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19. Coupling of A Microwave Cavity to A Waveguide
19. COUPLING OF A MICROWAVE CAVITY TO A WAVEGUIDE” 19. I . Problem A microwave cavity, designed to be resonant at 34 GHz, is to terminate a TE,,-mode rectangular waveguide. It is to be coupled to this guide by an iris in a thin wall separating the two. When supplied with power at resonance through this waveguide, 40% of the incident power is to be absorbed in the cavity, 60% is to be reflected. At this degree of coupling, the maximum frequency selectivity possible is to be maintained. Design an experiment for measuring the resonant frequency and unloaded Q of the cavity, and for selecting the correct iris size for the 40% coupling. T h e expected value of unloaded Q for this cavity is about 5000.
19.2. Solution There are a number of different procedures that can be used for solving this problem. We discuss here a particularly simple method that holds for the thin wall case, in which negligible coupling reactance is introduced. It follows the treatment by R. A. Lebowitz, IRE Trans. MicrowaTte Theory Tech. M’IT4, 51 (1956). We require a tunable oscillator, such as a reflex klystron, a slotted line, a directional coupler (20 dB), an attenuator, and an oscilloscope. T h e circuit is shown in Fig. 19.1. If the 20-dB attenuator shown in this circuit can be replaced by a circulator, more signal will be available, so that less care in the prevention of “hum” signals arising from ground “loops” will be required. We will assume, here, that the generator available for the experiment is a reflex klystron, which can be tuned mechanically over a considerable frequency range and tuned electronically (by repeller modulation) over a small range. The initial step is to find the resonant frequency. With no iris, i.e. with a solid wall between waveguide and cavity, nearly all of the energy incident
* Problem 19 is by
I. Kaufman. 135
136
I . KAUFMAN
on the wall is reflected. Almost lOOyo reflection occurs also with an iris if the frequency of the incident wave is not equal to or near the resonant frequency of the cavity. Consequently, to find the resonant frequency, we drill a very small hole in the thin wall separating waveguide from cavity, mechanically tune the klystron through the range of expected resonance
Wovemeter
FIG.19.1. Circuit for performing cavity coupling measurements.
FIG.19.2. Reflected power, as seen at the output of the detector following directional coupler: (a) with neither wavemeter nor cavity resonance within the klystron mode; (b) with cavity resonance dip on left and wavemeter dip on right; (c) with the two resonances superimposed. Since the oscilloscope is set for “exterior sweep input,” the horizontal coordinate is frequency, although the scale is usually not linear.
while the output frequency is being swept electronically by repeller modulation, and observe the pattern of reflected power at the output of the directional coupler. When the resonance is found, it will be observed as in Fig. 19.2. By superimposing the absorption dip of the wavemeter on the dip of the cavity, the resonant frequency is found from the wavemeter calibration. (In a more exact technique, another attenuator is inserted
19. COUPLING
OF MICROWAVE CAVITY AND WAVEGUIDE
137
between wavemeter and directional coupler, so that the two resonances will not interfere cooperatively and shift the resonance slightly.) It is possible that no cavity absorption dip will be observed on the first try. T h e coupling hole is then enlarged and the process repeated. ‘I’o adjust the coupling iris for 40% power absorption, we simply enlarge it progressively until the reflected power at the resonance dip drops to 60% of the value it would have if there were no iris (see Fig. 19.3). I n all of the measurements of relative power and relative fields, it must be remembered that when a crystal detector is used as a video detector, as in Fig. 19.1, it is a square-law device. Consequently, the vertical scale of the oscilloscope display is in units of relative radio-frequency power, or relative units of (E,,)’. T o ensure linearity in these units, the power incident on the crystal should not exceed a few microwatts.
FIG.19.3. Reflected power trace for 40% power absorption in cavity.
Since 40% absorption of incident power corresponds to a VSWR of 7.87, it would appear that we could also use the slotted line at the fixed frequency of resonance of the cavity and enlarge the coupling iris until the VSWR is 7.87. It will be found in the laboratory, however, that this technique is not feasible unless the signal source is highly stabilized in frequency by such methods as discriminator circuits, etc., for otherwise the frequency drift is too large. On the other hand, if a swept frequency source is used, the use of a slotted line for measurement of the percentage of power absorption becomes possible again. Furthermore, this technique will also enable us to measure the cavity Q. We will discuss the method of power measurement and from there lead into the scheme of measurement of unloaded Q. When a cavity is excited from a source whose frequency is the frequency of cavity resonance, the peak energy stored in magnetic fields exactly equals that stored in electric fields. There is thus no “reactive power’’ supplied to the cavity, only real power, which is dissipated in the cavity walls. As soon as the signal source is detuned from resonance, however, large amounts of reactive power are required. Accordingly, the cavity behaves exactly as a
138
I. KAUFMAN
lumped-constants resonant circuit. While it would appear that this situation is true only at the iris plane, because of the repetitive nature of a transmission line, this behavior is observed at a number of planes in the waveguide feeding the cavity. Specifically, there are two sets of such planes, referred to as the planes of the “detuned open” and “detuned short,” which are separated by quarters of a guide wavelength from each other. T h e equivalent circuits at these two sets of planes are shown in Fig. 19.4. T h e
FIG.19.4. Equivalent circuits of waveguide-cavity system at (a) detuned open and (b) detuned short. E and I here are taken to be RMS values. It is assumed that the output impedance of the source is also the waveguide characteristic impedance ZO.
-
Frequency
FIG. 19.5. Power detected by probe of slotted line at detuned open when cavity resonance is at center of klystron mode.
names “detuned open” and “detuned short” are used because of the equivalent impedance that the cavity presents when the signal frequency differs appreciably from the resonance frequency. If we have chosen a slotted line (Fig. 19.1) that is at least one half-guide wavelength long, at least one of each of the two sets of planes will be found along the line. Since the impedance at the detuned open when not at resonance is essentially an open circuit, then in this “of”’ resonance condition this plane is a position of maximum electric field in the standing wave pattern. When “on” resonance, however, the cavity absorbs some energy. As the klystron is swept through resonance, the output of the traveling probe located at a detuned open is therefore as shown in Fig. 19.5. I n fact, the position of a detuned open is found by moving the probe until a symmetric
’ 19. COUPLING OF MICROWAVE CAVITY AND WAVEGUIDE
139
dip is obtained. It will be noted that an increase of the coupling iris is accompanied by a deepening of the dip. Referring to Fig. 19.4(a), we conclude that an increase in the coupling corresponds to a reduction in Y in the equivalent circuit. We are now interested in the “incident power” in terms of the circuit parameters of Fig. 19.4(a). Since this is the power that would be absorbed by an infinite length of line or a matched load, it is obviously the quantity E2142,. If the power absorbed by the cavity is to be only 40% of the incident power, we find that Y is given by
(0.40E2) ( E Y / [ Y+ 20])2 ~-
42,
9
Y
so that we have either Y
or
=
0.1272,
r = 7.872,
when
V,,
when
VA, = 0.113E.
Z,
=
0.888E,
I : n
FIG.19.6. More complete equivalent circuit at detuned open.
Accordingly, there are two iris diameters that will satisfy the 40-60y0 condition. I n each case, of course, the VSWR is 7.87. It was required that of the two iris sizes we choose the one that results in the maximum amount of frequency discrimination. We therefore recall that the iris is actually a transformer; so that a more complete equivalent circuit at the detuned open is that of Fig. 19.6. Here, as the iris is altered, r’, C‘, and L‘ remain fixed, but n changes. The loaded Q , QL, which is a measure of frequency discrimination of the circuit, as calculated by using impedances of the “cavity” side of the transformer, is now QL = w,L’/ (&‘ + n2 2,). As n2 increases, QL decreases. For maximum frequency discrimination, QL should be as large as possible so that n should be as small as possible. Since at resonance Z A B= Y = y ‘ / n 2 , maximum QL is obtained when Y is as large as possible. Consequently, the iris is to be opened only to the smaller diameter, when
V,,
=
0,8883.
140
I. KAUFMAN
As stated above, the output \,ohage of the detector is proportional to the square of the rf voltage. Consequently, the correct iris size is that corresponding to a detected voltage at the notch of Fig. 19.5 of 0.79 of that which would exist if the resonance were missing. T h e final requirement of this problem is to measure the unloaded Q , Qo. This involves a measurement of the frequency response. I n the seriesrcsonant circuit which exists at the detuned open, this could only be obtained by a measurement of the relative current response. T h e probe of a slotted line does not respond to waveguide current, however, but waveguide electric field. We must therefore move to the detuned short, where the equivalent circuit of Fig. 19.4(b) holds. Here the frequency response is given by a voltage response, which will be detected by the slotted line probe. As seen from this circuit, the frequency response is a loaded Q (QL)response, for it is determined by the parallel combination of R and Z,. If Af is the frequency deviation from resonance at which the voltage VcD drops to 0.707 of its value at resonance, and if fo is the resonant frequency, then
Q
- -f.o
“-2Af
19.3. Exercises 19.3. I. Sketch the curve of detected voltage versus frequency in the measurement of QL. Where are the 0.707 points located on this curve? (Recall the method of response of the detector.) 19.3.2. In the equivalent circuit of Fig. 19.4(b), the unloaded Q, Qo, is Qo = H/2rtjoL.How is Qo determined from QL? 19.3.3. In the discussion above, we assumed that the source impedance was equal to the waveguide characteristic impedance. Is this of importance? 19.3 4. Sketch the reflected power curves that result as the iris is progressively opened from a very small to a very large diameter. Sketch the corresponding curves seen at the detuned open and detuned short.
19. COUPLING OF
MICROWAVE CAVITY AND WAVEGUIDE
141
19.3.5. Why is it important in these measurements to isolate the klystron by either a 20-dB attenuator or a circulator? 19.3.6. Suppose the cavity has been adjusted for 40% coupling. It is now desired to “match” the cavity to the guide at the resonant frequency. How can this be accomplished without changing the iris diameter, but with external components? 19.3.7.
If the directional coupler available for the measurements is a crossedguide type, which allows simultaneous observation of incident and reflected power, what precaution must be used to ensure correct measurements? 19.3.8. What test must be made to ensure square-law behavior of the detecting crystals? It is stated that 40% absorption of the incident power corresponds to a VSWR of 7.87. Show this. 19.3.9. Suppose you had a slotted line with loop coupling, instead of the usual probe coupling. How would this change the displays? 19.3.10. What could be the effect of inserting the probe of the slotted line too deeply into the waveguide? 19.3.1 I.
A cavity is to be used as a transmission cavity, i.e. as a filter inserted in a waveguide system. It is to be coupled to the two sides in equal amounts. If the unloaded Q of the cavity is 5000 and the loaded Q is to be 1000, how much is the signal attenuated in the cavity? Express the answer in decibels. 19.3.12. What procedure would you use for getting the correct iris sizes for the cavity of Exercise 19.3.11?
19.2. Solution There are a number of different procedures that can be used for solving this problem. We discuss here a particularly simple method that holds for the thin wall case, in which negligible coupling reactance is introduced. It follows the treatment by R. A. Lebowitz, IRE Trans. MicrowaTte Theory Tech. M’IT4, 51 (1956). We require a tunable oscillator, such as a reflex klystron, a slotted line, a directional coupler (20 dB), an attenuator, and an oscilloscope. T h e circuit is shown in Fig. 19.1. If the 20-dB attenuator shown in this circuit can be replaced by a circulator, more signal will be available, so that less care in the prevention of “hum” signals arising from ground “loops” will be required. We will assume, here, that the generator available for the experiment is a reflex klystron, which can be tuned mechanically over a considerable frequency range and tuned electronically (by repeller modulation) over a small range. The initial step is to find the resonant frequency. With no iris, i.e. with a solid wall between waveguide and cavity, nearly all of the energy incident
* Problem 19 is by
I. Kaufman. 135
136
I . KAUFMAN
on the wall is reflected. Almost lOOyo reflection occurs also with an iris if the frequency of the incident wave is not equal to or near the resonant frequency of the cavity. Consequently, to find the resonant frequency, we drill a very small hole in the thin wall separating waveguide from cavity, mechanically tune the klystron through the range of expected resonance
Wovemeter
FIG.19.1. Circuit for performing cavity coupling measurements.
FIG.19.2. Reflected power, as seen at the output of the detector following directional coupler: (a) with neither wavemeter nor cavity resonance within the klystron mode; (b) with cavity resonance dip on left and wavemeter dip on right; (c) with the two resonances superimposed. Since the oscilloscope is set for “exterior sweep input,” the horizontal coordinate is frequency, although the scale is usually not linear.
while the output frequency is being swept electronically by repeller modulation, and observe the pattern of reflected power at the output of the directional coupler. When the resonance is found, it will be observed as in Fig. 19.2. By superimposing the absorption dip of the wavemeter on the dip of the cavity, the resonant frequency is found from the wavemeter calibration. (In a more exact technique, another attenuator is inserted
19. COUPLING
OF MICROWAVE CAVITY AND WAVEGUIDE
137
between wavemeter and directional coupler, so that the two resonances will not interfere cooperatively and shift the resonance slightly.) It is possible that no cavity absorption dip will be observed on the first try. T h e coupling hole is then enlarged and the process repeated. ‘I’o adjust the coupling iris for 40% power absorption, we simply enlarge it progressively until the reflected power at the resonance dip drops to 60% of the value it would have if there were no iris (see Fig. 19.3). I n all of the measurements of relative power and relative fields, it must be remembered that when a crystal detector is used as a video detector, as in Fig. 19.1, it is a square-law device. Consequently, the vertical scale of the oscilloscope display is in units of relative radio-frequency power, or relative units of (E,,)’. T o ensure linearity in these units, the power incident on the crystal should not exceed a few microwatts.
FIG.19.3. Reflected power trace for 40% power absorption in cavity.
Since 40% absorption of incident power corresponds to a VSWR of 7.87, it would appear that we could also use the slotted line at the fixed frequency of resonance of the cavity and enlarge the coupling iris until the VSWR is 7.87. It will be found in the laboratory, however, that this technique is not feasible unless the signal source is highly stabilized in frequency by such methods as discriminator circuits, etc., for otherwise the frequency drift is too large. On the other hand, if a swept frequency source is used, the use of a slotted line for measurement of the percentage of power absorption becomes possible again. Furthermore, this technique will also enable us to measure the cavity Q. We will discuss the method of power measurement and from there lead into the scheme of measurement of unloaded Q. When a cavity is excited from a source whose frequency is the frequency of cavity resonance, the peak energy stored in magnetic fields exactly equals that stored in electric fields. There is thus no “reactive power’’ supplied to the cavity, only real power, which is dissipated in the cavity walls. As soon as the signal source is detuned from resonance, however, large amounts of reactive power are required. Accordingly, the cavity behaves exactly as a
138
I. KAUFMAN
lumped-constants resonant circuit. While it would appear that this situation is true only at the iris plane, because of the repetitive nature of a transmission line, this behavior is observed at a number of planes in the waveguide feeding the cavity. Specifically, there are two sets of such planes, referred to as the planes of the “detuned open” and “detuned short,” which are separated by quarters of a guide wavelength from each other. T h e equivalent circuits at these two sets of planes are shown in Fig. 19.4. T h e
FIG.19.4. Equivalent circuits of waveguide-cavity system at (a) detuned open and (b) detuned short. E and I here are taken to be RMS values. It is assumed that the output impedance of the source is also the waveguide characteristic impedance ZO.
-
Frequency
FIG. 19.5. Power detected by probe of slotted line at detuned open when cavity resonance is at center of klystron mode.
names “detuned open” and “detuned short” are used because of the equivalent impedance that the cavity presents when the signal frequency differs appreciably from the resonance frequency. If we have chosen a slotted line (Fig. 19.1) that is at least one half-guide wavelength long, at least one of each of the two sets of planes will be found along the line. Since the impedance at the detuned open when not at resonance is essentially an open circuit, then in this “of”’ resonance condition this plane is a position of maximum electric field in the standing wave pattern. When “on” resonance, however, the cavity absorbs some energy. As the klystron is swept through resonance, the output of the traveling probe located at a detuned open is therefore as shown in Fig. 19.5. I n fact, the position of a detuned open is found by moving the probe until a symmetric
’ 19. COUPLING OF MICROWAVE CAVITY AND WAVEGUIDE
139
dip is obtained. It will be noted that an increase of the coupling iris is accompanied by a deepening of the dip. Referring to Fig. 19.4(a), we conclude that an increase in the coupling corresponds to a reduction in Y in the equivalent circuit. We are now interested in the “incident power” in terms of the circuit parameters of Fig. 19.4(a). Since this is the power that would be absorbed by an infinite length of line or a matched load, it is obviously the quantity E2142,. If the power absorbed by the cavity is to be only 40% of the incident power, we find that Y is given by
(0.40E2) ( E Y / [ Y+ 20])2 ~-
42,
9
Y
so that we have either Y
or
=
0.1272,
r = 7.872,
when
V,,
when
VA, = 0.113E.
Z,
=
0.888E,
I : n
FIG.19.6. More complete equivalent circuit at detuned open.
Accordingly, there are two iris diameters that will satisfy the 40-60y0 condition. I n each case, of course, the VSWR is 7.87. It was required that of the two iris sizes we choose the one that results in the maximum amount of frequency discrimination. We therefore recall that the iris is actually a transformer; so that a more complete equivalent circuit at the detuned open is that of Fig. 19.6. Here, as the iris is altered, r’, C‘, and L‘ remain fixed, but n changes. The loaded Q , QL, which is a measure of frequency discrimination of the circuit, as calculated by using impedances of the “cavity” side of the transformer, is now QL = w,L’/ (&‘ + n2 2,). As n2 increases, QL decreases. For maximum frequency discrimination, QL should be as large as possible so that n should be as small as possible. Since at resonance Z A B= Y = y ‘ / n 2 , maximum QL is obtained when Y is as large as possible. Consequently, the iris is to be opened only to the smaller diameter, when
V,,
=
0,8883.
140
I. KAUFMAN
As stated above, the output \,ohage of the detector is proportional to the square of the rf voltage. Consequently, the correct iris size is that corresponding to a detected voltage at the notch of Fig. 19.5 of 0.79 of that which would exist if the resonance were missing. T h e final requirement of this problem is to measure the unloaded Q , Qo. This involves a measurement of the frequency response. I n the seriesrcsonant circuit which exists at the detuned open, this could only be obtained by a measurement of the relative current response. T h e probe of a slotted line does not respond to waveguide current, however, but waveguide electric field. We must therefore move to the detuned short, where the equivalent circuit of Fig. 19.4(b) holds. Here the frequency response is given by a voltage response, which will be detected by the slotted line probe. As seen from this circuit, the frequency response is a loaded Q (QL)response, for it is determined by the parallel combination of R and Z,. If Af is the frequency deviation from resonance at which the voltage VcD drops to 0.707 of its value at resonance, and if fo is the resonant frequency, then
Q
- -f.o
“-2Af
19.3. Exercises 19.3. I. Sketch the curve of detected voltage versus frequency in the measurement of QL. Where are the 0.707 points located on this curve? (Recall the method of response of the detector.) 19.3.2. In the equivalent circuit of Fig. 19.4(b), the unloaded Q, Qo, is Qo = H/2rtjoL.How is Qo determined from QL? 19.3.3. In the discussion above, we assumed that the source impedance was equal to the waveguide characteristic impedance. Is this of importance? 19.3 4. Sketch the reflected power curves that result as the iris is progressively opened from a very small to a very large diameter. Sketch the corresponding curves seen at the detuned open and detuned short.
19. COUPLING OF
MICROWAVE CAVITY AND WAVEGUIDE
141
19.3.5. Why is it important in these measurements to isolate the klystron by either a 20-dB attenuator or a circulator? 19.3.6. Suppose the cavity has been adjusted for 40% coupling. It is now desired to “match” the cavity to the guide at the resonant frequency. How can this be accomplished without changing the iris diameter, but with external components? 19.3.7.
If the directional coupler available for the measurements is a crossedguide type, which allows simultaneous observation of incident and reflected power, what precaution must be used to ensure correct measurements? 19.3.8. What test must be made to ensure square-law behavior of the detecting crystals? It is stated that 40% absorption of the incident power corresponds to a VSWR of 7.87. Show this. 19.3.9. Suppose you had a slotted line with loop coupling, instead of the usual probe coupling. How would this change the displays? 19.3.10. What could be the effect of inserting the probe of the slotted line too deeply into the waveguide? 19.3.1 I.
A cavity is to be used as a transmission cavity, i.e. as a filter inserted in a waveguide system. It is to be coupled to the two sides in equal amounts. If the unloaded Q of the cavity is 5000 and the loaded Q is to be 1000, how much is the signal attenuated in the cavity? Express the answer in decibels. 19.3.12. What procedure would you use for getting the correct iris sizes for the cavity of Exercise 19.3.11?