Figure 1. measurement of the kicker in the EM Lab shows the kicker line-source measurement being performed. The kicker is the stainless steel cylinder in the foreground.
The first kicker concept design for the Advanced Hydro Facility (AHF) was constructed to allow two plates to be driven thus directing, or ``kicking'', the electron beam into two subsequent beam lines. This bi-driven kicker contains two grounded plates and two driven plates (each with an input and an output port) allowing these two driven plates to be connected to external pulsers. The following electromagnetic measurements were performed on the bi-kicker during the Summer and Fall of 1997 and preceded the fabrication of the quad-kicker in the Fall of 1997. The quad-kicker contains input/output ports for all four plates and is designed to have two or four of its deflection plates driven with external pulsers.
This report covers the electromagnetic measurements performed on the bi-driven kicker (a four-port system with two ports per driven plate) and also measurements using a line current source to simulate a stiff electron beam (yielding a six-port system). The results include the cross coupling between ports, the resonances and anti-resonances due to the structure of the kicker, the electromagnetic coupling between the current source and the kicker plates, and the variation due to the line-current position inside the kicker.
The electromagnetic testing of the kicker was performed in the LLNL Electromagnetics Laboratory (EM Lab) and included port measurements of the kicker to identify resonances in the structure. These measurements served as the starting point for the characterization of the kicker because:
After adding the line current source (a simulated stiff beam), the next level of detail in the performance of the kicker was achieved:
These measurements were performed over the frequency band from 100 kHz to 7 GHz and also included a datum point at DC to illustrate the differences between the line conduction-current source used in these measurements and an actual electron beam. Additional measurements were made in the time-domain using an ideal pulse with the same pulse width as ETA-II to compare against the frequency-domain measurements.
Figure 1. measurement of the kicker in the EM Lab shows the kicker line-source measurement being performed. The kicker is the stainless steel cylinder in the foreground.
The bi-kicker consists of two driven deflection plates and two plates that terminate into the wall of the beampipe. Each driven plate contains an input and an output port. These ports were used for the matrix of linear electromagnetics port measurements that were initially performed on the kicker during the Summer of 1997. From a microwave standpoint for this configuration, the kicker looks like a standard four-port network. The input/output beampipe was covered with microwave absorber to prevent room-interference issues from corrupting the data. Without the absorber, these room-interference issues were observed increasingly above 1.6 GHz1. So any potential room effects are in the operating band of the absorber (1 - 8 GHz). The absorber was used for all of the port measurements to prevent room-interference from outside of the kicker.
During the Fall of 1997, the port measurements were repeated to insure that the kicker had not been damaged during it's time in the ETA-II beam line and that the beam line results that were observed were not the result of a mechanically damaged kicker. The latest data sets agreed with the previous data sets taken on the kicker in the Summer (see Table 3 for additional details).
Figures 2,3. a comparison between the transmission (in dB) vs. frequency (in Hz) of the ``right'' and ``left'' sides of the kicker indicating s21 (black) vs. s43 (red) shows only a slight change in the operating characteristics of the kicker at these frequencies. One physical change identified in the kicker was a slippage of one of the insulator pieces that occurred between the Summer and Fall measurement series.
This measurement sequence continued by including a line current source to simulate a stiff electron beam. This source was driven both on-axis and at a radius of 1/2 the pipe radius in 15° increments.
Finally, a series of time-domain measurements were performed to compare with the frequency domain measurements and to simulate an ETA-II pulse-width beam. These time domain measurements used both an impulse generator and a rectangular pulse generator.
The experiments were conducted on the kicker in the EM Lab in the frequency domain using a low frequency system (HP3577, 100 kHz - 200 MHz) and a high frequency system (HP8510, 45 MHz - 18 GHz). For the kicker measurements, the 100 kHz - 7 GHz band was used because it encompassed the regions of interest of the kicker. The high frequencies were included in the measurement because there are coaxial and waveguide TM modes for the kicker in the 1.8 GHz, 4 GHz, and 6 GHz bands (TM modes couple strongly to the electron beam.) The low frequency bands include the resonances of the kicker structure at multiples of 140 MHz corresponding to the electrical half-wavelength length of the plates in the kicker. There are anti-resonances spaced roughly half-way between the 140 MHz resonances. The specifics of these frequency measurements will be shown in the RESULTS section.
The time domain measurements used fast waveform generators to create idealistic waveforms to compare against the frequency-domain data and to simulate a typical beam pulse-width. Both a fast impulse source and a rectangular-pulse source were selected as two bounds on the problem. The impulse generator represents the leading edges of beam current pulses while the rectangular pulse source captures the main part of the pulse. These two allow different characteristics of the kicker to be isolated.
Figure 4. a canonical impulse waveform from the impulse source is shown in Volts vs. time (ns). The full-width-half-max (FWHM) of the pulse is 68 ps. The plot on the left shows the long-duration view of the pulse while the plot on the right is a zoomed in view of the main pulse.
Figure 5. a typical 50 ns rectangular pulse from the rectangular pulse source in Volts vs. time (ns) shows the stability of the main part of the pulse. The rise/fall times of this rectangular pulse are not representative of the ETA-II pulse but was used only to generate an idealistic rectangular current pulse. The rise and fall times used with this rectangular pulser were on the order of 500 ps and 1 ns respectively.
The RESULTS section contains more details about the pulse's traversal through the system based on the input impedance of the kicker at the excited ports.
Goals of the testing include the identification of the modes being exciting inside the kicker, determination of the coupling between a beam and the kicker ports, and identification of the strengths of monopole/dipole/quadrupole modes, and parallels with the frequency-domain data. The comparison to frequency-domain data is important because the S/N ratio in the frequency-domain measurements is much greater than the time domain measurement.
An important feature of the EM cold tests are that it was possible to electromagnetically predict many of the beam/kicker interactions. What is missing was the true simulation of an electron beam passing through the kicker. Results were obtained for a pure DC line conduction current running through the kicker and information on how to approach an appropriate beam simulation experiment.
The measurement of the kicker consisted of two very similar test configurations: one for the time-domain measurements and the other for the frequency-domain measurements.
The time domain measurements used Picosecond Pulse Labs pulsers (model 3500D for the impulse source and model 2600 for the rectangular pulse source). These sources drove 50
impedance coaxial cables that were then connected to the 50 ports of the kicker. In the case of the line current source, the input impedance varied as a function of frequency but no attempt was made at this time to generate a broadband match for the line source2.
Instead, the results of the mismatch were subtracted from the final measurement. The resulting waveform on the output port of the kicker was recorded using a Tektronix CSA803 digital sampling scope that was adjusted accordingly to resolve the impulse or the rectangular pulse.
Figure 6. test configuration for the time-domain measurements. These measurements consisted of one of two sources driving the transmission line ports of the kicker (or the line current source) and recording the results on the output.
Figure 7. test configuration for the frequency-domain measurements. These measurements used two network analyzers to cover the frequency bands of interest. The measurements were made between the six ports on the kicker (the 4 plate ports and the 2 line-current ports).
The frequency domain measurements used an HP8510 network analyzer and a HP3577 low frequency analyzer. The 3577 was limited by it's test set3 to 100 kHz - 200 MHz due to the inability to get a broadband match for the line-source measurement. The 8510 was used for the 45 MHz to 7 GHz bands; connector effects were observed at the higher frequencies for this band.
Figure 8. this plot shows the performance of the high frequency characteristics (dB vs. frequency) of the connectors/adapters used in making these measurements. There is also a resonance at 1.7 GHz (see Figure 10) caused by the GR connectors. The upper curve (red) is the transmission through the combination of SMA/GR/coax adapter pairs while the lower curve (black) is the reflection coefficient looking into the combination. The plot was (arbitrarily) cropped at -25 dB. The low frequency transmission and reflection data (not shown) is better than as shown here with the exception of the 1.7 GHz point mentioned above.
The network analyzers were calibrated to remove the cabling effects from the measurements. Otherwise, it was a direct measurement of the characteristics of the kicker and the associated connectors.
In both the time domain and the frequency domain line-current source measurements, a microwave absorbing section was used at each end of the beam pipe when the port-to-port measurements were performed. In the case of the line-current measurements, a metal backing end cap was provided to drive the line-current against in a monopole configuration. A comparison was made to determine if any additional longitudinal modes were established because of the metal end caps but none were observed as the result of performing a qualitative measurement comparing the two configurations. The lower order longitudinal modes due to the kicker plates are shown in Table 2 and are already quite extensive without the metal end caps.
Figure 9. this image shows the end caps in-place on the kicker for the line-current measurement configuration.
The kicker was used in a ``bench-top'' configuration in the EM Lab such that the four ports were easily accessible. Additionally, the input and output ends of the beampipe were either filled with absorber or covered with the metal end-plates (as appropriate). During the testing, the unused ports on the kicker were terminated in 50 broadband loads which extended from DC to several GHz.
Figure 10. s21 measurement in dB vs. frequency (Hz) of a pair of SMA-to-GR adapters (SMA-to-GR and GR-to-SMA) shows a 1.79 GHz resonance (10.0 dB amplitude) caused by the GR connectors (see Fig. 8).4
Data was acquired using either a computer-based data acquisition system (in the case of the Tek803 scope and the HP3577 analyzer) or the built in storage device (in the case of the HP8510 analyzer). Data was typically acquired over the GPIB bus using the READ3577 or TEKDIG programs as appropriate. For these measurements, the usual data acquisition software (VIPER and WINDIG) were not used since the former two met all of the needs of the data management system. In all cases, data was stored as ASCII frequency, real, imaginary or time, voltage pairs. Connections between the computer and the scopes were made using GPIB connections.
Figure 11. test setup showing the computer connected to the scopes through the GPIB cables (red). In these cases, the computer was used for storage only. The RF cables are shown in blue and the trigger line is shown in green.
To make the measurement, several different port measurements and frequency bands were used to cover the entire test matrix. The reflection and transmission parameters were measured for both terminals used in a particular configuration. This greatly reduced the size of the test matrix accordingly:
| i\ j | 1 | 2 | 3 | 4 |
| 1 | * | * | * | |
| 2 | * | * | ||
| 3 | * | |||
| 4 |
The reason for all of the port measurements was to determine the cross port coupling coefficients and resonances, to determine the characteristics that the kicker pulser would see, and to examine the fabrication issues of the kicker.
The transmitted waveforms for the line-current time domain measurements are shown below.
Figure 12. this time domain waveform (Volts vs. nanoseconds) was used to simulate an ideal ETA-II pulse. Notice the extremely sharp edges on the ideal rectangular pulse. This represents a composite of a DC excitation and two fast impulses.
Figure 13. spectrum of the rectangular pulse in Figure 12 vs. frequency in Hz.
Figure 14. this time domain waveform [Voltage vs. time (ns)] was used for the impulse testing. The motivation was to drive the kicker only with the leading/training edge components of the waveform shown in Figure 12. Note that this pulse is much faster than what will be actually be exposed to the operating kicker.
Figure 15. spectrum of the impulse shows that most of the energy is in the band that was included in the CW frequency sweeps. 37% of the energy is in the band below 2 GHz, 80% of the energy is below 7 GHz, 90% is below 10 GHz.
These waveforms were selected because they are highly stable and easy to characterize. Unfortunately, they are not representative of the current pulse that actually travels through the kicker. Although the rectangular pulse has an ``ETA-II like'' pulse width, the leading and trailing edges are much faster than what an operational kicker will see in practice. To evaluate this effect, a fast impulse was used to separately simulate the effects of the leading/trailing edges of the rectangular pulse. This allowed for the separate evaluation of two effects: (1) the DC part of the line-current pulse and (2) the leading and trailing edges of the pulse. The frequency components of the impulse were significantly higher in frequency than the frequency components of the rectangular pulse.
The frequency domain port measurements for the kicker were made using the input/output ports of the deflection plates. These ports form a typical four port network and were numbered as follows:
Figure 16. port assignments for the kicker. Ports 1 & 2 are on the first driven plate and ports 3 & 4 are on the second driven plate. Ports 5 & 6 were assigned to the input and output of the line-source. When not used, ports 5 & 6 were replaced with absorber plugs to prevent interactions with the surrounding room.
The scattering parameters6 for these measurements were obtained for each of the port combinations. The low frequency resonances correspond to the half-wavelength length of the deflection plates. The port-to-port electrical length of the kicker is 4.08 ns but the taper-to-taper length is 3.62 ns and corresponds to 138 MHz. These resonances are seen periodically in the data in half wavelength multiples.
Figure 17. the complete frequency sweep of the s21 transmission along the deflection plate (dB vs. frequency) shows the low frequency plate resonances and the high frequency over-moding. Subsequent figures show each band in more detail.
Figure 18. the low frequency parts of the s21 transmission spectrum (dB vs. frequency) show the strong periodic resonance roughly every 140 MHz. The two close peaks in the 140 MHz sweep represent the start and end of the tapered region of the plate.
The first higher order mode (TE11) is at 1.31 GHz and is seen as the point where the higher order cavity mode effects start to come in to play.
Figure 19. the higher frequency ranges of the s21 transmission plots show the higher order modes that are established in the beampipe. The plot is in dB vs. frequency (in Hz).
Table 1 shows the higher order modes for an ideal beam pipe. For the kicker, only the TM modes interact with the beam. Also note that the slots in the kicker's beam pipe break the circumferential current flow and also allows the establishment of TEM modes. Additionally, recall that the higher order coaxial modes7 in the region between the beampipe and the outer can of the kicker are at 2.2 GHz for the TM01 mode (the first TM mode) and 480 MHz for the TE11 mode (the first TE mode; and continues up in multiples of roughly 480 MHz.)
| TE11 | 1.31 GHz | TE22 | 4.78 GHz |
| TM01 | 1.71 GHz | TE02/TM12 | 5.00 GHz |
| TE21 | 2.18 GHz | TE61 | 5.35 GHz |
| TE01/TM11 | 2.73 GHz | TM41 | 5.41 GHz |
| TE31 | 2.99 GHz | TE32 | 5.71 GHz |
| TM21 | 3.66 GHz | TM22 | 6.00 GHz |
| TE41 | 3.79 GHz | TE13 | 6.08 GHz |
| TE12 | 3.80 GHz | TE71 | 6.11 GHz |
| TM02 | 3.93 GHz | TM03 | 6.17 GHz |
| TE51 | 4.57 GHz | TE42 | 6.61 GHz |
| TM31 | 4.55 GHz | TE81 | 6.88 GHz |
Table 1. the ideal higher order modes8 for a straight smooth beam pipe with a radius of 6.7 cm are provided for reference purposes.
There is a strong 960 MHz resonance in the data but it was not confirmed as a circumferential mode. The absence of a lower order circumferential mode makes this less plausible but the 960 MHz coaxial mode (TE21) does have a quadrupole structure which roughly matches the geometrical structure of the four plates (see the RECOMMENDATIONS section). The longitudinal resonances are functions of the plate lengths and physical dimensions. Table 2 describes the resonances seen in Figures 2, 3, 17-19.
| frequency | amplitude | manifestation |
| 132 MHz | -0.6 dB | plate size + taper length |
| 138 MHz | -4.0 dB | plate size =  /2 resonance |
| 277 MHz | -4.9 dB | 2 * /2 resonance |
| 415 MHz | -5.8 dB | 3 * /2 resonance |
| 540 MHz | -1.5 dB | |
| 553 MHz | -5.9 dB | 4 * /2 resonance |
| 591 MHz | -5.1 dB | |
| 666 MHz | -6.4 dB | |
| 690 MHz | -5.6 dB | 5 * /2 resonance |
| 759 MHz | -7.1 dB | |
| 825 MHz | -6.3 dB | 6 * /2 resonance |
Table 2. longitudinal resonances in the kicker. These frequency resonance points agree (to within 1 MHz) with data acquired during the Summer '97 experiments (see Table 3).
The cross coupling between plates shows the induced coupling between adjacent plates in the kicker. In this case, both measured ports are ``upstream'' with respect to the direction of the electron beam flow.
Figure 20. comparison between upstream / upstream (s31) and upstream / downstream (s41) cross coupling in dB vs. frequency(Hz). The differences are due to the near-field effects near the drive point and the establishment of the modes along the length of the kicker. The up/down- cross-coupling, s41, shown in red has greater isolation (-45 dB) over the low frequency band than does the up/up- cross- coupling, s31, shown in black (-25 dB).
Figure 21. cross coupling data (s31, dB vs. frequency) from plate-to-plate shows good isolation9 (-25 dB) over the span of the low frequency band except at the resonances. At the higher frequencies, the coupling due to the establishment of the higher order modes is as high as -4 dB. The normal operating band for the kicker pulser is expected to be below the 100 MHz point.
A comparison between the s31 and s42 data has excellent agreement to within 0.6 dB below 700 MHz and diverges above that point. The reasoning is the same as was used for the information presented in Figure 3.
The shift in the port performance from the Summer of 1997 data to the Fall of 1997 data was due to the slippage of the insulator section. This was known at the time but the small effect did not warrant the disassembly and repair of this kicker (kicker #1).
| status before insulator slippage (Summer '97) |
status after insulator slippage (Fall '97) |
||
| frequency | amplitude | frequency | amplitude |
| 132 MHz | -0.5 dB | 132 MHz | -0.6 dB |
| 139 MHz | -2 dB | 138 MHz | -4.0 dB |
| 277 MHz | -4 dB | 277 MHz | -4.9 dB |
| 416 MHz | -4 dB | 415 MHz | -5.8 dB |
| 540 MHz | -1.5 dB | ||
| 553 MHz | -5.5 dB | 553 MHz | -5.9 dB |
| 591 MHz | -5 dB | 591 MHz | -5.1 dB |
| 666 MHz | -7 dB | 666 MHz | -6.4 dB |
| 690 MHz | -6 dB | 690 MHz | -5.6 dB |
| 759 MHz | -7.5 dB | 759 MHz | -7.1 dB |
| 826 MHz | -4.5 dB | 825 MHz | -6.3 dB |
Table 3. shift in the resonances from Summer 1997 to Fall 1997 shows the effects of the insulator slippage. Note that there was no discernible 540 MHz resonance during the Summer 1997 examination of the kicker.
Figure 22 shows a TDR (time domain reflectometry) measurement using a fast rectangular pulse applied to kicker port #1.
Figure 22. the TDR of kicker port #1 shows the input and transition regions on the taper (in Volts vs. time(seconds)). The main rectangular pulse is the output from the TDR scope. The two peaks on the right are the actual reflections described in the text below.
The taper in the kicker is a flair from the center pin of the input coax to the 13 cm wide contoured deflection plate. The TDR plot shows the leading and trailing edges of the taper at each end of the deflection plate. The leading edge of the taper is difficult to see because of the near 50 match. The distance between each end of the tapered transition region is 0.5 ns (the distance between the transition-start and the transition-end is 0.5 ns). The reflection is 28% (-11 dB) for the high frequency components and agrees well with the -10 dB reflectivity obtained using the frequency domain measurements.
The line-current measurements consisted of using an (optionally) isolated wire to drive a current through the kicker to simulate the current from an electron beam. For this series of measurements, the line current followed a straight path through the kicker (see the RECOMMENDATIONS section). The line current was then rotated around the central axis of the kicker in 15° increments to examine the induced modal structure in the kicker.
Initial measurements using the line-current drove the current source down the middle of the kicker on-axis. The effects of the input reflection coefficient can be more easily seen in the next section (time domain line-current measurements). The kicker was measured as a six port network (see Figure 16). The test setup is shown in Figure 23 and the reflection and transmission coefficients, s55 and s65, through the wire system are shown in Figure 24.
Figure 23. the line source was driven at the r/2 radial point and stepped in 15° increments around the circumference. Notice the SMA cable (referred to as port #5) connected to the r/2 line terminal on the end cap10. Kicker port #1 is the brass connector on the left and port #3 is on the right.
From Figure 24, the impedance of the wire can be easily calculated at the points where the coupling to the other four ports is small.
Figure 24. the reflection and transmission coefficients, s55 (red) and s65 (blue), for the wire measurement for the r/2 radial point, frequency (in Hz) vs. amplitude (in dB). The resonances agree with those seen in the electron beam [3]; especially at 220 MHz. Note that Sqrt[1 - s552 - s652] is the power into the terminations.
The characteristic impedance, Z0, of an unknown transmission line of length, l, connected to a load, ZL, given the input reflection coefficient, 
, seen from an input transmission line with characteristic impedance, Zf, is calculated using:
Rearranging terms and grouping constants to solve for Z0 yields a solution of quadratic form, thus
were known to reasonable accuracy (more specifically, if the loss term 
were known or could be assumed). Unfortunately, due to the poor conductivity and small size of the wire, is not negliable which means that is a complex quantity instead of being purly imaginary.
Using the parameters for the experiment
The two techniques agree moderately well but the TDR measurement is a direct measurement using high frequencies and is probably a higher confidence measurement.11
Using the lower frequency points (below 140 MHz) to avoid the structural resonances in the kicker and to operate in the lower loss regime, the impedance of the line was found. To calculate the impedance of the line, four simultaneous equations are established using two frequency data points ( = s55 for 45 and 60 MHz) and the two components (real and imaginary) of the equation:
It is assumed in this case that R is not a function of frequency.12
Solving the four equations numerically yields the values shown below for the transmission line parameters for the r/2 offset wire used during the line-current measurement:
| R | 93/m*(f/f0)1/2 |
| L | 1.4mH/m |
| G | 0 |
| C | 11pF/m |
Table 4. characteristics for the wire transmission line used during the testing for the line-current at the r/2 radial point.
The Smith chart in Figure 25 shows the comparison between the measured input impedance of the wire in the kicker and the calculated input impedance from the values shown in Table 4. The resulting pure real termination for the wire measurement to reduce reflections would be 370 when in the r/2 radial position. The value for the ideal eccentric line [2]
Figure 25. comparison between the measured input impedance to the wire (red) and the calculated curve using the numbers in Table 4 (blue) show good agreement except for a stray inductive term.
It should also be observed that because the kicker is a four-port network (with two additional ports added by the wire measurement) then some of the power can couple to the other ports (as seen above) and be dissapated. At the frequency points where the fractional power absorbed in the lines and the terminations, 1 - s552 - s652, is less than 15% (the 15% figure was just selected to get minima across the entire lower frequency band below 500MHz and is probably reasonable for the determination of broadband terminations) and the coupling between the wire and the closest deflection plate (s25 in this case) is less than 10%, then the impedance may be calculated. For these purposes, the lower frequency points (below 85 MHz) were sufficient. Note however that higher frequency points also satisfy this criteria and are periodic (roughly every 140 MHz.
Figure 26. the coupling between the line-source and the adjacent port, s25 in dB vs. frequency (Hz), for the r=0 (red) and r=R2=3.175 cm (blue) cases. The increased amplitude in the r=R2 case was expected due to the closer spacing. The absence of structure shows that the wire was not significantly perturbing the cavity in this frequency range.
Figure 27a,b. coupling between the line-source and the adjacent port, s25 in dB vs. frequency (Hz) vs. angle (degrees) for 100 kHz to 50 MHz and 50 MHz to 500 MHz plotted logarithmically in frequency. Note the sharp resonances at 140, 280, and 420 MHz.
The set-up for the time domain line source measurements is shown in Figures 9 and 23. When the impulse source from the PPL3500 was applied to the line-source at port 5 and the waveform transmitted along the line-source was recorded at port 6, we get the following waveforms and spectra:
Figure 28. the time domain waveforms driving the line-current source through the kicker. The first waveform (Volts vs. time(seconds)) is just the impulse source seen in Figure 14 while the second waveform is the time domain response of the line-current through the kicker. The last plot is the spectra of each (amplitude vs. frequency(Hz).)
The reflection in the transmitted waveform that occurs 8 ns after the initial pulse is the down-and-back reflection along the line-current wire due to the mismatched impedance of the line w.r.t. the 50 SMA cable feeding the line-current. Removing the pulse spectra from the received waveform, yields a composite s65 for the wire in the center of the kicker:
Figure 29. an effective s65 for the line-current derived from the time domain impulse source, dB vs. frequency (Hz). The resonances are periodic roughly every 110 MHz and agree with the frequency domain data shown in Figure 24.
The ``pump-up'' time of the kicker from the excitation of the line-source is shown in Figure 30 for the rectangular pulse excitation. The 90% pump-up time is on the order of 80 ns and corresponds to an L/R term (1/e pump-up time) determined by
Figure 30. voltage vs. time (ns). The effects of the square wave (Figure 5) transmitted down the line-source (s65*Vin) show the pump-up time of the kicker system and is on the order of 80 ns and represents about 10 down-and-back cycles through the kicker -- the first few of which are clearly visible in the leading edge of the plot.
Figure 31. voltage vs. time (ns). The effects of the square wave between the line-source and one of the deflection plates (s25*Vin) shows the dI/dt coupling during the leading and trailing edges of the square wave. The response at each edge is similar to the response from the fast impulse (Figure 4) and represents an oscillation superimposed on top of the decaying portion of Figure 30.
Finally, observe that Table 4 is for the kicker structure itself while Figure 30 is for the simulated beam. This shows the tight relationship between the kicker structure and the beam characteristics.
For completeness, a purely DC conduction current was applied to the line-source wire. As expected, all of the return current was confined to the contiguous conducting surfaces (to within 1:1012 ). Two configurations were specifically examined:
It should be noted however that the cavities and surfaces do have intrinsic ``pump-up'' times required for the effects to stabilize. The previous time domain results discussed this time delay in more detail and basically corresponds to an L/R time.
This one-wire technique differed from other accelerator component measurements which used a two-wire measurement technique [4,5]. The goal in those tests was to measure the effects of the components on the beam. The one-wire technique described in this document uses one wire to simulate the electron beam. The goal of the one-wire tests was to measure the effects of the beam on the components.14 As will be shown shortly, the waveform used to drive the single wire is critical in its ability to simulate a beam.
The motivation in applying a DC current was to simulate the main part of the beam current pulse. But this underlines the difference between a current formed by isolated charged particles (which obey Lorentzian force equations) and a conduction current (which obey Faraday's Law of induction). So from Faraday's Law
If the flux through S is constant then 
,
the electromotive force is zero (emf = 0), and the current around the circuit in contour C is zero since [6]
As an example, observe the transition from DC to a time varying field (a sinusoidally temporally varying, uniform spatial field for example). It is found that
dependence.16
The goal then is to determine what time varying current17
with an associated field of So, using the inductive coupling relationship for the two conductors
For a DC return current that the ``flat top'' part of the beam would produce, I1 is -I2tR/M and is shown below:
Figure 32. current (amps) vs. time (ns). This is the wire-source drive current needed to simulate the effects of a 2 kA electron beam in its flat top region through the kicker structure. The maximum amplitude (2 kA) was selected arbitrarily to
match the beam current and the sweep time was selected to compensate and determine the coupling. In general, any current/time
combination will suffice as long as the correct slope is maintained.
The value for the expression M/R was determined from the experimental data using the low frequency sinusoidal coupling point at 50 MHz from Figures 24 and 26:
By solving the equation to simulate the constant return current equal to the beam current18
The above equation was derived only for the constant
e- beam simulated case. For a time varying current, the
current term, Ibeam, and the R/M
expression are functions of time(frequency). Do not attempt to
integrate just Ibeam to arrive at an expression.
Note that when the electron beam flows in the center of the kicker, the dipole field between the two driven plates (left and right plates) is zero but the quadrupole field between the two grounded plates and the driven plates is non-zero since the driven plates are still at non-zero potential since they are carrying the charge corresponding to the return current [3]. Note that if the slots between the plates had been connected with conducting straps, then the resulting equipotential surface would have been at a uniform potential of 90 kV iff the grounded ends of the plates had been removed -- and 0 Volts if all ends had been connected to ground.
The author would like to thank Jim Dunlap in the EM Lab for his efforts in helping make these measurements possible. Additional thanks go to Brian Poole and Judy Chen for their design and experimental activities on the kicker and discussions about the cold-test measurements.
Work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-ENG-48.
1. TE11 for the 6.7 cm radius beampipe has a cutoff starting at 1.3 GHz. See Table 1 for more details.
2. The impedance of the line-current varied in the complex plane from 48 3. Measurements were made using a power divider with the HP3577 but the bandwidth did not increase significantly because of the noise floor. To make the measurement using a power divider when driving a mismatched load (e.g. the line-current source), it is necessary to pad both outputs of the power divider so that mismatch reflections from the line source pass through multiple pads when returning to the receiver.
4. the design of the quad-kicker replaced the GR connector adapter terminations on the kicker ports with true coaxial terminations.
5. the advantage of this technique is more drammatic with more ports. In the case of the quad-kicker: out of a total of 64 possibilities, only 28 measurements need to be made per frequency band. See also Footnote 6.
6. the scattering parameter matrix ([s] matrix) consists of a series of Vout / Vin measurements between each of the ports. Thus, s21 is the transmission coefficient (historically called the insertion loss) through one of the deflection plates, s31 is the cross-coupling between ports on the upstream end, and s41 is the cross-coupling between and upstream port and a cross- downstream port. s11 is the input reflection coefficient (historically called the return loss). The complete [s] matrix contains 36 elements (from 4 kicker ports and 2 ``wire ports'') but since the structures are linear then sij = sji. In the case of the four port system, each port has an input impedance of 50 7. the first few higher order coaxial modes can be approximated by [1] (to within 4% for b/a<5, a and b are the inner and outer radii):
8. the higher order modes for a straight beam pipe were determined just by solving for the roots of the Bessel functions (which describe the azimuthal fields in the circular pipe) to satisfy the boundary conditions at the pipe wall. Thus, the TMnm modes are roots of:
9. recall that 20 dB is 10× in voltage. So for a coupling coefficient of -25 dB, an input of 5 kV to port #1 would induce 280V on port #3 for a suitably ``slow'' pulse having most of its energy below 100 MHz (centered in the range of 60 MHz). Faster edges on the pulse would appear as amplified spikes on the cross-coupling port since the higher frequency components have significantly greater cross coupling coefficients. On the other hand, the coupling between the upstream port #1 and the downstream cross component port #4 is significantly less and so a 5 kV input would only induce 28V on port #4.
10. absorber was placed in the ends of the kicker in front of each end-plate in an effort to reduce any cavity resonances set up between the two end-plates. But no appreciable end-plate resonances were observed so the absorber was removed. End-plate resonances would appear as longitudional resonances with a frequency lower than the deflection-plate longitudional resonances. The reason is that at the primary longitudinal resonant frequencies, the effective pipe formed by the deflection plates is still below its cutoff frequency. Additionally, surface currents circulate around the loop formed by the two grounded plates already and so the end plates do not add an appreciable path at these frequencies.
11. the ability to measure the phase at the low frequencies has greater error than at high frequencies. At 400 MHz, 10° = 21 mm.
12. in general, the R loss term goes as f1/2 and so a sufficiently narrow frequency band was selected to minimize this effect during the calculation. f0 in Table 4 is 60 MHz. Also note that these are the RF parameters. Compare these against the surface resitivity [1], Rs = 1.9*10-5*f1/2 for the steel ( 13. V = V0*(1 - e-1*t) = 2.3 Volts when (R/L)*t = 1. And (R/L)*t = 1 when t = 11.6 ns from the beginning of the pulse. So L/R is 11.6 ns. Notice that t0 = 18.2 ns in Figure 30 and that the e-1 point occurs at 18.2 ns + 11.6 ns = 29.8 ns on the graph.
14. another motivation for the one-wire technique was the ability and desire to have the wire follow a curved path inside the kicker or the ability to create a wire cage to simulate an expanding, contracting, or pinched beam inside the kicker. Unfortunately for this series of testing, none of those goals could be realized.
15. As Faraday noticed for a current in an (uncharged) wire, a time varying current is necessary to induce currents in adjacent conductors (i.e. there is no such thing as an air- or iron-core wire-wound DC transformer). This led directly to Faraday's Law of Induction. On the other hand, an electron beam represents a charged system and does produce induced return currents in adjacent conductors.
16. Observing that the above reduces to the circuit equation, I2R = emf = V = -M dI1/dt, then the induced current in a conductor from another conductor is a function of the inductance between them (the mutual inductance, M) and the current in that other conductor. For sinusoidal excitation, the voltage (and hence the induced current) is 90° out of phase with the driven current since (d/dt)sin(t) = cos(t). For DC excitation, the induced current is zero.
17. we have arbitrarily ruled out changing the current path, current direction, moving conductors, or changing the area that the B field intercepts.
18. Note that any peak magnitude excitation current can be used as long as the slope of the waveform is adjusted accordingly. In practice, it would not be attempted to drive the kicker to produce Ibeam on the output ports since this requires excessive currents being swept at a fast rate.
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is required to simulate the case for a line charge of charge density per unit length ql [C/m], which produces an electric field at a radius r from the line charge of
in addition to the magnetic field. So, a test charge qtc in the vicinity of the e- beam experiences a force
and as such the test charge (at a distance r=d from the e- beam) moving with a specific velocity in the +z direction could experience no net force. This +z moving negative charge is a -z current, I2, and forms the return current from the e- beam. For the wire measurements, the drive current, I1, is in the +z direction and is generated such that the induced current, I2, is correct.
and since to simulate the electron beam return current, I2 = -Ibeam is a constant. Then solving for I1
The wrap is included since it would be excessively difficult to continue ramping the current up during the entire pulse width. Notice that these wrappings (trailing edges) will cause spikes in the data due to the fall time and should be kept as short as possible.
then the (M/R) factor can be calibrated.
NOTES
CONCLUSIONS
RECOMMENDATIONS
range for the lower frequency band. For pulsed testing, the rise time should be selected such that it's impedance coincides with the low frequency impedance (the impedance circulates around on the Smith Chart based on the frequency).
ACKNOWLEDGMENTS
REFERENCES
S. Ramo, J. Whinnery, T. Van Duzer, ``Fields and Waves in Communication Electronics,'' John Wiley & Sons, 1984, pp 152-153, 429-430, ISBN 0-471-87130-3.
``Reference Data for Engineers: Radio, Electronics, Computer, and Communications,'' Howard W. Sams & Co., 7th Edition, 1985, pp 29.22-23, ISBN 0-672-21563-2.
B. Poole, Lawrence Livermore National Laboratory, private communication, 1997.
C. A. Avalle, ``FXR Accelerator Cavity Impedance Experiments,'' Lawrence Livermore National Laboratory, 1998, UCRL-ID-129493.
M. Ong, C. A. Avalle, R. Richardson, J. Zentler, ``LLNL flash x-ray radiography machine (FXR) double-pulse upgrade diagnostics,'' Lawrence Livermore National Laboratory, UCRL-JC-125879, June 26, 1997, http://www.llnl.gov/tid/lof/documents/pdf/231850.pdf
R. Wangsness, ``Electromagnetic Fields,'' John Wiley & Sons, 2nd Edition, 1986, pp 208, 242, 252-253, 263-265, ISBN 0-471-81186-6.
FOOTNOTES
to 1200 as a function of frequency. A few readily available loads were tried but each had similar effects on the reflections due to the wide variation in the actual impedance. 50 was ultimately selected based on the choices available given the time constraints.
This reduces the direct coupling from the source to the receiver but also causes the S/N to decrease. As a consequence of the rising noise floor, the results had a similar effective bandwidth as when using the test set. Note that this problem only occurs when driving a mismatched load using the HP3577 and bypassing the test set to utilize the 5 Hz - 100 kHz band. For these measurements, that band was skipped.
so a direct measurement could be performed. However, in the case of the ``wire ports'' (ports 5 and 6) the input impedance to the line-source is a strong function of frequency and was unfolded from the data.
fnm = 1/(2
(b-a) / m), m = 1,2,3... for TM
fnm = 1/(
(b+a) / m), m = 1,2,3,... for TE
Jn(pnm) = 0 , fnm = pnm / (2
a ), m = 1,2,3,...
while the TEnm modes are roots of:
Jn'(qnm) = 0 , fnm = qnm / (2
a ), m = 1,2,3,...
where 
and 
are the permeability and permittivity of the medium (vacuum) and a is the beam pipe radius. In the case of the kicker (which contains a slotted pipe) the assumption of a contiguous pipe is valid for lower frequencies when the gap is electrically small but starts to break down at higher frequencies. Also note the inability of circumferential currents to be established on the kicker beampipe. There are also modes established between the isolated plates.
r = 1000) yielding a high frequency resistance per unit length in the wire of (Rs*l/w)/l = 92/m at f0 (w is wire circumference).
Last Modified: Jan 21, 1998
UCRL-ID-129997