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Adaptive Feedforward Cancellation of Sinusoidal Disturbances
in Superconducting Radio Frequency Cavities

presented by

Tarek Hamdi Kandil

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Adaptive Feedforward Cancellation of Sinusoidal
Disturbances in Superconducting Radio Frequency
Cavities

By
Tarek Hamdi Kandil

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2005

t'0

'1”

’

ABSTRACT

Adaptive Feedforward Cancellation of Sinusoidal
Disturbances in Superconducting Radio Frequency
Cavities

By
Tarek Hamdi Kandil

A control method, known as adaptive feedforward cancellation (AFC), is applied
to damp sinusoidal disturbances due to microphonics in superconducting radio fre-
quency (SRF) cavities. AFC provides a method for damping internal and external
sinusoidal disturbances with known frequencies. It is preferred over other schemes
because it uses rudimentary information about the frequency response at the distur-
bance frequencies, without the necessity for an analytic model (transfer funCtion) of
the system. It estimates the magnitude and phase of the sinusoidal disturbance inputs
and generates a control signal to cancel their effect. AFC, along with a frequency
estimation process, is shown to be very successful in the cancellation of sinusoidal
signals from different sources. The results of this research may significantly reduce
the power requirements and increase the stability for lightly loaded continuous-wave

SRF systems.

To my parents, Hamdi and Hana

and to my siblings, Amro and Dahlia

iii

ACKNOWLEDGMENTS

I would like to thank my advisor Dr. Hassan Khalil, and Dr. John Vincent
of the National Superconducting Cyclotron Laboratory (NSCL) for giving me the
opportunity to conduct this research study, a subject that interested me immensely.
I gratefully acknowledge and thank both of them for their supervision, help and
advice. I am also thankful to Dr. Terry Grimm, and Mr. John popielarski for their

assistance in setting up and performing the experiments at the NSCL.

iv

TABLE OF CONTENTS

ABSTRACT .................................... ii
ACKNOWLEDGMENTS ............................. iv
LIST OF TABLES ................................. vii
LIST OF FIGURES ................................ viii
1 Introduction ................................... 1
2 Background ................................... 3
2.1 Particle Acceleration ........................... 4
2.2 Niobium Cavity Fabrication and properties ............... 6
2.3 Modelling the RF Cavity and Cavity Voltage .............. 7
2.4 Detuning in RF Cavities ......................... 9
2.4.1 Lorentz Force Detuning ..................... 9

2.4.2 Microphonics Detuning ...................... 11

2.4.3 Different Methods for Microphonics Mitigation ......... 13

2.5 Problem Formulation and Preliminary Work .............. 14

3 Adaptive Feedforward Cancellation ...................... 18
3.1 Introduction ................................ 18
3.2 Controller Design ............................. 19
3.3 Averaging Analysis ............................ 22
3.4 Average Model .............................. 27
3.4.1 Single Frequency Component .................. 27

3.4.2 Two Hequency Components ................... 31

3.4.3 Multiple Frequency Components ................. 35

3.5 Robustness to Errors in Modelling the Frequency ........... 37

4 Microphonics Control .............................. 42
4.1 Measurements ............................... 43
4.1.1 Bode Diagram ........................... 45

4.1.2 Fast Fourier Tiansform ...................... 50

4.1.3 Calibration ............................ 53

4.2 Experimental Demonstration ....................... 54

4.2.1 Experimental setup ........................ 55

4.2.2 Experimental Results ....................... 56
5 Conclusion .................................... 59
BIBLIOGRAPHY ................................. 62

vi

4.1

4.2

LIST OF TABLES

Numerator and denominator coefficients for the modelled transfer func-
tion. .................................... 48
Parameters for second-order poles and zeroes of the modelled transfer
function ................................... 49

vii

2.1
2.2
2.3
2.4
2.5
2.6

3.1

3.2

3.3
3.4

4.1
4.2

4.3

4.4
4.5

4.6
4.7
4.8
4.9

LIST OF FIGURES

Rare Isotope Production Process. ....................
A 6-cell Niobium RF cavity. .......................
Transmitted RF response of the cavity at 2 K. ............

Radiation pressure (Lorentz forces) shown acting on a single cell cavity.

Possible sources and transfer mediums for microphonics [17].

Transfer function of SRF Cavity. The left plot shows the magnitude
response, where V is the cavity voltage, while the right plot shows the
phase response, where «p is the phase angle between the driving current
and the cavity voltage. A disturbance causes the magnitude to change
from curve (1) to curve(2) resulting in a decrease in the amplitude,
which is then compensated in curve (3) by an increase in the input
power along with a phase shift in the opposite direction [17].

Overall system. . . . ' ...........................
Simulation results for disturbance cancellation with a precisely known
frequency. .................................
Simulation results for disturbance cancellation when wu > cud.
Simulation results for disturbance cancellation when wu < wd.

End view and section view of the 5 = 0.47 prototype cryomodule. . .
Top photo: Cavities inside the helium vessels. Bottom photo: Cold
mass after installation of thermal and magnetic shield. ........
Position of fine and coarse tuners while attached to the multi-cell cav-

ity. ............................ ' ........

Physical setup of the cavities with the control unit or lock-in amplifier.

The range of interest of the frequency response (Bode diagram) for the
system. ..................................
A comparison between the actual and the modelled Bode diagrams.

FFT of RF error 'signal at four different instances. ..........
Accelerometer measurements near the beating sources. ........
Implementation of the AFC on Simulink. ...............

4.10 Active damping of helium oscillations at 2K ...............

4.11 Active damping of external vibration at 2K. ..............

viii

10
11

'12
18

39
40
41

43
44

44
46

46
49
51
53
55
56
57

CHAPTER 1

Introduction

Control of the resonant frequency of superconducting radio frequency (SRF) cavities
is required in view of the narrow bandwidth of operation. Detuning of SRF cavities
is caused'mainly by the Lorentz force (radiation pressure induced by the high RF
.field) and microphonics (mechanical vibrations) [15, Chapter 19]. In continuous-wave
(cw) accelerators, microphonics are the major concern [15, Chapter 19]. It is natural
to think of using fast mechanical actuators to compensate for microphonics, i.e.,
attenuate the effect of mechanical vibrations on detuning. This concept was applied
successfully by Simrock et a1 [19] to a simple quarter wave resonator (QWR) with
a fast piezoelectric tuner. However, the high-gain feedback approach used in [19] is
too complex to apply to multi-cell elliptical cavities, which are the subject of this
work. In fact, in a previous work by Simrock [12] for elliptical cavities it is stated
that “the large phase shift over this frequency range makes it clear that feedback
for microphonics control using the RF signal will not be possible with the piezo
actuator.” To date, there has been no demonstration of microphonics control on

multi-cell SRF cavities, and the current thesis presents the first such demonstration.

In Chapter 2, a brief discussion of SRF cavities and different sources of detuning

is given. We formulate the microphonics control problem from a control theory view-
point and explore various standard control approaches. The measured spectrum of
cw systems in a reasonably quiet environment, as is the case with properly designed
accelerators, only exhibits limited narrowband sources (sinusoidal signals) of noise.
It will be shown that the AFC is the most appropriate for the task as it handles sinu-
soidal disturbances. AFC is developed for stable systems, as in the current case, and
it does not require an analytic model of the system to design a feedback controller. In
Chapter 3, we review the main elements of the theory of AFC, and in Chapter 4, we
present our experimental demonstration of the successful use of AFC in microphonics

control of elliptical cavities.

CHAPTER 2

Background

The Rare Isotope Accelerator (RIA) is a proposed Linear Accelerator (lines) that
is designed for accelerating heavy ions that would provide 400 MeV/neucleon beams
with power up to 400 kW. The driver linac is designed to accelerate any stable isotope
from hydrogen to uranium, onto production targets that typically consists of heavy
elements, which would produce a broad assortment of exo'.ic isotopes. As shown in
Figure 2.1, after the acceleration of isotopes onto a production target, the projectile

Production
= get

    
   
   
  

 

Accelerator
Gas ll

in:

Fragment
separator

Accelerator
Rare isotopes

Figure 2.1. Rate Isotope Production Process.

fragments, or any light particles ejected as the result of bombardment, pass through

a fragment separation process; then the rare atoms are captured in a gas (helium)
chamber in which they get ionized, re-accelerated, and delivered to different experi-
ments for the purpose of studying their properties. Rare Isotopes are very different
from the stable nuclei found on earth. The study of the exotic nuclei could help in
answering many questions about how stars live and die and what is the origin of
elements in cosmos. It can also lead to an understanding of the origin of the ele-
ments that constitute the earth and universe, as well as many other questions. By
studying the properties of those new isotopes, an entire arena for new medical and in-
dustrial applications could be created, such as cancer treatments, and advancements
in imaging‘tools for medical use.

The driver linac for RIA is about 500 m long, divided into two sections, the first
section uses low beta (beta is the ratio between the particle velocity and speed of
light) RF resonators that operate at frequencies ranging from 60-350 MHz, while the
second section uses high beta resonators operating at 805 MHz, which are typically
multi-cell elliptical cavities that may range from 1-9 cells. Figure 2.2 shows a 6-cell

RF cavity on which this research study has been conducted.

2.1 Particle Acceleration

The RF field inside the cavity is composed of both electric and magnetic fields that
are transverse waves. The charged particles are to be accelerated along the axis of
the cavity, and therefore we require the electric field to be coaxial with the cavity,

thus having the magnetic field transverse to the cavity axis. The most common and

 

Figure 2.2. A 6-cell Niobium RF cavity.

simplest transverse magnetic (TM) mode used for acceleration that satisfies these
requirements is TMmo Where the electric field is constant and travelling along the
cavity axis, in this case its intensity decreases by moving away from the axis. On the
other hand, the magnetic field is zero at the axis and increases by moving radially
away from the axis.

Now consider that half the wavelength of the RF field is equal to the length of
one cavity cell that is operating in 7r mode, where 7r mode means that the cavity is
operating at a certain frequency such that it produces a 180° (7r)‘phase shift between
adjacent cells. Since the charged particles are moving at a very high speed close to the
speed of light, if a negative charged particle is injected into the first cell of the cavity
when the RF field has just become positive, then the field will not change direction
during the transit time of the particle through the half wavelength sized cell, and
thus accelerating it through the first cell. When the particle enters the second cell,

the RF field has now completed the positive cycle and started the negative cycle,

but since the cavity is operating in the 1r mode, this means the field experienced a
180° phase shift inverting the field back to the positive cycle, hence accelerating the
particle again.

The energy qV gained by a particle, depends on both the particle charge and the
electric field voltage Vcav. Therefore to control the acceleration process, it is necessary
to keep the magnitude and phase of the cavity voltage at known set points. An RF
controller is needed to maintain accurate magnitude and phase for Vcav, which will

be discussed in the next chapters.

2.2 Niobium Cavity Fabrication and properties

A typical cavity generates a potential of over 1 million volts. A traditional copper
cavity dissipates as high as 1 million watts in the cavity walls. An SRF cavity on
the other hand, dissipates about 100,000 times smaller power than that of a copper
cavity. Therefore, SRF cavities are used. To have SRF cavities, we need special
material and extremely cold temperature. Niobium sheets of 4 mm are used with a
nominal Residual Resistivity Ratio (RRR) of 250 [6] at temperature of about -456 F

or 2 Kelvin (K), where

RRR _ resistivity at 300K
_ residual resistivity at low temperature (normal state)

 

(2.1)

and higher RRR provides the best insurance against thermal breakdown.
High—grade niobium is a soft metal, which is easily fabricated, and the high purity

of the sheet niobium has a reasonably high thermal conductivity, and can be electron

beam welded without introducing excess RF losses at the welded parts. The purity
is not considered only in terms of bulk purity, but also in terms of inclusions from
manufacturing steps, as they act as normal conducting sites for thermal breakdown
of superconductivity. Therefore, the niobium sheet is scanned initially for defects
by eddy-current scanning [2]. Other impurities, such as dissolved interstitial oxygen,
carbon, nitrogen, and hydrogen, act as scattering sites for the electrons, which lower
the thermal conductivity and enhance the chances of a thermal breakdown [14]
Cavities have been fabricated from sheet niobium by first deep draw or spin half-
cells. The half-cells are then electron beam welded under vacuum. Chemical etching
to a depth of 100—200 pm is done to the internal surface of the cavity to remove
mechanically damaged layers for best RF performance. The cavity is then placed in a
vacuum furnace for 10 hours at 600°C, 10.6 torr to prevent it from the Q degradation,
where Q is the quality factor of the cavity [6].1Following that it is rinsed with ultra-
pure water at high pressure to get rid of dust particles since microscopic particles stuck
to the surface of the cavity can degrade its high-field performance. The foregoing
processing takes place in a clean room of class 100 or better, which means that the
air must be filtered to have fewer than 100 particles larger than 1 am in size in a

volume of 100 cu.ft.

2.3 Modelling the RF Cavity and Cavity Voltage

The starting point in microphonics control is to develop a mathematical model that

describes how the mechanical vibrations and the control actuator determine the cavity

detuning. To obtain such a model, the cavity’s frequency response is determined first.
The cavity’s frequency response is generated using an RF voltage network analyzer
(VNA) that is connected directly to the cavity’s input current and output voltage.
The network analyzer sweeps the input frequency about the RF resonance, then
compares the output signal of the cavity to the input signal. Figure 2.3 shows the
transmitted RF amplitude and phase of the cavity at 2 K swept with the VNA,

where the frequency at which the gain response peaks is called the eigenfrequency or

Ref 31.821“

>1:Transmission Phase- 2Q.fl”/
5.0 dB/ Ref 'HZ.DQ dB

’2:Transmlssion

3-0 21:1 33

 

Center 805.125 873 MHz Span 0.633 500 MHz

Figure 2.3. Transmitted RF response of the cavity at 2 K.

resonance frequency of the RF cavity. Ideally when no disturbance is present, the

eigenfrequency should be at 805 MHz, with a zero phase at that frequency. It is easily
noticed that the frequency response shown in Figure 2.3 is similar to that of a parallel
RLC circuit, hence it is shown in [17, Section 3.2] that the relationship between the
cavity detuning Aw = wo — w and the phase angle 1,!) (between the driving current

and cavity voltage) can be approximated at steady state by

tam!) = 2QL (3:) (2.2)

where w is the RF generator frequency, we is the cavity eigenfrequency, and QL is the

loaded Q factor, defined by

Stored energy
Total power dissipation/ cycle

 

QL = 27f (2.3)

2.4 Detuning in RF Cavities

Since the SRF cavity is cooled down to 2 K and is made of thin niobium sheet (4 mm
thick), the cavity shape is susceptible to changes due to any force that might act on
it. As stated earlier in the introduction, detuning of SRF cavities is caused mainly

by Lorentz force and microphonics.

2.4.1 Lorentz Force Detuning

To have a more effective acceleration of the charged particles, it is desired to increase
the electromagnetic fields in the superconducting structure. However high fields in
the cavity cause radiation pressure known as Lorentz forces acting on the walls as

9

T K Vertical Forces due

to Magnetic Field
Horizontal
Forces due to
‘, Electric Field
* .
—> 4—

Figure 2.4. Radiation pressure (Lorentz forces) shown acting on a single cell cavity.

shown in Figure 2.4 This radiation pressure is given by [17]
1
B=flmWV-MEW as

where E and H are the electric and magnetic fields acting on the cavity walls. This
pressure exerted on the walls causes some deformation in the cavity’s shape and con-
sequently a change in the resonator’s volume by AV. In this case the eigenfrequency

is shifted by a volume change according to the following relation given in [17]:

 

a—wztxaflr—mmrmv
“’0 frx(<50|f’3[0l2 + Molfiopldv

where E0 and Ho are the unperturbed fields.
It is shown in Figure 2.4 that electric field causes axial contraction of the cell
(negative change), while the magnetic field causes radial expansion (positive change).

Both changes yield a decrease in the resonance frequency, as shown by Equation

10

13.53,

(2.5). However, operating in a continuous-wave mode rather than operating with a
pulsed accelerating field causes the Lorentz force to be insignificant, in addition to
having stiffening rings that are used to reduce radiation pressure effect along with
having relatively thick walls of 4 mm, which enhances the rigidity, yet is thin enough
to ensure the effectiveness in the cooling process as well as keeping a lower material

costs. Therefore our major concern is microphonics disturbances.

2.4.2 Microphonics Detuning

As mentioned in the previous subsection, microphonics is the main cause of detuning
Since the accelerator is operating in cw. Mechanical vibrations are always present
and appear in an uncorrelated manner. Possible sources of microphonics along with

the transfer medium are shown in Figure 2.5. However, not all of these sources are

Source Transfer Medium

 

 

 

P

 

 

Vacuum pumps Beam tubes ‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machinery
external vibration sources)
‘ SRF cavity
Traffic : Ground, supports ‘
Ground motion
Ocean waves
‘ Helium transfer line. ‘
Compressors Heat exchanger
Pumps
‘ Helium gas ‘

 

 

 

 

 

 

 

 

 

Figure 2.5. Possible sources and transfer mediums for microphonics [17].

11

      
   

  

 

 

.. . input power
.. V0 = . (3)
l
0\
x
\

 

 

 

q—Quooo.---:-O

 

0300, mRF=wo(to) a)

Figure 2.6. Transfer function of SRF Cavity. The left plot shows the magnitude
response, where V is the cavity voltage, while the right plot shows the phase response,
where 112 is the phase angle between the driving current and the cavity voltage. A
disturbance causes the magnitude to change from curve (1) to curve(2) resulting in a
decrease in the amplitude, which is then compensated in curve (3) by an increase in
the input power along with a phase shift in the opposite direction [17].

considered potential disturbances since a properly designed accelerator would be in

reasonably quite environment and adequately isolated from ground.

Effect of Detuning

As mentioned earlier in Section 2.1, the energy gained by a particle to be accelerated
depends on both the particle charge and the cavity voltage. Therefore, keeping the
voltage constant is a very important matter. Figure 2.6 [17] shows the cavity’s re-
sponse under three different conditions. The first curve (1) shows the response of the
cavity under the desired operating conditions when the resonator is initially excited
at resonance at the desired cavity voltage with zero phase (21) = 0). Curve (2) shows
the case when a disturbance is present that deforms the cavity walls and causes a shift

in the eigenfrequency of the cavity, hence operating at the same frequency results in

12

a voltage decrease as well as a phase shift (t/J aé 0). One way to compensate for this
change is to increase the RF power together with shifting the phase of the driving
signal in an opposite direction, and this is illustrated on the figure by curve (3), which
shows that despite of the shift in the eigenfrequency of the cavity, we still get the

desired voltage level with a zero phase.

2.4.3 Different Methods for Microphonics Mitigation

Electronic Compensation

This method is shown in curve (3) of Figure 2.6. Although it is considered one way of
solving the problem, increasing the power is not considered a practical solution due

to the high cost of this operation.

Structural Modification

Cavities typically have very thin walls and are susceptible to deformation easily;
hence using thicker sheets for the walls has been considered. As mentioned in Section
2.2, 4 mm niobium sheets were used, which are relatively thick compared to some
other cavities (~ 2.8 mm). The fact that the cavities are operated under very cold
temperatures puts a limit on how thick the walls could be, not to hinder the cooling

process. Stiffening rings are also used for increasing the rigidity of the walls.

Lowering the Loaded Q

It is desired to operate the SRF cavity at a high loaded Q factor to achieve very

high RF voltages at the eigenfrequency of the cavity, which results in a very narrow

13

bandwidth for operation. Lowering the loaded Q will result in a wider bandwidth,
hence operating at the initial RF frequency during the presence of disturbances will
not result in a dramatic drop in the voltage level. However the peak magnitude at
a lower-Q cavity is much lower than that of a high-Q cavity, therefore increasing the

RF power will be required, which again is not cost effective.

Mechanical Compensation

Mechanical compensation is based on the idea of changing the shape of the cavity to
indemnify for deformation that result in cavity detuning. Mechanical compensation
is typically done using either a fast tuner or a slow tuner or both. In our experiment
both kinds of tuners have been used, more detailed discussion will be addressed in

Chapter 4.

2.5 Problem Formulation and Preliminary Work

From (2.2), we see that detuning can be reduced by decreasing the phase angle 7/).
Towards that end, we develop a model for 1/2. Two basic assumptions in developing

this model are:

0 The system with input u and output i/J is linear and time-invariant. Hence, it

can be represented by a transfer function G (s) from u to 1,0 [12].

0 Mechanical vibrations, which affect the cavity in a distributed way, can be
modelled by an equivalent lumped disturbance that affects the system at the

same point where the control actuator is applied [11, Section 2.7]. In other

14

words, the input to the system can be represented as the sum u — d, where d is

the disturbance input and u is the control input.

The transfer function G (s) can be determined experimentally by applying a sinusoidal
input at u and measuring the steady-state phase angle 2p. Using a lock-in amplifier
to sweep the frequency of the sinusoidal input over the frequency band of interest,
we can determine the frequency response from the input u to the output 11), which
produces the Bode plots of the transfer function.

Ftom a control theory viewpoint, the problem reduces to designing the control u
to reject or attenuate the effect of the disturbance d on the output tb. Six different

control techniques for disturbance rejection have been examined. They are

1. Proportional (P)

2. Proportional-Integral (PI)

3. Proportional-Integral—Derivative (PID)

4. High-gain band-limited

5. Servocompensator design

03

. Adaptive Feedforward Cancellation (AFC)

The first four controllers are used for disturbance rejection of a wide class of dis-
turbance inputs. They do not require the disturbance input to have a special form,
other than being a bounded signal. The last two techniques work when the distur-

bance input can be represented as the sum of sinusoidal signals of known frequencies

15

but unknown amplitudes and phases. The six techniques were investigated in the
internal reports [13, 18] using simulation of an experimentally-determined model of
a single—cell copper RF cavity at room temperature. The simulation studies showed
that the traditional P, PI, and PID controllers would not achieve the desired level of
disturbance attenuation because the controller gains are limited by stability require-
ments. In the high-gain band-limited control design, a controller is designed to have
a high loop gain over the frequency band of interest, while rolling off the loop’s fre-
quency response rapidly at high frequency to ensure the stability of the closed—loop
system. In the low-frequency range the controller essentially inverts the system’s
transfer function, which is allowable in our case because the transfer function is sta-
ble and nnuimum phase. The drawback of this design is the relatively high order of
the controller, which may not be justified in view of the fact that such a controller
guards against a wide class of disturbance inputs that may not be present in the
current problem. It is worthwhile to note that this technique is used by Simrock et
a1. [19] for microphonics control of a quarter wave resonator with a fast piezoelectric
tuner. However, our investigation indicates that the complexity of the controller and
the demand on the control effort in such a design will be prohibitive for multi-cell
cavities because the order of the controller will be very high. Even in the simple
experiment of [19], the controller’s order is 20, i.e., the degree of the denominator
polynomial of the controller’s transfer function is 20.

A common cause of microphonics is mechanical vibrations that are almost peri-
odic; in particular, the disturbance signal can be represented as the sum of a finite

number of sinusoidal signals, such as disturbances caused by turning on nearby pumps,

16

“

motors,...etc. That are not well isolated from ground. For this type of disturbance,
the techniques of servocompensators, e.g. [3, 8], and adaptive feedforward cancella-
tion, e.g. [1, 20], are more appropriate because they are designed to work with this
particular class of signals. The servocompensator approach includes an internal model
of the disturbance signal as part of the controller in such a way that the loop gain at
the frequencies of the disturbance is infinite; hence rejecting the disturbance asymp-
totically. Adaptive feedforward cancellation uses an adaptive algorithm to learn the
magnitudes and phases of the sinusoidal disturbances and synthesizes the control to
cancel them. Both approaches performed satisfactorily in the simulation study [18],
but the AFC has the advantage that the only information about the transfer function
C(s) that is needed is its magnitude and phase at the input frequencies, which are
easily obtained from the measured Bode plots. We will see in the next section that
we can tolerate up to 90 deg error in determining the phase and that errors in deter-
mining the magnitude will affect the speed of convergence of the adaptive algorithm
but will not alter its stability. Although [1] showed equivalence between the AFC and
a special design of the internal model for the servocompensator approach, we must
still obtain an analytic model of the system in the form of a rational transfer function
to use in designing the compensator. Because of the simplicity of the AFC method,
we have adopted it in the experimental part of our work. The method is explained

in more detail in the next section.

17

CHAPTER 3

Adaptive Feedforward Cancellation

3. 1 Introduction

The purpose of this chapter is to introduce Adaptive Feedforward Cancellation (AFC).
AFC is a control technique for disturbance attenuation that is based on the compar-
ison of an error signal to an estimated signal, which is adjusted continuously to drive
the error asymptotically towards zero. Convergence of the error to zero ensures that
the estimated parameters converge to the true ones.

Consider a linear stable system represented by the transfer function 0(3). Let y
be the output of the system and suppose the input is the sum of two signals u — d, as

shown in Figure 3.1, where u is the control input and dis an unknown disturbance that

0(8)

U(S) é 9(3) Y(S) :

AFC

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.1. Overall system.

18

can be modelled as the sum of sinusoidal signals of known frequencies, but unknown

amplitudes and phases, that is,

d = Z A,- sin(w,~t + 51') (ii-2f 21a,- cos(w,-t) + b,- sin(w,-t)] (3.1)
i=1 i=1
in which w,, for 2' = 1, - - - , n, are known but a,- and b,- are unknown.

The controller design will be shown in Section 3.2 using an online adaptive law for
parameter estimation. In Section 3.3, analysis supporting the controller design will be
shown by following the analysis done in [16, Chapter 4], then some extra work deriving
an average model for the system will be demonstrated in Section 3.4. Robustness of
the method to uncertainties in the frequencies cal to can will be discussed in the last

section.

3.2 Controller Design

The goal is to design the control input so as to attenuate the output y in the presence
of the disturbance (1. Had we known the amplitudes and phases of the sinusoidal

signals, we could have cancelled the disturbance by the control

21. = Z[a,~ cos(w,-t) + b,- sin(w,-t)]

i=1

19

To cope with the uncertainty in the parameters a,- and b,, we use the control

it = Z[&,~(t) cos(w,t) + (3,-(t) sin(w,-t)] (3.2)

i=1

where [1,-(t) and f),(t) are estimates of a, and b,, respectively, obtained by the adaptive

algorithm [7, Chapter 4]

6,-(t) = —7,-y(t) cos(w.-t + 0,) (3.3)
(3.0:) = —'y,-y(t)sin(w.~t + 0.) (3.4)
for 2' = 1,...,n, where the adaptation gains 71,...,’y,, and the phase advances
61, . . . , 6,, are chosen to ensure the convergence of d,- and f),- to a,- and b,, respectively.

The adaptive algorithm can be rewritten as

2U) = —Fy(t)wa (3'5)

20

where

 

P d1(t) — a1 ' - cos(w1t + 01) '
(31(t) — b1 sin(w1t + 61)
(12(t) — a2 cos(w2t + 62)
Z“) = (32(t) — b2 a 11),, = sin(w2t + 62)
61,,(25) — an cos(wnt + 6n)
(3,,(t) — bn sin(wnt + 6n)

 

 

 

P = diag[’71,’71,"i‘2, '72) ' ' ° 3,771,771]

It will be shown in section 3.4 that by choosing 0,- : AG(jw.-) and choosing 7,- suffi-
ciently small, we can ensure that

lim z(t) = 0 (3.6)

t—roo

It will be also shown that we can tolerate up to 90 deg error in determining the phase
of the transfer function at 0),. The limit (3.6) implies that limtxoo y(t) = 0. We
can then conclude that the adaptive algorithm ensures convergence of the parameter
estimates ti,- and f), to the true parameters a,- and b,, respectively, and convergence of

the output y(t) to zero.

21

3.3 Averaging Analysis

We start by expressing wa in terms of

~ 1
cos(wlt)

sin (W1)

cos (wn)

 

 

sin(wn)

. —

using the trigonometric identities

cos(wt + 6) = cos wt cos 6 — sin wt sin 6

sin(wt + 6) = cos wt sin 6 + sin wt cos 6

which can be written in matrix form as

cos(wt + 6) cos 6 — sin 6 cos wt

sin(wt + 6) sin 6 cos 6 sin wt

we have

wa(t) = E1005)

22

where

cos 6,- — sin 6,
E = blockdiag[E1, E2, . . . , En], E, 28‘

sin 6,- cos 6,-

State—Space Model

Let {A,B,C} be a minimal realization of the transfer function C(s). Then, the

overall system can be represented in state space as

:i:(t) = Ax(t)+B[u(t)-—d(t)] (3.8)

y(t) = Ca:(t) (3.9)
where

u(t)—d(t) = :[(&,(t )—a,-) )cosw,-t+(f),-(t——) b)sinw,-t]

 

 

 

 

- T - -
ale) — a1 cos(wlt)
tie) — b1 sin(wlt)
age) — a2 cos(wgt)

= 132(t)— b2 sin(wgt)
Mt) - an cos(wnt)

_ 13,,(t) — b, _ _ sin(wnt) _
= 2T(t)w(t)

23

Fiom equations (3.5), (3.7), and (3.9)
2(t) = —I’y(t)Ew(t) = —FEw(t)y(t) = —I‘Ew(t)Cr(t) (3.10)
Augmenting (3.10) with (3.8), we obtain the closed-loop model

2(t) = —FEw(t)C:r(t) (3.11)

d:(t) = Aa:(t)+BzT(t)w(t) (3.12)

System 'Ii‘ansformation

Equations (3.11) and (3.12) take the form of [16, equations 4.4.14 and 4.4.15 ]. We

will follow that book in applying averaging. Define
z
u(t, z) = / eA(‘“T)BzT’w(T)dT
0

Consider z to be frozen (treated as a fixed parameter), then V represents the steady-

state value of :r.

t
V(t,z) = [eA(’"T)BwT(T)de
o

t
@- = [BwT(t) + A/ 8A(t-T)BZT’LU(T)dT 2
8t 0
= BwT(t)z + Au(t, z)
t
Q = / eA(t-T)BwT('r)dT déf n(t)
62 0

24

Thus, u(t, z) = n(t)z. Consider now the following change of variables

= A1: + Bsz — Bsz — AV — h;(t)z'

2 Ah + rtI’Ew(t)C(h + u)

Let 5 = max,- 7, and write P as

P=EP1

where the elements of the diagonal matrix P1 satisfy 761 S 1. Then the transformed

system can be written as

2 = —5I‘1Ew(t)C[h+rs(t)z] (3.13)

it = Ah + ers(t)I’1Ew(t)C[h + K(t)z] (3.14)

Averaging Theory

Equations (3.13) and (3.14) take the form of [16, equations 4.4.1 and 4.4.2] with

f(t,z,h) = —I‘1Ew(t)C[h+rt(t)z]

g(t, 2, h) = rt(t)P1Ew(t)C[h + rc(t)z]

25

Noting that the eigenvalues of A have negative real parts, because C(s) is stable,
we conclude from [16, Section 4.4.1] that by choosing the adaptation gains '7,- small
enough, z(t) would be much slower than w(t) and a:(t) and we can apply the averaging
theorem [16, Theorem 4.4.3] to conclude that z(t) can be approximated by the solution

of the (time-invariant) average system

1 to‘f‘T

2(t) = 57111205; to f(T,Z,0)dT

to+T
= lim —1-/ [—FEw(T)CK.(T)z]dT

to+T
2 —FE lim — [10(T)CI€(T)]dT z

where

1 to+T
F = —I‘E lim T [QU(T)CK(T)]dT

T—ooo t
O

1 to+T T
= —I‘E lim — {w(r)C/ eA(T—”)BwT(0) d0} (17
to 0

Tfioo

Assuming that F is a Hurwitz matrix (all Eigenvalues have negative real parts), it

follows from [16, Theorem 4.4.3] that

lim 2(t) = 0, lim :z:(t) = 0

t—roo t—roo

which shows that limb...” y(t) = 0. We conclude that, in the absence of measurement

noise, the adaptive algorithm ensures convergence of the parameter estimates EL,- and

26

f),- to the true parameters a,- and b,, respectively, and convergence of the output y(t)
to zero. In the presence of bounded measurement noise, we can invoke standard
perturbation analysis, e.g., [10, Chapter 9], to Show that, after finite time, z(t) and

y(t) will be of the order of the amplitude of the measurement noise.

3.4 Average Model

By following the analysis of [16], an expression for F has been derived. Now we will
carry out some extra computations to further simplify F, obtain conditions under
which it will be Hurwitz, and see how error in measurements would affect the con-
ditions for stability. To obtain a general expression for the matrix F, we will start
our computations with the case of a disturbance with single frequency component wl;
then we will carry on with the calculations for the case of two frequencies wl and wg,

at which point a general formula for multiple frequencies can be easily concluded.

3.4.1 Single Frequency Component

Assume that the disturbance signal has one frequency wl. Then,

coswlt 71 0 cos 61 — sin 61
w(t)= 7 P: , E=E1=

sin wt 0 '71 sin 61 cos 61

CK.(T) = [CeA(T—°)BwT(0)da
0

C(3) = C(sI—A)_’B

27

For averaging analysis, Crt(7') will be taken as the steady-state response of C(jw) to

the input w(t). Let

GUM) = Rl+jll

where

R1 = |G(jw1)|cosq51 (3.15)

11 = |G(J'w1)| sin cm (3.16)

and 621 = 40(jw1). Therefore at steady state

CK“) = Rlcos(w1t) — Ilsin(w1t), Ilcos(w1t) + Rlsin(w1t) (3-17)

Then w(t)Crs(t) will be a 2 x 2 matrix such that

(1,1) = R1 cosg(w1t) — Ilsin(w1t)cos(w1t)
(1,2) = [1 c032(w1t) + Rlcos(w1t)sin(w1t)

(2,1) = Rlcos(w1t)sin(w1t)-‘Ilsin2(w1t)

(2,2) = 11 cos(wlt)sin(w1t)+R1 sin2(w1t)
Since
1 to‘f‘T
F = —PET1im — [w('r)CK.(7')]dT

28

= —FE{’UJ(t)CK(t) }average

we will need to average each element of the w(t)Cn(t) matrix. Since

{sin(wit)cos(w,t)}avemge = 0 (3.18)
1
{sin2(w,-t)}a,,emge = 2 (3.19)
1
{cos2(w,-t)}a,,emge = 5 (3.20)
we have
1 R1 11
{w(t)c,.(t)},,..,,. = 5
—11 R1
and
F = —FE{W(t)CK(t)}average
71 cos61 —sin61 R1 11
= "5
sin 61 COS 01 —11 R1

p

.71 R1 c0361 + 11 sin61 11 c0361 — R1 sin61

R1 sin61 — 11 c0361 11 sin61 + R1 cos 61

 

h

From equations (3.15) and (3.16) we see that the elements of F are further simplified

to

(1,1) = %|G(jw1)|[cos61cosd>1+sin6lsin¢1]

29

= -;-|0(jw1)lcos(61— a)

(1,2) = %|G(jw1)| [cos 61 sin c151 — sin 61 cos Q51]
= %IG(jw1)|Sin(91 — (151)

(2, 1) = %|G(jw1)| [— c0861 cos a. + sin 9, sin a]
= ‘é‘lGUMllSiDWI — $1)

(2, 2) = %|G(jw1)| [cos 61 cos $1 -l- sin 61 sin 051]

= élcewolcosw. —¢.)

Then

0 —- ' 6 -
F: _121|G(jw1)| cos( 1 (b1) sm( 1 (I51)
—sin(61—¢1) cos(61—¢1)

The eigenvalues of F are the roots of
72 ,72
det()\I — F) = [A + ZIIG(jw1)|2c082(61— qf>1)]2 + EIIG(jw1)|2sin2(61— $1): 0
which are given by

a. = $1962.): cos(91 — (m i j-‘gl-Iaijwm sin(o. — a)

30

Hence
Re{/\(F)} = —%IG(jw1)l cos(01 - e1)

Choosing 61 to satisfy

[61 — AG(jw1)[ < 90deg (3.21)

ensures that the eigenvalues of F have negative real parts at —12‘—|G ( jw1)| cos(61— (151).

The best choice would be
61 = ZGUM)

which yields multiple real eigenvalues at -3’2l|G(jw1)|. In this case, (3.21) shows that
we can tolerate up to 90 deg error in determining the phase of the transfer function

at wl.

3.4.2 Two Frequency Components

To calculate F with two frequency components present in the disturbance signal, we

will go through the same steps as done in section (3.4.1), but with two frequencies.

31

Consider

 

 

 

 

r' ' r' -
cos wt 71 0 0 0
sin wt 0 71 0 0
w(t) = , I‘ =
cos wt 0 0 72 0
sin wt 0 0 0 72
E1 0 cos 6.,- — sin 6,-
E = , E,- =
0 E2 sin 6, cos 6,-
Then, at steady state,
- - T

R1 cos(wlt) — 11 sin(wlt)

Il cos(wlt) + R1 sin(wlt)
Cit(t) =

R2 COS(w2t) — 12 sin(wgt)

 

 

12 COS(w2t) + R2 sin(wgt)

where G(jw,-) = R, + jI,. The 4 x 4 matrix is given by the following elements

(1, 1) = R1 cos2(w1t) — Ilsin(w1t)cos(w1t)
(1,2) = 110082(w1t) + Rlcos(w1t)sin(w1t)
(1,3) = R2 cos(wlt) COS((U2t) — 12 cos(wlt) sin(wgt)
(1,4) = 12 cos(wlt) COS((.U2t) + R2 cos(wlt) sin(wgt)
(2, 1) = R1 COS(LU2t)SlIl(UJ1t) — 11 sin2(w1t)

32

Ilcos(w1t)sin(w1t) + R1 si112(w1t)
R2 sin(wlt) cos(wgt) — 12 sin(wlt) SlD(Ld2t)
12 sin(wlt) COS(LU2t) + R2 sin(wlt) sin(wgt)
R1 COS(UJ2t)COS(Ld1t) — 11 cos(wgt)sin(w1t)
Ilcos(w1t)cos(w2t) + R1 COS(LU2t) si11(w1t)
R2 cos2(w2t) — 12 SlIl(UJ2t) COS(UJ2t)
[2 cos2(w2t) + R2 cos(wgt) si11(w2t)
R1 sin(wgt)cos(w1t) — 11 sin(wlt) Slfl(td2t)
11 sin(wgt)cos(w1t) + R1 sin(wgt)sin(w1t)
R2 COS(LU2t) sin(wgt) — 12 sin2(w2t)

12 COS(Ld2t) Sifl((.d2t) + R2 sin2(w2t)

To calculate {w(t)Crt(t)}avemge we first need to calculate the averages of cross
product terms with two different frequencies. Since these terms are not periodic, we

use the generalized form limT__.oo % Lia-VT [f (t)] dt to calculate the averages.

{cos(wlt) cos(wgt)}ave,age

to+T
= T121100 T to [cos(wlt) cos(wgt)] dt
1 t0+T1
= lim — —{cos((w1 — w2)t) + cos((wl + w2)t)}dt

T—’°° T to 2

= %{cos((w1 — my) + cos((w1 + w2)t)}.....g. = 0 (3-22)

33

{sin(wlt) sin(wgt)}avemge

[sin(wlt) sin(wgt)] dt

1 fto+T
to

%{cos((w1 — w2)t) — cos((w1 + w2)t)}m,emge = 0

{sin(wlt) cos(wgt)}avemge

to+T
[sin(wlt) COS(UJ2t)] dt

lim —

éfsinflwl + w2)t) + sin((wl — w2)t)}avmge ._. 0

Along with Equations (3.18), (3.19) and (3.20), we conclude that

to+T
T133100 %/ -:-{cos((w1 —- w2)t) — cos((w1 + w2)t)}dt

(3.23)

1 tO+T 1
lim -— / —{sin((w1 + w2)t) + sin((wl — w2)t)}dt
T—--—*OO T t 2

(3.24)

 

 

 

 

F = —I‘E{w(t)Cr~t(t)}avemge
F cos 61 — sin 61 0 0 R1 11 0 0
F sin 61 cos 61 0 0 —11 R1 0 0
— 2 0 0 cos 62 — sin 62 0 0 R2 12
L 0 0 sin 62 cos 62 0 0 —12 R2
cos(6,- — (15,) sin(6, — (1),)

blockdiag -%|G(jwi)l
— sin(6, — 653') 008(9i — (bi)

34

For i = 1,2. Again by choosing 6,- to satisfy
[6, -- 100.600] < 90 deg

for i = 1,2, we ensure that the eigenvalues of F have negative real parts at
—12-‘-|G(jw1)| cos(61 — Q51) and —123|G(jw2)| cos(62 — (152). The best choice results
from taking 61 = AGle) and 62 = 1G ( ng), which ensures that F has real negative

eigenvalues at —% [G(jw1)| and —323 |G(jw2)|.

3.4.3 Multiple Frequency Components

Repeating the steps of driving F for single and double frequency components, it can

be easily seen that in the general case
F = blockdiag[F1, F2, . . . , Fn]

where

. cos(9i — 952') sin(6, ‘ (61')
F.- = glam-n

— sin(6, — (61) 008(0i "' ¢i)

and <25,- 2 ZG(jw,-). Choosing 6,- to satisfy

(0,- — AG(jw,-)| < 90 deg (3.25)

35

ensures that the eigenvalues of F, have negative real parts at -—3,121|G(jw,-)| cos(6, — 69,).

The best choice would be

03' = ZG(]LU¢')

which yields multiple real eigenvalues at —32i|G(jw,-)|. In this case, (3.25) shows that
we can tolerate up to 90 deg error in determining the phase of the transfer function
at w,.

The eigenvalues of F corresponding to different diagonal blocks would vary with
the magnitude of the frequency response [G ( jw,)]. The range of these variations can
be limited by choosing 7,- : c,/|G(jw,-)| for some positive numbers c,-, which results
in eigenvalues at ~%, for 2' = 1,. . . , 72. By appropriate choice of the constants C), we
can have n eigenvalues of the same order of magnitude. It is clear that errors in
determining [G(jw,-)| will not be crucial, as they affect the location of the eigenvalues
of F but do not change the fact that their real parts will be always negative.

The notation used in this chapter is retained in the experimental section, with the
exception that the output y is taken to be the phase angle 16- The state x does not

appear in the experimental section as the state model is used only for analysis.

36

3.5 Robustness to Errors in Modelling the Fre-

quency

In the previous few subsections, the averaging analysis showed that the control al-
gorithm keeps the system stable, and drives the output signal to zero, provided the
disturbance frequencies are precisely known. In this section, we use simulation to
investigate the effect of having errors in modelling the frequency components of a

disturbance. Consider C(s) to be

10000

0(3) = s2 + 803 + 10000

 

and consider the presence of a disturbance signal d, oscillating at wd = 10 rad/ sec

with a magnitude of 1V such that

d = sin(10t)

The simulation was done on Matlab/Simulink by implementing Equations (3.3) and
(3.4) to construct the control signal u as shown in Equation (3.2), and 6 is chosen
such that 6 = AG(jwu), where wu is the frequency used for constructing the control

signal it, and

wu=wd+Aw

Figure 3.2 shows simulation results when the frequency of the disturbance signal

37

is precisely known. Figure 3.2(a) shows the output y of the system that is desired
to be zero. Figure 3.2(b) shows the disturbance signal d that is used throughout the
entire simulation, and Figure 3.2(c) shows the output of the system after applying a
cancellation signal at a frequency w,, = wd = 10 rad/sec, which results in driving the
phase angle 1,6 to zero as expected from the foregoing analysis. The control signal it is
shown in 3.2(d), while the last figure shows the input to the system (u — d) converging
to zero, which means that the control signal it is converging to the disturbance signal
(1.

Now we will check for the robustness of the system by running different simulations
using wu 75 wd (i.e. Aw # 0). Figure 3.3 shows results of using a cancellation signal
when wu > wd for three different values Aw, while Figure 3.4 shows three more
results for wu < wd. The left column of plots in Figures 3.3 and 3.4 represent the
system’s output, 3;, with an applied control signal, while the right column of plots
shows the error between the estimated disturbance signal (control signal) and the
actual disturbance signal, i.e., u — d.

In Figure 3.3, Plots (a) and (b) Show the system’s output, 3;, and the input,
u - (1, respectively using wu = 11 rad/sec, Plots (c) and ((1) Show the case when
wu = 15 rad/sec, and Plots (e) and (f) Show the case when wu = 50 rad/sec. In
Figure 3.4, Plots (a) and (b) show the case of using wu = 9 rad/sec, Plots (c) and
(d) show the case when wu = 5 rad/sec, and Plots (e) and (f) show the case of using
wu = 1 rad/ sec. It is clear from these figures that the steady-state amplitudes of y

and u - d are proportional to small frequency error Aw.

38

Cavity Output (y) with No Cancellation Disturbance Signal (d)

 

 

1 _ .§_,.(a). 1. _ _ .- . . ..;...(b....

05... .. ...... ...... f ...... , o.5~ ....... . .......

Magnitude (V)
O
Magnitude (V)
O

 

 

 

 

 

 

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Time (sec) Time (sec)

Cavity Output with Cancellation Control Signal (u)

r r

 

 

0_5 _. . . . . . . . .....

 

Magnitude (V)
o
I
Magnitude (V)

 

 

 

 

 

 

 

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Time (sec) Time (sec)

 

05 ......... g ........ ........ g .......

 

Magnitude (V)
O

 

 

 

0 0.1 0.2 0.3 0.4 0.5
Time (sec)

Figure 3.2. Simulation results for disturbance cancellation with a precisely known
frequency.

39

Magnitude (V)

Magnitude (V)

Magnitude (V)

0.5

Output y

 

 

....... ....(a)jw=.1,1radlsec..

 

 

 

0.2 0.3
Time (sec)

 

 

 

T .(cfmeis. fad/seci

 

 

0.2 0.3
Time (sec)

0.1 0.4 0.5

 

 

7 f (e) u) = 50 fad/sec

.......................................

 

 

0.2 0.3
Time (sec)

Magnitude (V)

Magnitude (V)

0 0.1

Magnitude (V)
O

 

 

Input u - d

 

r

........ (b)w=.11rad/sec..

.......................................

 

 

 

0.2 0.3
Time (sec)

 

(dim.=15.fadlsec-

...................

 

 

 

0.2 0.3
Time (sec)

 

l i (T) u) = 50 fad/sec

 

 

 

0.2 0.3
Time (sec)

Figure 3.3. Simulation results for disturbance cancellation when wu > w.

Magnitude (V)

Magnitude (V)

Magnitude (V)

Output y

 

1 ........ E ....... f. .(a)m.=9.radlsec...

 

 

 

0.2 0.3
Time (sec)

 

0.4 0.5

 

1.. . , .2 f. , . .(d)m=.5.radlsec..

 

 

 

 

0.2 0.3
Time (sec)

0.4 0.5

 

r

1 ..... f ........ 3 ..... (e) u) = 1. rad/sec. .

 

 

 

 

0.2 0.3
Time (sec)

Magnitude (V)

Magnitude (V)

Magnitude (V)

Input u - d

 

1.. . ...(b)w=9rad/sec..

 

 

 

 

0.2 0.3
Time (sec)

 

 

 

 

 

0.2 0.3
Time (sec)

0.4 0.5

 

1 ........ E ........ E ....... (f) (n = 1. rjadlsec, .

 

 

 

 

0.2
Time (sec)

Figure 3.4. Simulation results for disturbance cancellation when wu < wd.

CHAPTER 4

Microphonics Control

A prototype 805 MHz cryomodule has been tested to demonstrate the required per-
formance for the Rare Isotope Accelerator [4, 5, 9]. The prototype cryomodule has
two multi-cell cavities. Figure 4.1 shows the end view and section view of the 6 = 0.47
prototype cryomodule, while Figure 4.2 shows the cavities before and after the instal-
lation of thermal and magnetic shields. Each multi-cell cavity has an external tuner
actuated by a piezoelectric actuator that operates in room temperature for ease of
maintenance, while the cavities are cooled down to 2 K under cryoplant temperature
regulation. Figure 4.3 shows that the cavity’s RF resonance frequency can be tuned
using both a coarse (slow) tuner and a fine (fast) tuner (Piezo—electric actuator).
The coarse tuner has a linear effect with a span of 1 MHz that is only used at the
beginning before operation, by either tightening or loosening on a screw that would
push or release some pressure off the tuner rocker arm, which consequently squeezes
or relaxes the shape of the cavity, thus tuning the RF resonant frequency within the
desired range. The piezo-electric actuator receives a voltage signal from the Adaptive
Feedforward controller. For an input range of about 10V, which corresponds to 10
kHz of detuning, the actuator behaves linearly. In the next section, different mea-

surements that have been conducted in preparation for the damping process will be

42

   
  
  
 
 
  

LN2 Supply/Return

  
  

Instrumentation Port. \

. LHe Reservoir

LHe Supply ,,

Support Link

Magnetic
Shields

 

 

         
  

Alignment

mm“ in Ill—wmm—[NV'J
Tim... M 7: ,wuiL.‘

Tuner

 

 

 
 
   
   
  

 

       

 

 

 

gr- .-

  

II
In:-
I

 
 

Inpul. Coupler

Figure 4.1. End view and section View of the 6 = 0.47 prototype cryomodule.

discussed while Section 4.2 shows the experimental setup as well as the results of the

cancellation process.

4.1 Measurements

Figure 4.4 shows the experimental setup, where an RF signal generator is used to
drive the cavity, through an antenna placed at one terminal of the multi-cell cavity,

while another antenna at the other terminal is used to pickup the output signal, where

S
a
l

[Vinl sin(wnpt + (Pin)

Vout = [Vent] Sin(wRFt + (Pout)

43

 

Figure 4.2. Top photo: Cavities inside the helium vessels. Bottom photo: Cold mass
after installation of thermal and magnetic shield.

 

   
 

 

 

‘ . Tuner
Piezoelectric rocker arm

Mechanical fast tuner

slow tuner

Figure 4.3. Position of fine and coarse tuners while attached to the multi—cell cavity.

44

The signal V,,, is passed through a phase shifter that compensates for the phase
introduced due to cables, probes, and amplifiers and adds a 90" phase shift so that

Vin would take the form
l/in = IVEnl 008(prt + so...)

These two signals are then mixed together through a mixer to generate a signal
containing both the sum and difference of the frequency components of the two mixed
signals. The mixer’s output is passed through a low pass filter to obtain the low
frequency component signal, which is proportional to the cavity detuning through
the relation given by equation (2.2). This setup provides us with a real time error

signal that is analyzed by different methods to obtain
1. Frequency response of the system (Bode Diagram)
2. Etequency components present in the signal due to disturbances (FFT)
3. Disturbance cancellation using AFC

which will be discussed in detail in the next three subsections.

4.1.1 Bode Diagram

For some physical systems, the structure of the model and its parameters can be
easily determined using laws of physics, properties of materials, etc. In many other
cases, the plant model and parameters have to be obtained by identification experi-

ments, namely, observing the response of the system to known inputs. If the system

45

___.______.____________.‘

  
    

RF signal

Piezo
Actuator

 

Figure 4.4. Physical setup of the cavities with the control unit or lock-in amplifier.

 

 

-20

 

 

 

 

 

 

 

 

a
8
3
“E —60 ,
3
E
‘800 20 4o 60 80 100
Frequency (Hz)

Phase (deg)

     

20 40 60 80 100
Frequency (Hz)

Figure 4.5. The range of interest of the frequency response (Bode diagram) for the
system.

parameters are fixed then one can easily derive an analytical model that can be used
with different control techniques. But our plant parameters are not fixed; they keep
changing quite often that the plant model might change from one day to another,
due to changes in operating conditions, pressure, temperature, etc, which makes it
hard to model the plant by a fixed transfer function. As mentioned earlier, one of the
advantages of the Adaptive Feedforward Cancellation algorithm is its independence
on an analytical model for the plant. However, knowing the frequency response of
the system at the disturbance frequency ensures stability as shown in chapter 3, and
speeds up the disturbance cancellation process as well. Figure 4.5 shows the Bode
diagram that has been used in the cancellation process. Although the actual Bode
diagram was measured from 1 Hz to 1000 Hz with a phase rolling down to about
—3500 deg, Figure 4.5 only illustrates the range of interest where disturbances have
been observed. The Bode diagram was generated from a lock-in amplifier that sends
a sinusoidal signal to the piezo—electric actuator through an amplifier. This signal
is swept through the desired range of frequencies, step size, and sampling rate, then
the error signal is fed back into the lock-in amplifier to be compared to the swept
sinusoidal signal and produce a frequency response (Bode diagram) of the system.
This process is done off-line, simply by replacing the controller by the digital lock-in
amplifier. The data is then saved in the form of a look—up table, from which the
phase and magnitude values that correspond to the disturbance frequencies are used
as inputs to the controller for better response.

An analytical model (transfer function) of the Bode diagram shown in Figure 4.5

can be approximated by the following 14th order transfer function, which is obtained

47

Table 4.1. N umerator and denominator coefficients for the modelled transfer function.

 

 

 

Num coeff Coeff values Den coeff Coeff values
al 2152 [31 1
a2 1.6446005 fig 130.4
cm 2.1466009 03 1.2656006
(1.; 1.3756011 [34 1.3816008
(15 8.846014 [35 6.6786011
as 4.5636016 [36 5.9666013
a7 1.9256020 67 1.9156017
a8 7.4996021 58 1.3496019
019 2.3396025 09 3.2286022
0110 6.1056026 310 1.6876024
all 1.5026030 1311 3.2076027
0112 1.976031 012 1.1076029
0213 3.9876034 513 1.7416032
[314 29826033
[315 3.987e036

 

 

 

 

 

by trial and erro

G(s)

1',

_001 e(—0.001 s) (

01512 + 622311 + . ..

 

61314 + 62313 + . ..

+ (11281 + C113)

+ 61451 + 615 (4.1)

where the numerator (num) and denomenator (den) coefficients (coeff) are given in

Table 4.1. The transfer function (4.1) has 6 second-order zeroes, and 7 second-order

poles whose parameters are shown in Table 4.2. This model is only derived to show

the complexity of the system’s transfer function and how the changing nature of the

parameters of the plant will make it hard to keep modelling the transfer function.

Figure 4.6 shows the actual Bode diagram obtained experimentally using a lock—in

amplifier, along with the Bode diagram calculated from the transfer function (4.1).

48

Table 4.2. Parameters for second-order poles and zeroes of the modelled transfer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

function.
Poles’ center Corresponding Zeroes’ center Corresponding
frequencies (rad / sec) damping ratios frequencies (rad / sec) damping ratios
333 0.018 351.86 0.0128
364.4 0.0137 370.71 0.0202
383.3 0.0365 389.56 0.03
395.84 0.0299 408.4 0.0208
414.69 0.0145 446.1 0.0022
452.39 0.0111 464.96 0.0107
578 0.0299
Cavity’s Bode diagram
'20 F . T 3
e-so-v
3 -4o 3
3 t : . 2
3’ - ' -- Modelled plot 3
2 ”60"" — Actual plot . ‘
”700 20 4o 60 so 100
Frequency (Hz)
200 _ . ,
19‘ 100 _ .. .............................
B ; ; .
m 0. ........ .
g ._ - -' Modelled plot ; g
'2000 20 40 so so 100
Frequency (Hz)

Figure 4.6. A comparison between the actual and the modelled Bode diagrams.

49

4.1.2 Fast Fourier Transform

The cavity frequency shifts due to vibrations and pressure fluctuations that could

deform the cavity walls. Typical sources of vibration are
1. Fluid fluctuation

0 Boiling
o Cavitation

o Turbulent flow
2. Rotating machinery

o. Motors

0 Pumps
3. Ground motion

The disturbances caused from fluid fluctuation and rotating machinery are typi-
cally narrow band sinusoidal disturbances, while those generated from ground motion
are usually broadband vibrations. A properly designed accelerator only exhibits lim-
ited narrowband sources of noise, while ground motion should have negligible effect
because of adequate ground insulation of the cryomodule. The mechanical frequen-
cies of concern are usually low, less than 200 Hz. In our experiments, the observed
disturbances were only below 100 Hz.

Figure 4.7 shows the frequency domain of the RF error signal at three different

instances. Instances (a) and (b) take place at the final stage of the cooling process,

50

Amplitude (V)

Amplitude (V)

0_05 ..................................... _,
004 ....................................... -
0.03 - - ..............
0.02 ... _. .......

0.06

0.05

.0
o
A

P
o
o:

.0
0
re

0.01

 

1i

instance (a)

 

 

 

 

 

Frequency (Hz)

 

i T l r

........................................

 

 

 

 

0 . . . .
0 20 40 60 80 1 00
Frequency (Hz)

0.06 j i i .
Instance (b)
0.05 .................................. V ....... _
E
0
'D
2
i
E
4:
Frequency (Hz)
0.06 . . . f
instance (d)
005...... .................... I ....... _
0.04 .. . . . .................................. _.
E
0)
'0
g 003 _ ..................................... _
D.
E
<
0.02 .........................................
0.01 ......................................... _
0 . .' - - i~ .
0 20 40 60 80 100
Frequency (Hz)

 

 

 

 

 

 

 

 

 

Figure 4.7. FFT of RF error signal at four different instances.

51

 

where liquid helium is introduced rapidly filling a helium reservoir to cool the cavities
down. Once the helium reservoir is filled, thermo—acoustic oscillations appear at
about 6.5 Hz due to trapped gas volumes in the liquid helium space, sometimes this
disturbance appears along with its second harmonic as shown at instance (a). These
oscillations at 6.5 Hz start decreasing in magnitude as the helium level decreases,
allowing the release of trapped gas volumes, and this is shown throughout instances
(a), (b) and (c). It is worthwhile mentioning that these oscillations will not be present
under normal operating conditions, as the helium reservoir will be empty. Instance
(d) takes place after the helium reservoir has been emptied and the thermo—acoustic
oscillations are no longer present. We can see that at instance ((1) the main source
of disturbance is at about 60 Hz, in addition to a small peak at 54 Hz, which most
probably corresponds to some rotating machines. It is noticed that the magnitude of
the disturbance at 60 Hz varies at different instances, which suggests the possibility
of having more than one driving term, rotating at close frequencies, and thus causing
frequency beating. Indeed by zooming in onto the 60 Hz, we observed the presence of
two components causing that disturbance, one at 59.5 Hz and the other at 59.7 Hz.
By taking floor measurements using an accelerometer, and by tracking the signals, the
sources were identified to be two cryoplant screw compressors. The accelerometer’s
readings assured that there are two disturbances one at 59.5 Hz and the other at
59.7 Hz as shown in Figure 4.8. The cryoplant was designed to cool superconducting
magnets, and no isolation from the floor or piping were done as vibration was not an

issue.

52

 

x 10' Accelerometer Readings

\

 

1.5

59.70 Hz
Cryogenics MECH room 1.453 "W
Floor near door
23 Apfil 2004
is"
>
/'
0.5 ~ 59.50 Hz / -
611.6 pV

 

 

 

 

 

LA— AJI AM‘ *A

Frequency

 

Figure 4.8. Accelerometer measurements near the beating sources.

4. 1.3 Calibration

In this subsection, we start by using an example of a mechanical disturbance to have
a better understanding of how to relate the RF error signal to the cavity detune,
and how to interpret the information obtained from the FFT as to how far does the
eigenfrequency of the SRF cavity shift. Then, the calibration measurement that was
done for this particular experiment will be discussed.

The RF error signal contains all of the needed information on how the vibra-
tions affect the cavity’s RF resonant frequency. The magnitude of the error signal
corresponds to how far does the RF resonant frequency wo shifts off resonance. Let
us consider the following example for clarification. Assume that we have a nearby
pump operating at 60 Hz that is not well isolated from the floor, which would act

as a mechanical disturbance. The error signal (detuning) will then be a sinusoidal

53

signal of frequency 60 Hz, i.e., Aw = M sin ((27r x 60t) + (p). This will cause the RF
resonance frequency (wo) to oscillate between (wo + M) and (wo — M) at 60 Hz.
Calibration measurements were done to relate the error signal to the frequency
deviation using a Voltage Network Analyzer (VNA) in the continuous wave (cw)
mode. The error signal was centered around 0 V with a peak voltage of about 75
mV and the RF signal was swept over a band of 200 Hz centered at the resonance
frequency, corresponding to :l:100 Hz of frequency detune, i.e., for this experiment a

disturbance signal of a peak voltage of 100 mV causes the RF resonant frequency to

detune from wo by i100 Hz.

4.2 Experimental Demonstration

As shown in Figure 4.4, the estimated noise signal is added to the system by directly
shaking an SRF 6-cell elliptical cavity, cooled to 2K, using a piezo—electric actuator
(PI, model P—842.60). The controller can also be replaced by a lock-in amplifier to
generate the Bode plot of the system.

The block diagram of the AFC algorithm is shown in Figure 4.9 for the case of a
single-frequency disturbance. It is an implementation of Equations (3.2), (3.3) and
(3.4).

In Figure 4.9, w is the angular frequency of the disturbance signal that is calculated
from an FFT of the RF error signal, 6 is a phase advance introduced to ensure
maximum stability of the system, and 'y is the adaptation gain. Both 6 and ”y are

determined from a measured Bode diagram, where 6 is the phase at the frequency to

54

 

Sin(a) t)

Unknown Sine
Sinusoidal Sin((u t + 6)
Disturbance

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sine

 

 

 

 

 

 

 

 

 

I/S ‘ X

 

Plant *9—f
L (3(5)
11 - al
[Cosine X I: “S X

Cos(w t + 6)

 

 

 

 

l"

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[Cosine

Cos(w t)

 

Estimated noise signal

 

 

 

Figure 4.9. Implementation of the AFC on Simulink.

be cancelled and '7, as discussed in Chapter 3, can be calculated from the magnitude
information such that its value is large at small magnitudes and relatively small at

large magnitudes.

4.2.1 Experimental setup

An external PC is used for modelling the controller in MATLAB/Simulink, which is
then built in dSPACE CONTROLDESK developer version that communicates with
an external hardware (dSPACE RT11104 board), with 16 I/ O ports. The user’s inter-
face is through dSPACE CON TROLDESK developer version for real-time adjustment
of the variable parameters.

The setup for obtaining the Bode diagram was discussed earlier in Section 4.1.1
using a lock-in amplifier (SRS digital lock-in amplifier model SR850). The FFT of the

RF error signal is generated from the LeCroy Waverunner LT342 digital oscilloscope,

55

 

60 l i i T fii r

- - 1 - Undamped
— Damped _

Cavity Detune (Hz)

 

 

 

 

Figure 4.10. Active damping of helium oscillations at 2K.

from which the largest frequency components are picked for damping to acceptable

levels.

4.2.2 Experimental Results

We observed two types of microphonics vibration: internal (helium oscillations) and
external (motors, pumps, etc.). The results of applying AFC to both types are shown
in Figures 4.10 and 4.11.

Figure 4.10 shows an FFT of the detuning for the undamped and damped re-
sponses of thermo—acoustic oscillations that was addressed in Section 4.1.2. During
this experiment the effect of the oscillation at 6.5 Hz was accompanied with its sec-

ond harmonic at 13 Hz. After applying a cancellation signal at 6.5 Hz, the effect of

56

 

(a)
01
1

 

----~Undamped
— Damped .

(A)
O
I

 

 

 

to

01
I
U!
.‘l
.01
IN

Cavity Detune (Hz)
s

—I
O
I
. V-'-I-'-'I-'-l-’-*-'
- - — o - - - - - -
l

 

 

 

 

"535 40 45 50 55 60 65 70 75 so
Disturbance Frequency (Hz)

Figure 4.11. Active damping of external vibration at 2K.

the oscillation was clamped at that frequency; however the internal energy causing
this oscillation was still present, and its effect was observed to have shifted up to the
oscillation’s second harmonic at 13 Hz increasing the disturbance at that frequency,
where another cancellation signal was applied. The first peak at 6.5 Hz was reduced
by a factor of 6 from a cavity detune of 59 Hz to 10 Hz, while the second peak at 13 Hz
was reduced to a cavity detune of 4 Hz. These oscillations will not be present under
the operating conditions as mentioned before. However, testing the active damping
for different kinds of disturbances was done to check the performance of the control
algorithm.

Figure 4.11 shows the undamped and damped responses due to external vibration

from a motor that was turned on purposely for demonstration. The noise appeared

57

 

at 57.5 Hz, and it was successfully damped by a factor of 7.4 from a cavity detune of

31 Hz to 4.2 Hz.

58

 

CHAPTER 5

Conclusion

A microphonics control algorithm has been developed to control the microphonics
problem opposing the particle acceleration process through 6-cell SRF niobium cav-
ity. It is the prototype for a possible RF control system for the Rare Isotope Ac-
celerator project. The adaptive feedforward cancellation (AFC) controller has been
implemented at the National Superconducting Cyclotron Laboratory at Michigan
State University (E. Lansing, MI, USA). The challenge is to design and develop an
RF control that handles disturbances present during continuous wave operation of
superconducting cavities. Since Lorentz forces have very small effects on detuning
during continuous wave operation, detuning is mainly caused by microphonics.

To date, there has been no demonstration of microphonics control on multi-cell
SRF cavities, and the current work presents the first such demonstration, where we
have demonstrated the successful use of piezo—electric actuators and the adaptive feed-
forward cancellation control to damp sinusoidal disturbances due to microphonics in
SRF cavities. In the next step, a digital low level radio frequency (LLRF) controller
has to be demonstrated working in conjunction with the adaptive feedforward cancel-
lation on the SRF cavities, where the AFC control would attenuate the cavity detune

down to low levels such that the LLRF controller could handle. The LLRF would

59

then apply further cancellation by controlling the RF driving signal of the cavity
by continuously adjusting the RF amplitude and phase to maintain a steady output

signal that meets the desired amplitude with zero phase.

 

60

BIBLIOGRAPHY

61

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63

   

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