A Magnetic Trap for Evaporative Cooling of Rb Atoms
Benjamin Deissler
Schifferstadt, Germany
Vordiplom, Universität Kaiserslautern, 2001
A Thesis presented to the Graduate Faculty
of the University of Virginia in Candidacy for the Degree of
Master of Science
Department of Physics
University of Virginia
August 2003
i
A man will be imprisoned in a room with a door that’s unlocked
and opens inwards; as long as it does not occur to him to pull
rather than push.
Ludwig Wittgenstein, Culture and Value
ii
Acknowledgements
I realize that I am indebted to a number of people, and I would like to use this opportunity
to mention and thank them.
First of all, I would like to thank the German Academic Exchange Service (DAAD) for
its generous financial support, without which my year here at the University of Virginia
would not have been possible.
I would also like to thank my high-school physics teacher, Mr. Zschernitz, for showing
me the beauty of physics, as well as my professors at the University of Kaiserslautern, Prof.
Hotop and Prof. Fleischhauer, for keeping my fascination alive and kindling my interest
in AMO physics. A big nod of appreciation also to my teachers here at the University of
Virginia – Prof. Fowler, Prof. Fishbane and Prof. Poon.
I cannot begin to thank my advisor, Prof. Sackett, for all that he has done for me. His
support and helpfulness stand unmatched, I can only hope to have the pleasure of working
with more such people in my career. Thanks to Prof. Jones for the helpful comments and
for being on my committee. Jessica and Ofir also deserve appreciation for their help in the
lab, and the same goes to Ken, Jeramy, Jessica, Patrick and Patipan, who were here in the
summer.
Were I to list all of the people I wish to thank in the Ballroom Dance Club at UVa,
this section would probably be longer than my thesis. So here’s a general thank-you to all
of you for keeping me sane and for the wonderful times we’ve had together – the last year
would not have been as fun without you!
My girlfriend Kristen deserves my gratitude for her interest, understanding, and patience
in hearing me complain when my project wasn’t working (again) and for believing in me
that it would work.
Finally, my parents have always been there for me, supported me in all possible ways
and helped me in ways I cannot express in these few lines. Thank you.
Contents
1 Introduction
1.1 Magnetic Traps . . . . . . . . . . . . . .
1.1.1 Quadrupole Trap . . . . . . . . .
1.1.2 Ioffe-Pritchard Trap . . . . . . .
1.2 Evaporative Cooling . . . . . . . . . . .
1.2.1 A Model for Evaporative Cooling
1.2.2 Cooling Sequence . . . . . . . . .
1.3 Goals . . . . . . . . . . . . . . . . . . .
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1
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10
10
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14
15
3 Computer Simulation
3.1 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Effects of Varistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
19
21
22
4 Technical Details
4.1 Making the Coils . . .
4.2 Subcircuits . . . . . .
4.2.1 Debouncer . .
4.2.2 LEDs . . . . .
4.3 Connection Diagrams
4.3.1 Front Panel . .
4.3.2 Board 1 . . . .
4.3.3 Board 2 . . . .
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24
24
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5 Performance
5.1 Measurements with “Dummy” Coil . . . . . . . . . . . . . . . . . . . . . . .
5.2 Measurements with Quadrupole Coils . . . . . . . . . . . . . . . . . . . . .
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
34
35
39
2 Description of Circuit
2.1 Main Circuit . . . .
2.1.1 Switching Off
2.1.2 Switching On
2.1.3 MOT2 Field
2.2 Control Circuit . . .
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iii
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iv
A MATLAB code
A.1 sc.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 V.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
40
41
B Part Specifications
B.1 Main Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Board 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Board 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
42
42
43
List of Figures
1.1
1.2
Coil configuration in quadrupole trap . . . . . . . . . . . . . . . . . . . . . .
Coil configuration in Ioffe-Pritchard trap . . . . . . . . . . . . . . . . . . . .
3
4
2.1
2.2
2.3
Main circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Control circuit generating V1 , V 1 and V2 . . . . . . . . . . . . . . . . . . . .
Control circuit generating Vcontrol . . . . . . . . . . . . . . . . . . . . . . . .
11
16
17
3.1
3.2
3.3
3.4
3.5
Simulation circuit . . . . . . . .
Current vs. time . . . . . . . .
Voltage vs. time . . . . . . . .
Current vs. time with varistor .
Voltage vs. time with varistor .
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19
21
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23
23
4.1
4.2
4.3
4.4
4.5
4.6
Magnetic field of the coils . . .
Switch debouncer . . . . . . . .
Subcircuit to drive LEDs . . .
Connections on the front panel
Connection diagram of board 1
Connection diagram of board 2
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25
27
28
29
30
31
5.1
5.2
5.3
5.4
Turning
Turning
Turning
Turning
with “dummy” coil
with “dummy” coil
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35
36
37
38
on
off
on
off
the
the
the
the
magnetic
magnetic
magnetic
magnetic
field
field
field
field
v
List of Tables
2.1
2.2
Components of main circuit . . . . . . . . . . . . . . . . . . . . . . . . . . .
Switching sequence for Vcontrol . . . . . . . . . . . . . . . . . . . . . . . . .
12
17
4.1
4.2
Connections on first board . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connections on second card . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
33
vi
Chapter 1
Introduction
The experimental achievement of Bose-Einstein Condensation (BEC) was one of the holy
grails of physics before successful observation of BEC in the mid 1990’s in rapid succession
by three groups at JILA [1], MIT [2], and Rice [3]. We are attempting to produce a BoseEinstein Condensate for use in atom interferometry experiments [4]. For example, since the
phase shift in a Sagnac type interferometer is given in general by [5]
φrot =
4π
ΩA ,
λv
(1.1)
where λ is the wavelength of the interfering field, v the velocity, Ω the rotation frequency
and A the area enclosed by the interferometer beams, using atoms instead of light gives a
factor
mc2
h̄ω
greater sensitivity
dφ
dΩ
[4]. Using a BEC allows the area enclosed to be increased,
giving an additional boost in sensitivity.
In our experiment, we use a magneto-optical trap (MOT) to capture and pre-cool
87 Rb
atoms. However, the temperature and density that can be reached by optical traps are
limited by the effects of spontaneous emission heating, radiation trapping effects and inelastic excited-state collisions [6], [7]. To reach lower temperatures and higher densities,
we therefore use evaporative cooling. Making the magnetic trap for evaporative cooling is
therefore a crucial part of the process of producing a Bose-Einstein Condensate.
1
1 Introduction
1.1
2
Magnetic Traps
Magnetic traps take advantage of the magnetic moments of neutral atoms and the forces
exerted on these by inhomogeneous magnetic fields. The energy of an atomic level with
angular momentum F~ and magnetic quantum number mF in a magnetic field of strength
B along the z-axis is [7]
E(mF ) = gµB mF B ≡ µm B ,
(1.2)
where g is the gyromagnetic moment and µB the Bohr magneton. This shows that the
potential that an atom sees will be proportional to the strength of the magnetic field. If we
use this to trap and cool atoms, the depth of the trap will be on the order of
T =
µm (Bmax − Bmin )
,
kB
(1.3)
where T 1K for magnetic fields that are conveniently generated1 . The atoms must
therefore be pre-cooled to µK temperatures by the MOT in order to be loaded into a
magnetic trap for further cooling. From equation (1.2), we see that if gmF > 0, that we
have a weak-field seeking state that needs a minimum of B in order to be trapped, while for
gmF < 0, we have a strong-field seeking state, being trapped at a maximum of B. It can
be shown that no local maxima of B are possible, so only weak-field seeking states can be
trapped [8],[9]. Two main field configurations are used for magnetic traps – the quadrupole
trap and the Ioffe-Pritchard trap [9].
1.1.1
Quadrupole Trap
The quadrupole trap, originally suggested by Wolfgang Paul [10] and first used in 1985 to
trap neutral atoms [11], consists of two identical coaxial coils carrying currents in opposite
directions (Anti-Helmholtz coils) (Fig. 1.1). A simple configuration is an axially symmetric
trap, which gives a linear potential with
x̂ ·
1
~
~
~
dB
dB
dB
= ŷ ·
= −2ẑ ·
,
dx
dy
dz
The depth of our trap is estimated in section 4.1.
(1.4)
1 Introduction
3
I
I
Figure 1.1: Coil configuration in quadrupole trap
where the condition ∇ · B = 0 from Maxwell’s equations produces the factor of 2 in the ẑ
term. A magnetic field
B(x, y, z) = B 0 [xx̂ + y ŷ − 2z ẑ]
(1.5)
satisfies this condition and provides the desired trap. Such a trap has equal depth in the
radial and longitudinal directions when the separation of the coils is 1.25 times their radius
[12].
Though the implementation of this kind of trap is extremely simple, it does have one
major problem. The field at the center of the trap is zero, so when the atoms reach this
point, they can flip their spin in a Majorana transition [13] and thus fall out of the trap.
Simply adding a spatially uniform time-independent bias field to this arrangement does not
eliminate the zero-point, but simply shifts it, so a more elaborate setup is needed. This can
be acheived by adding a time-dependent bias field
Bbias (t) = B0 [x̂ cos Ωt + ŷ sin Ωt] ,
where the rotation frequency Ω is faster than the atomic orbital frequency ωtrap =
with M being the mass of the atoms and rD =
2
B0
B0
(1.6)
q
µm ∇B
M rD
the radius of the “circle of death”2 . For
The “circle of death” is the trajectory of the zero-point of the magnetic field. If trapped atoms reach
1 Introduction
4
a typical setup, ωtrap /2π ≈ 20Hz [6]. We therefore choose Ω/2π ≈ 20kHz. The zero-point
will then move continuously, and the atoms will seek the “hole” but not reach it. Adding
this additional field changes the field configuration, so that in the time average the potential
is now a so-called Time-Orbiting Potential (TOP), where3
U = µm B0 + Bρ00 ρ2 + Bz00 z 2
with ρ2 = x2 + y 2 and Bρ00 =
B 02
2B0
=
Bz00
4
(1.7)
[9].
We will use this trap configuration for our experiment.
1.1.2
Ioffe-Pritchard Trap
(b)
(a)
I
-
+
+
-
I
x
z
y
Figure 1.2: Coil configuration in Ioffe-Pritchard trap. (a) side view (b) front view
An alternative coil configuration where the minimum of the magnetic field is not zero
is the Ioffe-Pritchard trap. This consists of four current-carrying bars plus two pinch coils
(see Fig. 1.2). Near the origin, this gives the magnetic field
 



0
−xz
x

 


00
 



0
~

 B 
B(x,
y, z) = B0 
−yz
 0  + B  −y  + 2 
 



1
0
z 2 − 12 (x2 + y 2 )



,


the zero-point, they can fall out of the trap as explained above.
3
Ref. [9] neglects the constant part of the field B0 . Compare [14], Eq. 21 and [17], Eq. 2.
(1.8)
1 Introduction
5
or in cylindrical coordinates





 −zρ
 0 
 cos 2φ 
00 
 


B
~ φ, z) = B0  0  + B 0 ρ  − sin 2φ  +

B(ρ,
0

 


2

 


2
z 2 − ρ2
1
0



.


(1.9)
Here, the first term describes the uniform B-field along the z-axis, the second term is a
quadrupole field with a minimum at the origin in the ρ − φ-plane while the last term is a
“bottle” magnetic field with a minimum at the origin along z and a maximum at the origin
in the ρ − φ-plane. Altogether, for kB T |µm |B, we get a harmonic potential4
U≈
with Bρ00 =
1.2
B 02
B0
−
µm 2B0 + Bρ00 ρ2 + B 00 z 2
2
(1.10)
B 00
2 .
Evaporative Cooling
Evaporative cooling, like cooling in a cup of coffee, consists of continuously removing the
high-energy tail of the thermal distribution. In doing this, more than the average energy
per atom is removed, so the remaining atoms, after rethermalization, are cooled.
1.2.1
A Model for Evaporative Cooling
In this section, we use a simple model (described in [6], [15]) to understand how evaporative
cooling works. The model assumes that elastic collisions dominate over inelastic ones, that
thermalization is much faster than the rate of cooling, and that escaping atoms do not
interact with the atoms in the trap. The cooling process is here described as happening in
one step, although in reality it is a continuous process.
We consider a trapping potential
s2
s3
s1
x
y
z
U = 1 + 2 + 3 a1
a2
a3
4
Ref. [9] again neglects the constant part of the field 2B0 .
(1.11)
1 Introduction
6
where the ai are some characteristic length and si the power in a given direction i. Then
the volume scales as V ∝ T ξ with
ξ≡
1
1
1
+
+
.
s1 s2 s3
(1.12)
The model then describes evaporative cooling with only two parameters,
ν≡
N0
,
N
(1.13)
which is the fraction of the atoms remaining in the trap after cooling, and
γ≡
log(T 0 /T )
log(T 0 /T )
=
,
log(N 0 /N )
log ν
(1.14)
so that the temperature changes as T 0 = T ν γ .
It can be shown that the average reduced energy ¯ =
Ē
kB T
before truncation of the
high-energy tail is given by
¯ = ξ +
3
.
2
(1.15)
For each evaporated atom, the energy carried away is
3 1 − ν γ+1
,
out = ξ +
2
1−ν
(1.16)
so for ν ≈ 1,
γ=
out
−1,
ξ + 32
(1.17)
which is just the excess energy above the average energy.
Choosing the speed for evaporation is a trade-off between efficiency and speed. If the
atoms are cooled too quickly, then thermalization will not be completed and cooling will
not be efficient. On the other hand, if cooling is too slow, then the loss of particles through
inelastic collisions becomes important. Finding the speed of evaporation uses the principle
of detailed balance to give
dN
= −N f ( > η)kel = −N ηe−η nσv̄ ,
dt
(1.18)
where if we lower the trap depth to a value ηkB T , f ( > η) is the fraction of atoms in the
thermal distribution with > η, and kel = nσv is the elastic collision rate 5 .
5
The elastic scattering cross-section is given by σ = 8πa2 with a = 5.4nm for
87
Rb [16].
1 Introduction
1.2.2
7
Cooling Sequence
In our experiment, we hope to achieve Bose-Einstein Condensation with the following steps.
1. Transfer pre-cooled atoms from MOT1 to MOT2. The MOTs both have magnetic
G
field gradients of B 0 ≈ 10 cm
.
2. Laser cool the atoms by turning off the magnetic field and lowering the intensity of
the lasers for about 2ms (this is the same cooling procedure as in MOT1).
3. Optical pumping for about 1ms. We want the atoms to be in the correct (weakfield seeking) states for evaporative cooling, these are the |F = 1, mF = −1i and
|F = 2, mF = 1, 2i states in our case. We will pump to one of these states.
4. Turn the magnetic trap on at a fraction of full strength, matching the trap depth to
the temperature of the atoms. This gives us a potential with its width on the order
of the size of the atom cloud6 .
5. Adiabatically increase the field strength, leading to a decrease in the trap width. This
causes the atom density as well as the temperature to increase.
6. Evaporative cooling
We have two possibilities for the actual evaporative cooling
(a) rf-induced evaporation
This type of evaporation uses radio frequency radiation to flip the atomic spin,
so that the attractive potential becomes a repulsive one [17]. If the rf-radiation
is of frequency ωrf , then a resonance will occur at |g|µB B = h̄ωrf . In a trapping
potential U = mF gµB (B(r) − B(0)), only atoms with E > h̄|mF | (ωrf − ω0 )
will be removed, where ω0 is the frequency that induces spin-flips at the bottom
of the potential. However, at the origin, the magnetic field is zero, so the atoms
6
We can estimate at what fraction of its maximum value the field is turned on when we know the geometry
of the coils. This is done in section 4.1.
1 Introduction
8
can flip their spin randomly, causing them to leave the trap (Majorana transition
[13]) 7 .
(b) time-dependent bias field
We can apply a rotating magnetic field so that the zero-point of the magnetic
field rotates [9]. This creates a “circle of death” since if the atoms reach the
zero-point, they can flip their spin and fall out of the trap. If the zero-point is
not too far outside of the cloud of atoms, the hottest atoms can leave by getting
to the B = 0 spot. This setup was explained in more detail in section 1.1.1.
We are thinking about using the rf-induced evaporation for a while, then switching to the
time-dependent bias field. We think that the density at the origin is not so high at the
beginning of the cooling as to allow many atoms to leak out.
7. Turn the magnetic trap off, transferring the atoms to a smaller magnetic trap with a
weak potential. The density will decrease, which is what we desire in order to ultimately do the planned interferometry experiments. We need to do this very carefully
in order to make sure that the atoms remain in a condensate.
1.3
Goals
In designing the magnetic trap and the circuit that goes along with it, the main problem
arises when switching the trap on and off. We want to switch extremely quickly so that the
atomic cloud does not expand too far or fall under the influence of gravity. If the magnetic
field does not turn on or off fast enough, these influences will disturb the cloud and our
experiment.
When turning the field on, we want the atoms to move much less than the size of the
cloud during the switching time ts . We can estimate the temperature of the atoms before
loading into the magnetic trap to be approximately 50µK, their velocity is thus about 10 cm
s
7
For details on rf-induced evaporation on Rb atoms, see Ref. [17].
1 Introduction
9
and the size of the atom cloud 1mm. So we want
d = vts 1mm = L
ts 0.1cm
= 10ms .
10cm/s
At a time of 500µs, the atoms will have moved
d
L
(1.19)
≈ 5% of the cloud size.
We want to preserve the velocity information of the atoms, e.g. for imaging, when
turning the field off. Non-adiabatic processes preserve this information, so we want the
field to turn off fast compared to the period of trap oscillations. For a frequency of about
20Hz, this means that the switching must take less than 50ms.
A problem arises from extremely fast switching times – a quickly changing magnetic
field gives a large voltage according to the law of induction
V = −L
dI
,
dt
(1.20)
where the magnetic field B is proportional to the current I. According to Lenz’s rule,
this voltage works to keep the magnetic field from decreasing when switching off and from
increasing when turning on. The large expected voltage can also damage components in
our circuit.
We thus aim to design a circuit controlling our magnetic trap with the following properties:
• fast switching times less than 1ms
• ability to switch between field configurations for magnetic trap, MOT, and no field
• ability to control the strength of the magnetic trap
• an emergency shut off if the temperature of certain components gets too high
It is beneficial to generate the magnetic field needed for the magneto-optical trap (MOT)
with the same pair of coils. This has the advantage that the MOT and the magnetic trap
are already aligned as we need them to be.
Chapter 2
Description of Circuit
The main task in achieving our goals is to design a fitting circuit. We purchased a 750A/20V
power supply and then had to build a switch that accomplishes the goals we have described
above. This was done, and the result will be explained in two parts: the “main circuit” and
the “control circuit”, which provides inputs to the IGBT switches of the main circuit and
to the power supply.
2.1
Main Circuit
Fig. 2.1 shows an overview of the main circuit. Table 2.1 gives the values of the components.
For power and voltage ratings, see Appendix B.
IGBT1 works as the main switch to send the current from S1 through the coils, it is
controlled by the gate voltage V1 . We want the current source S1 to be on when no current
is flowing through the coils (or rather, have it shut off slowly so that the shunt resistors
R1 do not overheat), so we need an alternate path for the current to flow when IGBT1 is
not conducting. This is implemented by applying a conjugate voltage V 1 to IGBT2, with
a shunt resistance R1 equal to the resistance RL of the coils in front of the switch.
The rest of the main circuit can itself be subdivided into three different parts, each
having a different task.
10
S1
D1
V1
Figure 2.1: Main circuit
IGBT2
2x
R1
6x
V1
S3
Varistor
IGBT1
3x
L
+
V2
R8
R6
C1
R2
R7
R5
T1
Switch off
C3
C4
R3
IGBT3
MOT 2
D2
C2
5x
R4
S4
S2
T2
Switch on
+
2 Description of Circuit
11
2 Description of Circuit
12
Power Supplies
S1
20V / 750A
S2
500V / 25mA
S3
12V battery
S4
20V / 40A
Resistors
R1
16.7mΩ
R2
8Ω
R3
10kΩ
R4
20kΩ
R5
500Ω
R6
1kΩ
R7
100kΩ
R8
100kΩ
Inductance
L
83.3µH, 17.1mΩ
Capacitors
C1
10µF
C2
250µF
C3
1µF
C4
0.33µF
Table 2.1: Components of main circuit
2.1.1
Switching Off
The main problem in switching the magnetic field off is that the inductance tries to keep
the field up according to Lenz’s rule. With
V = −L
dI
dt
(2.1)
2 Description of Circuit
13
and the fact that the magnetic field B is proportional to the current through the coils I, we
see that a quickly changing field will cause a large voltage. Our goal is to limit the voltage
to below the voltage rating of the IGBTs, 600V. We achieve this with two devices:
• A varistor, which works like a bi-directional Zener diode in that it lets a current flow
when the voltage reaches a breakdown voltage characteristic of the varistor – in our
case, this is 400V.
• The C1 - R2 combination, which damps once the varistor stops conducting. There
will be oscillations in the voltage after switching off, and we would like the resistor
R2 to dissipate the energy in these. The capacitor C1 works as a DC block – we do
not want to dissipate power when we are sending current through the coils.
In addition, the capacitor C3 increases the time for the anticipated voltage spike to reach
its maximum after switching off.
2.1.2
Switching On
Lenz’s rule also hinders quickly switching on the magnetic field. The induced current will
try to prevent the magnetic field from building up. To counteract this, we want to apply a
large voltage when the switch is closed.
The source S2 is used to charge the capacitor C2 . We can estimate the charge as
Q = C · U = 250µF × 500V = 0.125C. We can now estimate the maximum current that
flows during the discharge. The voltage will oscillate as V = V0 cos ωt with ω =
√1
LC
≈
6.9 × 103 rad
s . Then with
I=
dQ
dV
=C
= CV0 ω sin ωt
dt
dt
it follows that the maximum current will flow at a time t =
(2.2)
π
2ω
≈ 225µs. The maximum
current is then
r
Imax = V0
C
L
s
= 500V ·
250µF
≈ 860A .
83.3µH
(2.3)
2 Description of Circuit
14
When the switch (IGBT3) is closed, a high voltage is applied, causing diode D1 to be
reverse biased, thus not allowing the high current of source S1 through. Since it is again
related by V = −L dI
dt , a high voltage allows a fast buildup of current to flow through the
coils. As the strength of the current rises, the voltage will drop, and at some point the
diode will open up, allowing the high current to get through and flow through the coils. We
need IGBT3 to switch shortly after IGBTs 1 and 2 switch, otherwise the capacitors will
discharge through the shunt resistor R1 , which is not desired.
Our switch will be IGBT3, which needs a constant difference between gate and emitter
voltages. This is not given in our case because the voltage is changing when the capacitors
discharge. To fix this problem, we use the transistor T1 with its gate controlled by a signal
from the optocoupler. Since its collector is at 12V, the gate of IGBT3 will be 12V higher
than the emitter when transistor T1 is switched on. The Zener diodes were placed between
base and emitter to prevent voltage spikes above 15V to cause damage to IGBT3.
2.1.3
MOT2 Field
We want to be able to run a current backwards through the coils to generate the field
necessary for the magneto-optical trap (MOT). If the transistor T2 , working as a switch, is
closed, then the current generated by power source S4 just flows off to ground. When T2 is
open and the 750A current from source S1 is not flowing through the coils, then the 40A
current from S4 can flow through the coils and generate the desired magnetic field.
In the switching process, the voltage is negative for part of the time. In this case, the
transistor works as a diode, since the emitter of the transistor is at a lower voltage than
the collector, leaving the voltage above the transistor to be a diode drop (about 0.5V)
above ground. Were the voltage to drop below ground, the current would flow through the
diode/transistor, an undesired result.
2 Description of Circuit
2.2
15
Control Circuit
We want our circuit to have three different configurations. “On” means the current is
flowing through the coil, “Off” means the current source S1 is off, and “Ready” means that
the current source is on, the current flows through the shunt resistor.
Starting with the current source off (“Off”), we first turn the source on, letting the
current flow through the shunt resistor (“Ready”). Next, we let the current flow through
the coils (“On”), turning the current source off some time later (“Off*”). Finally, we can
turn “Ready” off, not changing anything (“Off”).
These transitions need to be realized in our control circuit. “Switch On” has the capacitor C2 discharge through the coils, the high current from S1 following thereafter. After
10ms, we want to start recharging C2 .
V1 −→ 15V
V 1 −→ 0V
V2 −→ 15V, 10ms delay then V2 −→ 0V
“Switch Off” redirects the high current through the shunt resistor R1 and turns off the
power supply by turning off the control voltage.
V1 −→ 0V
V 1 −→ 15V
Vcontrol −→ 0V
Finally, “Get Ready” lets the power supply warm up while the main switch is still off, so
no current flows through the coils. If V1 = 0V, the current flows through the shunt resistor
R1 .
Vcontrol −→ Vin
The control circuit consists of two parts. The first part uses the digital input on/off
to generate the outputs V1 , V 1 and V2 . The second part uses digital inputs on/off and
ready and analog input Vin to generate the analog output Vcontrol , which controls the
2 Description of Circuit
16
15V
15V
15V
2k
IC7a
ON/OFF
2.5V
20
V1
15V
15V
15V
IC7b
2k
20
V1
2.5V
15V
15V
IC8
IC7c
IC9
2.5V
3.3k
IC10
V2
Figure 2.2: Control circuit generating V1 , V 1 and V2
2 Description of Circuit
17
IC2
Vcontrol
Vin
IC1
READY
ON/OFF
J
CP
K
Q
_
Q
IC3
IC4a
IC5a
IC6
IC4b
IC5b
Figure 2.3: Control circuit generating Vcontrol
power supply S1 . Vin is a voltage 0 . . . 10V which corresponds to a current 0 . . . 750A put
out by the current source. In the second part, the switching sequence at the flip-flop will
follow Table 2.2.
J (ready)
K (on/off)
Q
XOR (clk)
ON
1
1
1
0
OFF*
1
0
0
1
OFF
0
0
0
0
READY
1
0
1
1
ON
1
1
1
0
Table 2.2: Switching sequence for Vcontrol
A deviation from the given switching sequence makes it possible for Q to be in the
opposite state of what we want. We therefore installed a LED to show the state of Q. If
we are in the wrong state, we need to switch on/off twice to get into the desired state.
2 Description of Circuit
18
We put a delay in before the J and K inputs using a one-shot, because we want to be
sure that the J and K signals arrive at the flip-flop after the clk input. Both are determined
only by the ready and on/off inputs, the clk being the exclusive OR of these two.
For technical details on the circuit, see chapter 4.
Chapter 3
Computer Simulation
In order to find the theoretical behavior of the circuit, we did a computer simulation with
MATLAB. We were especially interested in the behavior when switching off the magnetic
field since we expected the highest voltages to occur here and we wanted to make sure the
chosen values for C1 , C3 and R2 were optimal.
3.1
Differential Equations
We need to consider only the part of the circuit that is relevant to the process of switching
off. The effective circuit therefore looks like Fig. 3.1.
L
I1
RL
VC
R2
C1
Ceff
I2
Figure 3.1: Simulation circuit
19
3 Computer Simulation
20
We do need to remember that the IGBTs as well as the Varistor have some intrinsic
capacitance. They are given by Cvar ≈ 16nF and CIGBT ≈ 160nF for each IGBT. The
effective capacitance Cef f is then given by Cef f = Cvar + 3 · CIGBT + C3 .
Looking at the voltages in various parts of the circuit gives us a system of differential
equations:
V̇C
V̇C
VC
I1
+ R2 I˙1
C1
I2
=
Cef f
= V0 − L I˙1 + I˙2 − RL (I1 + I2 ) .
=
(3.1)
(3.2)
(3.3)
Inserting (3.2) into (3.3) gives us:
VC = −LI˙1 − LCef f V̈C − RL I1 − RL Cef f V̇C ,
(3.4)
and setting the time derivative of this equal to (3.1) gives:
...
I1
−LI¨1 − LCef f V C − RL I˙1 − RL Cef f V̈C =
+ R2 I˙1 .
C1
(3.5)
Finally, inserting (3.1) into this and simplifying gives us an ordinary differential equation
for I1 :
...
Cef f
−R2 LCef f I 1 − L + L
+ RL R2 Cef f I¨1
C1
Cef f
1
− RL + R L
+ R2 I˙1 −
I1 = 0 .
C1
C1
(3.6)
To solve this differential equation numerically, we need to find the initial conditions. We
assume that the voltage VC (0) is zero and so is the initial current I1 (0). The initial current
I2 (0) through the IGBT is 750A. Using this, we equate (3.1) and (3.2) to get I˙1 (0):
I˙1 (0) =
1
I2 (0) .
R2 Cef f
(3.7)
We can find I¨1 (0) by equating the time derivatives of (3.1) and (3.2) and inserting I˙2 (0)
from (3.3) to get:
1
I¨1 (0) =
R2
!
V0
1
RL
1
−
I2 (0) .
2 I2 (0) + R L I2 (0) − R C C
LCef f
R2 Cef
2
2 1 ef f
f
(3.8)
3 Computer Simulation
21
We can also find the voltage:
VC
3.2
= V0 − L I˙1 + I˙2 − RL (I1 + I2 )
C
ef
f
= V0 − R2 LCef f I¨1 − L + L
+ RL R2 Cef f I˙1
C1
Cef f
− RL + RL
I1 .
C1
(3.9)
Numerical Results
Solving these differential equations with MATLAB (see Appendix A for program code)
yields the results seen in Figs. 3.2 and 3.3.
400
300
I1 [A]
200
100
0
−100
−200
0
0.5
1
1.5
2
2.5
t [s]
3
3.5
4
4.5
5
−4
x 10
Figure 3.2: Current vs. time
The switching time of 100µs for both current and voltage is very good. As can be seen
in Fig. 3.3 however, the voltage rises to a maximum of 3500V, which is much higher than
the 600V voltage rating of the IGBTs. We must therefore find a way to limit the peak
voltage.
3 Computer Simulation
22
3500
3000
2500
VC [V]
2000
1500
1000
500
0
−500
−1000
0
0.5
1
1.5
2
2.5
t [s]
3
3.5
4
4.5
5
−4
x 10
Figure 3.3: Voltage vs. time
3.3
Effects of Varistor
Now we want to examine the effect of the varistor, which will conduct current once the
voltage reaches the threshold of 400V. To simulate this situation, we can assume that the
voltage VC (0) is and always has been at 400V. Then V̇C (0) = V̈C (0) = 0 and I1 (0) = 0. From
this and using VC (0) =
I2 (0)
Cef f
and from V̇C (0) = 0, I¨1 (0) =
it follows that I˙2 (0) = I¨2 (0) = 0. We also find I˙1 (0) = − VCL(0)
RL
L VC (0).
The results can be seen in Figs. 3.4 and 3.5.
The voltage is now limited to under 500V while the current reaches approximately 63A.
The switching time for the voltage is still about 100µs. It has gotten slightly higher for the
current (circa 250µs), but this is still acceptable. This time does not include the time in
which the varistor conducts.
3 Computer Simulation
23
0
−10
−20
I1 [A]
−30
−40
−50
−60
−70
0
0.5
1
1.5
2
2.5
t [s]
3
3.5
4
4.5
5
−4
x 10
Figure 3.4: Current vs. time with varistor
500
400
300
C
V [V]
200
100
0
−100
−200
0
0.5
1
1.5
2
2.5
t [s]
3
3.5
4
Figure 3.5: Voltage vs. time with varistor
4.5
5
−4
x 10
Chapter 4
Technical Details
4.1
Making the Coils
As part of the trap, we made two coils out of 1/4 x 1/4-inch square copper tubing. This
somewhat unconventional tube shape allowed tight packing while maximizing the crosssection of the tube to get as low a resistance as possible. Coil 1 (see markings) was made
first. The coil has about 17 turns, a resistance of 8.5mΩ and an inductance of 50µH. We
made coil 2 later, trying to match these values as closely as possible. This coil also has
approximately 17 turns, a resistance of 8.6mΩ and an inductance of 49.9µH. The inductance
of the two coils together is approximately 83.3µH.
We then measured the magnetic field between the coils at a current of 10A and a
separation of the coils of about 68mm. This separation is close to how the experiment will
be set up. The results of the measurement with the Hall probe are seen in Fig. 4.1.
As can be seen in the figure, the zero point of the magnetic field is about 0.5mm from
the center, which is good enough. The gradient in the region we are interested in is constant
to a good approximation.
Given the properties of the coil, we can now estimate the magnetic field gradient. In
cylindrical coordinates, the components of the magnetic field generated by a quadrupole
24
4 Technical Details
25
2.0
1.5
1.0
0.244z - 8.393
B [G]
0.5
0.0
-0.5
-1.0
-1.5
-2.0
26
28
30
32
34
36
38
40
42
z [m m ]
Figure 4.1: Magnetic field of the coils
trap can be described by an expansion as given in [12]:
X
3
3
Bz =
bn Bzn = b1 z + b3 z − zρ + . . .
2
n
X
X
Bρ =
bn Bρn +
ρn [cn cos((n + 1)φ) + dn sin((n + 1)φ)]
n
(4.1)
n
1
3 2 3 2
= −b1 ρ − b3
ρz + ρ + . . .
2
2
8
X
+
ρn [cn cos((n + 1)φ) + dn sin((n + 1)φ)] ,
(4.2)
n
where b1 =
3µIAR2
(R2 +A2 )5/2
and b3 =
4(4A2 −3R2 )
b .
6(A2 +ρ2 )2 1
The coils are positioned at z = ±A, their
radius is given by R and in vacuum, µ = µ0 = 4π × 10−7 in units A, m, T and µ0 =
4π
10
in
units A, cm, G.
We now consider the gradient in the direction of z. The expansion gives us
dBz
3 2
2
= b1 + b3 3z − ρ + . . . ,
dz
2
(4.3)
so for small values of z and ρ = 0, the gradient is just equal to N · b1 if our coils have
N turns. Inserting the values of our coil geometry I = 750A, N = 17, A ≈ 3.5cm and
4 Technical Details
26
R ≈ 8cm gives us
dBz
3µIAR2
G
G
≈N 2
≈ 211
.
≈ N · 14.3
5/2
2
dz
cm
cm
(R + A )
(4.4)
We can compare the measured value for the gradient with the calculated one. In Fig. 4.1,
G
the gradient is approximately 2.4 cm
. Using Eq. 4.4 with a current I = 10A gives us a
G
. The difference can be attributed to the approximations made in the
gradient of 2.8 cm
calculation as well as the fact that the coils do not have a fixed radius or separation, but
that these vary from turn to turn.
Using the values for the coil geometry, we can now estimate the depth of our trap as well
as the fraction of its maximum strength when turning the field on. For both estimates, we
need to know the magnetic moment of the Rb atoms. For Rubidium, the contribution of the
nuclear magnetic moment is negligible, so we use the values of the electronic gyromagnetic
moment g ≈ 2 and the magnetic quantum number mS = 21 . We can therefore use µm = µB
in Eq. 1.2.
For the depth of the trap as described in section 1.1, given by the magnetic field at the
edge of the smaller magnetic trap (radius z ≈ 1cm), we assume a linear potential with the
gradient as given by Eq. 4.4 so that Eq. 1.3 becomes
T =
1
dBz
µB
z ≈ 14mK .
kB
dz
(4.5)
We now estimate at what fraction of its maximum strength the field should be turned on
in order to match the temperature and size of the atom cloud (cf. section 1.2.2). Assuming
the cloud has a diameter of 1mm and a temperature of 50µK, we get
dB
kB T
50µK
T
G
=
≈
≈ 0.149 = 14.9 ,
dz
µB z
0.6717KT−1 · 0.5mm
m
m
which is about 7% of the maximum value given by Eq. 4.4.
(4.6)
4 Technical Details
4.2
Subcircuits
4.2.1
Debouncer
27
The manual switches that we used for testing and manual control have the tendency to
“bounce” when switching, i.e. they toggle between on and off several times before settling
into the desired state. To prevent this, [18] suggest a debouncer as in Fig. 4.2(a). The
debouncer can be realized with a J-K flip-flop, as can be seen in Fig. 4.2(b). We put these
debouncers in front of the on/off and ready signals that come from the manual switches.
The debouncer is on board 1.
(a)
(b)
+5
A
Q
A
GND
B
Q
Q
____
C L R 1 _ V_ C_ C_
CLR2
J_ 1_
__
J
2
K1
K2
C_
LK1
_
P R 1 C _L _K _2
Q_1_
PR2
Q2
Q1
__
Q2
GND
B
+5
Figure 4.2: (a) Switch debouncer. (b) Connections on J-K flip-flop.
4.2.2
LEDs
In order to see what state the computer inputs on/off and ready are, we want to include
LEDs on the front panel that light up when the inputs are high. Since the computer cannot
supply the current needed to drive the LEDs, we need to add a subcircuit that does this.
4 Technical Details
28
We would also like to make sure that the signal is not affected by noise on the line. We
achieve this by adding two Schmitt trigger inverters. The part of the circuit that does this
is seen in Fig. 4.3.
Signal in
Signal out
5V
LED
Figure 4.3: Subcircuit to drive LEDs
The resistor in series with the LED must be chosen to give the current required for the
LED. In the case of the green LEDs we used for the computer inputs on/off and ready,
this current is I = 2mA. Given a voltage of U = 5V, the resistance must be R =
U
I
= 2.5kΩ.
We would also like to see the state of Vcontrol . Since the Schmitt trigger produces
unsatisfactory results here, we use just the transistor and LED, driven by the output Q̄ of
the J-K flip-flop on board 2.
4.3
Connection Diagrams
On the following pages are connection diagrams for the front panel as well as for the two
cards with the control circuit on them. The tables list the connections between them.
4.3.1
Front Panel
See Fig. 4.4.
4 Technical Details
29
Board2
Z
A
Board1
22
22
1
1
Manual/
computer
ready
on/off
+15V
5V -15V
Connectors
to/from coils
Vcontrol
V1
To battery
IGBT3 out
V1
battery in
battery out
ready
Vin
on/off
Figure 4.4: Connections on the front panel
4.3.2
Board 1
See Fig. 4.5 and Table 4.3.2.
4.3.3
Board 2
See Fig. 4.6 and Table 4.3.3.
The delays on the one-shot are set by the combination of a resistor and a capacitor. In
front of the J and K̄ inputs of the flip-flop, we want a short delay. Here, the delay time is
given by
0.7
tw = 0.28 × RT × Cext × 1 +
RT
(4.7)
where tw is measured in ns, RT in kΩ and Cext in pF. Using RT = 5kΩ and Cext = 10nF
4 Technical Details
30
74ACT14
I1
O1
I2
O2
I3
O3
GND
VCC
I4
O4
I5
O5
I6
O6
____
C L R 1 _ V_ C_ C_
CLR2
J_ 1_
__
J
2
K1
K2
C_
LK1
_
P R 1 C _L _K _2
Q_1_
PR2
Q2
Q1
__
Q2
GND
B
1
Y
22
Figure 4.5: Connection diagram of board 1
gives a delay tw ≈ 16µs.
Before the optocoupler, we want a pulse of about 10ms. In this case
tw = 0.33 × RT × Cext
(4.8)
with the values in the units as before. If we use a 330nF capacitor and 100kΩ resistor, the
4 Technical Details
31
pulse time will be tw ≈ 10.9ms.
IC5
74LS08N
74LS08N
A1
B1
Y1
A2
B2
Y2
GND
A1
B1
Y1
A2
B2
Y2
GND
VCC
B4
A4
Y4
B3
A3
Y3
VCC
B4
A4
Y4
B3
A3
Y3
IC9
IC4
74LS123
74LS123
1A
VCC
1_
B__ 1R/C
1_
CLR
1C
_
1Q
1Q
_2
_Q
_
2Q
2C
2CLR
2R/C
2B
GND
2A
1A
VCC
1_
B__ 1R/C
1_
CLR
1C
_
1Q
1Q
_2
_Q
_
2Q
2C
2CLR
2R/C
2B
GND
2A
IC2
IC8
ADG417BN
S
NC
GND
VDD
D
VSS
IN
VL
IC7
____
C L R 1 _ V_ C_ C_
CLR2
J_ 1_
__
J
2
K1
K2
C_
LK1
_
P R 1 C _L _K _2
Q_1_
PR2
Q2
Q1
__
Q2
GND
LM339N
O2
O1
V+
I1I1+
I2I2+
O3
O4
GND
I4+
I4I3+
I3-
IC1
411C
BAL
INV+15
IN+
OUT
V-15
BAL
IC10
CNY17
ANODE
CATH
IC6
4070BE
A1
B1
Y1
Y2
B2
A2
VSS
B
C
E
Figure 4.6: Connection diagram of board 2
VDD
B4
A4
Y4
Y3
B3
A3
IC3
4 Technical Details
32
A,Z
ground
green
1
computer on/off in
brown
2
computer ready in
blue
4
+5V
yellow
6
manual on/off A
brown
7
manual on/off B
brown
8
manual on/off out
brown
11
manual ready A
blue
12
manual ready B
blue
13
manual ready out
blue
14
battery in (to +)
purple
15
optocoupler in
purple
17
battery out (to -)
purple
18
IGBT3 out
purple
B
computer on/off to board
brown
C
computer ready to board
blue
D
on/off LED
brown
E
ready LED
blue
K
shorted to L
L
shorted to K
R
LED Vcontrol (Q) out
orange
S
LED Vcontrol (Q) in
orange
Table 4.1: Connections on first board
4 Technical Details
33
A,Z
ground
green
2
+15V
red
3
battery in (to +)
purple
4
+5V
yellow
6
V1 (output for IGBT1)
orange
8
V 1 (output for IGBT2)
orange
10
V2 (optocoupler output)
orange
12
Vcontrol
orange
13
Vin
gray
15
ready
blue
18
on/off
brown
21
-15V
black
22
Q out
orange
Table 4.2: Connections on second card
Chapter 5
Performance
5.1
Measurements with “Dummy” Coil
The first test of the circuit was performed with a “dummy” coil, since the real coils were
already in use as MOT coils in the experimental setup. This “dummy” coil had a similar
inductance of around 100µH, but a larger resistance of R ≈ 0.6Ω. Since the resistance was
higher and the flow rate of water through the coils lower, it was only possible to increase
the current up to values of around 400A before the coil got too hot.
In order to measure the changing magnetic field, we installed a probe coil with its axis
parallel to the axis of the “dummy” coil and measured the voltage induced in it. Again
because of the law of induction V = −L dI
dt , this voltage is a measure of the change of the
magnetic field.
Fig. 5.1 shows the effect of turning on the magnetic field with the current from S1 at
400A and the voltage of S2 at 500V. As can be seen, VBE rises to about 11.5V, meaning
that the switch of IBGT3 is now closed. Approximately 300µs later, the capacitors start to
discharge and the probe coil measures a change in the magnetic field. The whole process of
switching on is completed after about 700µs.
In Fig. 5.2, the voltage at the collector of IGBT1, VC , is clamped by the varistor to
about 350V and dies down after about 200µs. Similarly, the voltage in the probe coil peaks
34
5 Performance
35
500
Voltage [V]
400
300
200
100
0
-100
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Voltage [V]
10
0
t [ms]
Figure 5.1: Turning on the magnetic field. The upper diagram shows the voltage at the
capacitors. In the lower diagram, the solid line is VBE at IGBT3 and the dotted line the
voltage at the probe coil divided by a factor of 5.
sharply and then decays after 200µs. The total switching off time is thus about a third of
the time for turning on.
5.2
Measurements with Quadrupole Coils
The circuit was then connected to the actual quadrupole coils positioned on the optical
table for the BEC experiment. These coils were able to handle a current of 750A through
5 Performance
36
350
300
Voltage [V]
250
200
150
100
50
0
-50
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Voltage [V]
0
-50
t [ms]
Figure 5.2: Turning off the magnetic field. In the upper diagram, the solid line is VC at
IGBT1 and the dashed line is VBE at IGBT3. The lower diagram shows the voltage at the
probe coil.
effective water cooling. The magnetic field was now switched on and off at a voltage of
500V at S2 and a current of 750A generated by S1 . The voltage at S1 when the current
was at its maximum value was 17.2V.
Since the coils were already in place around the chamber, it was not possible to install
the probe coil with its axis parallel to the axis of the quadrupole coils. We therefore placed
it with its axis perpendicular to that of the quadrupole coils and thus measured the change
5 Performance
37
500
Voltage [V]
400
300
200
100
0
-100
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
15
Voltage [V]
10
5
0
-5
t [m s]
Figure 5.3: Turning on the magnetic field. The upper diagram shows the voltage at the
capacitors. In the lower diagram, the solid line is VBE at IGBT3 and the dashed line the
voltage at the probe coil.
in Bρ instead of Bz . As we were primarily interested in the time of change and not the
absolute values of the field, this was not a problem.
Fig. 5.3 shows the measurement of switching on the current through the coils. Again,
there is a delay of approximately 200µs between switching (IGBT1 switches on, IGBT2
switches off) and the capacitors discharging. The whole process of switching is again finished
after 600µs.
5 Performance
38
400
350
300
Voltage [V]
250
200
150
100
50
0
-50
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Voltage [V]
15
10
5
0
-5
t [m s]
Figure 5.4: Turning off the magnetic field. The upper diagram shows VC at IGBT1. In the
lower diagram the solid line shows the voltage at the probe coil and the dashed line is VB
at IGBT1.
The measurement of turning off the magnetic field can be seen in Fig 5.4. The voltage
is clamped by the varistor to the same value as before. Since the current is higher and
thus the magnetic field is stronger, this clamped high voltage persists for a longer time, ca.
250µs. The change in the magnetic field is also on the order of 250µs. The switching times
thus do not depend dramatically on the current flowing through the coils.
5 Performance
5.3
39
Conclusion
We have implemented a strong magnetic field that can be used to trap neutral atoms for
evaporative cooling. We have shown that the switching times of the circuit controlling the
field are under 1ms when running at its maximum power, as desired and predicted by the
computer simulation. Since the final plan calls for switching the field on at around 7% of
its maximum strength, the time should be somewhat less in practice.
An emergency interrupt has already been made to shut off the current source S1 if
temperatures at critical points exceed 100◦ C. This could happen at the shunt resistors,
2
which dissipate P = RI 2 = 0.1Ω × 750A
≈ 1500W of power for short periods of time,
6
which is almost twice the rated power. Overheating these components would unquestionably
lead to their destruction.
In addition, the computer control for the current source S1 will be implemented, allowing
us to vary the magnetic field such that it follows the optimum path of evaporation calculated
by a computer simulation.
As of now, current source S4 has not been integrated into the circuit and will need a
separate control circuit controlling it, preferably also with computer inputs.
When these activities are finished, the achievement of BEC and the onset of experiments
with an atom interferometer should be only a matter of time.
Appendix A
MATLAB code
A.1
sc.m
% program sc
% solves ODE of switching circuit
clear;
L = 100e-6;
RL = 16.7e-3;
R2 = 8;
C1 = 10e-6;
Ceff = 496e-9+1000e-9;
% use only one set of these initial conditions
% initial conditions
I2 = 750;
I1 = 0;
dI1 = I2/(R2*Ceff);
ddI1 = 1/R2*(1.2/Ceff/L - I2/R2/Ceff^2 + RL*I2/R2/L - I2/R2/C1/Ceff);
% initial conditions after varistor shutdown
V = 400;
I1 = 0;
dI1 = -V/L;
ddI1 = RL/L^2*V;
% solve ODE for current
[T,Y] = ode45(’V’,[0 0.001],[I1;dI1;ddI1]);
figure(1);
plot(T,Y(:,1));
%title(’Current vs. time in switching circuit’);
xlabel(’t [s]’);
40
A MATLAB code
ylabel(’I_1 [A]’);
% solve ODE for voltage
VC = 1.2 - L*R2*Ceff.*Y(:,3) - (L+L*Ceff/C1+RL*R2*Ceff).*Y(:,2)
- (RL+RL*Ceff/C1).*Y(:,1);
figure(2);
plot(T,VC);
%title(’Voltage vs. time in switching circuit’);
xlabel(’t [s]’);
ylabel(’V_C [V]’);
A.2
V.m
function dy = V(t,y)
L = 100e-6;
RL = 16.7e-3;
R2 = 8;
C1 = 10e-6;
Ceff = 496e-9+1000e-9;
a = R2*L*Ceff;
b = L+L*Ceff/C1+RL*R2*Ceff;
c = RL+RL*Ceff/C1+R2;
d = 1/C1;
dy = [y(2); y(3); -b/a*y(3)-c/a*y(2)-d/a*y(1)];
41
Appendix B
Part Specifications
B.1
Main Circuit
• IGBT: Powerex CM600HU-12F
max. values:
ICE = 600A
VCE = 600V
VBE = 20V
IGBT1: 3 in parallel; IGBT2: 2 in parallel; IGBT3: 1
• Diodes D1 :
6 diodes IRKE320-08 in parallel. Forward voltage drop per diode: ca. 0.8V at 80A.
• Varistor: Littelfuse V151BA60
Clamping voltage: 400V
Max. energy dissipated in single pulse (2ms): 530J
• Resistors R1 :
6 resistors ISOTEK IRV800P in parallel at 0.1Ω each. Maximum power dissipated:
800W each.
• Resistor R2 : RH-50 8Ω. Maximum power dissipated: 50W.
• Capacitor C1 : GE 97F5300S, 10µF. Maximum voltage: 1000V
• Capacitors C2 :
5 capacitors GE 97F5211S in parallel at 50µF each. Maximum voltage: 1000V.
• Capacitor C3 : GE A28F5604, 1µF. Maximum voltage: 2000V
B.2
Board 1
• J-K flip-flop: 74LS109
• Hex inverter with Schmitt trigger input: 74ACT14
42
B Part Specifications
B.3
Board 2
• Quad 2-input AND gate: 74LS08
• Dual One-Shot: 74LS123
• Quad Comparator: LM339
• Optocoupler: CNY17
• J-K flip-flop: 74LS109
• Quad 2-input XOR gate: 4070
• Op Amp: LF411
• CMOS Switch: ADG417
43
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