 line section moves the load impedance 0.25 clockwise from point A. Consequently, the point where the VSWR circle crosses the right-hand side of the horizontal centerline (point B) corresponds to the normalized impedance seen by the transmission line. This impedance is j0. The actual impedance is Therefore, the impedance seen by the 100- transmission line is 100. Indeed, it is as if the line is matched to the load, thereby permitting the effective transfer of power from the line to the load. 3-73

24 Figure Matching a 100- transmission line to a 25- resistive load, using a /4 line section. 3-74

25 It is obvious that the impedance-transformation and impedance-matching properties of a quarter-wavelength line section are valuable only around a single, unique frequency, that is, the frequency at which the line section is /4 long. Half-Wavelength ( /2) Line Sections Line sections that are exactly /2 long can be used to achieve impedance transformation and matching. /2 line sections are often used to create DC (direct current) and RF (radio-frequency) short circuits: /2 open-ended line sections can be used to create short circuits to RF signals, while remaining open to DC (direct current). /2 shorted line sections can be used to create short circuits to DC, while remaining open to RF signals. Corrections on the Smith Chart for Lossy Lines In the above examples, ideal conditions were assumed: a lossless line of constant VSWR throughout the line. With a lossy line, the VSWR decreases as we approach the generator, so that the VSWR is represented as a spiral, rather than a circle, on the Smith Chart. Because of this, corrections must be made to impedance measurements, using the ATTEN. [db] horizontal scale at the bottom of the chart. For example, the Smith Chart of Figure 3-46 shows the effect that a 2.1 db insertion loss has on the impedance of a 50- open-ended line having a length of /4. Without losses, the VSWR circle of the line would correspond to the 0- R circle. Due to the insertion loss, however, a correction must be made to find the actual impedance of the line. This is performed by using the following steps: 1. From the right-hand extremity of the 0- R circle, drop a vertical down to the 0-dB mark of the ATTEN. [db] scale. 2. Moving toward the generator (to the left), locate the point on the ATTEN. [db] scale which is 2.1 db away (point B). Measure the distance between points A and B on this scale. 3. The measured distance, carried over the horizontal centerline of the chart, specifies the normalized line impedance seen by the generator (point C): j0. This gives an actual impedance of Thus, the open-ended /4 lossy line, which would normally appears as a short circuit to the generator, acts as a resistive load of about

26 Figure Correction made to find the actual impedance of a lossy line. 3-76

27 Procedure Summary In the first part of the procedure, you will observe the effect that adding a /4 line section to an open-ended line has on the location of the loop and node of the standing wave. In the second part of the exercise, you will verify that a /4 line section can be used to match the Thevenin impedance of a generator to a load impedance. In the last part of the exercise (optional), you will solve a theoretical problem, using a Smith Chart. The problem will consist in determining the impedance at a given point on a lossless line. PROCEDURE Impedance Transformation Using a /4 Line Section 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 3-47, 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 to the O (OFF) position. This places the impedance of the load at the receiving end of TRANSMISSION LINE A in the open-circuit condition ( ). The connections should now be as shown in Figure

Figure 3-47. TRANSMISSION LINE A in the open condition (before impedance transformation). G 3. Make the following settings on the oscilloscope: Channel 1 Mode........................................ Normal Sensitivity.

28 Figure TRANSMISSION LINE A in the open condition (before impedance transformation). G 3. Make the following settings on the oscilloscope: Channel 1 Mode Normal Sensitivity V/div Input Coupling AC Channel 2 Mode Normal Sensitivity V/div Input Coupling AC Time Base s/div Trigger Source External Level V Input Impedance M or more G 4. In the SIGNAL GENERATOR section, set the FREQUENCY knob to the fully counterclockwise (MIN.) position. 3-78

Slowly turn the FREQUENCY knob clockwise and stop turning it as soon as the amplitude of the voltage at the sending end of the line [0-meter (0-foot) probe turret of TRANSMISSION LINE A] becomes

29 Slowly turn the FREQUENCY knob clockwise and stop turning it as soon as the amplitude of the voltage at the sending end of the line [0-meter (0-foot) probe turret of TRANSMISSION LINE A] becomes minimum. The frequency of the voltage should now be nearly 2 MHz (T 0.5 s), as Figure 3-48 shows. Consequently, TRANSMISSION LINE A is approximately /4 long. Since this line is open-ended, a node occurs at the sending end of the line (that is, at /4 from the receiving end), causing the amplitude of the voltage (and therefore the impedance) at that point to be minimum; a loop occurs at the receiving end of the line, causing the amplitude of the voltage (and therefore the impedance) at that point to be maximum. Measure the peak (positive) amplitude of the voltage at the sending end (V S ) and receiving end (V R ) of TRANSMISSION LINE A. Record your measurements below. V S (BEFORE IMPEDANCE TRANSFORMATION) = V R (BEFORE IMPEDANCE TRANSFORMATION) = V V Figure A node occurs at the sending end of line A, while a loop occurs at the receiving end of this line. G 5. Perform impedance transformation of TRANSMISSION LINE A. To do so, connect the receiving end of this line to a /4 open-ended line section (i.e., 3-79

30 TRANSMISSION LINE B), in order to obtain the circuit shown in Figure Use 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 all the toggle switches in this section set to the O (OFF) position. Leave channels 1 and 2 of the oscilloscope connected to the sending end and receiving end of TRANSMISSION LINE A. The connections should now be as shown in Figure Figure Impedance transformation of TRANSMISSION LINE A, using a /4 line section. G 6. On the oscilloscope, the frequency of the voltage at the sending end of the TRANSMISSION LINE A should still be set to 2.0 MHz approximately (T 0.5 s). 3-80

With the additional /4-line section, the voltages at the sending and receiving ends of the line should now look like those shown in Figure 3-50.

31 With the additional /4-line section, the voltages at the sending and receiving ends of the line should now look like those shown in Figure Observe that a loop now occurs at the sending end of TRANSMISSION LINE A instead of a node (you may want to slightly readjust the FREQUENCY knob to confirm this). a node now occurs at the receiving end of this line instead of a loop. Figure With impedance transformation using a /4 open line section, a loop now occurs at the sending end of TRANSMISSION LINE A, while a node occurs at the receiving end of this line. G 7. Measure the peak (positive) amplitude of the voltage at the sending end (V S ) and receiving end (V R ) of TRANSMISSION LINE A. V S (AFTER IMPEDANCE TRANSFORMATION) = V R (AFTER IMPEDANCE TRANSFORMATION) = V V G 8. Compare the voltages measured at the sending and receiving ends of the transmission line, before and after impedance transformation. From your comparison, did impedance transformation, through the addition of a /4 open-ended section to TRANSMISSION LINE A, cause the node to be replaced by a loop at the sending end of this line, and the loop to be replaced by a node at its receiving end? 3-81

32 G Yes G No G 9. Disconnect all the cables and probes. Proceed with the exercise. Impedance Matching Using a /4 Line Section G 10. Measure the Thevenin voltage, E TH, at the SIGNAL GENERATOR 100- BNC output: Connect the SIGNAL GENERATOR 100- BNC output to the BNC connector at the LOAD-section input, using a short coaxial cable. Connect the SIGNAL GENERATOR 50- BNC output to the trigger input of the oscilloscope, using a coaxial cable. Using an oscilloscope probe, connect channel 1 of the oscilloscope to the probe turret just next to the BNC connector at the LOAD-section input. In this section, make sure all the toggle switches are set to the O (OFF) position. G 11. Make the following settings on the oscilloscope: Channel 1 Mode Normal Sensitivity V/div Input Coupling AC Time Base s/div Trigger Source External Level V Input Impedance M or more G 12. Make sure the frequency of the SIGNAL GENERATOR output signal is 2.0 MHz approximately (T 0.5 s), as Figure 3-51 shows. Measure the peak (positive) amplitude of the voltage on the oscilloscope screen. This is the Thevenin voltage, E TH, at the SIGNAL GENERATOR 100- BNC output. E TH = V 3-82

Figure 3-51. SIGNAL GENERATOR 100- output signal set to 2.0 MHz approximately. G 13. Remove the coaxial cable between the SIGNAL GENERATOR 100- BNC output and the LOAD-section input.

33 Figure SIGNAL GENERATOR 100- output signal set to 2.0 MHz approximately. G 13. Remove the coaxial cable between the SIGNAL GENERATOR 100- BNC output and the LOAD-section input. Connect the SIGNAL GENERATOR 100- BNC 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. 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 R2 (25 ). Using an oscilloscope probe, connect channel 1 of the oscilloscope to the sending end of TRANSMISSION LINE A [0-meter (0-foot) probe turret]. The connections should now be as shown in Figure

34 Figure Impedance matching using a /4 line section. G 14. Measure the peak (positive) amplitude of the voltage at the sending end of TRANSMISSION LINE A on the oscilloscope screen. Record your result below. V S = V G 15. Compare the voltage measured at the sending end of the line, V S, to the Thevenin voltage of the STEP GENERATOR 100- output, E TH, previously measured. You should observe that V S is approximately equal to half V TH. This indicates a relatively efficient transfer of power between the generator and the load, in spite of the impedance mismatch between the generator Thevenin impedance (100 ) and load impedance (25 ). This occurs because TRANSMISSION LINE A, which is 24-m (78.7-ft) long, makes a /4 section when operated at f 2.0 MHz, as Figure 3-53 shows. 3-84

35 Consequently, the impedance of this section (50 ) and that of the load (25 ), when taken together, is approximately equal to 100 when seen from the generator's point of view: where Z IN = Impedance of the /4 line section and load taken together, as seen by the generator; Z /4 = Characteristic impedance of the /4 line section. Z L = Impedance of the load. Note: The above equation is derived from that used to calculate the impedance that a /4 line section must have to properly adapt the impedance of a generator, Z TH, to that of a mismatched load impedance, Z L, as previously seen in the DISCUSSION section of this exercise: The 100- impedance seen by the generator therefore forms a one-half voltage divider with the 100- Thevenin impedance of the generator, as Figure 3-53 shows. This is the reason why the voltage at the sending end of TRANSMISSION LINE A is approximately half the Thevenin voltage of the 100- generator output. Figure TRANSMISSION LINE A is used as a /4 section for matching the 100- Thevenin impedance of the SIGNAL GENERATOR to the 25- impedance of the load. Determining the Impedance at a Given Point Along a Lossless Line G 16. The Smith Chart in Figure 3-54 will allow you to perform the steps to follow. 3-85

36 Figure Smith Chart. 3-86

37 G 17. A 50- lossless line is terminated by a load Z L = 60 + j15.0. Using the Smith Chart of Figure 3-54, find the impedance at a point X located 1.4 near the load. a. Calculate the normalized load impedance: b. Plot the normalized load impedance on the chart (point A) of Figure c. Draw the VSWR circle corresponding to the plotted impedance (point A), assuming no losses. d. Draw a line from the center of the VSWR circle through point A to intersect the wavelength circle at 0.18 on the CW scale, since impedance toward the generator (from the load) is to be measured. Mark the point where the drawn line intersects the CW scale as "B". e. The point where the impedance must be determined (point X) is 1.4 from point B. Locate point X on the wavelength circle at a distance (CW) corresponding to 1.4 from point B, and mark point X on the chart. f. Draw a line from point X to the center of the SWR circle. The point where this line crosses the VSWR (X') gives the normalized impedance value at point X. This impedance is approximately a j0.24 b j0.35 c j0.24 d j0.35 g. The actual impedance at point X is a j12 b j17.5 c j12 d j17.5 G 18. Figure 3-55 shows computation of the problem performed in the previous step. The data on this chart should resemble the data you plotted on the chart of Figure

Figure 3-55. Determining the impedance at a given point along a lossless line.

38 Figure Determining the impedance at a given point along a lossless line. G 19. Turn off the Base Unit and remove all the connecting cables and probes. 3-88

39 CONCLUSION When a transmission line is not terminated by its characteristic impedance, the impedance seen at the input of the line varies depending on the electrical length of the line and, therefore, on the frequency of the generated signal. The Smith Chart is a graphical computation tool that permits evaluation of the VSWR and impedance at any point along a line, for various electrical lengths and load impedances. It consists of a set of impedance coordinates used to represent normalized impedance. The "R" coordinates are represented by circles of constant resistance values. The "±jx" coordinates are represented by arc of constant reactance values. Applications of the Smith Chart include: determining the VSWR of a line, determining the impedance at any point along a line, correcting mismatch conditions, etc. Quarter-wavelength ( /4) line sections can be used to perform impedance transformation and impedance matching, in order for the generator to transmit the maximum possible power to the load (for resistive loads only). REVIEW QUESTIONS 1. What is the normalized value of the impedance 80 + j40, given a characteristic impedance of 75? a j1.88 b j0.53 c j0.8 d j The center point of a Smith Chart represents a purely resistive, normalized value of a. 0 b. c. 1 d When looking at a Smith Chart, the circles that are all tangent at the right end of the horizontal centerline each represent a a. constant resistance (R) value. b. variable resistance (R) value. c. constant reactance (±jx) value. d. variable reactance (±jx) value. 4. Assuming that the normalized load impedance on a lossless line is j0.3, the VSWR on this line is approximately 3-89

40 a. 1.1 b. 1.4 c. 2.2 d If the line of review question 4 were a 150- line /4 long, its input impedance for the load condition and VSWR stated would be a j34.3 b j15.2 c j13.1 d j

Standing Waves and Voltage Standing Wave Ratio (VSWR)

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