Physics 401 Classical Physics Laboratory. Experiment 44 MICROWAVE CAVITIES. Table of Contents

Transcription

1 University f Illinis at Urbana-Champaign Physics 401 Classical Physics abratry Department f Physics Experiment 44 MICROWAVE CAVITIES Table f Cntents Subject Page References Resnant Frequency The Q f a Cavity Cupling t a Cavity aded and Unladed Q Apparatus Experimental Prcedure I. Wavelength Measurement II. Cavity Resnance III. Magnetic Field Directin and Cupling IV. Electric Field Distributin V. Cavity Q Reprt Revised 4/01. Cpyright 01 The Bard f Trustees f the University f Illinis. All rights reserved.

2 Physics 401 Expt 44 Page /17 Micrwave Cavities References 1. W. M. Schwarz, Intermediate Electrmagnetic Thery, pp F. E. Terman, Electrnic Measurements, pp M.I.T. Radar Schl Staff, Principles f Radar, 1946, p E. M. Purcell, Electricity and Magnetism, Berkeley Physics Curse, Vl., pp C.G. Mntgmery, Techniques f Micrwave Measurements, pp.86 89; C. G. Mntgmery, R. H. Dicke and E. M. Purcell, Principles f Micrwave Circuits, pp D. W. Prestn and E. R. Dietz, The Art f Experimental Physics, pp Resnant frequency The resnant (angular) frequencies,, f the TE mdes in a rectangular cavity f dimensins; a, b, and c are mnp m n p v. (1) a b c mnp In the abve expressin m, n, and p are integers, n mre than ne f which can be zer at the same time; and v is the phase velcity in infinite space f the medium filling the cavity. v is the speed f light in vacuum fr an empty cavity. (Since we use c fr ne f the cavity dimensins, we cannt als use it fr the speed f light in vacuum.) If b is taken as the shrtest dimensin, the lwest mde (lwest resnant frequency) is btained fr m 1, n 0, and p 1. This mde is called the TE mde. Fr this mde, Eq. 1 takes the simpler frm 1 1 v. () a c The electric and magnetic field cnfiguratins in the TE mde can be represented schematically as in Fig. 1 where a, b, and c are in the x, y, and z directins, respectively.

3 Physics 401 Expt 44 Page 3/17 Micrwave Cavities Y X Y X X Z Z TOP VIEW SIDE VIEW AONG X OR Z Z H FIED X = 0 r Z = 0 X = a r Z = c E FIED Fig. 1. Field cnfiguratins in the TE mde The Q f a cavity Resnant cavities will lse energy by pwer dissipatin in the walls and by radiatin. The lsses are usually discussed in terms f the quality factr Q defined in the same way as fr a lumped RC resnant circuit: ttal stred energy Q. (3) decrease in energy per perid Recall that the Q f a RC circuit is Q. (4) R

4 Physics 401 Expt 44 Page 4/17 Micrwave Cavities where the resnant frequency,, is 1 R C. Fr small damping (high Q) the res- 1 nant frequency. It can be shwn (see fr example, Berkeley Physics Curse) C that fr a high-q lumped circuit, the Q is als given by Q (5) where ( 1) is the difference between the tw frequencies, 1 and, at which the absrbed pwer is reduced by a factr f tw, r the current is reduced by a factr f rt frm their maximum values. The same relatin is valid fr a resnant cavity. Thus, Q can be determined frm the frequency respnse curve. Other methds f determining Q include standing wave measurements as a functin f frequency, and the measurement f the decay f the fields as a functin f time. Cupling t a cavity In rder t excite fields in a cavity and t make measurements n it, ne r mre transmissins lines must be cupled t the cavity. This cupling may be dne with an pening, knwn as an iris diaphragm, an electric-diple antenna, r a magnetic-diple antenna. The latter is called a cupling lp. The cupling is usually discussed in terms f the equivalent circuit f Fig.. The cupling cefficient describes the strength f cupling. In defining, it is cnvenient t use the circuit f Fig. 3, in which the cavity is fed by a line f impedance Z. Z is the impedance f the real line transfrmed t the cavity terminals by the ideal transfrmer represented by the inductances, (, ), in Fig.. Z is essentially a pure resistance. Here is defined as Z R, and may have any value between 0 and.

5 Physics 401 Expt 44 Page 5/17 Micrwave Cavities M R INE O C OOP CAV ITY Fig.. Equivalent circuit f a cavity and cupling lp Z RC CAV ITY Fig. 3. Transfrmed Cavity Cupling System The Pwer Transfer Therem tells us that the greatest pwer is transferred t a lad when the lad impedance equals the surce impedance. If we apply this therem t the circuit f Fig. 3 at resnance where the cavity is resistive, we see that maximum pwer transfer t the cavity ccurs when R Z, r 1. This cnditin is called critical cupling. At critical cupling the standing wave rati at resnance takes n a minimum value, which will be clse t unity if the cupling lp is small enugh t be nn-inductive. The cupling cefficient can be changed by rtating the lp abut the axis f the line. aded and unladed Q

6 Physics 401 Expt 44 Page 6/17 Micrwave Cavities When the cavity is cupled t an external pwer surce and detectr in rder t measure the characteristics f the cavity, sme f the lsses in the external circuit are added t thse in the cavity. The Q measured is therefre less than the value characteristic f the cavity alne. The intrinsic Q f the cavity is called the unladed Q, ften designated as Q, and the measured value is called the laded Q, ften designated as crrespnd t a series resistance Q. At resnance, the ttal lsses in the system R Z in Fig. 3, and the laded Q is therefre given by Q R Z. This may be written as Q Q Z. (6) 1 R1 R If the cupling is adjusted fr maximum pwer transfer ( 1), then Q 1 Q. (7) Apparatus T study such a cavity it will be necessary t excite electrmagnetic waves inside the cavity. T d this we will set up the apparatus shwn in Fig. 4. Micrwave Oscillatr Prbe & Stub Tuner Prbe detectr (scpe) Cavity Detectr (DMM A meter) Sltted ine Adjustable ine Cavity

7 Physics 401 Expt 44 Page 7/17 Micrwave Cavities Fig. 4. Apparatus Micrwaves are generated by the vltage cntrlled scillatr frm Mini-Circuits and prpagate dwn the caxial line and adjustable line t the terminatin lp in the cavity. An expanded view f the cavity and waveguide terminatin is shwn in a side crss sectin in Fig. 5. Y OUTER CONDUCTOR INNER CONDUCTOR COUPING OOP CAV ITY WAS (X axis int paper) Z Fig. 5. Side view f the cavity Ntice that currents in the caxial line will prduce magnetic fields in the cavity in planes parallel t the xz plane. If these fields are scillating at the prper frequency, they will satisfy the bundary cnditins and standing waves will be bserved. Experimental prcedure I. Wavelength and frequency measurement Recrd the mdel numbers, serial number, the frequency range f the micrwave scillatr, and the relatinship between the frequency and the tuned vltage. Observing the crrect plarity, cnnect the micrwave scillatr int the 6 VDC pwer surce n the terminals near the bench. The shrted tuning stub n the prbe shuld have already been adjusted. In case it is necessary, it shuld be inserted apprximately half way int its munt. Its psitin shuld be adjusted t prvide maximum signal n the scpe.

8 Physics 401 Expt 44 Page 8/17 Micrwave Cavities The scpe is cnnected t the dide detectr which is psitined inside the caxial sltted line t detect the electric field. The dide rectifies the vltage and prvides a DC current. Since the scpe has large input impedance, yu shuld be able t see the crrespnding vltage n the screen. The minimum is defined as the vltage clsest t the grund level and the ppsite is true abut the maximum. Adjust the hrizntal and vertical sensitivities apprpriately. It is cnvenient t use the measurement ptin t read the vltage. Yu may mve the prbe alng the sltted line t a maximum f the detectr current t adjust fr the ptimum scpe setting. If it is necessary, reduce the sensitivity f the detectr by shifting the psitin f the tuning stub n the prbe. Micrwave Oscillatr Prbe & Stub Tuner Prbe detectr (scpe) Sltted ine Open End Fig. 6. Wavelength measurement with the sltted line. Determine the scillatr frequency, f, by measuring the wavelength in the sltted line. Remember that the wavelength in a caxial line is equal t the wavelength in air and is equal t twice the distance between successive minima. Find the minima n either end f the sltted line, and cunt the number f minima in between them t find the number f half wavelengths in the line. Make the measurement f wavelength carefully, since an accurate value f the frequency is needed. It shuld be pssible t measure wavelength t 0.1%. Calculate the frequency and angular frequency f the scillatr and their uncertainties. II. Cavity Resnance Examining Eq. 1, we see that the cavity resnance may be bserved by varying the frequency fr a cavity f fixed dimensins, r (as we d here) it can be bserved fr a fixed frequency by changing the physical dimensins f the cavity. (We btain resnance by varying the right-hand side f Eq. 1, keeping the left-hand side fixed.)

9 Physics 401 Expt 44 Page 9/17 Micrwave Cavities T the pen end f the sltted line add the adjustable line with rtating jint, the cupling lp, the cavity and the detectr lp, as shwn in Fig. 7. Cavity Detectr (DMM A meter) Cavity X Z Adjustable ine Cupling p Mvable Plunger Fig. 7. Tp view f the cavity The plane f the cupling lp shuld be vertical t prduce maximum cupling t the directin f the H fields we expect at resnance. The detectr lp shuld be parallel t the cupling lp. The lp f the cavity field detectr acts as a small inductive lp which prduces an EMF when placed in an alternating magnetic field. The EMF is rectified by the dide detectr, and the current is measured with a sensitive ammeter. This lp is sufficiently far frm the input cupling lp s that, except near resnance where large fields are prduced in the cavity, the mutual inductance between them is very small. First lcate the psitin f the TE resnance apprximately by mving the plunger ut frm the cupling lps in large steps. Use Eq. 1 t shw that the TE mde will resnate when the length f the cavity, c, is abut 7.5 cm and the TE 10 mde will resnate when c is twice this value. After the psitin f the resnance is lcated apprximately, mve the plunger in a few millimeters. Devise an algrithm which gives the length f the cavity, c, in terms f the psitin f the plunger. The dimensins f the cavity and plunger are printed n the utside f the cavity. Then the plunger shuld be slwly and smthly withdrawn by turning the knurled knb in very small steps. Resnance will be bserved as a very sharp peak in the respnse f the cavity field detectr. The steps must be small t determine accurately the width f the resnance. Nte that near resnance the fields are very strng and this prduces large EMFs. If the

10 Physics 401 Expt 44 Page 10/17 Micrwave Cavities meter ges ff scale, decuple the detectr by rtating the plane f the detectr lp away frm the vertical t decrease the effective area f lp. Determine the length f the cavity, c, at tw resnances, TE and TE 10, and cmpare yur measurements t the value calculated using Eq. 1, and the scillatr frequency f measured abve. III. Magnetic Field Directin and Cupling As was discussed abve, the H field f the TE and TE 10 mdes lies entirely in planes parallel t the xz plane. The magnetic field prduced by the current in the lp will have sme cmpnent in this directin and will therefre tend t excite that mde in the cavity. With the cavity in resnance fr the TE mde, measure the current in the cavity field detectr while changing the angle f the input cupling lp plane frm the vertical, 0, thrugh 360 in steps f 10. Explain the shape f the curve. See part V fr a discussin f the respnse f the field detectr t the field. IV. Electric Field Distributin The electric field, E y, in the TE and TE 10 mdes varies in the z directin is sin zc as shwn in Fig. 1. We will demnstrate this dependence with the fllwing technique. (A mre sphisticated versin f this same methd is used in practice fr accurate mapping f the fields f a cavity.)

11 Physics 401 Expt 44 Page 11/17 Micrwave Cavities Fig. 8 Side view f cavity A thin dielectric sheet is placed in the cavity with the plane f the sheet parallel t the xy plane, see Fig. 8 abve. The sheet is placed a distance frm the end f the cavity at z 0, the end with the cupling lps. The sheet is affected nly by the electric field in the y directin, E, at that psitin, since E E 0 in the TE and TE 10 mdes. It can be shwn that the x z dielectric shrtens the effective guide wavelength in the z directin, and the amunt f shrtening is prprtinal t the strength f the electric field at the dielectric psitin. The full expressin cntains an integral f the electric field strength squared ver the entire cavity vlume, s that a simple change in the verall magnitude f the driving micrwaves des nt als change the effective length f the cavity. The dielectric may be viewed as an additin f a lumped capacitance in parallel. Hence, the c dimensin f the cavity when in resnance will be shrter as the sheet is placed in prgressively strnger fields. et c be the length f the cavity at resnance when n dielectric is present, and let c be the length f the cavity at resnance when the dielectric sheet is at. et the difference, c c c. This is difference in the length f the cavity at resnance due t the presence f the dielectric. If we measure c as a functin f, we will get an idea f the variatin in the field in the z directin. y The dependence f the electric field n psitin in the cavity is mre easily seen in the TE 10 mde than in the TE mde. Fr each psitin f the dielectric sheet we must pen the cavity, psitin the sheet, clse the cavity, and the mve the plunger t find the resnance. The dielectric sheet can be cnven-

12 Physics 401 Expt 44 Page 1/17 Micrwave Cavities iently psitined with a simple jig. Open the cavity, insert the dielectric with the feet f the sheet tward the cupling lps, see Fig. 8, and then use the jig t push the dielectric sheet int the cavity t a psitin which is 0.5 cm less than 1 c. Clse the cavity and mve the plunger until resnance is bserved. Recrd the psitin f the plunger, c, and calculate the difference in the 1 length f the cavity due t the presence f the dielectric, c. Since the dielectric is in a weak field, this difference will be small. Next pen the cavity, and use the jig t push the sheet an additinal 0.5 cm tward the cupling lps. Recrd this psitin,. Clse the cavity, and again mve the plunger until resnance is bserved. Recrd psitin c, and calculate c. Always mving the dielectric tward the cupling lps, repeat the prcess, mving the dielectric by apprximately 0.5 cm in each step, until the dielectric is at the cupling lps. Then plt c versus. 1 V. CavityQ resnance curve P versus their maximum resnance values. (Equivalently, the pwer falls t 1/ f its maximum resnance value.) As was discussed earlier, the Q f the cavity can be measured as Q f f, where f is the resnant frequency and f is the difference between the (angular) frequencies at which the fields fall t 1 f A resnance curve btained frm varying the frequency (at fixed cavity length) is shwn in the figure n the left belw. Pwer is pltted versus (angular) frequency. The resnant frequency f is identified, and the frequency difference at half pwer f is identified. In ur experiment we measure the laded Q, Q. By sweeping the frequency f the scillatr, we identify Q with the measured f f as in the figure abve n the left. Q Recall that Q. Then Q 1 Q 1 f0 / f. It is clear frm these 1 expressins that as the cupling f the cavity t external lads is decreased, i.e. as 0, the measured laded Q, Q, appraches the Q f the cavity, i.e. Q Q. The measured laded Q is always smaller than the Q f the cavity. In the case f critical cupling, i.e. 1,

13 Physics 401 Expt 44 Page 13/17 Micrwave Cavities 0 Q 11 Q 11 f f f f ; hwever, we cannt assume critical cupling in ur experiment. Assuming that ur calculatin f the cavity Q frm Eq. 11 is accurate (we nly have a handbk value fr the resistivity, and we neglect imperfectins in the cavity), we can infer the cupling parameter fr ur experiment frm the expressin Using Eq 11, we find that Q 1. Q Q is abut Values fr Q vary between 300 and 000, depending n the rientatin f the antenna and detectr lps, the quality f the cntact between the plunger fingers and the cavity walls, etc. Fr these values f Q, then is between 6 and 7 far frm critical cupling. Frequency sweep f the Micrwave scillatr using the Wavetek functin generatr. t a gd apprximatin, I V In the cavity field detectr, the vltage develped in the lp, V, is prprtinal t the magnetic field strength, V H. Hwever, the dide is very nearly a square-law detectr, i.e.. Thus the current develped in the dide, I, is prprtinal t the square f the field, I H. At resnance where we bserve maximum current, we have maximum field, I H. At the half-pwer pints where the field is reduced by a factr f max max 1, I H half pwer max. The current at half pwer is equal t half f the maximum current, Ihalf Imax. The frequencies, f and 1 f, at which the current falls t half f the pwer maximum current, give the cavity bandwidth, f f f1, needed fr ur calculatin f Q. T bserve the resnances in cavity by varying the scillatr frequency, first we need t find the apprximate psitins f the plunger crrespnding TE and TE10 mdes. This is necessary because f the limitatin in the tunability f the scillatr (typical frequency range is.95-

14 Physics 401 Expt 44 Page 14/17 Micrwave Cavities 3.5GHz). The frequency will be swept by applying a vltage t the mdulatin input f the scillatr. Frequency f the scillatr is cntrlled by the vltage V m (DC r slw changing) applied t V_Tune IN cnnectr. The range f the mdulatin vltage is 0-10V (DO NOT EXCEED THIS RANGE). Mdulatin vltage V m is supplied by Wavetek functin generatr Typical FM mdulatin calibratin curve fr the Micrwave scillatr. DC ffset vltages we can btain (FG). By adjusting amplitude and frm FG triangular wavefrm f psitive plarity. Scanning frequency (frequency f FG) is nt very imprtant parameter and it can be chsen in a range Hz. Frequency sweep range prvided by Micrwave scillatr (f~ ghz) is usually t large fr recrding f the resnance curve. But by changing f the amplitude f the utput signal f FG, we can adjust the frequency range and vary the FG ffset vltage change the mean frequency f the Micrwave Oscillatr. T bserve the resnance the scillscpe shuld be switched int the X-Y mde and detectr utput signal shuld be cnnected t channel (Y) and V_Tune OUT t channel1 (X). Bth channels shuld be set t 1MΩ terminatin. Trigger the scpe using Chan 1. Measure the respnse frm the dide detectr using the Tektrnix TDS301B scillscpe and dwnlad the resnance curve t the cmputer. The dependence f the utput frequency f the Micrwave Oscillatr n the mdulatin vltage (V_Tune IN) is very clse t linear and crrespnding equatin is supplied t each Micrwave Oscillatr (sticker n the scillatr case). This analytical equatin (f(v_tune IN)) will allw yu t cnvert the x-axis f yur vltage data t the frequency. Plt yur data as dide respnse as a functin f frequency and include data fr the TE and TE10 mdes. Remember that the dide is a square law detectr, thus yur measured respnse is prprtinal t the square f the electric field amplitude. Calculate f f f1, and Q.

15 Physics 401 Expt 44 Page 15/17 Micrwave Cavities Sample data taken fr the TE and TE10 mdes. In this measurement, the frequency f the drive is held nminally fixed (near 3GHz) and the length f the cavity is adjusted in rder find the resnance. The prcedure fr finding these mdes is as fllws. First calculate using Eq. (1) the length f the cavity that crrespnds t the TE and TE10 mdes. Place the plunger t be clse t the expected lcatin fr a given mde. Next, begin sweeping the frequency f the dide and mnitr the respnse frm the detectr using the prcedure described abve. In rder t quickly find the resnance, yu can carsely mve the plunger by hand near the lcatin that yu expect the resnance t ccur and bserve the respnse. The resnance will g by quickly s it will take a little practice be patient. After yu lcate the resnance, yu can fine tune its psitin by turning the thumb screw n the plunger. Measure the length f the cavity and cmpare yur results with the predicted value. Cmpare the measured value f Q with the value f the unladed Q frm the calcula- tin carried ut by M. Schwartz in the intermediate electricity and magnetism text, Principles f Electrdynamics, n pages fr the TE mde, Q abca c 3 3. (11) b a c ac a c In this expressin is the skin depth at angular frequency,. The skin depth is, where the resistivity f the cavity made f red brass, H/m m and the permeability,. (T be cmplete, nte that Schwartz has several errrs in his derivatin, and his expressin differs frm Eq. 11 by a factr f. Eq. 11 is crrect. See, fr example,

16 Physics 401 Expt 44 Page 16/17 Micrwave Cavities Technique f Micrwave Measurements, Vl. 1, edited by C. G. Mntgmery, sectin 5-4, fr expressins fr the Q f a rectangular cavity in an arbitrary TE mde.) VERY IMPORTANT: The Q f the cavity depends critically n the cupling. In rder t be able t bserve the intrinsic Q f the cavity (the unladed Q), yu shuld minimize the cupling t the cavity fr this part f the lab. Yu can d this easily by rtating BOTH the drive and the pickup lps s that the plane f the lp lies near the xy-plane. Nte if it were t lie exactly in the xy-plane, there wuld be n cupling t the TE mdes and yu wuld nt bserve any signal. Yu will bserve that as yu increase yur cupling by rtating the lp int the yzplane, yur signal grws but the Q decreases. Yu shuld ptimize the cupling s that yu have a nice measureable signal, but with as high a Q as pssible. lmn

17 Physics 401 Expt 44 Page 17/17 Micrwave Cavities REPORT Include data and sample calculatins fr each part f the experiment. Include data analysis where necessary. Discuss briefly abut yur results and bservatins in each part. 1. Part I: (a) What is yur wavelength in the sltted line? (b) What is the scillatr frequency? (c) What is the per cent uncertainty btained frm yur measurement?. Part II: (a) What are the values f c fr bth the TE and TE 10 mdes? (b) Estimate the per cent uncertainties fr the c. 3. Part III: (a) Plt the cavity field detectr current versus the angle f input cupling lp plane. (b) Draw a sketch f the magnetic field lines inside the cavity and discuss hw the current varies when the cupling lp is rtated. (c) Frm the shape f the plt, can yu explain whether the detectr current is directly prprtinal t the square f the magnetic field? 4. Part IV: (a) Plt c Discuss the nature f the tw plts. versus. (b) On the same graph, plt sin c versus. (c) 5. Part V: (a) Frm yur measurements, calculate Q. Estimate its uncertainty. (b) Calculate the unladed Q frm Eq. 11. Cmpare the tw values.