1. Project Report: A Control Based System of
Mechanical Loss Measurement for High Quality
Factor Oscillators
Louis Gitelman
11/10/2015
1 Introduction
In this project a controls based system of measuring the quality factor or Q of an
oscillator was developed. This system works by locking the phase between the
oscillator's exciter and normal mode to
π
2 and locking the oscillator's amplitude
with control loops. This allows one to assume that the rate at which energy
goes into the oscillator is equal to the rate at which the oscillator loses energy
or it's mechanical loss[1]. This system was rst proposed by Nicolas Smith in
his paper A Technique for Continuous Measurement of the Quality Factor of
Mechanical Oscillators and it's main intended use is for measuring the quality
factor of lenses and lens samples that may be used in the LIGO interferometer
[1].
So far I have made a proportion Integral Derivative (PID) controller system
capable of locking both the phase and amplitude of the oscillator to fractions
of a percent on a test oscillator with Q of 5 ∗ 104
± 104
. I have also been able
to lock the phase between the exciter and the LIGO lens sample to
π
2 for optics
with quality factors of around 8∗105
±105
with only a few degrees of error with
a proportional controller. Currently I am redesigning the PID based amplitude
and phase controller to run on the 8∗105
±105
Q system. It is important to note
that this project was a collaboration between Greg Harry, Jonathan Newport,
Isaac Jafar, Nicolas Smith, Matt Abernathy and myself. Finally, I have only
mentioned the most pertinent lessons, experiments and tests of this project in
this paper and it is meant to be a representation of these things not a complete
representation of all work I have done on this project.
2 Background
The Laser Interferometer Gravitational Wave Observatory (LIGO) is a National
Science Foundation funded collaboration that aims to detect gravitational waves
with an interferometer. Unfortunately these waves are weak in comparison to
1

2. the noise sources in the environment including thermal noise[2]. This thermal
noise is a negligible noise source in many optical systems but is a key source of
noise in the high precision LIGO interferometer. Low thermal noise is correlated
to a high quality factor. As such, lens samples with high quality factors are
desired for use in LIGO as are systems to eciently measure the mechanical
loss of lens samples. Quality factor or Q is essentially the rate at which an
oscillating system loses its amplitude of oscillation and mechanical loss is the
material specic analog of this concept [10]. Having ecient systems of quality
factor measurement are especially important as alterations to LIGO lens samples
are often made and altered to combat other noise sources such as shot noise and
Brownian noise and resultant decreases in the quality factor of the lenses must
be checked and controlled[6].
The current system of quality factor measurement employed by many LIGO
lens sample testing operations depends on driving lens samples at resonance
until they reach a maximum resonance amplitude and then turning o the driver
and allowing the lens sample to ring down while recording amplitude decay over
time. Specically, in the current system employed at American University, when
nding the quality factor of an lens sample we generate and AC signal whose
frequency is the same as the frequency with which we wish to drive the lens
sample. This signal originates in the function generator and passes through
a voltage amplier and into a comb exciter, which is simply a long piece of
wire coiled tightly around a piece of metal and stationed physically close to
the lens sample. As the high AC voltage moves through the wire it creates an
electromagnetic eld whose amplitude oscillates at the frequency of the function
generator. This causes the lens sample to sinusoidally ex or oscillate at the
said frequency. This oscillation is observed with a birefringence meter and is
monitored with a spectrum analyzer and oscilloscope. The drive frequency
is altered until one believes they have found a mode based on the frequency
response. At this point they turn o the driver and a computer script analyzes
the decrease in the LIGO lens sample's amplitude over time to nd the Q. These
components are organized as seen in gure 1.
The mathematics that the computer uses to equate decrease in oscillation
amplitude over time to Q is described below:
Where E is energy and A is amplitude of an oscillation[5]:
E =
1
2
kA2
(1)
Where E is energy after time t, E0 is initial energy when the exciter is turned
o, T is period and Q is quality factor[5]:
E = E0e
−t2π
T Q (2)
Plugging these two results together we get:
1
2
kA2
=
1
2
kA2
0e
−t2π
T Q (3)
This equation can be simplied as shown:
2

3. Figure 1: A diagram of the components used in American University's system
for computing Q values by the ring down technique.
3

4. A = Ae
−t2π
T Q
0
A = A0e
−t2π
T Q
A
A0
= e
−t2π
T Q
ln(
A
A0
) =
−t2π
TQ
(4)
Q =
−t2π
T ∗ ln( A
A0
)
(5)
Although this system is functional it has some aws. It requires the high
Q lens samples to be driven at a resonance frequency which can be challenging
given the inverse proportionality of bandwidth and quality factor, Q as shown
in equation 6.
Where FWHM=The width of the frequency band at half the maximum
amplitude[5]:
FWHM =
w0
Q
(6)
Also for some LIGO lens samples whose Qs can run in the millions having
to wait for a lens sample to lose most of its amplitude can take a long time.
For instance, for a lens sample that has a rst mode frequency of 2690 which is
about average for LIGO lens samples and Q of 8 ∗ 106
to decay its oscillation to
5 percent of its original maximum oscillation amplitude, making a measurement
with a minimum of 5% fractional uncertainty possible, will take:
From Equation 4:
ln(
A
A0
) =
−t2π
TQ
simplifying
1
t
=
−2π
T ∗ Q ∗ ln( A
A0
)
and simplifying further, where A0 = 1 and A = .05, or 5% of the original,
and T = 1
f = 1
2850 = .000350877 is
t ≈
.000350877 ∗ 3 ∗ 106
∗ ln(.05
1 )
−2π
≈ 502(seconds) ≈ 8.4(minutes)
4

5. This is troublesome given that this system can only measure lens samples
with quality factors in the millions with at best 10 or 20% error[8]. Necessitating
repeated ringing.
The slowness and inaccuracy of this system of Q measurement necessitates
the system of control loops which may be quicker and more accurate and is
highly automated making automation of this measurement process less labor
intensive.
The controls system of Q measurement depends on several other key ideas.
First is phase, which in this case refers to the phase between the harmonic
driver, and harmonic oscillator or lens sample. At the point at which phase
equals
π
2 or the phase where driver and oscillator are completely out of phase
all the energy of the driver goes to the oscillator[5].
It is common for mode frequencies of LIGO tests mass modes to drift so
quickly that if one wants to repeatedly measure the Q of a lens sample, by the
time the lens sample has been rung down so the Q may be recorded the mode
frequency has shifted. As one can see from Equation 6 the full width at half
max of a mode frequency of a LIGO lens sample, which often have Qs upwards
of 106
and a rst mode frequency of 2690 may be very small [8]. Because of this
small bandwidth even a slight shift in ω0 can mean one observes a much lower
frequency response if one drives at the old ω0. The feedback based system will
continuously close this error and track ω0 and continuously drive at this mode.
One positive of this system is that it is a second system of Q measurement
and it can give conformation for the current system of Q measurement we al-
ready have. This constant amplitude also means that the signal strength is
constant, which will give a constant signal to noise ratio. This feedback system
also allows one to simultaneously derive Q values from more than one mode.
This is possible with a ring down but is very hard. This feedback system also will
allow people to investigate the eect of oscillation amplitude on mechanical loss.
In the ring down technique system of quality factor measurement the mechan-
ical loss must be measured as a reection of a variety of dierent amplitudes,
because a ring down is a change in amplitude. Feedback Q measurement would
make amplitude dependent mechanical loss measurement possible. Although
no one has ever made a continuous Q measuring system capable of measure the
Q of oscillator whose Qs are in the 105
range Mathew Winchester created a
continuous Q measurement system capable of measuring Qs of magnitude 104
using Nicolas Smiths method.
Another idea key to understanding this controls system of Q measurement
is knowledge of a proportion, Integral, derivative (PID) controller. A PID con-
troller takes the error signal which is the dierence between the set point or
desired system value and the actual state of the system to create the error sig-
nal. A PID controller then performs three operations with this error signal.
Firstly, it multiplies this error by a given proportion and sends this signal as a
correction factor to the actuator, which in this case is an oscillator, altering its
behavior to bring its output to the set point. Simultaneously the integral term
which is the integral of error with respect to time over either all time or a set
period of time, which is essentially an average error is calculated and multiplied
5

6. by a constant. This signal is also sent to the actuator to appropriately alter its
behavior to bring its output to the set point. Also simultaneously the derivative
term of the PID which is the derivative or rate of change of error over all or
some of the system's run time is calculated and multiplied by a constant and
sent to the actuator to appropriately alter its behavior to bring its output to
the set point. These mechanisms are mathematically described below in the
equation. Where e(t) is the error signal, u(t) is the total correction signal, τ
is the time over which the error is intergraded, Kpis the constant altering the
level of eect of the proportional term, Kiis the constant altering the level of
eect of the integral term and Kdis the constant altering the level of eect of
the derivative term. [3]:
u(t) = Kpe(t) + Ki
t¢
0
e(τ)dτ + Kd
de
dt
(7)
A last key concept to understanding the methods and results found in this
lab is a brief overview of a lock-in amplier. A lock-in amplier can be used
to nd a signal whose amplitude is below the noise oor and nd the phase
between a reference signal and the experimentally derived input signal. In my
case the Vac, the experimentally derived signal is the signal determined by the
birefringence meter's interaction with the lens sample and Vref, the reference
signal is the drive signal.
Where: V0 is the amplitude of the unknown signal,Vac is the electric signal
coming from the experiment with known gain from the lock-in's preamplier
ofGac, amplitude V0, and angular frequency w0,Vref is a experimenter gener-
ated and dened signal with amplitude V1, angular frequency w1and a phase
dierence of φ from Vac, Equation 1 shows the experimentally generated signal
V0Sin(w0t) multiplied by a gain from the lock-in's preamplier Gac
Vac = GacV0Sin(w0t) (8)
Vref = V1Sin(w1t + φ) (9)
The two signals are sent into a multiplicative mixer.
Vref Vac = GacV0V1Sin(w0t)∗Sin(w1t+φ) =
GacV0V1
2
[Cos(w0t−w1t−φ)−Cos(w0t+w1t+φ)]
(10)
Where:
w0 = w1 (11)
GacV0V1
2
[Cos((w0−w0)t−φ)−Cos((w0+w0)t+φ)] =
GacV0V1
2
[Cos(φ)−Cos((2w0t)+φ)] =
6

7. GacV0V1
2
Cos(φ) −
GacV0V1
2
Cos((2w0t) + φ) (12)
With an integrator or low pass lter of high capacitance AC signals are
attenuated to ground leaving:
GacV0V1
2
Cos(φ)
V1, Vref and Gac are known. Therefore by multiplying these known terms
through wecan nd the hitherto unknown experimental signal amplitude V0.
GacV0V1
2
∗
2
GacV1
= V0Cos(φ) = X (13)
Simultaneously in a 2nd branch of the circuit the reference signal is phase
shifted 90 degrees resulting in a 2nd reference signal, Vref2 = V1Sin(w1t +
(φ + 90)). Vref2 and Vsig are combined as described for Vref and Vsig and the
resulting in the output is:
GacV0V1
2
Cos(φ + 90)
and
GacV0V1
2
Cos(φ + 90) ∗
2
GacV1
= V0Cos(φ + 90) = Y
Geometrically wecan see the phase φ is:
φ = tan−
(
Y
X
) = tan−
(
V0Cos(φ + 90)
V0Cos(φ)
)
And the signal amplitude is:
R = X2 + Y 2 = (V0Cos(φ))2 + (V0Cos(φ + 90))2
3 Method
In his paper Nicolas Smith set Q in terms of the output of a driven amplitude
and phase locked system as:
ϕ = Q−1
=
2ωU {a}
cHU ω0
(14)
Where ϕ = mechanical loss, ω0= natural frequency, HU =feedback gain at
unity gain frequency, c =amplitude set point, ωU = unity gain frequency and
{a}= the integral or average of the control signal of amplitude over time.
I approached this project as three distinct tasks. The rst task was to create
a phase locked loop, which locked the phase between the lens sample and the
7

8. Figure 2: A diagram of the circuit inside the crystal oscillator simulator.
driver to
π
2 . The second task was to create an amplitude locked loop that chose
a set point and locked the amplitude of the lens sample's oscillation while it's
phase was locked to
π
2 . The third task was to nd a system of continuously
measuring unity gain frequency and feedback gain at unity gain frequency.
Accomplishing each task made accomplishing the next task possible and pro-
vided more of the parameters needed to compute Q with equation 13. Locking
phase to
π
2 made it possible to nd the rst mode natural frequency ω0 or be-
cause ω0 is the drive frequency where phase equals
π
2 . It also made locking
amplitude pertinent because until phase=
π
2 not all of the driver's energy is be-
ing transferred to the oscillator and one should not expect that the energy into
the system is equal to the energy out of the system. Locking amplitude made
it possible to nd the error of the amplitude lock and c, the set point of the
amplitude. Locking the amplitude makes it possible to nd the feedback gain
of the system at the unity gain frequency because the feedback system needs to
exist before one can nd it's feedback gain at its unity gain frequency.
Before any actual construction could begin it was necessary to create a theo-
retical model for how the components of this system of feedback controls should
interact to lock phase and amplitude of a LIGO lens sample. This concept has
stayed fairly constant throughout the project. The essential components consist
of a plant which creates a signal which functions both as a drive signal and a
reference signal. The displacement as a function of time output signal derived
from the driven oscillator is then sent to the amplitude and phase detector where
it is compared to the reference/drive signal to see the phase between the oscil-
lator and the drive signal and nd the driven oscillator's amplitude. The phase
and amplitude detector then outputs the phase and amplitude signals which
are compared to the set point values to create an amplitude error signal and
a phase error signal. As discussed above once the phase has been locked to
π
2
the amplitude lock may begin to act but not before. The error signals are then
processed by the amplitude feedback function and phase feedback function to
create phase and amplitude correction signals. These signals are then used to
reset the plant's output frequency and phase. This process is shown in Figure
3.
The next task was to translate this ideal system into a physical reality using
the components I have at my disposal. Comparing the components of the old Q
measurement system seen in gure 1 to the components of my new theoretical
8

9. Figure 3: A block diagram of theoretical components of this feedback system.
system the function generator; amplier and electromagnetic comb actuator
function as my plant. The signal the LIGO lens sample makes in conjunction
with the birefringence meter, and band pass lter function as the oscillator and
the lock-in is my phase and amplitude detector. The computer functions as the
set point holder, error signal generator and feedback function generator. This
next set up is shown in gure 4. Much like the theoretical infrastructure of this
project the physical incarnation of this system has not changed much over the
course of this project.
4 Results
Some of the most informative results of this project was the development of
the feedback functions of the phase lock and the amplitude lock. Throughout
the project my phase and amplitude locks functioned as single input single
output, feedback controls whose inputs were amplitude and phase from the lock-
in amplier and returned new voltages and frequencies to which the computer
set the function generator's output[11]. I also found that the phase at which all
energy was being transferred from the actuator to the oscillator was −π
2 instead
of
π
2 which also transfers all of the driver's energy to the oscillator as one can
see from gure 5 [5]. This meant that as I approached ω0 from 0 driving at each
frequency and observing frequency
response wewould have a negative error signal because:
Where x −90:
−90 − x 0
And where x −90:
−90 − x 0
This means when my error signal was less than 0 I needed to add a positive
frequency correction signal to my current drive frequency and when error signal
was greater than 0 I needed to add a negative correctional signal to my current
frequency.
9

10. Figure 4: The physical system used in this project.
10

11. Figure 5: Frequency response and phases at various drive frequencies.
11

12. Figure 6: This is the pseudo code of the phase locking function which shifted
phase by shifting frequency by a set increment.
The rst attempts to create a phase lock were conducted on the low Q
test oscillator as outlined in the methods section. This controller had a simple
mechanism in which if phase error was negative the frequency was increased
by an assigned increment and if phase error was positive the frequency was
decreased by the assigned increment. A sample of this code can be seen in
gure 6 and the resultant phase lock can be seen in gure 7. As one can see
from the two order of magnitude dierence in band width between the low
Q oscillator and the LIGO lens sample, the increments of frequency change
needed to be much smaller to lock to phase to −π
2 . Also the frequency response
was much slower at this order of magnitude higher Q as one might expect from
equation two. As one can see from gure 5 as phase approaches −π
2 each change
in frequency has more of an eect on phase. This means that it is impossible to
create a controller that is both quick to get to −π
2 quickly and accurate enough
to drive directly at frequency −π
2 .
Two strategies were tried to address this issue. First every time the fre-
quency was changed the old increment of change was decreased or increased by
a constant as can been seen in the code in gure 8, the resultant phase error on
a low Q system can be seen in gure 9. This failed to work on the LIGO lens
sample because the corrective ability was not a direct function of the amount of
correction needed. The more eective strategy was to redene the increment of
frequency change in terms of the components of equation 6, ω0 and Q and phase
error every time the frequency was readjusted to reect both the band width of
the mode and the amount of correction needed for the driver to drive at
π
2 . The
old frequency that the function generator has been assigned to generate was
stored and the newly dened change increment was added to the old frequency
and this new frequency was assigned to be become the output of the function
generator and the process was repeated. This error and bandwidth based term
was then multiplied by a constant which was altered and tested many times
until the system behaved satisfactorily. An example of this can be seen in code
in gure 10. This strategy resulted in a phase lock capable of locking the phase
within 4 degrees of −π
2 on the LIGO lens sample. A graph of the output of this
phase controller can be seen in gure 11.
12

13. Figure 7: Phase with constant increment of phase shift
Figure 8: This graph shows the code informing a variable frequency shift incre-
ment controller the code for which can be seen in gure 8. The resultant phase
error can be seen in gure 9.
13

14. Figure 9: This graph shows phase error of a variable frequency shift increment
controller, operating on a low Q oscillator the code for which can be seen in
gure 8.
Although this was a step forward I knew that in order to maximize the accu-
racy of the input parameters used to compute Q I needed to tighten our phase
lock. Also this version of the phase lock loop was incapable of simultaneously
locking the amplitude. In order to make a PID controller and thereby a stronger
phase lock, integral and derivative functions returning variables equal to the in-
tegral and derivative of error over the number of error steps were written into
my script. The output of the proportional feedback was assigned a variable,
then these proportional, integral and derivative variables were then added to-
gether to create the control signal. The integral function as computed using the
mathematics of Taylor's Equation for average:
xm =
x1+x2+···+xN
N
=
xi
N
(15)
Where N is the number of entries in a number set, xm is the mean of the
number set and x1toN and xi are all the entries in the data set [7].
And the derivative function was developed with a mathematical basis from
Taylor's equation for slope, Where x is the x axis x coordinate of a given point,
and y is the y coordinate of a given point [7]:
Slope =
N xy − x y
N x2 − ( x)2
(16)
Figures 12 and 13 show the code of the integral and derivative functions.
14

15. Figure 10: Phase error with a proportional controller on a high Q LIGO lens
sample. Although the phase error may not look much better than that shown
in gure 9 generated with the variable increment phase lock. Figure 9 was
generated on the low system with a band width two orders of magnitude greater.
Figure 11: Proportional feedback code used to generate the phase error on the
high Q LIGO lens sample.
15

16. Figure 12: The code of the integral function.
A PID controller has 5 parameters. These are the constants by which each
term is multiplied putting a gain on each terms eect on the overall feedback
and the amount of time over which the derivative and integrals are computed.
In my case instead of computing the error over time I computed error over steps
or changes in frequency which happened once every time my computerized loop
completed its run. In order to do this a list of errors was established and this
list was maintained at a prescribed length. The integrals and derivatives were
then computed from the data points in these lists. These list lengths were
this project's analogue of the amount of time over which the derivatives and
integral were computed. Although PID controllers are highly eective, tuning
all ve of these parameters to work eectively in concert is a hard task. These
functions were then used in the phase lock loop and amplitude lock loop feedback
functions, allowing the low Q oscillator to lock both phase and amplitude with
only minimal error as one can see in gure 14. Currently I am reconguring
the parameters of the amplitude and phase lock loop so the PID controllers can
work eectively on the LIGO lens sample.
Some of the other work that was done as part of this project was testing the
lock-in time constant that would allow the feedback loops to most eectively
control the phase. my most successful locks were seen with a time constant of one
second. However, these tests took place while I was using incremental correction
control. On some later tests when phase was within 30 degrees either side of
π
2 and the time constant was set to one second, phase increased and decreased
as much as 30 degrees while the drive frequency was held constant. While an
overall decrease or increase in phase may be indicative of a high Q system with a
slow frequency response, as per equation two, coming into phase with the driver,
seemingly random uctuations seem to be more indicative of a noisy signal. So
16

17. Figure 13: The code of the derivative function.
17

18. Figure 14: Error in phase and amplitude of the working amplitude and phase
lock loop running on the low Q oscillator. Note the amplitude set point was
.0407946 V.
Figure 15: Percent error in amplitude lock loop working on the low Q oscillator.
Notice that it is very low at all points.
18

19. I believe moving forward it would be good to test the controllers at higher time
constants which means more noise will be averaged out but updates on phase
and amplitude will come less quickly[4].
Another important result of this research was the creation of a piece of
script that detected modes. The script set the function generator to various
frequencies and recorded the amplitude response. This script maintained a list
of frequency responses excluding the most recent ones and took the average of
this list and compared it to the current frequency response. When the current
frequency response became suciently greater than the average of past responses
the phase lock loop was triggered to being locking. Although this was a good
piece of code, on tests of the phase and amplitude locks in which I am roughly
aware of the modes frequency, which take place often, it is entirely irrelevant.
In future work I would like to encapsulate this part of the script so it is easier
to turn o.
Because I still have to tune the PID until the amplitude lock loop and phase
lock work eectively on the LIGO lens sample and I still need to develop a system
of nding unity gain frequency and thereby nd the feedback gain this project
can't really be called an unmitigated success. Non the less a lot of progress has
been made. Hopefully it has set the ground work for actually measuring the Q
of a LIGO mirror in spring of 2016 when I will continue this work as I begin my
role as a research assistant for my advisor and supervisor Professor Greg Harry.
Appendix: Propagation of Error
According to equation 3.8 of Taylor [4]:
q =
x ∗ . . . ∗ z
u ∗ . . . ∗ z
,
δq
|q|
≈
δx
|x|
+ . . . +
δz
|z|
+
δu
|u|
+ . . . +
δw
|w|
and
δq = |q|
δq
|q|
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