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https://doi.org/10.1109/TIM.2013.2240954
Robust Control of a Two-Terminal Cryogenic Current Comparator
M. E. Bierzychudek, R. S. Sánchez–Peña and A. Tonina
Abstract—A digital H controller for a two-terminal cryogenic current comparator is designed. To this end, a set of mathematical models covering the actual system is proposed. Simulation results compare the open and closed loop systems based on the proposed controller and the traditional integral control. According to these results, the new controller can significantly reduce the noise in the SQUID sensor.
Index Terms—Current comparators, current measurement, H-infinity control, resistance measurements, Superconducting Quantum Interference Device (SQUID).
T I. INTRODUCTION HE cryogenic current comparator (CCC) bridge is widely used in high accuracy metrology. It is the chosen system for many national metrology institutes (NMIs) to calibrate standard resistors against the quantum Hall resistance. In this system, a SQUID sensor (Superconducting Quantum Interference Device) is used to fix the current ratio. A SQUID can detect extremely small changes of magnetic field and it is a non-linear device. This combination of characteristics can produce rectification, flux jumps or saturation of the SQUID controller. These non-desired effects can increase the standard deviation, produce systematic errors [1] or even impede the measurement. Since its origin, the SQUID stabilization and a low null error were obtained using an analog integral controller. Recently, more complex control strategies have been applied. For example, some NMIs have included digital filters in the current sources [2] or have used feed-forward techniques [3] to decrease the effect of the time constants difference between the arms of a four-terminal CCC. In addition, a national metrology institute has presented a dynamic model of the CCC and a control strategy [4]. In a more recent publication, a high frequency control loop has been applied to decrease the effect of noise in the SQUID [5]. Besides these works, little has been published on the dynamic model of the CCC and its control.
Our work presents analysis, synthesis and simulations that contribute to understand the CCC dynamic behavior and obtain a better stabilization. Here, an H controller is applied to a particular case, a two-terminal cryogenic current comparator [6]. The goal of the H control technique is to synthesize a controller that stabilizes and keeps the output of the system below a given threshold for a class of exogenous disturbances and a set of models that represent the physical system. This is achieved minimizing the infinity norm of the transfer function from the disturbance to the output [7], [8]. Robust control theory was
Manuscript received July 29, 2012; M. E. Bierzychudek and A. Tonina are with the Quantum Standards Laboratory, Electricity Division, Instituto Nacional de Tecnología Industrial, San Martín B1650KNA, Argentina (e-mail: marcosb@inti.gob.ar; atonina@inti.gob.ar). R. S. Sánchez–Peña is Investigador Principal from CONICET working at the Buenos Aires Institute of Technology (ITBA) (e-mail: rsanchez@itba.edu.ar).
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developed in the early 1980s and has been applied to many complex systems since then.
II. TWO-TERMINAL CCC MODEL The two-terminal CCC has been fully described in [6] and [9]. Each resistor (R1,2) to be compared is connected in series with a winding (N1,2) forming the two arms of the bridge, see Fig. 1. A direct voltage source (v(t)) is connected in parallel. The magnetic fluxes produced by the resistor’s currents (i1,2(t)) have opposite directions and the flux difference is measured with a dc SQUID. The bridge is balanced using the SQUID output voltage to drive a feedback current (iF(t)) in a third winding (NF).
Fig. 1. Diagram of the two-terminal CCC. Note that if i2(t)=0 the system will be quite similar to an ultra-low current amplified based on a CCC.
As a first approximation, the cryogenic current comparator can be modeled like an adder element [4]. The CCC takes the advantage of the magnetic shielding properties of superconductors. Using Ampere’s law, the relationship between the winding currents and the screening current (iscreen) in the superconductor shield is obtained [10].
iscreen t i1t N1 i2 t N2 iF t NF
(1)
At high frequencies, the effective currents in the coils are different from the currents at the resistors due to the stray
capacitances [11]. The coils have their own parasitic capacitance and a coupled parasitic capacitance with its neighbors; each
coil-capacitance combination has an associated resonant frequency. To study this problem we propose the following simplified
electrical model, see Fig. 2. We take only three windings, the primary, the secondary and the feedback winding. The first two
windings are resistive whereas the last one is superconducting. A more complex model can be used, but this would increase the
complexity of the controller.
Some assumptions were considered:
a- The primary winding has the maximum value of turns and the largest stray capacitance (C1) in the CCC. b- The control bandwidth is limited by the self-resonant frequency of the winding with maximum number of turns.
c- Based on the previous points the effect of the stray capacitances of the windings, except for the primary, can be
neglected.
d- The superconducting shield reduces the effective inductance of the windings due to the image effect [12].
e- The SQUID detector is considered with the flux lock loop on. The inputs to the model are small in order to maintain
this feedback locked, e.g. no SQUID’s output voltage saturation.
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f- The simplified electrical model is considered to be linear and with concentrated parameters. With this hypothesis the superposition principle can be applied.
Fig. 2. (a) Simplified electrical model of the two-terminal CCC. To study this circuit we applied the superposition principle, the sources are turning on one at a time, (b) and (c).
In this simplified model, the primary coil is replaced by a capacitor (C1), a resistor (RW1) and an inductor (L1), whereas the secondary winding is replaced with a resistor (RW2) and an inductor (L2), due to assumption c-. Since the feedback winding is superconducting, it is represented with an ideal inductor (LF). The current iL1(t) in the ideal inductor L1 will depend on the primary resistor current, secondary and feedback currents. Applying the superposition principle, iL1(t) can be calculated as the sum of the current in L1 produced by i1(t), i2(t) and iF(t), one at a time.
The current flowing through the secondary and feedback windings will induce a current in the primary coil. This behavior can be modeled with a transfer function and it can be calculated applying Faraday’s law and solving the electrical circuit. First, we calculate the transfer function from the feedback current, Fig. 2(b). The mutual inductance between the primary and feedback windings is represented by M1F. The voltages vC1(t) and vL1(t) are the ones across C1 and L1, respectively.
vL1 t
L1
diL1 t
dt
M 1F
diF t
dt
(2)
iL1 t
C1
dvC 1 t
dt
vC 1 t
R1
0
(3)
vC1t iL1t RW 1 vL1t
(4)
To solve the set of linear equations we use the Laplace transform, where ‘s’ is a complex variable represented by s=j·ω
(Fourier transform), with ω the frequency. We include a subscript in the primary inductance current to indicate that it is
produced by the feedback current. To distinguish between model representations, Laplace transform variables are written in
uppercase and the time variables in lowercase.
I L1F s TL1F s I F s
(5)
TL1F
s
C1 L1 s 2
s
M1F s L1 R1
C1s
1 R1
C1RW 1
RW 1
R1
1
(6)
TL1F(s) is the transfer function from the feedback current to the current in the primary winding. We can calculate the transfer function from the secondary winding (TL12(s)) in the same way, where the mutual inductances and the respective subscript has to be replaced. Also, the current flowing in L1 as a function of I1(s), as in Fig. 2(c), can be calculated. Again, a subscript in the
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primary inductance current was included to indicate that it is produced by the current through the primary resistor. The resultant transfer function is called TL11(s).
IL11s TL11s I1s
(7)
TL11 s
C1 L1 s 2
1 s C1RW 1
1
(8)
Comparing the equations (6) and (8), for large R1 the denominators become identical. Now, we write the Laplace transform of
equation (1) and replace the primary current by the effective winding current.
Iscreen s I L1s N1 I2 s N2 I F s NF
(9)
Next, the effective current in the primary winding is replaced by equations (5) and (7). For simplicity, the complex variable is
omitted in the following equations.
Iscreen I L11 I L1F I L12 N1 I2 N2 I F NF
(10)
I screen
I1 TL11 N1 ΦI2I sN 2 TL12 N1
ΦF s
I F (NF TL1F N1 )
(11)
Variable ΦF(s) represents the total feedback flux and ΦI(s) the flux generated by the primary and secondary currents, both in
ampere-turn (AT) units. The screening current is sensed with a dc SQUID. The relationship (GCCC(s)) between iscreen and the
input flux in the SQUID (ΦSQ(s)) can be calculated like the inverse of the current linkage.
ΦSQ s GCCC sΦI s ΦF s
(12)
Finally, the SQUID – from the flux input to the output voltage (VSQ(s)) – is simulated like a low pass filter TSQ(s), by assumption e-.
VSQ s TSQ s GCCC s ΦI s ΦF s
(13)
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III. SET OF MODELS
A block diagram of the whole system is shown in Fig. 3. Input ‘D(s)’ is a voltage perturbation that is connected to the
SQUID´s output. It represents the Johnson noise, mechanic vibrations, external noise or SQUID noise. Variable ‘ΦI(s)’ is the reference input and it depends on the resistor’s values, the applied voltage and the dynamics of each arm of the two-terminal
CCC. These dynamics are outside the feedback loop and cannot be controlled. For this reason, we are not interested in a precise
determination of TL11(s) and TL12(s) transfer functions. Only for simulation purposes, they can be calculated at ambient temperature. A different situation holds for transfer function TL1F(s) because it is inside the feedback loop. This function is affected by the resistor under test and by the image effect of the superconducting shield. In addition, by assumption c-, the stray
capacitance of all other coils, except for the primary, were neglected. This could be unrealistic, i.e. feedback induces a current in
each winding which will decrease the total feedback flux. Robust control approaches this problem by representing the physical
system behavior with a set of models, instead of a nominal one, and designing a controller that will provide stability and
performance to this whole set [7], [8]. Performance is defined here as the rejection in output Y(s) of a set of exogenous
disturbances D(s), i.e. bounded norm of Y(s) for all disturbances D(s) in the set
d (t)
2 2
d
t
2
dt
1.
Here,
the
norm
of
Y(s)
is
also measured as the energy integral and d(t) is the signal that corresponds to the Laplace transform D(s). It is a more
conservative solution but it guarantees (robust) stability of the entire model set and hence of the actual system, contained in it.
To define the model set, first rewrite equation (13) where input ‘ΦI(s)’ has been eliminated.
VSQ s TSQsGCCCs(GNF TL1F sN1) IF s
(14)
The set of models used to synthesize the controller can be described as a set (Ψ) of transfer functions G(s) centered in the
nominal model G0(s), which is the negative of the SQUID’s transfer function times the CCC gain, figure 4. Note that NF is equal to 1.
G(s) : G(s) G0 s 1 W s , 1
(15)
To this end, all possible transfer functions N1·TL1F(s) in equation (14) for the envelope of possible parameters, are covered by
Fig. 3. Complete block diagram of the two-terminal CCC, it is equivalent to equation 13. The controller box K(s) represents the original integral (Ki(s)) or the H∞ controller (KH(s)). In the feedback loop, a switch is included to indicate where the loop is opened.
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a weight WΔ(s) that represents the frequency distribution of model uncertainty times the bounded uncertainty , with 1 .
This structure is called multiplicative dynamic uncertainty in the robust control jargon and quantifies the set of models that covers the physical system. In robust control, model uncertainty is the (frequency dependent) radius around the nominal model and represents a deterministic quantification of the lack of knowledge of the nominal model with respect to the physical system. This should not be confused with uncertainty in the metrological sense.
The uncertainty weight is obtained by simulating the transfer function N1·TL1F(s) in extreme situations, as illustrated in Fig. 5. The two-terminal CCC under study has been specially designed to measure high value resistor (HRCCC) and it has five ratio windings: one of 3100 turns, another of 310, two of 31 and one of 4 turns. The first four windings have been made with resistive wire of phosphor-bronze. For simulation purpose, we considered the primary winding with 3100 turns, the secondary with 310 turns and the superconducting feedback winding of 1 turn. The first two windings have nominal resistances of RW1=2800 Ω and RW2=280 Ω, respectively. The current linkage was measured during the system setup, 3.9 µAT/Φ0. The SQUID’s cutoff frequency was obtained from its specification – 50 kHz [13] – and the DC gain was measured. This value is usually called flux sensitivity and in our case it is 0.72 V/Φ0. The inductance of the primary winding was selected equal to the theoretical value 1.14 H and the capacitance value C1 was chosen in such a way that the resultant resonant frequency agrees with that obtained in the measurement of the spectrum. Proportional values were used in others windings. The mutual inductance between the feedback or secondary coil and the primary were calculated with the worst case coupling factor, 1.
First, the value of the primary resistor can attenuate the resonant peak and increase the low frequency gain of the function N1·TL1F(s). This is shown in Fig. 5, the dotted line is the function N1·TL1F(s) with R1 equal to 1 GΩ and the dash-dot line with R1 equal to 1 MΩ. In addition, the effect of other winding resonances can be simulated; the fluxes produced by the induced currents in each winding will have the same direction. The dashed gray line shows this situation with four ratio windings, the resistance connected to each winding was the estimated isolation resistance, 10 TΩ. Also, the behavior of N1·TL1F(s) with others parameters, e.g. inductance, was studied but no significant changes were obtained. Finally, the solid blue line represents the proposed uncertainty weight. To improve closed loop performance, the proposed weight does not completely cover all curves of the transfer function N1·TL1F(s) above 0 dB. This is possible because 0 dB represents 100 % relative model uncertainty and at that value, from a control point of view, it is irrelevant if the design uncertainty represented by WΔ(s) is larger or not, while it remains above 0 dB. In some sense, weight WΔ(s) is used as a tuning parameter: a better performance will be obtained at frequencies where its absolute value is lower.
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Fig. 5. The dash-dot and dotted lines are the simulations of function N1·TL1F(s) for different values of the primary resistor. The dashed gray line shows a
situation with four windings. The proposed weight is in solid blue line. Fig. 6 shows the calculated transfer function from the primary and feedback current to the SQUID’s voltage. This simulation and the following were performed with the same primary inductance and resonant frequency that before, R1 equal to 1 GΩ and R2 equal to 100 MΩ. Clearly, both transfer functions are dominated by the lower resonant frequency corresponding to the larger inductance. This result is coherent with the literature [4] and with experimental observations. The transfer function from the feedback current amplified above 100 dB signals below 10 MHz and the DC gain is proportional to the winding number. Hence, this huge gain is the key feature of the CCC, although it also amplifies undesirable signals that can decrease the SQUID performance. For this reason, the main objective in the CCC design is the attenuation of any type of noise. Fig. 7 shows the response of the open loop system to a current step of amplitude 1 μA in the feedback winding.
Fig. 4. Simplified block diagram of the set of models used to synthesize the controller. CCC-based amplifier and four-terminal CCC can be represented with this diagram with small modification.
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Fig. 6. Calculated transfer function of the open loop system from the primary current (TVS1(s)) and from the feedback current (G(s)) to VSQ(s). The SQUID’s voltage spectrum, without feedback, is showed in the inset.
Fig. 7. Step response of the open loop system. The primary RLC circuit produces high frequency oscillations that can decrease the SQUID performance.
IV. CLOSED LOOP BEHAVIOR WITH INTEGRAL CONTROL
In the HRCCC the original controller is a low pass filter followed by an integrator and a Howland current source. We
calculate the controller’s transfer function from the circuit scheme. The low pass filter has a cut-off frequency of 122 Hz, the
Howland source has a fixed gain of 1·10-4 Ω-1 and the calculated transfer function (KI(s)) has to be multiplied by the number of turns of the feedback coil, in this case only one turn.
KI
s
s
s
0.67
766.67
(16)
The following figures show the step, impulse and frequency responses of the transfer function from the primary current input
to the ‘Y(s)’ output. Clearly, the integral control attenuates low frequencies and this produces that the SQUID output voltage
tends to zero for the step response, as seen in Fig. 8. On the other hand, the CCC is working as a bandpass filter with a huge gain
of 200 dB near the resonant frequency (see Fig. 9). This produces that high frequency components at the input are amplified. For
example, a spike in the voltage source can produce a flux jump or saturation of the SQUID controller and this can be studied
with the impulse response, as illustrated in Fig. 8.
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Fig. 8. Response to an impulse and step current of 0.1 pA in the primary resistor with integral control. The system amplifies more than 5·1013 times a spike in the voltage source.
Fig. 9. Frequency response of the transfer function from the primary current input to the SQUID’s output with integral control (dash-dot line) and with H∞ control (solid line). The system with integral control amplified more than 100 dB signals between 0.1 Hz and 100 kHz. In addition, the resonant frequency is amplified by 200 dB. The new controller has better attenuation of undesirable signals from DC to 1.3 kHz.
V. CLOSED LOOP BEHAVIOR WITH H∞ CONTROL Based on the previous sections we find a main problem in the HRCCC, the wide bandwidth of the CCC and the slow integral feedback produce a huge amplification of the resonant frequency. In the literature we can find only one work where this problem was considered [5]. There, a high frequency loop in the SQUID was established to decrease the effect of high frequency noise and the SQUID sensor was specially designed for that purpose. Here we propose an H∞ controller that closes the loop in the traditional feedback winding. This optimal method has been used successfully for more than 20 years and a numerically robust toolbox is available to design controllers. In this case it adequately fits the statement of the problem and improves over previous techniques by considering sets of models and disturbances. A trade-off design was achieved in order to find an optimal controller that maintains the closed loop system (robustly) stable despite uncertainty in some parameters for a class of inputs, while increasing disturbance rejection. This controller was designed based on a mixed sensitivities structure using the -iteration algorithm [7], [8]. The following equation shows the transfer
10 function of the H∞ controller, KH(s), which at low frequencies has an integral action.
KH s
1.19 s 1.26106 s 1.00106 s 3.14105
(17)
s 5.801010 s 1.00106 s2 4.18104 s 4.52108
Fig. 9 shows the frequency response of the closed loop system. Compared with the integral control, the new controller
decreases the effect of high frequency noise more than 20 times and the resonant peak by a factor of 3. It also has better step and
impulse responses, as seen in figures 10 and 11. These simulations show that the H∞ control reduces the effect of undesirable
signals in the CCC inputs. For example, for power line frequencies (50 or 60 Hz) the new controller attenuates 21 dB more than
the traditional integral control.
The transfer function from perturbation D(s) to output Y(s) is compared in the integral and H∞ controller cases in Fig. 12.
Here again, the new controller attenuates the perturbation 20 times more and therefore the effect of the noise generated or
coupled to the CCC or SQUID will be much lower.
Fig. 10. Response to a step current of 0.1 pA in the primary resistor with integral control (left) and H∞ control (right). These simulations were made with the same parameters than before but the inductance of the primary coil was (from top to bottom) 2 H, 1.14 H and 0.1 H. With the new controller the settling time drastically decreases but the amplitude increases by 10 %.
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Fig. 11. Response to an impulse current of 0.1 pA in the primary resistor with integral control (left) and H∞ control (right). These simulations were made with the same parameters than figure 6 but the inductance of the primary coil was (from top to bottom) 2 H, 1.14 H and 0.1 H. With the new controller the settling time drastically decreases and the amplitude remain constant.
Fig. 12. Transfer function in log scale from the perturbation D(s) to the output Y(s) with the H∞ controller (solid line) and the integral controller (dash-dot line).
VI. CONCLUSIONS AND FUTURE RESEARCH The simulations show that the robust control can be successfully applied to a two-terminal cryogenic current comparator. Measurement systems with better noise rejection can be designed and this can impact in lower standard deviations and systematic errors. Simulations show that the noise in the voltage source or in the SQUID is attenuated 26 dB more than with the original (integral) controller. The two-terminal CCC is quite similar to a CCC-based current amplifier [14]. In this way the results presented in this paper can be applied, with small modifications, to Single Electron Tunneling (SET) measurements and quantum metrological triangle. In addition, this work can be extended to a four-terminal CCC. The resulting H∞ controller has to be implemented with digital technology and a higher frequency bandwidth will result in the
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feedback loop. To calculate the resistor’s ratio, the feedback current has to be measured at DC. High frequency components could affect this measurement therefore extra filtering could be necessary. To avoid this, two feedbacks loops could be used, one for low and the other for high frequencies (see also [5]).
In this work we assumed a complete lack of knowledge in the model above 1.5 kHz. If a model identification process of the physical system is performed, the uncertainty weight could be decreased and as a consequence, the closed loop performance could be increased even more.
REFERENCES
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comparator,” IEEE Trans. Instrum. Meas., vol. 54, no. 2, pp. 525-528, Apr. 2005. [7] R. S. Sánchez Peña and M. Sznaier, Robust Systems: Theory and Applications, John Wiley & Sons, 1998. [8] K. Zhou, J. Doyle, K. Glover, Robust and Optimal Control, Prentice-Hall, 1996. [9] M. Bierzychudek and R. E. Elmquist, “Uncertainty evaluation in a two-terminal cryogenic current comparator,” IEEE Trans. Instrum. Meas., vol. 58, no. 4,
pp. 1170-1175, Apr. 2009. [10] J. M. Williams, “Cryogenic current comparators and their application to electrical metrology”, IET Sci. Meas. Technol., vol. 5, iss. 6, pp. 211-224, 2011. [11] K. Grohmann and D. Hechtfischer, “Self-calibrating cryo current comparators for AC Applications”, IEEE Trans. Instrum. Meas., vol. 33, no. 2, pp. 91-96,
Jun. 1984. [12] J. Sesé, F. Lera, A. Camón, and C. Rillo, “Calculation of effective inductances of superconducting devices. Application to the cryogenic current
comparator”, IEEE Trans. Appl. Supercon., vol. 9, no. 1, pp. 58-62, Mar. 1999. [13] DC + RF SQUID Systems Brochure, Quantum Design. Available: http://www.qdusa.com/sitedocs/productBrochures/squid3.pdf. [14] F. Piquemal and G. Genevès, “Argument for the direct realization of the quantum metrological triangle,” Metrologia, vol. 37, pp. 207–211, 2000.
M. E. Bierzychudek received the electronics engineer degree from the Universidad de Buenos Aires, Buenos Aires, Argentina, in 2006. In 2005, he joined the Electricity Division, Instituto Nacional de Tecnología Industrial, Argentina, and he has worked since then in the voltage and resistance laboratory; involved in measurements of the Josehpson effect and the Quantum Hall effect.
R. S. Sánchez Peña (SM’00) received the Electronic Engineering degree from the University of Buenos Aires in 1978, and the M.S. and Ph.D. degrees from the California Institute of Technology, in 1986 and 1988, respectively. From 1977 to 2004 he worked in several research institutions in Argentina. He also was Professor at the University of Buenos Aires (1989-2004), ICREA Research Professor at UPC (Barcelona, 2004-09) and Visiting Researcher and Professor at universities in the USA and EU. He worked in aerospace projects with NASA, DLR (Germany), CTA/INPE (Brazil), and ZonaTech (Arizona). He published 3 books and more than 120 journal and conference papers. Since 2009, he directs the Ph.D. program at the Buenos Aires Institute of Technology as a CONICET Principal Investigator. He is a Full Member of the International Academy of Astronautics, Senior Member of AIAA, and was Policy Committee member of IFAC (2005–2008). His research interests include the applications of Robust Identification and Control to practical problems in Engineering and Medicine.
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Alejandra Tonina was born in Buenos Aires, Argentina, in 1962. She received the Licenciatura and Ph.D. degrees in physics from the University of Buenos Aires in 1993 and 1998, respectively. Since 1998, she has been with the Instituto Nacional de Technologia Industrial, San Martín, Argentina, where she is currently in charge of the quantum electrical metrology laboratory.
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