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Investigation of dynamic properties of superconducting quantum interference devices

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Investigation of dynamic properties of superconducting quantum interference devices
Creator:
Hines, Bruce A
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English
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vii, 63 leaves : ; 28 cm.

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Subjects / Keywords:
Superconducting quantum interference devices ( lcsh )
Array processors ( lcsh )
Amplifiers (Electronics) ( lcsh )
Amplifiers (Electronics) ( fast )
Array processors ( fast )
Superconducting quantum interference devices ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (M.S.)--University of Colorado Denver, 2009.
Bibliography:
Includes bibliographical references (leaves 60-63).
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Bruce A. Hines.

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|University of Colorado Denver
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Full Text
INVESTIGATION OF DYNAMIC PROPERTIES OF
SUPERCONDUCTING QUANTUM INTERFERENCE DEVICES
by
Bruce A. Hines
B.S., University of Colorado Denver, 2006
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2009


This thesis for the Master of Science
degree by
Bruce A. Hines
has been approved
by
Miloje Radenkovic
Bob Grabbe


Hines, Bmce A. (M.S., Electrical Engineering)
Investigation of Dynamic Properties of Superconducting Quantum
Interference Devices
Thesis directed by Professor Martin E. Huber
ABSTRACT
I report on investigations into the dynamic properties of Superconducting Quantum
Interference Device (SQUID) Series Array Amplifiers (SSAAs) and their use in a
feedback configuration to form a broadband current-to-voltage transducer. This
investigation used analytical modeling, SPICE simulation, and experimentation with
Helium 4 dewars, refrigeration systems, a function generator and an oscilloscope.
The investigation was both general in nature and specific in relation to distortion that
occurs in the frequency response of the SQUID amplifier system used in the
Cryogenic Dark Matter Search (CDMS) as a part of the signal detection and
amplification chain. I have derived the transfer function for an ideal SQUID
amplifier using low-temperature DC SSAAs in feedback configuration (flux-locked
loop). I have examined, both theoretically and experimentally, non-ideal behavior of
such amplifiers. Experimentation and modeling indicate that the peaking in the
SQUID amplifier frequency response is a result of inductive coupling between the
input and feedback coils of the SQUIDs in the SSAA, which reduces the phase
margin to the point of instability. Low impedance of the input circuit leads to induced
current in the input coil, of sufficient magnitude and phase to destabilize the input
signal in the frequency range of the peaking. This information will be applied in
improving future designs of such amplifiers.
This abstract accurately represents the content of the candidate's thesis. I recommend
its publication.
Signed
Martin E. Huber


TABLE OF CONTENTS
List of Figures................................................... vi
Chapter 1. INTRODUCTION.............................................1
1.1 Broadband amplifiers and linearity..............................1
1.2 SQUID amps..................................................... 1
1.3 SQUID amps in CDMS..............................................2
1.4 This work.......................................................3
Chapter 2. USE OF SSAAs IN CDMS....................................5
Chapter 3. SIMPLIFIED DESCRIPTION OF SQUID OPERATION...............10
Chapter 4. IDEAL TRANSFER FUNCTION IN CLOSED LOOP..................17
4.1 Derivation.....................................................17
4.2 SPICE simulation of ideal SQUID amp............................22
Chapter 5. ANALYSIS OF NON-IDEAL RESPONSE..........................26
5.1 Inductive coupling between coils...............................26
5.2 Delay in feedback line.........................................38
5.3 Impedance mismatch of transmission lines.......................40
5.4 Capacitive loading.............................................40
Chapter 6. EXPERIMENTAL INVESTIGATION...........................48
IV


6.1 Results at UCD...............................................48
6.2 Separate Transmission lines for returns......................49
6.3 Frequency response with capacitor in parallel................52
6.4 Delay in feedback line...................................... 54
Chapter 7. CONCLUSION............................................59
REFERENCES.......................................................60
v


LIST OF FIGURES
Fig. 1 Experimental results from Berkeley...............................2
Fig. 2 Depiction of our galaxy..........................................6
Fig. 3 Illustration of WIMP scattering..................................6
Fig. 4 TES behavior.....................................................7
Fig. 5 Phonons break Cooper Pairs.......................................9
Fig. 6 Cartoon of SQUID................................................ 10
Fig. 7 Biased SQUID....................................................11
Fig. 8 SQUID current to voltage conversion.............................11
Fig. 9 Periodicity of SQUID............................................12
Fig. 10 Array of SQUIDs.................................................13
Fig. 11 SQUID as a washer...............................................14
Fig. 12 SQUID geometry with separate washer.............................15
Fig. 13 General negative feedback diagram...............................17
Fig. 14 Feedback diagram for SQUID amplifier............................19
Fig. 15 Example of SQUID amp circuit....................................22
Fig. 16 Ideal SQUID amp model schematic.................................23
Fig. 17 SPICE simulation of ideal SQUID amp.............................24
Fig. 18 Analytical graph of idal transfer function......................25
Fig. 19 Transformer circuit for input and feedback coils................26
Fig. 20 Graph of induced current magnitude in feedback coil.............29
Fig. 21 Graph of induced current in input coil with high resistance.....30
vi


Fig. 22 Graph of induced current in input coil with low resistance.......31
Fig. 23 Graph of current phase shift in transformer......................34
Fig. 24 Phase-shifted induced current in input coil..................... 35
Fig. 25 Idealized flux-locked loop circuit.............................. 38
Fig. 26 Transfer function with delay term................................40
Fig. 27 Graph of feedback line impedance versus frequeny.................42
Fig. 28 Schematic for SPICE simulation ..................................43
Fig. 29 Results of SPICE simulation......................................44
Fig. 30 Schematic with capacitive loading................................46
Fig. 31 Simulation of SQUID amp with capaciive load......................47
Fig. 32 SQUID amp output at UCD, large resistance on input...............49
Fig. 33 SQUID amp output with separate transmission lines................50
Fig. 34 Comparison of SQUID amp output..................................51
Fig. 35 Analytical graph of SQUID amp, two delay times...................52
Fig. 36 Output with 4.7 pF capacitor.....................................53
Fig. 37 Output with 10 nF capacitor......................................54
Fig. 38 Comparison of SQUID amp output...................................55
Fig. 39 Schematic with inductive coupling................................57
Fig. 40 SPICE simulation with inductive coupling.........................58
vii


1. INTRODUCTION
1.1. Broadband amplifiers and linearity
In this thesis, I investigate various aspects of the dynamic behavior of
Superconducting Quantum Interference Devices (SQUIDs) and SQUID Series Array
Amplifiers (SSAAs). These devices have many uses, one of which is amplification.
Issues that pertain to all kinds of amplifiers also apply to SQUIDs and their associated
bias and readout electronics. In general, one usually strives to achieve an amplifier
with a specified gain over some range of frequencies. Typically an amplifier
magnifies a pulse of certain duration and shape in the time domain. If this pulse were
a step function (i.e. a rise-time of zero) the amplifier would ideally need to be able to
amplify all frequencies. The shorter the rise time, or the shorter the duration of the
features of interest in the pulse, the wider the bandwidth of the amplifier must be. The
amplified pulse can also have overshoot and ringing, which appears as peaking
(localized increase in gain) in the frequency response. In our case, we seek a
broadband amplifier with a constant gain from DC up to a frequency of about 600
kHz, which is the limit on the overall bandwidth imposed by the input circuit. SSAAs
lend themselves to this task quite well.
1.2. SQUID amps
As will be described in more detail later in this paper, SQUIDs are very sensitive to
magnetic fields, or, more specifically, to the magnetic flux passing through the
SQUID loop [1,2]. In one common configuration, this magnetic flux is created by
current through an inductor that is coupled to the SQUID. A very small fluctuation in
the current (as small as a few nA [2]) results in a measurable fluctuation in the
voltage across the SQUID. While the relation between the current and voltage is not
1


linear (it is in fact periodic), the transfer function can be made linear by operating the
SQUID in a feedback configuration. We will see how this is accomplished.
Throughout this paper we will refer to the SSAA with its associated room-
temperature electronics as a SQUID amplifier, or SQUID amp.
1.3. SQUID amps in CDMS
SSAAs, designed and fabricated at the University of Colorado Denver and at the
National Institute of Standards and Technology, have been in use by the Cryogenic
Dark Matter Search (CDMS) for many years [3], While they have been more than
adequate in providing linear amplification of signals in the frequency range of
interest, it has been observed that peaking occurs in the frequency response, lessening
the usable bandwidth. In Figure 1, below, is a graph of the output of a CDMS SQUID
amp that shows this.
As we shall see later in this paper, ideally one would expect a single-pole roll-off in
the frequency response.
2


SQUID amp output, FEB at UCB
Fig. 1. Experimental results from UC Berkeley of the SQUID amp output, showing peaking.
The y-axis is the log of the magnitude of the output of a CDMS SSAA set up for
feedback operation. The x-axis is time in seconds. The trace is the result of a
sinusoidal current through the input of the SQUID amplifier with the frequency swept
logarithmically from 100 Hz to 4 MHz in 0.1 seconds. The peaking occurs around
200 to 300 kHz.
1.4. This work
In this work, I have investigated the dynamic behavior of SQUIDs with an aim of
understanding the cause of the peaking in the SQUID amp frequency response. This
led to a study of resonances occurring in SQUIDs in general, of feedback systems, of
general amplifier stability, and of the limitations of electronics.
This investigation has focused on several possible explanations for the SQUID amp
peaking. These include:
3


1) mutual inductance between the input and feedback coils in the SQUIDs in the
array;
2) a delay in the feedback loop, due to length, inductance and/or capacitance of
the transmission lines, leading to a delay term in the transfer function and a
frequency characteristic with very similar peaking;
3) impedance mismatch between the transmission lines that run from the room
temperature electronics to the SSAA and the terminations of these lines
causing reflections that resonate at certain frequencies;
4) capacitive loading on the final op amp in the amplification chain of the room-
temperature electronics;
5) phase shift in the overall amplification chain, sufficient to reduce the phase
margin to a point where the SQUID amp becomes unstable.
By experimentation, circuit analysis and the use of SPICE [38] modeling, I have
compiled evidence that shows that inductive coupling between the input and feedback
coils of the SSAA results in a phase-shifted induced current in the input coil that
increases the SQUID amp gain over the frequency range of the peaking.
Chapter 2 introduces the use of SSAAs and a SQUID amplifier system in CDMS.
Chapter 3 covers the basic operation of SQUIDs and SSAAs. Chapter 4 contains the
derivation of the transfer function for an ideal SQUID amp. An analysis of non-ideal
responses, including possible theoretical explanations for the resonance seen in the
frequency response of the CDMS SQUID amplifier system, is presented in Chapter 5.
Chapter 6 consists of experimental results and Chapter 7 contains the conclusions and
summary.
4


2. USE OF SSAAs IN CDMS
CDMS is a collaboration of many institutions in a major experiment with the purpose
of identifying the composition of dark matter [3,4,5], Various other investigations
have shown that there is about five times as much matter that is dark (i.e. it does not
interact with electromagnetism) than there is matter that we can see (such as stars,
galaxies, planets, interstellar gas, etc.) [4], A leading theory from the discipline of
particle physics holds that this dark matter is composed of Weakly Interacting
Massive Particles (WIMPs) [3,4,5], These particles, according to the theory, would be
more massive than a proton and would respond only to the gravitational force and to
the weak nuclear force [10], They would be found, for example, in galactic halos [See
Fig, 2] [5,10,11], The WIMPs would pass through Earth at calculable velocity, rate,
and density distributions. Most WIMPs would pass right through the Earth with no
interaction at all. CDMS, in an attempt to validate the theory, expects a certain
number of these WIMPs to recoil off of atoms in detectors made of single-crystal
silicon or germanium [3,4,5], The experiment takes place in the bottom of an inactive
iron mine in northern Minnesota about a half mile underground. The overburden
filters out cosmic radiation that would interfere with the possible detection of
WIMPs. The detectors are brought to ultra-cold temperatures (~50 mK) to reduce
vibrational energy (thermal noise). The scattering of a WIMP off of a detector is
illustrated in Fig. 3.
5


The Milky Way
Fig. 2. Depiction of our galaxy and its halo, theoretically containing hypothetical particles
called WIMPs.
detector
/
Si/Ge
energy transferred appears in the
wake (as phonons) of the recoiling
nucleus
WIMP-Nucleus Scattering
Fig. 3. Illustration of WIMP scattering in a CDMS detector.
6


Highly sensitive thermometers, called transition- edge sensors (TESes) [3,4,5], are on
the surface of the detectors. A current flowing through a TES also flows through the
input of an SSAA, which is wired in series. When the temperature of the detector, and
thus the TES, is slightly raised due to a recoil in the crystal, the resistance of the TES
markedly increases. The current through the SSAA input then drops, which results in
a sudden change of the SSAA output voltage. In turn, this change in output voltage of
the SSAA gets amplified by associated room-temperature electronics. The rise time
and duration of this pulse is such that a bandwidth of less than a few hundred kHz is
sufficient for adequate resolution [9],
Key to the detection of such an interaction, in addition to the SQUID itself, is the
transition-edge sensor. A material that becomes superconducting does so at a certain
critical temperature, Tc. It transitions over a very narrow temperature range from a
normal metal (or a compound in so-called high-temperature superconductors) with a
finite resistance to a superconductor with zero resistance [6], This is shown here in
Fig. 4.
A Tc
Fig. 4. TES behavior. Graph of resistance
versus temperature for a TES. Tc is the
critical temperature of the TES, or the
temperature at which it becomes
superconducting. A bias current through the
TES raises its temperature and keeps its
resistance at a point where it is transitioning
from normal to superconducting. Then the
slope of the curve is steep, meaning that a
small change in temperature of the TES
results in a relatively large change in
resistance. Since the biasing is done with a
constant voltage, the current through the TES
drops as its temperature rises. [7,8,9]
7


The WIMP-nucleus recoil in the detector imparts thermal energy in the form of
phonons [3,4,5], These are quanta of acoustic or vibrational energy (sound),
considered in solid state physics to be a discrete particle rather than a wave,
analogous to photons. Aluminum fins (small strips) on the surface of the detector are
at a temperature such that the fins are superconducting. This means that the electrons
in the aluminum have formed into Cooper pairs, which are bound pairs of electrons,
of opposite spin, that are the basis of superconductivity in classic superconductors [6],
The phonons generated within the germanium crystal move into the aluminum fins
and break the Cooper pairs with their additional energy. Quasiparticles (electrons
from the broken Cooper pairs) then diffuse into the tungsten transition-edge sensor
which is in contact with the aluminum, thereby raising the temperature of the TES.
The TESes used in CDMS are called Quasiparticle-trap-assisted Electrothermal-
feedback Transition-edge sensors (QETs). This is simpler than it might sound. The
TES traps quasiparticles with a consequent rise in temperature and drop in current.
The drop in current results in a cooling of the TES, since it is heat-sunk to the
refrigerator in which it sits, until it stabilizes at its original temperature as determined
by the constant bias voltage. Hence, we get electrothermal feedback [7-9], Below in
Fig. 5 is a cartoon of the arrangement.
8


Fig. 5. Phonons break Cooper pairs in aluminum fins, quasiparticles diffuse into tungsten
TES, temperature of TES rises. TES is voltage-biased on its superconducting-normal
transition. Rising temperature leads to higher resistance causing lower current through the
TES. The pulse is then measured as a change in current with SQUID readout.
The TES operated in this way becomes a very sensitive thermometer. How the
SQUID system detects and amplifies this change in temperature is described in
Chapter 3.
9


3. SIMPLIFIED DESCRIPTION OF SQUID OPERATION, INCLUDING
ARRAYS
A Superconducting Quantum Interference Device (SQUID) is often used as a
sensitive amplifier of tiny signals [15]. These investigations have used SQUIDs that
are employed in the Cryogenic Dark Matter Search (CDMS). These are, more
specifically, an array of 100 low-temperature DC SQUIDs wired in series [5], This
configuration is called a SQUID Series Array Amplifier (SSAA) [12-15], These
arrays are fabricated on 0.2 x 0.2 inch chips.
Each individual SQUID is a superconducting ring, made of niobium, interrupted by
two nominally identical thin insulating layers, or Josephson junctions [16-18],
When it is biased with a constant DC current of sufficient magnitude, it develops a
voltage that is sensitive to magnetic fields, or, in this case, to currents through an
inductor [2,17], The inductor is, in CDMS SQUIDs, a superconducting planar,
rectangular coil surrounding the SQUID.
Fig. 6. Cartoon of a SQUID. Red
areas are insulating layers, typically
aluminum oxide, called Josephson
junctions. Grey areas are
superconducting metal (eg.
niobium). Current flows through
junctions by tunneling.
10


0applied
^0
Fig. 7. Representation of a SQUID with bias
current, i. The xs are the Josephson junctions.
The two junctions are nominally identical. The
currents, i0, are the critical currents for each
junction, or the maximum current that the
junctions can sustain and still remain
superconducting (i.e. no voltage drop across
the junctions). In the SQUIDs used for CDMS,
the magnetic flux, 0appiied, is induced by the
input and feedback coils that are coupled to the
SQUID.
The result is a current-to-voltage converter with a certain transresistance.
Fig. 8. is converted to Voul. M
is the mutual inductance between
the SQUID and the inductor. 0^
Is the magnetic flux through the
SQUID loop created by i^g
through the coil.
The output of the SQUID, Vout, is periodic relative to 0apPUed, the period being one
flux quantum, small fluctuations in current, on the order of a few nA up to about 2 pA (based on
characterization of CDMS SQUIDs at UCD).
11


Fig. 9. Periodic relationship between
applied flux and output voltage of a
SQUID. If a DC bias current is put
through the coil such that the applied
flux is ~o/4, the responsivity of Vout to
small changes, 6(pappi,ed, is maximized.
At this point the slope of the curve is
steepest, resulting in maximal change
in Vmt.
applied
When the SQUIDs are connected in series, the output voltage is amplified by a factor
of 100, while the voltage noise increases only by a factor of 10. Since the noise from
each SQUID is uncorrelated with the others, and the SQUIDs in the array are
nominally identical, the noise adds in quadrature [13,14], We see a cartoon of an
SSAA in Fig. 10.
00
12


Fig. 10. An array of SQUIDs, or an
SSAA. The inductances of each
coil add in series, as do the mutual
inductances. The same holds for
the output voltages of the SQUIDs.
Since they are nominally identical,
simply multiply by the number of
SQUIDs in the array, N, to get the
total output voltage, Varray, total
mutual inductance, AfnA, and total
input inductance, LmA. The
currents, 7fc,aJ. and ij,g, are the same
for each individual SQUID.
The output of SQUIDs is periodic in relation to the magnetic flux through the
superconducting ring [4], However, SQUIDs lend themselves to linearization by
means of negative feedback. The change in flux through the SQUID coming from the
input coil is cancelled by flux from a separate feedback coil, which is wound in the
opposite direction [19,20,21], We see cartoons of possible geometric relationships of
the SQUID, input coil, and feedback coil in Fig. 11 and Fig. 12.
I add the following comment in regards to parasitic capacitances in the CDMS
SQUIDs. In many past papers [22,23,24,25], considerable study was done on the
effects of parasitic capacitance on SQUID performance. Many of these studies were
done on a SQUID design developed by Jaycox and Ketchen [22] that had the purpose
of tightly coupling magnetically the input coil to the SQUID. In this configuration,
13


the SQUID itself has the form of a broad, planar washer and the input coil is
fabricated on top of it with a thin insulating layer between them. This is depicted in
the cartoon in Fig. 11.
This design relies on the so-called Meissner Effect [2,6], which states that a magnetic
field cannot penetrate a superconducting metal beyond a very thin skin depth (about
Fig. 11. SQUID as a washer with tightly coupled input coil on top. SQUID is green. Input
coil is blue. Red x's are Josephson junctions. Metals (green and blue) are superconducting.
40 nm for Niobium) [1], The magnetic flux lines generated by the currents in both the
input coil and the SQUID itself are focused through the rectangular hole in the middle
of the SQUID. This means that the mutual inductance coupling between the SQUID
and the input coil is high, about 0.95 [22], The conductors in cross-section then have
the following configuration:
14


The blue areas represent the turns of the input coil, the orange an insulator (SiC>2,
which has a dielectric constant of 4.5), and the green the SQUID. This geometry
results in a relatively high parasitic capacitance between the input coil and SQUID.
The SQUIDs used in the CDMS SSAAs have a different design that significantly
reduces the parasitic capacitances [12], This is shown in Fig. 12.
Fig. 12. Geometry of conductors in a CDMS SQUID. Blue is the input coil. Pink is the
feedback coil, a single turn around the outside. Dark green is the SQUID. Light green is the
electrically isolated focusing washer.
\ /
V /
/ \
15


Since the focusing washer is now electrically isolated, there is no path for any net
current to flow between it, via capacitance, and either of the SQUID, input coil, or
feedback coil. In this planar geometry, the capacitances between the conductors
making up the loops in the SQUID are small, since the traces are oriented "edge to
edge" and are thin (a few hundred nm in thickness), and are neglected for our
purposes.
16


4. IDEAL TRANSFER FUNCTION IN CLOSED LOOP
When the SSAA is operated in feedback mode, its ideal behavior is that of a linear
broadband amplifier with a single-pole roll-off.
4.1. Derivation
First, I briefly review feedback theory. We see a general negative feedback diagram
below in Fig. 13, where A is the (possibly complex) gain and is the (possibly
complex) feedback fraction.
k
Fig. 13. General negative feedback diagram.
From this we get:
xou, = MXi -pxottt)
By re-arranging, we get the general transfer function, or closed-loop gain:
17


A
1 + Afi
The quantity, A, is called the open-loop gain, and is the gain of the system if the
feedback loop is open. Since, generally, |A|1, the closed-loop gain is
approximated by \/fi. The quantity, Afi, is sometimes referred to as the loop gain.
In a SQUID amp, Xin, Xe, and Xf in Fig. 13 represent values of magnetic flux. Xm and
Xf are produced by currents through the input and feedback coils of the SSAA,
respectively. Xe is the difference between them. Here the SQUID itself is the
summing node, where the currents through the coils have opposite signs, as
determined by the direction of the windings of the coils. In our case, Xout is a voltage,
which, in series with a feedback resistor, produces the current through the feedback
coil [19],
In an analogous way, the flow chart in Fig. 14 shows the feedback arrangement of an
SSAA with the room-temperature electronics, with the SQUID itself functioning as
the summing node [20],
18


Fig. 14. Feedback diagram for SQUID amplifier with room-temperature electronics in
feedback mode.
This is called a flux-locked loop, as the flux through the SQUID is kept constant
(locked) at some integral multiple of derive the ideal transfer function of a SQUID amplifier.
Min is the mutual inductance between the input coil and the SQUID. magnetic flux from the input coil through the SQUID loop. 0p, is the magnetic flux
from the feedback coil through the SQUID. and it is the error signal in this feedback system. D(m) = dVsq/d (= 80 mY/@o) [20],
which is the slope of the V<1> curve at the SQUIDs operating point. Vsq is the output
voltage of the SSAA. Ap(co) is the gain of the preamp of the room-temperature
electronics. Vp is the output voltage of the preamp. A/coJ is the gain of the integrator
of the room-temperature electronics. V0 is the output of the system. Rp, is the
resistance of the feedback line. Ip is the current in the feedback circuit. Mp is the
mutual inductance between the feedback coil and the SQUID.
We have the following relations:
19


% =
dVsa-D-M'
K V*
V. = a7vp
V
b-
R
fb
Vo-A,ApD( ^,-0^)
= A,ApD(IinMin -IpMfi)
= A/ApD(/1,M,,-V0Myi//?yb)
From these relations we conclude:
Vo A,ApDMin
/,, 1 +A,ApDMfb/Rfb
The terms Ai and Ap are frequency dependent. For our purposes here, we will consider
Ap to be a constant, as the frequency response of the preamp does not roll of until it is
well outside the bandwidth of SQUID amplifier system. The integrator in the circuit
is a type of loop shaping to provide dominant-pole compensation. It rolls off at a
frequency, a)mt, low enough to ensure that the gain of the overall transfer function is
less than unity when the phase shift reaches -180. Since the integrator in our room-
temperature electronics has a single-pole roll-off, we can express its frequency-
dependent gain as Ar(0) m/(aw + s), where s=jco [26,27], We shorten this to
Ao)int + s). For small signal measurements we consider D to be constant.
This leads to:
20


1
Kji, (*+<>!)
)
M pDApA,wM
This is the ideal transfer function of the SSAA in feedback mode, relating the input
current to the output voltage. It is a linear amplifier with a single pole roll-off, since
R/b and the mutual inductances are taken to be constant. The bandwidth, or -3dB
point, is the frequency where:
Vo II V,
hn hn
1
which gives:
fidB 2n (1 + MjbDA,Ap IRfl,)
A typical value of (Oi/2n is 2200 Hz. M^D works out to be approximately 50 Q,
though this value varies depending on the SSAA chip and the biasing parameters. Ap
is taken to be 100, and A/ can be varied, depending on the adjustable parameter
selection, from 4 to 28. Rp is 1 k£l If we choose Aj to be 20, the resulting bandwidth
is 222 kHz. The bandwidth can be varied by adjusting the values of the feedback
resistor and the gain of the amplification chain.
An example of one circuit that fits the above transfer function is shown below in Fig.
15. Here, the functionality of an integrator is accomplished by a variable-gain op amp
followed by an RC pole. Additionally there is an output amplifier with a gain of 4 by
which one can select the polarity of the output. These elements are combined into the
Ai(co) term. This particular circuit was part of a prototype PCB that was designed and
fabricated at FermiLab for CDMS.
21


f *
| 3.7Sk
Prearrp
cair. :::
i

Front Ena Gain
varlaole,
Ult
U 03Uz
_o*?ass
> . Gain 4
E> W
:.322nrt
-L- 4
Output Gain
Variable,
Signs. Generator ^ j,
<*'-o
C.ltti |2N
3:0:112
~owr *
DC feedback Gam
- 400 to 28CC
Fig. 15. Example of SQUID amp circuit (courtesy of D.Seitz, UC Berkeley).
4.2. SPICE simulation of ideal SQUID amp
Below we see a circuit that simulates an ideal SQUID amplifier. It is based on the
circuit in Fig. 15, above. The SQUID itself is taken to be a current controlled voltage
source. In this schematic there are two such voltage sources. One is voltage generated
by current through the input coil with a gain of 500 and the other, with opposite
polarity, is from current through the feedback coil with a gain of 50. The gain of 500
represents a 500 ohm transresistance of the SQUID array (typical value). The ratio of
500 to 50 is based on the 10:1 turns ratio of the input coil to the feedback coil. The op
amps in the feedback chain are modeled as voltage controlled voltage sources one
22


for the preamp with a gain of 100 (Epreamp), one for the variable-gain amp with a
gain of 5 (Evg) and one for the output with a gain of 4. The integrator is modeled as a
resistor in series with a capacitor to ground, which forms an RC pole at about 2200
Hz. The purpose of this is to provide loop shaping by means of dominant-pole
compensation to prevent peaking.
Fig. 16. Ideal SQUID amp model schematic.
A SPICE simulation of the above circuit results in the graph in Fig. 17. This does
show, as expected, a single-pole roll-off (as evidenced by the slope of the roll-off at
20 dB per decade), with the -3dB point at the expected frequency.
23


JB< out)
AC: Graph: Generated by Schematics Ptus. Project Name: SQamptdeaiOSOSI 2 Date 7/3/09 Time S:04 PM
Fri, Jul 03. 2009 17:05:17
Fig. 17. SPICE simulation of ideal SQUID amp, generated by modeling software using the
schematic in Fig. 16.
An analytical graph of the derived transfer function gives the same curve, as seen in
Fig. 18. It is included here as a separate figure to show that the SPICE model and the
transfer function give matching results.
24


Fig. 18. Analytical graph of ideal transfer function for a SQUID amp.
25


5. ANALYSIS OF NON-IDEAL RESPONSE
5.1. Inductive coupling between the SQUID input and feedback coils.
As seen in Chapter 4, the input coil and the feedback coil both couple magnetically to
the SQUID. However, we see that, due to the geometric arrangement, there is also
inductive coupling between the two coils that could affect the transfer function.
In Fig. 19 below, we see a schematic of how the input and feedback coils around the
SQUID act as a transformer. In this configuration, the input coil is the primary and
the feedback coil is the secondary.
Kinfb
0.5
Rin
IK
Rqet
1.2

Rfb
IK
Fig. 19. Transformer circuit for input and feedback coils.
In this circuit, Vin is a varying voltage source that, through an equivalent input
impedance (Rin + Rsh\\(Rqet+sLir,)), produces a current /, through the input coils of the
SQUIDs in the SSAA. Rsh is the shunt resistor and Rqet represents the resistance of the
26


TES. The input coils, 100 of them in series, have a total self-inductance, Lin (about
0.2 uH in the CDMS SSAAs [2]). Similarly, the feedback coils have a self-inductance
Lfl, (about 2nH [2]). Minfb is the mutual inductance between the input coils and
feedback coils. Its value is:
where k,nj, is the coupling constant of the mutual inductance, reported to be about 0.5
Using Kirchoff s Voltage law in the two loops in the circuit above, we get [27]:
The negative signs in these equations are a result of the fact that the mutual
inductance with varying currents produce the equivalent of voltage sources, in series,
in both loops.
If we assume that the AC voltage source is sinusoidal, it will help us understand the
effects of the mutual inductance between the two coils. Then:
[28],
27


b
dL
dt

= \slin\ = dt

By substitution and some algebra we arrive at:
(oM. A.
/in_jb in



where:
^fb ~ \Rfl> + S^Jb\
| - + (UiLfb)
Likewise, since the feedback resistor and coil also have a voltage source in series
with them (what we have been calling Vout), we can write:
jn_]b I ft,
W+K.)2
These expressions give the amount of induced current in each coil in relation to the
current in the other coil. These add (by superposition since the induced voltages are in
series) to the currents produced in each coil by their respective voltage source.
Here in Fig. 20, we see a graph of the magnitude of versus frequency, as in
the above equation for the induced current in the feedback coil. One can view this as
the current-to-current transfer function, for the inductive coupling. It is clear
from the graph that over the frequency range of interest, up to about 800 kHz, the
current induced in the feedback coil is extremely small, especially considering that
this occurs in a flux-locked loop with a 10:1 input to feedback turns ratio. The current
in the feedback coil would change by about 0.005%, which we can neglect.
28



--------------u£-------------_--- . ----------
Fig. 20. Graph of the current-to-current transfer function for the inductive coupling from the
SSAA input coils to the feedback coils with a resistor of 1 kQ in series with the input coil.
We get a similar result when plotting the transfer function for the induced current in
the input coil when a 1-kQ resistor is in series with the AC voltage source to produce
input signal. This was the setup used for the initial tests at UCD on the SQUID amps
frequency response. Similarly, the change in the current in the input coil due to
inductive coupling from the feedback coil is negligible, as can be seen in the graph in
Fig. 21, below.
However, we get a significantly different result when the input coil is biased with a 1-
Q resistor, as in the schematic in Fig. 19. A shunt resistor of 0.02 Q is wired in
parallel with the coil. This has the effect of putting a constant voltage, or an effective
voltage source, in series with the 1-Q resistor and the input coil. The resistor value of
1 Q, Rqei, approximates that of the TESes used in the CDMS experiment. It should be
noted that this creates an L/R pole with a cutoff frequency of about 600 kHz
29


(.R/(2jcL)), which limits the frequency range over which the SQUID amp needs to be
linear. It should also be noted that this pole occurs before the summing node (the
Fig. 21. Graph of the current-to-current transfer function for the inductive coupling from the
SSAA feedback coils to the input coils, with a 1 kQ resistor in series with the input coil.
SQUID itself), so it does not contribute to the phase shift of the amplifier system.
The graph in Fig. 22 shows how the induced current in the input coil, which adds to
input signal current, relates to the current in the feedback coil. (The possibility of the
induced current subtracting from the input signal is considered below.) Because the
current in the input coil is about 10 times as great as that in the input coil, the transfer
function in Fig. 22 is multiplied by a factor of 10. We see that at a frequency of 600
kHz, the transfer function has a value of about 0.30. The significance of this will
30


become apparent. This means that at this frequency, the current in the input coil
increases by as much as 30%,. This logic is based on the following considerations:
5^.10-S/,,
U1in total UIin +
= <*>/. +10-b/.
W+ks
wM,
in^Jb
VV+K*.)1
/
= d/._
1+ 10-
toM
in^Jb
AJ+K.)!
Fig. 22. Graph of the current-to-current transfer function for the inductive coupling from the
SSAA input coils to the feedback coils with a resistor of 1 Q in series with the input coil.
In order to understand how the current induced in the input coil, due to the
transformer action, adds to (or subtracts from) signal current, it is necessary to
31


examine the relative phases of the currents in the two coils and the phase shifts that
occur in the SQUID amp.
If the SQUID amp were infinitely fast, including the SQUID itself, the connecting
cables and the room-temperature electronics, the currents in the input and feedback
coils would be 180 out of phase [26], This is true for any negative feedback
amplifier. Another way to look at this is that the directions of the respective currents
through their loops around the SQUID must be opposite, in order that the magnetic
flux through the SQUID produced by the feedback coil cancel will cancel, or null that
produced by the input coil. Then the current values have opposite signs. This means
that if the current in the input coil has a value of sin(arf), then the current in the
feedback coil would be proportional to sin(atf -180).
The SQUID amp, however, is not infinitely fast. So, the phase shift is greater than
180 and we can write the current in the feedback coil as sin(cut 180 (f>amp), where
amP is the additional phase shift caused by the delay in the signal from the system
components. If in phase, which is to say that they add together. Instead of negative feedback we have
positive feedback. It is this situation, when the gain of the system is greater than
unity, that results in the amplifier being unstable or resonating.
To see what happens in a SQUID amplifier system, it is helpful to look at
transformers in general. In order to conform to Lenzs Law, there is a phase
difference between the currents in the primary and secondary coils. Whether the
phase shift is positive or negative depends on the relative directions of the windings
of the two inductors in the transformer [29], In the circumstance where we are
looking at the feedback coil as the primary and the input coil as the secondary, the
32


phase shift is given by ^ras = -arctan[i?9e/(a>L,)]. This calculates to a phase shift of -
90 up to a frequency of about 10 kHz, and is about -53 at 600 kHz.
Below is Fig. 23 we see the situation where the phase of the current in the secondary
coil is shifted by -90. If, for example, the current through the primary coil is
increasing in a positive direction, then the magnetic flux through the secondary coil is
also increasing. This induces a current in the secondary coil in the opposite direction
(i.e. the sign of the current will be negative) to produce flux opposed to that from the
primary coil (Lenzs Law). It is this situation that exists in a SQUID, as the currents
in the input and feedback coils produce respective magnetic flux from each coil that is
opposed.
Now if we view the feedback coil as the primary and the input coil as the secondary,
we have the situation described above of the additional current induced in the input
coil. In light of this discussion, its phase becomes sin(arf 180 (f>amp 53) at 600
kHz.
Therefore, if the amplifier cutoff frequency, resonance would occur in the frequency response. In
a SQUID amp with an integrator with a pole at 2.2 kHz, such as the example circuit
shown in Chapter 4, a phase shift of -90 occurs at 600 kHz from the integrator in the
amplification chain. If an additional shift of -37 results from other components in the
SQUID amp at 600 kHz, the phase margin is then zero at that frequency. Commonly,
phase margin is kept greater than 45 to ensure amplifier stability [26], With a phase
margin of 45, there is peaking to a value 1.3 times the gain of the amplifier system
at low frequencies [26],
33


Below in Fig. 24 we see graphically how the induced current in the input coil, due to
its phase shift, could add to and increase the amplitude of the input signal, thus
increasing the SQUID amp output.
Currents in primary and secondary coils
----current current in ieconoary
Fig. 23. Currents in the two coils of a transformer with a 90 phase shift in the secondary.
34


currents, input and feedback coils, phase shifts
10
u
D
3
Its
f-800
b
800
b *
primary input
feecDdc<
addition to r>put
total nput current
10 -
-IS J
angle (degrees)
Fig. 24. Phase-shifted induced current adds to the input signal, which would increase the
magnitude of the output.
In the above ideal analysis, one must first consider that there are three inductors
interacting in the SQUIDs used in the CDMS SSAAs. These are the input coil, the
feedback coil and the SQUID itself. What of the self-flux from the SQUID itself?
Are we justified in neglecting it?
The inductance of the SQUID itself comes from the fact that it is a loop that generates
circulating currents in order to maintain the quantization condition of magnetic flux
through the loop [17], Corresponding to this we have, then, a 3 by 3 magnetic flux
matrix [30]:
35


in = LiJin + Min_sqIsq + Mm_fblfb
^5? = M in _sqhn + ^sq^sq + l^sq_fl.Ifb
jb ~ ^in_fb^in + ^sq_fb^sq + fb
Here, the Os represent the magnetic flux through the three inductors, the Ms are
mutual inductances, the U s are self-inductances and the Fs are currents through the
inductors. In particular, Isq is the circulating current around the SQUID loop, as
opposed to the bias current across the SQUID to power it.
When a SQUID is operating in a feedback amplification system, a DC bias current
through the input coil sets the SQUID at its working point on the steepest slope of the
V0 curve, as covered in Chapter 3 [17,19], This creates a constant flux through the
SQUID loop. In order to maintain the quantization condition of magnetic flux
through the SQUID loop, a circulating current forms around the SQUID, creating a
flux through the SQUID such that the total flux, Om + account by the room-temperature electronics. The SSAA output voltage is connected
to the inverting terminal of the preamp and an equal voltage is applied to the non-
inverting terminal. We then amplify small-signal fluctuations in the current through
the input coil, which we signify by dlin. The middle equation above then becomes:
<50 = M. 61. + L <5/ + ^-61,
sq in_sq in sq sq sq_Jb Jb
Because of the flux nulling brought about by the quantization condition, we know that
6Isq will adjust in response to changes in 61 in or 61^, so that <50= 0. This leads to:
6I Min.iq-Vin + M^-dlp,
sq
sq
36


The electronics in the SQUID amp are set up so that the magnetic flux created by the
input coil and feedback coil are opposed. So if we decide to call the positive direction
of current as that which produces magnetic flux in the direction of the positive normal
vector to the planar loop, it is evident that dlin and dip have opposite signs. We also
know:
N-10'W
because of the turns ration between the two coils.
Furthermore,
Min_Sq = kl-^LinLsq
and
Msq_jb = k2-\jLS(]Ljh
where ki and k2 are the respective coupling coefficients. This gives
leading to
"10
Due to the geometry of the SQUIDs and input and feedback coils on the SSAA chip,
and the presence of the superconducting flux-focusing washer, we expect that kj^,
so that dlsq becomes negligible. This implies that we can reduce the 3 by 3 flux
matrix, above, to a 2 by 2 matrix:
(i>,n = LJ>n +
= ^ in in + Lft, I ft,
37


This is the same as treating the mutual inductance between the input and feedback
coils as one would for a transformer.
5.2. Delay in feedback line
This possibility is covered in papers authored by Dietmar Drung [19,31,32], The
following discussion uses his notation and borrows figures used in his papers.
Simplified SQUID electronics are depicted in Fig. 25.
Since the SQUID amplifier system has the same frequency response as a first-order
lowpass filter, one can view it ideally as an integrator with a constant DC gain and a
single-pole roll-off. Then, the open loop gain is described by
G,(f)
1
OC-----
jf
Then, the normalized frequency response is
c(/) =
1
jf/A
38


where/; is the unity-gain frequency of the feedback loop. In this circumstance, the
ideal (i.e. zero delay) closed-loop frequency response rolls off with a -3dB cutoff
frequency given as
The effects of the delay between the SQUID and the room-temperature electronics
can be expressed as
where U is the total delay time. In actuality this quantity includes the propagation
time of the signals on transmission lines plus delays caused by components in the
circuitry. From this expression we derive the magnitude of the closed-loop frequency
response to be
In this way of looking at the system, the product/;/,; determines whether the amplifier
will be stable or not, and it also changes the bandwidth. When there is no delay time,
i.e. when Q=0, the frequency response is that of a single-pole lowpass filter with
fi=fidB- Another way to say this is that the phase margin is then 180. The delay term
adds a second order term into the denominator of the transfer function, meaning the
system acquires a second pole. Interestingly, if the fjtj product is 0.08, the transfer
function exhibits no peaking and the bandwidth is increased by a factor of 2.25.
Above this point, the phase margin is reduced to a point where peaking appears.
Below in Fig. 26 are graphs of the transfer functions of this possibility showing the
results for different values of the/;U product. If the bandwidth of the CDMS SQUID
amp is 600 kHz (see Chapter 6), then U would have to be greater than about 140 ns
for peaking to appear in the transfer function. The cables used in the experimental
G(/) = ^777exP(-^2^)
jf Ifi
39


setup at UCD uses 50-Q coax of type RG-58/U, which has a propagation time of
approximately 1.5 ns/foot [33], In our setup there is a total of roughly 12 feet of coax
between the SQUIDs in the array and the feedback coils, which contributes about 18
ns of delay time. From this we see that this is a relatively small fraction of the total
delay time needed to produce peaking. A frequency of 600 kHz equates to about 1.7
ps/cycle. Since 360 make up a full cycle, a delay time of 18 ns is a phase shift of
only -4.
5.3. Impedance mismatch of transmission lines.
From the output of the SSAA to the non-inverting input of the room temperature
preamp there is a transmission line, or actually a series of them. In CDMS, this
consists of a so-called flyover cable, a stripline, and an I/O cable (twisted pairs).
These each have a characteristic impedance of maybe 75 £2 to 125 £2. The input of the
op amp used for the preamp is high impedance (1 M£2 or higher). The returns for the
40


three signal lines (one for the SQUID itself, one for the input coil and one for the
feedback coil) also go via transmission lines to a common ground at room
temperature. The operating resistance of the SSAA (i.e. on the SQUID bias line) is
about 70 80 Q [2], Thus, there are impedance mismatches between the transmission
lines and the components both at room temperature and at the cold end.
The input impedance, Zin, of a transmission line with characteristic impedance Zq, that
has on it a load of impedance ZL, is given by [34]:
ZJ-l,() = Z0
|^ + y-tan(f)
w-frMf)
where / is the length of the transmission line, c is the speed of light in a vacuum, and j
is the square root of -1. Here, the load is taken to be at position x=0 and the input at
x=-l. Clearly the impedance seen by the SQUIDs and the coils of the SSAA depend
on both the frequency and the length of the lines.
Below in Fig. 27 is a graph of this equation showing the frequency response for a
value of / = 1.3 meters, the length of the cables in our experimental setup at UCD. Zo
is set at 50 Q, since we are using 50-ohm coax in our lab at UCD, and Zi at (1 kQ +
sLfb), the feedback resistor and feedback coil in series. This shows that the resistance
across which Vout drops, Rjbeff, decreases starting at about 400 kHz, which would lead
to an increase in current through the feedback coil. Substituting Rfl, ef in the equations
for the ideal transfer function and -3dB frequency in Chapter 4 show that the effect
would be to lessen the SQUID amp output amplitude and increase its bandwidth. It
does not lead to peaking in the frequency response.
41


Fig. 27. Graph of feedback line impedance versus frequency. This is for a cable length of
1.3 meters and a transmission line characteristic impedance of 50 ohms.
We also examine the voltage in the transmission line between the SSAA and the
room-temperature electronics by means of a SPICE simulation. The circuit schematic
is shown in Fig. 28.
The results of the SPICE simulation are displayed in Fig. 29. The transmission lines
have no effect on the voltage until about 2 MHz, which is outside of our frequency
range of interest. This indicates that this does not contribute to the peaking in the
SQUID amp frequency response.
42


Fig. 28. Schematic capture for SPICE simulation of the effect of transmission lines on the
SQUID bias for the SSAA. It uses a normalized AC voltage source, an SSAA resistance of
80 Q, two 50-Q transmission lines (one to ground and one to the input of an operational
amplifier represented by a large resistor = 5 MQ), and propagation delays of 6 ns (the
approximate propagation time for a 1.3 meter 50-Q coax).
43


V
frQuttftey Hz
ACGracft : Gnnttd tty Schemata Plu*. Project Nan*: T1iwSQ090705 Date 7/7/09 T* 3:17 PM
TM.M07. 200* 13:1 8:31
Fig. 29. Results of SPICE simulation of the effects of transmission lines on the SSAA output
and return.
How additional phase shifts introduced by the transmission lines via parasitic
capacitance might affect the output is examined separately in section 5.4, below.
5.4. Capacitive loading
When a capacitor is connected at the output of an op amp, in parallel with the load,
ringing can occur [35]. An unwanted parasitic capacitance, such as that of a
transmission line on an op amps output, can result in unexpected behavior in an
amplifier system. Op amps have an inherent output resistance, Ro, which, in
conjunction with a capacitive load, forms an additional pole in the amplifiers transfer
function. If Cl is the capacitive load, a pole is formed at a frequency of 1/(2ji RoC'l)
However, this is complicated by the fact that the output impedance of the op amp is
actually frequency-dependent, generally increasing with frequency.
44


Here we briefly discuss the concepts of phase margin and feedback amplifier
stability. The feedback signals of all amplifying systems have a time delay in relation
to their input signals. This delay is mainly caused by internal resistances, current
limits and capacitances within the amplifier. The result is a phase difference between
the amplifier's input and feedback signals. If there are enough stages in the amplifier,
at some frequency, the output signal will lag behind the input signal by one
wavelength. In this situation, the amplifier's output signal will be in phase with its
input signal though lagging behind it by 360. This lag is important in amplifiers that
use feedback. The amplifier will oscillate if the fed-back output signal is in phase
with the input signal at the frequency at which its open loop voltage gain equals its
closed loop voltage gain and the open loop voltage gain is unity or greater. The
oscillation occurs because the fed-back output signal then reinforces the input signal
at that frequency. An operational amplifier operated with negative feedback (through
the inverting input) introduces a phase shift of -180 by itself. So the critical output
phase angle is -180, as the total phase shift becomes -360 when added together.
The data sheets of op amps often show graphs of the output resistance versus
frequency. For example, the op amp connected to the feedback coil currently used in
CDMS, an AD8001, has an output resistance of about 0.6 Q at a frequency of 600
kHz. This means that there would have to be a capacitive load of 4.4 pF to create a
pole at 600 kHz. This would result in a phase shift of -45 at that frequency. This is
relatively a very large capacitance.
For our purposes the question becomes whether there is sufficient parasitic
capacitance in the CDMS cabling connected to the Vout node in the system to create a
pole at a low enough frequency to cause peaking in the output. We shall see in
Chapter 6 below that, experimentally, this is not the case.
45


Fig 30. Schematic used in a SPICE simulation of the effect of capacitive loading on the
SQUID amp.
In Fig. 30 and Fig. 31, we see the results of a SPICE simulation of a SQUID amp
with a 10 nF capacitive load. This simulation uses an LM741 op amp, which has its
own internal dominant-pole compensation. For this reason an integrator or RC
46


lowpass filter is not used in the circuit. While this will not match the results of the
CDMS SQUID amp, it still shows the effect of capacitive loading. We see the
peaking at about 700 kHz. There is a smooth single-pole roll-off when the 10-nF
capacitor is not added to the circuit.
^dB( out)
ACrGraph : Generated by Schematics Plus. Project Name: 5Qnpldeal0A090705 Orta 7/5/09 Tim* 4:46 PM
Sun. Mi OS. 2009 16:47:09
Fig. 31. Simulation of ideal SQUID amp with added capacitive loading.
47


6. EXPERIMENTAL INVESTIGATION OF THESE POSSIBILITIES
6.1. No peaking with 50-ohm coax and no shunt resistor at VCD
In our initial setup at UCD, the SQUID amp using a CDMS SSAA did not have
peaking in its frequency response. Below in Fig. 32 is a graph of measurements made
with a function generator and an oscilloscope. A sinusoidal voltage, 80 mVp-p, was
put through the input coil via a 100 kQ resistor in series with a 1 kQ resistor that is on
the probe. I used a liquid helium dewar with a probe designed for the CDMS SSAAs.
The probe has 50-Q coax that runs from room temperature to the SQUID at 4K, with
50-Q SMA connectors. For the input coil, the SQUID, and the feedback coil, there
was one coax each i.e. the center conductor was the signal and the shield was the
return. The feedback line has a 1 kQ resistor at the cold end, with the coax shields
connected to ground at the cold end. The SQUID output has a preamp with a gain of
100 on its output, which in turn is connected to a feedback circuit, which includes
additional amplification and an integrator. The peak-to-peak voltage of the signal
generator and resistor values were chosen to produce a small-signal current
fluctuation in the input coil of 790 nA.
We see the results of this test below in Fig. 32. It shows the output voltage
magnitude in dBm versus frequency.
48


SQUID amp output 090529
Fig 32. SQUID amplifier output at UCD with a large series resistance on the input coil.
We observe that there is no peaking with a bandwidth of about 700 kHz. The roll-off
looks to be about 20 dB per decade, but then steepens between 1 and 2 MHz,
indicating an as-yet-unidentified pole.
6.2. Frequency response with separate transmission lines between cold chip and
room-temperature electronics
The same setup was used with a different wiring of the cables that run between room
temperature and the SSAA chip. At the cold end, the shields were left unconnected.
Three additional coaxial cables were added, so that each signal had a coax with the
49


shield grounded only at the warm end, as did each return. This wiring scheme is used
in the CDMS experiment (i.e. the returns have separate cables grounded only at the
warm end). The idea of this test was to see if adding transmission lines from the
SSAA to ground changed the frequency response. Repeating the same measurements
resulted in the chart below in Fig. 33.
SQUID amp output with T 090601
25.0 -----------------------------------------------------------
-10.0 ----------------------------------------------------------------1-------------
frequency (Hz)
Fig. 33. SQUID amp output with separate transmission lines for signals and returns.
We see again that there is no peaking. Putting the results from both sets of
measurements on the same graph, for easier comparison, gives the following in Fig.
34.
50


SQUID Amp Output
ground at cold end
ground at warm end
Fig. 34. Comparison of SQUID amp output with and without the returns being transmission
lines to a common ground at the warm end.
We see that the graphs are virtually identical up to about 800 kHz, at which point the
output values for the setup with separate transmission lines are higher. A possible
explanation is that the added transmission lines increase time delay, either in the form
of propagation time through the additional coax, or added capacitive loading in
parallel to the feedback loop, or a combination of the two. This is supported by the
graph shown in Fig. 35. This is the equation for SQUID amp frequency response
from Section 5.2, with a difference of 20 ns time dealy between the two curves. The
bandwidth for zero time delay (-fi) in these equations is 700 kHz, close to the value
in the results in Fig. 34.
51


Fig. 35. Graph of analytical equation for the SQUID amp frequency responses with two
different delay times in the electronics.
6.3. Frequency response with capacitor in parallel
Measurements similar to those above were made with a capacitor added in parallel as
an additional load on the amplification chain. This was done once with a relatively
large capacitor (4.7 pF) and another time with a smaller one (10 nF). (See the
discussion on capacitive loading in Chapter 5.) The results are seen below in the two
graphs in Fig. 36 and Fig. 37. Peaking does occur from this in both cases, with the
expected lessened bandwidth caused by the additional pole at a frequency below the
unity-gain frequency. These figures also show the fact of no peaking when no
capacitor was added in parallel to the feedback line.
52


CDMS SQUID Amp Output, at UCD, with and without 4.7 uP cap
1000
Fig 36. Output with a 4.7 pF capacitor in parallel with the feedback line.
SQUID amp output, with and without lO nF cap, at UCD
Frequency (kHz)
w-th lOnF cao
ro ca a
Fig. 37. Frequency response with 10 nF capacitor.
53


Tests conducted at UC Berkeley indicate that the total capacitance of the transmission
lines used in the CDMS experiment is about 700 pF [28], Since capacitors with
values well over an order of magnitude higher than this were required to cause
peaking in the SQUID amp frequency response, I conclude that capacitive loading is
not the cause of the CDMS peaking.
6.4. Delay time in feedback line
Below in Fig. 38 are graphs showing results from Berkeley and Denver using three
different cabling schemes that result nearly identical peaking. One cabling scheme at
Berkeley used a relatively short (about 1 foot) cables inside the test dewar to connect
the SSAA to the room-temperature electronics. Another scheme used the same
cabling inside the test dewar plus an 8 foot cable of twisted pairs added in. The UCD
scheme is described above in Section 6.1.2, but with one significant change. The
shunt resistor (as in Fig. 19 above), is wired in, plus a 1-Q resistor in place of the
QET. In the graphs below, the two Berkeley cabling schemes are used with two
different sets of room-temperature electronics one is called the DCRC for Detector
Control and Readout Card, and the other is called FEB for Front End Board. At UCD,
warms electronics with yet another design was used.
These results, along with the lack of peaking when the shunt resistor was not included
in the circuit at UCD, indicates that the propagation time delay is not the cause of the
SQUID amp peaking, as the cable lengths and cable characteristic impedances in the
three schemes are considerably different. Capacitive loading is also ruled out.
54


SQUID amp output OCMC UCS
squid amp output t at UCS
SQUID amp output with Rah and Rqet !*, at UCD
SQUID amp output, log magnitude,
with three different warm and cold
cabling schemes. All three show the
same peaking in the frequency
response. In common are the shunt
resistor and IQ series resistor on
input coil (in place of TES).
Fig. 38. Graphs indicating that the SQUID amp peaking does not depend on the feedback
delay time or capacitive loading.
This leaves the mutual inductance between the input and feedback coils and the
resultant additional phase shift as the most likely explanation. The details of this are
covered in detail in Section 5.1.
To corroborate this, I did the same SPICE simulation as was described in Section 5.4,
except that instead of the capacitive load, I added inductive coupling between the
input and feedback coils. The schematic is shown in Fig. 39 and the results of the
simulation is shown in Fig. 40.
55


We see that peaking does in fact result in this simulation.
The peak occurs at a frequency of about 1.5 MHz. This higher frequency is likely
caused by an amplification chain that is different from that used in CDMS. The
dominant-pole compensation is done internally in the op amp (an LM741), so there is
no integrator or lowpass filter in the chain. In principle the circuit design is the same
however. We would not expect the same bandwidth, as the dominant pole does not
have the same frequency, and the bandwidth is directly proportional to this frequency
(see equation for bandwidth in section 4.1).
However, these results do support the hypothesis that the peaking we have observed
in the CDMS SQUID amp is caused by the inductive coupling between the input and
feedback coils.
56


Fig. 39. Schematic capture used in SPICE model of a SQUID amp with inductive coupling
between the input and feedback coils.
57


""* **; im .
*
u
*rfa.n^y 7
WG'jpM O'W'atfricty brNwAlrt *V. N.l Krf UJ^rurf-pnAWOW n*t V m -' VS1 PV
Tv*, wl 07 2009 17:51 36
Fig. 40. Results of AC simulation of SQUID amp model with inductive coupling between the
input and feedback coils.
58


7. CONCLUSION
I have investigated Superconducting Quantum Interference Device Series Array
Amplifiers and their use in a feedback configuration to form a broadband amplifier.
This investigation used analytical modeling, SPICE simulation, and experimentation
with Helium 4 dewars, refrigeration systems, a Vector Network Analyzer, a function
generator and an oscilloscope. The investigation was both general in nature and
specific in relation to peaking that occurs in the SQUID amplifier system used in the
Cryogenic Dark Matter Search. I described in general how SQUID amplifiers
operate, how they are used in CDMS, and derived their ideal transfer function. I
evaluated theoretically and experimentally possible causes for the SQUID amp
instability, including the effects of transmission lines with mismatched impedances,
propagation delay, capacitive loading, phase shift and phase margin, and mutual
inductance between the input and feedback coils. I have concluded that of these
possibilities, only the inductive coupling between the coils and the resulting
additional phase shift can cause the observed SQUID amp instability. A SPICE
simulation has supported this conclusion.
The next stage of this investigation will be to test other designs of the room-
temperature electronics and/or the SSAA to stabilize the SQUID amp.
59


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