Proceedings of IBIC2012, Tsukuba, Japan

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SIGNAL EQUALIZER FOR SPS ECLOUD/TMCI INSTABILITYFEEDBACK CONTROL SYSTEM∗
K. Pollock† , J. Dusatko, J.D. Fox, C. Rivetta, D. Van Winkle, SLAC, Menlo Park, USA
R. Secondo, CERN, Geneva, Switzerland
The 4 GS/sec Ecloud/TMCI instability control system indevelopment for the CERN SPS requires 1.5 GHz of processing bandwidth for the beam pickups and signal digitizer. An exponentially tapered stripline pickup has sufficient bandwidth, but has a phase response that distorts thebeam signal in the time domain. We report on results fromthe design and implementation of an equalizer for the front
end signal processing with correction for the pickup andcable responses. Using a model of the transfer functions
for the pickups and the cabling, we determine a desired
frequency response for the equalizer. Design for the circuitry and component value fitting is discussed as well asboard construction and reduction of parasitic impedances.
Finally, we show results from the measurement of an assembled equalizer, and compare them with simulations.

DEFINITION OF THE PROBLEM
Electron clouds and transverse mode coupling induce
intra-bunch instabilities for high intensity beams in circular accelerators [1]. A feedback instability control system
is in development to sample the vertical displacement of
the bunch at 4 GS/sec such that the transverse head-tailmodes can be detected and the necessary correction signal
applied [2].

this information will be used to apply a specific correction
signal centered on the bunch.

Figure 2: The beam shape is expected to be close to Gaussian, however the pick-up and cabling cause distortion in
the time domain. Shown above is the modeled pick-up signal (red) and a measured signal (green).
The responses of the pick-up and cabling have beenmodeled and measured for the system at the SPS [4]. Thetransfer function of the pickup is modeled asHp (s) = s+sca ∗ (1 − e−(a+−2Ls)c)

2L

where L is the pick-up length and equals 0.375 m, ais the coefficient that describes the exponential taper andequals 2.48, and c is the speed of light. This is plotted in
Figure 3.
The cable transfer function is represented byFigure 1: Instability control system diagram.
In order to measure the higher order transverse modes
we need a pick-up with a bandwidth of up to 1.5 GHz. Exponentially tapered stripline couplers have sufficient bandwidth and flat frequency response at high frequencies, however, the phase response of the pick-ups distorts the beam
signal [3]. Also, the long cables between the beam tunnel and signal processing introduce further distortion of thebeam signal. The combination of these responses results inthe deviation from the Gaussian as seen in Figure 2, andequalization is needed to recover original signal. An accurate time domain picture of the bunch is critical because∗ Work supported by the US-DOE under Contract DE-AC0276SF00515 and US-DOE LARP program† kmpollock@stanford.edu√s1s
Hc (s) = e 2 (−a0 (1+j)0.707 π −a1 2π )The coefficients a0 and a1 are the measured cablecoefficients. For the current cables a0 = 1.05×10−4 and a1= 3.5×10−10 . The transfer function magnitude and phaseare plotted in Figure 4.
The equalization can be done with either software orhardware. A digital signal processing approach is attractive because any desired transfer function can be used andit can be changed easily, however, for our system this approach becomes too computationally intensive. For a software implementation the processing cost increases by n2
whereas with a hardware equalizer circuit it is possible to
do n channel parallel processing. Also, the cabling andpick-ups are unlikely to change often. For these reasons,an analog equalizer circuit has been designed.

ISBN 978-3-95450-119-9
Feedbacks and Beam Stability

1c 2012 by the respective authors — cc Creative Commons Attribution 3.0 (CC BY 3.0)
Copyright ○Abstract

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Proceedings of IBIC2012, Tsukuba, JapanINITIAL CIRCUIT TOPOLOGY AND
OPTIMIZATION
A low pass filter can be implemented with a variety ofpassive components. A distributed element solution is often used for high frequency filters, however it is impracticalfor our bandwidth due to the large areas needed for eachelement. Instead, we use a lumped element implementation with surface mount components to try to minimize unwanted parasitic elements. Various low pass filters weremodeled and fit using a simple optimization algorithm.
Figure 3: Model of exponentially tapered stripline pickup
shows a frequency response similar to a differentiator.

c 2012 by the respective authors — cc Creative Commons Attribution 3.0 (CC BY 3.0)
Copyright ○Figure 5: Simple low pass topology capacitor implementation, with frequency response results shown in Figure 6.

Figure 4: Model of cable between beam pipe and control
room shows a frequency response similar to a low pass filter.

DETERMINATION OF IDEAL
EQUALIZERThe ideal equalizer would have a frequency domain
behavior equal to the inverse of the transfer function of thepick-up and cable. The current system, which monitors thebeam position without feedback, uses this inverse transfer
function in an offline software equalization. In designing
an analog implementation, we first fit a polynomial tothe ideal equalizer transfer function which suggests a
circuit topology. System functions of linear, lumped, finite
networks are real rational functions and can be represented
as the ratio of two real polynomials, where the zm are thezeros and the pn are the poles of G(s) [5].

1 )(s−z2 )···(s−zm )G(s) = H ∗ (s−z(s−p1 )(s−p2 )···(s−pn )A polynomial fit with one pole and two zeros fits wellup to 500 MHz, and suggests an easy implementation witha simple passive low pass filter. If it is determined thatgreater than 500 MHz is necessary, a more complicated circuit can be designed.

Figure 6: Frequency domain fit results for simple low pass
topology as shown in Figure 5.
Our initial circuits had problems with parasitic components causing resonances within the desired bandwidth.
Each circuit element can be modeled as a resistor, capacitor and inductor, the values of which can be measured andincluded in the model (see Figure 7).
Figure 8 shows the results from including these parasitics elements in the modeling step. The original circuit without parasitics included fits the ideal polynomial fit
well, however, when actually building the circuit there wasresonance at 100 MHz. Then, by adding the parasitics into
the model before optimizing the component values the fit
is again good and resonances are not found in the desired
bandwidth.
Additionally, added to the optimization code was theoption to put multiple resistors or capacitors in parallel,ISBN 978-3-95450-119-9
2Feedbacks and Beam Stability

Proceedings of IBIC2012, Tsukuba, Japan

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Figure 9: Circuit Design with component values chosen
by optimization algorithm and three resistors in parallel toreduce the effect of parasitic inductance.

Figure 10: Constructed circuit with one amplifier on either
end of the circuit. Input is on the left, output on the right,
and DC biasing for the amplifiers at the bottom. Circuitcomponents are 0805 surface mount.

CONCLUSIONS
Figure 8: Component optimization with and without parasitics included in the model.
thereby reducing the overall parasitic impedances. Finally,
we used a matching T network at the output in order towork into a higher load resistance, which helps in getting
realistic component values. Using the optimization codeand mocking up a variety of solutions led to a topologythat achieved a good fit of the polynomial.

A hardware equalizer has been designed, constructed
and tested for use on the front end of a feedback instability
control system, replacing the current software implementation and allowing the system to operate in feedback mode.
In designing an analog equalizer we found that the parasitic
elements associated with lumped element circuits made circuit design difficult, and proceeded by including the parasitic elements in the modeling. The resulting equalizer cir-CONSTRUCTED CIRCUIT RESULTS
The circuit design was chosen to balance the requirements of a good fit and low loss of signal. Amplifiers withlow noise figure were chosen to recover the magnitude lostin the equalizer. The frequency domain results can be seenin Figure 11.
To test the equalizer a measured pick-up signal was discretized, played out through the system 4GS/sec DAC, run
through the equalizer and measured with an oscilloscope asshown in the top and bottom panels of Figure 12. To detect
the higher order modes the system must have a bandwidthof 1.5 GHz, but for a first implementation, a bandwidth of
500 MHz will be able to measure the beam offset and thelower order modes.

Figure 11: Frequency Domain Measured Circuit Results.
The modeled and measured magnitudes have amplification
included.

ISBN 978-3-95450-119-9
Feedbacks and Beam Stability

3c 2012 by the respective authors — cc Creative Commons Attribution 3.0 (CC BY 3.0)
Copyright ○Figure 7: Models used for resistors (top) and capacitors(bottom) with parasitic impedances included.

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Proceedings of IBIC2012, Tsukuba, JapanACKNOWLEDGEMENTS
We thank John Cesaratto and Ozhan Turgut for discussion and technical support. Also, thanks to Thai Nguyen
and Lucio Tan from the Stanford Undergraduate Summer
Research Program. And finally, we thank the SLAC ARD,
CERN AB RF Departments and the US LARP program fortheir support.

REFERENCES[1] J.D. Fox et. al., “Excitation of Intra-bunch Vertical Motionin the SPS - Implications for Feedback Control of Ecloudand TMCI Instabilities” IPAC’12, New Orleans, May 2012,
MOEPPB015, p. 112 (2012)

c 2012 by the respective authors — cc Creative Commons Attribution 3.0 (CC BY 3.0)
Copyright ○Figure 12: Pick-up signal discretized and played out
through a 4GS/sec DAC (red), and then through equalizer(green).
cuit has a low pass topology and has a transfer function thatfits well to 500 MHz.
Next we will need to determine whether a higher bandwidth equalizer is needed to resolve higher order modes
within the bunch. The same methods developed for thisinitial equalizer can be used to design higher bandwidth
equalizers, or equalizers with a different frequency response for new front end pick-ups or cabling. Finally, thesame procedure could be used to create a pre-equalizer forthe back end of the system. The kicker has a low pass frequency response with a cut off at 200 MHz, so a high passequalizer could be designed to correct the time domain signal seen by the bunch.
This initial version of the equalizer will be tested at
CERN on the SPS in the fall of 2012.

[2] J.D. Fox et. al., “A 4.2 GS/s Synchronized Vertical Excitation System for SPS Studies - Steps Toward Wideband Feedback” IPAC’12, New Orleans, May 2012, WEPPR090, p.
3144 (2012)[3] T. Linnecar, “The High Frequency Longitudinal and Transverse Pick-ups Used in the SPS,” CERN-SPS/ARF/78/17,
CERN, Geneva, (1978)[4] R. de Maria et. al., “Analysis of the Performance of the SPS
Exponential Coupler Striplines Using Beam Measurements
and Simulation Data” IPAC’10, Kyoto, Japan, May 2010,
WEPEB054, p. 2812 (2010)[5] A. Budak, Passive and Active Network Analysis and Synthesis, (Boston: Houghton Mifflin Company, 1974)

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