Experiment to Produce A Spinning Electromagnetic Field in a Cylindrical Cavity

 

Mode and Frequency Used

 

On this page are details of the method used and the results obtained when carrying out the experiment to produce a spinning field. Also described are the problems encountered, so if you wish to carry out similar work I hope this information will help avoid some of the potential difficulties. The cavity used was chosen to be resonant in the 70cm amateur radio band which covers 430-440MHz and for which a transmitting license was available. The cavity was of cylindrical form as this was easier to construct than a sphere and was intended to be resonant at 435MHz in the centre of the band. It was decided to use a TM mode, so the lowest frequency spinning mode (TM110) was selected as this allowed the physically smallest size of cavity to be constructed. If a TE mode were used TE111 would be even smaller and have fewer possible interfering modes although the required measuring probe positions would be different from those described here. The resonant frequency of the TM110 mode is independent of the cavity height so the diameter can be chosen solely for resonance at the desired frequency and works out to 84.06 cm diameter. (See Tables of Cylindrical Cavity Resonant Modes for the formula used). As the resonant frequency of many of the other modes also depends on the cavity height it is important to choose a height which does not bring another resonance near to the desired one as this would make setting up extremely difficult. The resonant frequencies of the other modes for the cavity used are also shown in the Tables of Cylindrical Cavity Resonant Modes. Mode charts are available which show graphically all the resonant frequencies of a cavity and these are ideal for a rapid frequency selection if a different frequency is to be used. A typical Mode Chart is shown in Fig 30 below:-

 

 


 

 

In this case a height of 50 cm was chosen as this gives a ratio of diameter to height of around 1.68 ( = 2.82 ) and is clear of other modes as shown by the “X” in the above chart.                                                              

 

Cavity Construction

 

The cavity was fabricated from 18 s.w.g. half hard copper sheet. The maximum length sheet available was 8ft (2.44metres) which meant that in order to give the required 2.64 metre circumference a join was necessary. For this a butt joint was used with an exterior 18 s.w.g. backing strip just 1.3 cm wide added for strength. It was decided to braze the cavity using high silver content brazing alloy. This was done in order to give minimum cavity losses and maximum Q. Before brazing the cylinder joints were bolted together and end plates temporarily held in place so that a check of the resonant frequency could be made. It appeared that the frequency was about 2.5MHz too low so 1.6 cm was removed from the circumference to bring the frequency back to 435 MHz. It was subsequently realized that it was the poor bolted joints that had caused the frequency reduction and it would have been better to have constructed the cavity to the exact calculated size. Fortunately even with the reduced diameter of 83.4 cm the cavity was still resonant within the 70 cm band at just under 438MHz.

 

At the required brazing temperature of 6500 C, copper of this thickness goes very pliable and will not hold its shape when heated. As the gap for a brazed joint must be kept between say .02 - .15 mm it was necessary to rivet the joints at least every 5 cm using copper rivets. The circular cavity end plate was annealed and given a 1cm flange by bending the edges around a circular wooden former. The end plate was riveted in the cavity prior to brazing. After brazing it was necessary press the wooden former through the cylinder to maintain the cavity’s circularity. For anyone wishing to construct a similar size cavity I would suggest soft soldering as this could be done using temporary clamps instead of rivets as the soldering temperature is lower and the joint gaps are not so critical. The end plates could be left flat and soldered straight onto the cylinder ends without the need for a flange as the copper would distort much less. Also the copper would not loose its hardness at soft solder temperature which would help keep it circular. To keep the Q high just keep the joints as tight as possible and remove any surplus soft solder from inside the cavity.

 

Despite having gone to such lengths to get maximum Q it was finally decided to only braze the bottom cylinder end plate and at the other end a flat copper plate was just placed in position on top of the cavity. To make better contact it was weighted down all around the edges with a total of 50 kilo in weight. The loose end plate was initially used in order to maintain access to the cavity for fitting probes and as it did not give any major problems it was then left like that (so much for the high Q construction!!). The theoretical Q for a cavity of this size is about 50,000 and a loaded Q of 20,000 was actually measured which is adequate for this particular experiment.

 

Measuring Equipment

 

Initially the cavity was energized in the conventional stationary mode. A Yaesu FT-100 transceiver was used which provided a 12 watt radio frequency (r.f.) power source of high frequency stability and with a built in digital frequency readout. This source fed a small 4.5 cm diameter copper wire loop projecting into the cavity half way up the side.   The r.f. was fed through the cavity wall using a 50 ohm BNC socket onto which the loop was directly connected. By not quite fully tightening the BNC socket it was possible to rotate it and the loop from outside the cavity. This provided a very useful means of matching the loop impedance to the r.f. source as the effective area of the loop is the area that is normal to the magnetic field and the loop impedance is proportional to the area of the loop squared. For the modes used in these tests the maximum loop impedance is with it positioned vertically and as it is turned to the horizontal the impedance falls to a low value.

 

For the TM110 mode the maximum value of electric field occurs at a distance of half the cavity radius away from the cavity centre axis. A ring of probes was therefore placed in the top cylinder end plate on a circle about the cavity centre (the circle being the same diameter as the cavity radius). The probes were positioned at 45 degree intervals around the circle. Each probe comprised a 3 cm length of copper wire connected to a BNC socket which provided the feed through the cavity top end plate. The voltage on the probes was measured using a Tektronix 2445A, 150MHz oscilloscope. Unfortunately a higher bandwidth oscilloscope was not available but tests showed that although both channels suffered a 10 times reduction in sensitivity at 440MHz, deflection was linear provided 60% deflection was not exceeded. The phase shift difference was within 0.18nSec and constant and unaffected by changing input voltage amplitude within this range. So, although not ideal, this instrument was adequate for the measurements to be made provided it was operated within the voltage limitations and the constant phase error allowed for. This instrument did have the advantage of having four channels. Two of these were of limited sensitivity and had different calibrations from the main channels at 440 MHz so they were not used for measurement. However, they were still useful for monitoring the voltage on extra probes which assisted in initially setting up the rotating field.

 

 It was found that use of the standard x10 oscilloscope probes was not possible due to signal pick up which gave totally unreliable readings. Instead a number of small metal boxes, each fitted with a BNC input plug output socket were made. An 82kohm resistor was fitted in each box, connected between input and output, to reduce the oscilloscope loading. It was found that fitting one of these isolation boxes to a cavity probe significantly detuned the cavity but fortunately connecting the oscilloscope to an isolation box output did not further affect the cavity. Because of this eight isolation boxes were made and one connected to each cavity probe. These boxes should be identical and low tolerance, matched value, carbon resistors used. Connection from these isolation boxes to the scope was with about a metre of 50 ohm RG58C/U coax cable fitted with BNC connectors. Four coax cables were used and all cables were initially made the same length. The cables and isolation boxes were numbered and a particular cable and isolation box was used only with an assigned oscilloscope channel. The lengths of the cables were then adjusted slightly so that all oscilloscope channels gave identical phase readings. Measurements could then be made on a probe by swapping its isolation box for one with a monitoring oscilloscope channel attached and this would have a negligible effect on the cavity tuning. Using screened connections and leads throughout in this arrangement gave fairly stable and repeatable voltage and phase readings.  The numbering used for the cavity probes and energizing loops is shown in a plan view of the top of the cylindrical cavity in Fig 31 below. This also shows the second energizing loop which was used for the later spinning field tests:-

 


Tuning The Cavity (Conventional Mode)

 

To tune the cavity the r.f. source frequency was adjusted to give maximum oscilloscope deflection with the oscilloscope connected via an isolation box to the cavity end plate probe number three, positioned opposite to the input loop. Only input loop 1 was fitted for these experiments and it was turned for best matching of the 50 ohm r.f. source to the cavity. This was determined by measuring the standing wave ratio (SWR) between the source and the loop using a cheap commercial SWR bridge. It is necessary to retune the r.f. frequency for maximum oscilloscope deflection each time the orientation of the loop is altered as turning the loop even slightly detunes the cavity. Best match was an SWR of about 2.0 and usually occurred with the loop turned about 20 degrees from the vertical for the 4.5 cm diameter loop. From theory the loop should have a low inductive reactance and that is why a 1.0 SWR is not obtained. It was found that if the r.f. source was tuned slightly away from peak oscilloscope deflection then a 1.0 SWR could be obtained. In this condition the cavity is being tuned off resonance and the cavity reactance produced as a result of this is tuning out the loop reactance. However, a 2.0 SWR was perfectly adequate and was considered preferable to having the cavity detuned. Because of this most tests were carried out with a reduced power of only 2.5 watts in order to protect the source from the high r.f. voltages which can arise with a raised SWR. This still gave sufficient signal for measurement and also protects from the high SWR which results if the cavity is further mistuned. It would be possible to get a lower SWR by using a matching circuit but the additional complication was again considered unnecessary at these power levels.

 

For a small coil the magnetic field is a maximum passing through the centre of the loop. However, there is also a magnetic field produce at right angles to this off the sides of the loop. As the size of the loop is increased the proportion of field off the side increases relative to the field through the centre. These two loop fields energize the same cavity mode (ie TM110 in this case) but the two modes are at right angles to each other. If the cavity is not exactly circular the two modes will have slightly different resonant frequencies and the two resonant peaks in the cavity probe voltages may be observed. The main mode being measured here has a maximum electric field adjacent to the loop (i.e. probe 7) and also 180 degrees around the cavity diameter from this (i.e. probe 3). This can be checked by measuring the voltage on all the cavity end plate probes. In all tests the peak to peak voltage was measured on the scope and is what has been used in the typical results shown below:-

 


Fig 32A(a) also shows the theoretical shape of the voltage readings it would be expected to obtain at the cavity probes. It has the shape of an |A sin(ф-ε)| curve where  ф is the phase angle around the cavity measured from the point of minimum field and the amplitude (A) and phase error (ε) have been selected to best fit the experimental results. Each probe is spaced 45 degrees so the phase error can be obtained from the zero of the best match theoretical curve, which is at 4.85 instead of 5.00, so the error is 45 x (5.00 – 4.85) = 6.75 degrees. It could be due to misalignment between the end plate probes and the energizing loop. However it is most likely due to a small voltage being present from the field off the end of the loop which is adding to the main field and altering the null position. Apart from this the measured voltages are in reasonable agreement with the theoretical values. The frequency was 438.39870MHz.  Fig 32A(b) below is the plot of the phase measured between probe 3 and each of the other probes. This was obtained by measuring the time difference in reaching the peak voltage and this is also as expected for a TM110 cavity mode:-

 

 


 

If the cavity is tuned to the second mode which is being energized by the field off the end of the loop this will be when the field at the probes 90 degrees from the loop (probes 1and 5) is a maximum. In this case it was at a frequency of 437.3621 MHz, just over 1 MHz below the main mode. Depending on the shape of the cavity it could easily have been up to 1 MHz or more above. A typical set of peak to peak probe voltages and phase readings is:-

 

 


 

 


The theoretical voltage curve is now of |A' sin(ф-90-ε)|. Despite the phase measurements not fitting the theoretical curve so well the readings clearly confirm that the theoretical TM110 mode is still being energized and that it is at right angles to the previous main mode. In this case probe 1 was used as reference as probe 3 has a low voltage so it would be more difficult to take accurate phase readings to this probe without switching ranges on the oscilloscope which was not permissible. The SWR for this mode was 5.5 and was with the coil vertical which indicates that there is a poorer impedance match. However, even so, the maximum cavity probe voltage was almost identical to that obtained for the main mode.

 

It was found that both modes went to a minimum when the energizing loop was rotated to the horizontal position. This provided a useful means of confirming the loop position when the cavity was completely assembled.

 

 How to Produce the Spinning Mode in the Cavity

 

The standard method of producing a rotating field is to use two energizing loops spaced 90 degrees apart around the circumference and fed with signals time displaced 90 degrees. This is similar to the way a two phase motor produces a rotating field. It is just as applicable at r.f. frequencies and is the best method found to produce a purely spinning field. The voltage plots in Fig 32A and 32B for the single loop stationary modes confirm that the field in the ф direction is sinusoidally distributed and it is also known that it is varying in amplitude sinusoidally. We can show mathematically that a spinning field is produced by writing the field from the single loop as:-

 

       E1 = F.

 

Where F is a function representing the field distribution in the radial and z (height) directions. NB. The expressions used for the phases are just representative, no attempt has been made to match them to actual loop positions used.

If there is another loop 90 degrees apart around the circumference and the current is 90 degrees phase shifted then the field from this loop will be:-

 

       E2 = F.

 

The combined field with the two loops energized will be the sum of these two fields:-

 

       Et = E1 + E2 = F(+)

 

Using the standard trig relationships:-

 

2cosAcosB = cos(A+B) +cos(A-B)

   and

2sinAsinB=cos(A-B)-cos(A+B)

 

This gives:-

 

    Et = (cos(ф-ωt) –cos( ф+ ωt) + cos( ф+ ωt) + cos(ф-ωt))

 

         = F cos(ф-ωt)                                                 (1)

 

In some respects this is just a different way of writing the addition of two stationary fields but it also represents a completely new field system which is physically spinning in the +ф direction. The reality of the spinning field in the similar two phase motor case is apparent from the rotational speed and torque developed by the motor and is just as real, although not so obvious, in this r.f. field.

 

The simplest way to produce the required current phase difference is by feeding the two loops in parallel and using a quarter wavelength longer feed line to one of the probes. Ideally, if the loop impedance is 50 ohm then connecting them in parallel would present the r.f. source with a 25 ohm impedance. However, a quarter wavelength of 75 ohm line will convert a 50 ohm impedance to 100ohms so this was used in each loop feed line to convert the loop impedance to 100 ohm so the parallel impedance would be 50 ohm. A diagram of the phasing line used is shown in Fig 33 below:-

 

 


 

The phase delay produced by a length of line does not normally just depend on its length as the reactive impedance of the load also affects it considerably. Fortunately for lines which are exact multiples of a quarter wavelength, such as are being used here, the phase delay is independent of reactive impedance. If the two loops have the same reactance the extra quarter wavelength in the loop 2  line will convert it’s reactance from inductive to capacitive so when the two are connected in parallel there will be a tendency for the reactances to cancel, although not normally exactly. However, this is the ideal case and one problem is mutual inductance between the loops and this will be opposite for each loop as the voltage from one loop will be in phase by the time it  reaches the other but the voltage from the other loop will be in antiphase at the other.  It was thought that this might be a problem and in practice the best spinning field was obtained by tilting the two loops to different angles so the above theoretical perfectly matched condition was not used. However, using this phase delay line it was found possible in practice to adjust the currents in the loops to obtain a spinning field provided an SWR of about 2 was acceptable at the r.f. source, which it was.

 

A quarter wavelength in free space is about 70/4 = 17.5 cm long. This is not the length required for a quarter wavelength of coaxial cable as the wave in the cable only travels at about 0.65 of the free space velocity. This is known as the velocity factor and varies depending on the cable construction.  A quarter wavelength of coax is therefore approximately 17.5x0.65 = 11.4 cm long. Experimental methods to obtain an exact electrical length of coax cable rely on the fact that an open circuit quarter wavelength of line (also ¾, 1¼…etc wavelength) will appear to be a short circuit if the impedance is measured at one end. Conversely a short circuited half wavelength (also 1, 1½, 2…etc wavelength) of cable will appear to be open circuit if the impedance is measured at the opposite end to the short. If you have an r.f. impedance bridge you can measure the cable impedance directly. If not there are various simple techniques available which make use of these characteristics. The main error is due to additional impedance caused by the measuring system and cable terminations used. The method adopted here was with the r.f. source, set to the resonant frequency of the cavity, used to apply an r.f. voltage to the cable under test. The test length of cable was energized through a resistor network so that when the cable was a quarter wavelength long the voltage across it, which was monitored using the oscilloscope, would be a minimum. The circuit used is illustrated below. The two resistors ensure the source always feeds into a reasonable SWR load:-

 

 

 


 

 

The earths were connected to a small copper plate and made as short as possible and the resistors were carbon. The coax line under test is cut, removing about 3mm at a time and the voltage on the scope recorded each time. The minimum voltage will be reached when the line is a quarter wavelength long. It is likely you will overshoot first time but the required length will now be known. The same technique can be used to find a ¾  wavelength. (and if it’s not three times the length of the quarter wavelength cable you have a problem with the technique!). Half wavelength could be found by repeating the procedure but with a short circuit on the far end of the line. However due to the inconvenience of fitting the short it is easiest to just use a cable twice as long as the measured quarter wavelength.

 

Tuning the Cavity for the Spinning Mode.

 

To tune the cavity the two loops are first energized using the phasing line and the voltage on cavity probes one and three monitored on the oscilloscope. If the r.f. source frequency is adjusted either side of the calculated cavity resonant frequency it is likely that two peaks will be observed on the oscilloscope. These will be the main and second modes measured above. To bring the two resonances to the same frequency a substantial octagonal wooden frame was built around the cavity. This nominally touched the cavity in eight places around the diameter although the frame was constructed slightly oversize so that there was about 2mm clearance between the frame and the cavity at these points. The frame was fitted with four simple screw jacks each with a vertical load spreader bearing against the top half of the cavity circular sides at 45 degree intervals around the periphery. Two were on the same axis as the two energizing loops, one midway between them and one 45 degree away (i.e. adjacent cavity probes 4,5,6 and 7). The load spreaders were simple hardwood strips about 23x5x0.7cm, each with wooden block about 5x5x2cm attached which had a hole drilled through it to locate the screw of the jack. Four notches about 5.5x1cm each were cut in the octagonal frame to make room for the load spreaders. This arrangement is shown in Photographs 1, 2 and 3.

 

Adjusting the circularity of the cavity with the screw jacks allows the main and second resonant peaks to be brought to the same frequency. The tilt of the loops can then be adjusted until the voltages on the cavity top plate probes are as identical as possible and the phase difference between each adjacent probe is 45 degrees. This is the condition which is required for a spinning field to be produced. For this adjustment it was useful to have the two additional oscilloscope channels available which were temporarily connected to probes 2 and 4 to approximately confirm that the voltage and phase here appeared to be correct before a full set of all probe readings were taken using the two calibrated oscilloscope channels. To reach the correct condition requires repeated small adjustments to the r.f. frequency, cavity circularity and loop angle as the adjustments are very critical and inter-related. Because the cylinder top plate was not soldered in position the tuning could also be disturbed by a change in top plate contact resistance particularly as the cavity top plate probes were interchanged to take scope readings. So after taking a complete set of readings the initial probe readings were again checked and if these had changed significantly the results had to be discarded and a new set of readings taken. An example of typical readings obtained is shown in Figs 36(a) and 36(b) below:-

 


         


 

Although the probe voltages are not quite identical the results are in reasonable agreement with theory and indicate that a spinning field has been produced in the cavity. This is the best technique found for producing a pure spinning field in the cavity.

 

Effect of Change in Phase or Amplitude in the Field of an Energizing Loop – A Simple Theory

 

The affect of tuning the cavity can be analyzed mathematically using a similar method as was done previously. If we consider just the main mode field and assume that these fields from the two loops have different amplitudes and a phase difference between them which is not exactly 90 degrees then the equation for the field from loop 1 is:-

 

       E1 = F.

 

Where F is again a function representing the field distribution in the radial and z (height) directions.

If the current in loop 2 is 90 degrees phase shifted then the field from this loop will be:-

 

       E2 = F.A.

 

Where A is the ratio of the field amplitudes and a represents the phase error from the correct 90 degree phase difference. The combined field with the two loops energized will be the sum of these two fields:-

 

       Et = E1 + E2 = F(+A)                    (2)

 

The variation of Et with time at different cavity top plate probes (ie values of phi) can be plotted for various values of A and a. First assuming the phase difference is constant at 90 degrees (ie a is zero) the effect of a change in A, the relative amplitude of the current through one of the coils is shown in Fig 37:-

 

 

 


 

It can be seen from these plots that with equal field amplitudes from the two loops the voltage measured at all the top plate cavity probes will be the same and the phase difference measured between them will be the same as their physical angular separation (i.e. 45 degrees in this case). This is the purely spinning field which was also created experimentally as shown previously in Figs 36(a) and 36(b).

 

As the field produced by one of the loops reduces the voltage on the cavity probe adjacent to the loop and directly opposite it also reduce. The phase difference between the maximum and minimum voltage probes (which are either adjacent or opposite the loops) stays fixed at 90 degrees but the phase difference between a probe at maximum voltage and a probe 45 degrees from it reduces considerably.

 

If the amplitudes are the same but the phase difference varies from 90 degrees, by the angle alpha, then the measured fields will be:-

 

 


 

In this case when the phase difference between the energizing coils deviates from 90 degrees it will affect the amplitude as well as the phase of the voltages measured at the top plate probes.

 

The above two figures show the sort of effects that are observed when tuning the cavity for a spinning field. If just two cavity probes are to be observed for the initial tuning then one technique is to use two probes physically spaced 45 degrees apart (probes 1 and 2) and adjust for equal voltages and 45 degree electrical phase displacement. Then check two probes spaced 90 degrees apart and opposite the energizing coils (probes 1 and 3) to check that each of the two component fields are the same magnitude.

  

The above analysis illustrates the difficulty in obtaining a set of readings in which all probe voltages are exactly the same magnitude and the phase difference between them is the same as their physical angular separation. This would only occur if the two component fields were absolutely identical in amplitude and have exactly 90 degrees phase difference. If this condition were not exactly met then a spinning field would still be produced but in addition there would be a small conventional stationary field superimposed on it.

 

Obtaining a Spinning Field With Just One Loop Energized

 

The above explanation for obtaining a spinning field from two loops has for simplicity assumed that each loop only produces a main cavity mode. In practice the loops also produce a substantial second cavity mode at right angles to the main one. It was observed that the optimum spinning field was obtained not with the two loops rotated to the same angle but with loop 2 almost vertical and loop 1 almost horizontal. This means that loop 1 would have been making only a minor contribution to the total cavity field. The most likely reason for this is that the second cavity mode produced by loop 2 is providing a large portion of the required field it has previously been assumed loop 1 main mode would produce. The spinning field is still being produced by two fields at right angles and with a 90 degree phase difference but the source of each of these two fields is not solely its associated loop.

 

To test this a further experiment was done with just loop 1 energized but the cavity circularity was adjusted to bring its main and second modes to the same frequency to see if a spinning field could be produced. For a spinning field it would require the main and second modes to be equal amplitude. Looking at Fig 32A(a) and Fig 32B(a) it can be seen that for the 4.5cm diameter loop used they very nearly are. There must also be a 90 degree phase difference between them. Typical results obtained are shown in Fig 39(a) and Fig 39(b) below:-

 


 

 


 

The above figures confirm that just one loop is able to produce a substantial spinning field. Fig 39(a) indicates that out of a peak probe voltage of 140mV the spinning field is just over 70mV.  It is possible to re-plot equation 2, for the total field (i. e. Et = F(+A) ), to give the calculated phase difference and voltage at the probes. This has been done on the above figures and is shown by the green crosses for a phase error (α) of 33 degrees. This is equivalent to a phase difference of 57 degrees between the main and second fields, assuming that they are the same amplitude. This phase error gives the best fit to the actual readings and so it is reasonable to assume that this is the phase error which existed when the test results were taken. It was found difficult to improve on this with just one loop energized as the only adjustment available was the circularity of the cavity.  It does, however, illustrate the ease with which a limited amount of spinning field can be produced.

 

 

 

 

 

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