Digital feedback loops for d.c. SQUIDs

Digital feedback loops for d.c. SQUIDs D. Drung University of Karlsruhe, Instit0t for Elektrotechnische Grundlagen der Informatik, Hertzstrasse 16, D - 7500 Karlsruhe 21, FRG Received 23 July 1986

Digital feedback loops for d.c. SQUIDs using pulse-rate modulation are proposed and investigated by simulations and measurements. The design of a d.c. SQUID magnetometer yields a simulated energy resolution of ~- 11 h (h = Planck's constant) where ~ 3 h corresponds to additional noise due to the feedback loop. Measurements at Josephson sampler prototype chips showed that the digital feedback loop is very stable and fast enough to obtain a flicker-free display on an oscilloscope. In contrast to conventional feedback loops an integration of sensor and feedback loop on one or several chips with Josephson junctions seems feasible. Keywords: SQUIDs; digital feedback loops

D.c. SQUID magnetometers or gradiometers are usually read out with the flux-locked loop circuit developed by Clarke et al 1. An a.c. flux with a peak-to-peak amplitude of ~o/2 and a frequency of typically 100 kHz is applied to the SQUID. The output voltage drives an inductancecapacitance resonant circuit 1 or a transformer2 at liquid helium temperature and a low noise amplifier at room temperature. The amplified 100 kHz signal is lock-in detected, integrated and fed back as a current into the modulation coil of the SQUID. The feedback current is proportional to the flux applied to the SQUID. This readout scheme is very useful for small values of the maximum Josephson current, i.e. for SQUIDswith U((I)) characteristics strongly rounded by noise. Ultra-low noise SQUIDs with small noise rounding require improved readout schemes3. Samplers with Josephson junctions may be read out with a pulse-rate modulation scheme 4. Both signal and sampling pulse are coupled to a comparator, i.e. a Josephson junction or a SQUID without shunt resistors, A trainoftrapezoidalpulseswith a repetitionfrequencyof typically a few MHz switches the comparator into the voltage state with a certain probability depending on the sum of signal and sampling pulse current. The series of voltage pulses is integrated and fed back into the comparator keeping the switching probability constant at 5 0 % . A simplified and improved version comprises a semiconductor comparator, a stretcher and an amplifier as integrators where the feedback current is proportional to the signal current. In this Paper a digital feedback loop is proposed for magnetometers and samplers. In contrast to the above described conventional feedback loops an integration of the sensor and the digital feedback loop on one or more chips in Josephson technology seems feasible for two reasons: (1) the digital feedback loop does not need integrators, i.e. amplifiers with a high d.c. gain and a large range of output voltage; (2) the output signal of the digital feedback loop is available digitally allowing for an undisturbed transmission from helium bath to room 0011-2275/86/110623-04 $03.00 (~) 1986 Butterworth & Co (Publishers) Ltd

temperature electronics. The proposed digital feedback loop has low intrinsic noise. It can be used for low and for high Josephson currents, i.e. for Josephson junctions with large and small noise rounding, respectively.

Description of the digital feedback loops The digital feedback loop in Figure 1 comprises a feedback circuit and a sensor (e.g. magnetometer or sampler) with inductive4'5 or direct ° coupling to a d.c. SQUID comparator. The direct coupling between the sensor and comparator is preferred for samplers due to the higher speed 6.7. For magnetometers the inductive coupling shown in Figure 2 is advantageous since the sensitivity of the comparator can be increased by using an input coil with many turns instead of a single control line. Furthermore, the influence of the comparator on the sensor may be smaller for inductive coupling. The comparator switches into the voltage state if the sum of the signal current from the sensor and the feedback current and the pulsed gate current from the clock generator exceed the threshold curve of the comparator SQUID. The feedback circuit driven by the voltage pulses of the comparator generates a feedback current either fed 7,. . . . -1-----~

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1986 Vol 26

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during the next clock pulse due to positive components of the plasma oscillation and the comparator switches into the voltage state without input signal rise, i.e. the switching probability may be increased up to P = 100%. To obtain zero switching probability again, the control flux has to be lowered to a value * c : . The P(*c) characteristic therefore exhibits hysteresis in the range

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into the comparator (sampler) or into the sensor (megnetometer), The integrating feedback circuit consists of an n-bit binary up/down counter and an m-bit D/A converter with m -< n. The comparator SQUID controls the counter: if the comparator switches into the voltage state during a clock cycle the counter decreases, otherwise the counter increases. The mean contents of the counter do not change if the switching probability of the comparator is P = 50%. If the switching probability differs from 50% the counter increases or decreases until the value P = 50% is regained. The content of the counter represents the binary

output signal and is converted into the feedback current, i.e. the analogue output signal via a D/A converter. The D/A converter should have a low glitch. For magnetometers, the flux range of the D/A converter should be + ~o. If the signal flux, ~ , exceeds the limits + ~ , the counter is reset and may be started again, The described feedback loop has the advantage that, due to the integration function of the counter, sensor and comparator are always biased at the same working points, i.e. the characteristics of the sensor and comparator do not affect the static accuracy. The digital output signal is proportional to the analogue input signal within the accuracy of the D/A converter. The slewing rate (i.e. the maximum possible output signal charge per unit time) is usually limited by the speed of the room temperature electronics, especially by that of the D/A converter. Since integrated counters s'9 and D/A converters in Josephson technology should be feasible, an integration of the feedback loop on the same chip as the sensor can no longer be excluded. A dramatic increase of the slew rate is expected at the expense of a larger number of integrated circuits per chip. For low noise the clock frequency should be chosen as high as possible to improve the averaging of the voltage pulses at the comparator output. However, an upper limit of the clock frequency exists, when hysteresis in the dependence of the switching probability, P, versus control flux, ~ c , i.e. the P(~c) characteristic becomes important. The following example may clarify this: assume that the noiseless comparator is biased at a working point with zero switching probability and that the control flux, ~c, is increased slowly. At a flux Oc~ the comparator switches for the first time into the voltage state. At the falling edge of the clock pulse the comparator switches into the zero voltage state. If the plasma oscillation has not died out before the next clock pulse, the gate current is increased

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Hysteresis in the P(~c) characteristic increases the low frequency noise in a very similar way to that in the U (IG) characteristic although it may be hidden by thermal noise 10"11. Therefore, for very high clock frequencies the plasma oscillations should be damped by resistors parallel to the Josephson junctions. The shunt resistance has to be chosen carefully - if it is too large the plasma oscillations are damped insufficiently, if it is too small the noise of the comparator increases due to the thermal noise of the shunt resistors. More critical than the plasma oscillations are oscillations in parasitic resonant circuits (e.g. the input coil resonant circuit of an inductively coupled comparator) because the corresponding resonance frequencies are often much smaller than the plasma frequency. To avoid additional noise due to input coil resonances the input coil circuit must have a sufficiently large damping. In any case the upper limit of the clock frequency may be affected by resonancesof the comparator or magnetometer input coils. For special purposes the functions of the magnetometer and comparator may be merged in coupling the signal flux directly into the input coil of the comparator =2.

Simulation of a magnetometer feedback loop The noise performance of the digital feedback loop has been simulated for the design data of a magnetometer prototype. The equivalent circuit is shown in Figure 2 with the RCSJ (resistively capacitively shunted junction) model of a Josephson junction. The junctions of the magnetometer SQUID have a maximum Josephson current Io = 30 ~tA, a junction capacitance C = 0.45 pF and a voltage dependent quasiparticle current lqp shown in Figure 3. The junctions are damped via resistors to obtain a McCumber 80

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Digital feedback loops for d.c. SQUIDs: D. Drung damping parameter 13 = 2~tR2Clo/@(, ~ 0.25. The flux, coupled into the magnetometer, ~CM, is approximated by a control current, ICM = OcM/LM • The gate current, I G M . is inserted asymmetrically, The output voltage of the magnetometer is coupled into the comparator input coil inductance L2 = 3.6 nH via the coupling resistance RK = 2R. Transmission lines between magnetometer and comparator are approximated in Figure 2 by the discrete coupling inductance LK = 4 nH and the discrete coupling capacitance CK = 1.6 pF. The parasitic distributed capacitance Cx = 4 pF between the input coil and the comparator inductance L~ = 17 pH is approximated by two discrete capacitances Cx/2 at the input coil ends It. The input coil resonant circuit consisting of L2 and Cx/4 in parallel may be damped by RD to decrease hysteresis in the P(@c) characteristic of the comparator SQUID for large clock frequencies, fc- The coupling coefficient of the comparator input coil is oc = 0.8 and the mutual inductance M = 0¢(LIL2) ~/2 = 0.2 nil. A control current Ic = ~c/L~ produces a control flux in the comparator, A low characteristic phase ~, = 2rtLild~o = ~x/2has been chosen for a small overlap of the flux quantum states, To obtain a sharp transition of the switching probability, P, i.e. a high d/c/d/c, the Josephson junctions have been chosen unequally large, a = 3, and the bias current is inserted asymmetrically into the larger Josephson junction 13. The comparator voltage is tapped with a load resistance of RL = 50 Q. The nonlinear differential equations corresponding to the equivalent circuit in Figure 2 are solved on a PACER 600 hybrid computer ~. The simulation time has been chosen to be 108 times the real time of the circuit, Thermal noise of the resistors is taken into account by noise current sources parallel to the resistors. Thermal noise of the quasiparticle current is approximated by a noise current source with a spectral density Sl = 4kTIR i (Rj see Figure 3). The flux noise spectral density, So, is determined by means of a second-order band-pass filter 14 with a centre frequency of 1-2 MHz and a quality factor of 4.5. The average time is at least 100 times the reciprocal centre frequency of the band-pass filter. The accuracy of the simulated spectral densities and energy resolutions is +_ 20%. The pulsed gate current, IG, of the comparator is simulated by an ideal square-wave pulse train after passing through a second-order unity gain low-pass filter with a cut-off frequency of 1 GHz and a quality factor of 1.1 corresponding to an overshoot of 20%. The 20% overshoot has been chosen to get nearly equal mean duration of the comparator output and clock pulses. A clocked semiconductor comparator senses the SQUID comparator state. A dead time, td, between the falling edge of the clock pulse and the corresponding change of the semiconductor comparator state is taken into account in approximating transmission line delays between the helium bath and room temperature electronics and delays inside the room temperature electronics. The intrinsic noise of the comparator SQUID without magnetometer (RE = ~ ) is simulated with the integrating feedback loop. An 8-bit binary up/down counter and an 8-bit D/A converter on the hybrid computer are used to represent the lowest 8 bit of the n-bit up/down counter and the m-bit DIA converter. The output signal of the D/A converter is applied to the comparator as a control current, If, after passing through a second-order unity gain low-pass filter with a cut-off frequency of ]00 MHz and a quality factor of 1.1. The intrinsic low frequency flux noise density, So, is shown in

Figure 4 for n = m = 11 with the coupling capacitance, Cx, as parameter. For Cx ~ 0 the comparator input coil resonant circuit is shunted by a resistance. So decreases with increasing clock frequency, fc. For Cx = 0 the simulated spectral density may be al~proximated by the dashed line, So = 6.4 x 10 -5 @~/fc. The coupling capacitance increases the flux noise, probably due to the input coil resonant circuit of the comparator leading to hysteresis in the P(q~c) characteristic. For small clock frequencies, fc < < 100 MHz, the input coil capacitance is expected to have no influence on the flux noise. The dead time, td, slightly increases the flux noise as shown in Figure 4 for fc = 700 MHz, td =10 ns (O)and td = 1.4 ns (O), respectively. The energy resolution, e = 0.5 Sa,/LM, of the magnetometer with the integrating feedback loop is simulated for different clock frequencies and plotted in Figure 5 versus the number of binary digits, n, of the up/down counter. A proper working point of the magnetometer SQUID yields a nearly optimum intrinsic energy resolution e~ ~ 8.5 h represented by the dotted line in Figure 5. For the chosen working points, the flux gain of the magnetometer is d ~ c / d ~ c M = (M/RK)dUM/d~cM ~ 4 where U M is the mean magnetometer voltage. Due to the large flux gain the 'intrinsic flux noise of the comparator is smaller than the flux noise caused by the magnetometer for clock frequencies larger than = 100 MHz. The energy resolution depends only weakly on the clock frequency for fc >_ 200 MHz. It decreases with increasing n, approximately as rdh = 1600×2 -"/2, shown in Figure5 by the dashed line. The dotted line intersects the dashed line for n ~- 15 indicating that no improvement of the energy resolution is expected for n > 15. It is noteworthy that the energy resolution depends on n, the number of binary digits of the up/down counter and not on m, the number of binary digits of the D/A converter, provided that m is large enough, i.e. that the lowest significant bit (LSB) of the D/A converter (~Lsn = (I)0/2" - ~ ) is smaller than the flux change, A(I~M, corresponding to the transition of the switching probability of |

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Figure 5 Energy resolution, E, of a magnetometer with the digital feedback loop versusnumber of binary digits, n, of the binary up/down counter for different clock frequencies, ft. ©: fc = 200 MHz; 0, m: fc = 700 MHz. The number of binary digits, rn, of the D/A converter is m = n (O, O) and m = n - 4 (i). - - - : dependence, dh = 1600x2-~v2; .....: intrinsic energy resolution, e~ ~ 8.5 h of the magnetometer SQUID

the comparator. The increase of the energy resolution for small n may be due to high frequency noise from the feedback loop coupled into the magnetometer since the slew rate of the feedback loop increases with decreasing n. It is also possible that the quantization error due to the finite number of binary digits causes an increase of e for decreasing n.

Measurements with a sampler feedback loop A digital feedback loop has been tested with Josephson sampler prototype chips 6 for n = m = 12 and a clock frequency of 4 MHz. The clock frequency was limited by the D/A converter speed. For all junctions the maximum Josephson current Io ~ 0.8 mA, the junction capacitance C = 4.4 pF and the quasiparticle resistance R~ = 13 f~ at 2 mV. The inductances of sampling pulse generator and comparator are = 0.5 pH. The sampling pulse generator is coupled to the comparator via a small coupling resistance RK = 1.4 ~ . Signal and feedback current are coupled inductively into the comparator, The intrinsic noise of the comparator SQUID has been determined without applying trigger pulses to the sampling pulse generator. A peak-to-peak current noise of 1.5 laA for a 4.7 kHz bandwidth and a 0.1 s recording time has been measured, corresponding to a peak-to-peak flux noise of -~ 2 x 10-4~o. This low flux noise is in the order of magnitude expected from hybrid simulations, With trigger pulses applied to the sampling pulse generator the noise was dominated by the sampling pulse generator noise, The feedback loop was fast enough to obtain a flicker-free display on an oscilloscope. It was very stable, i.e. no unlocking event occurred during the measurements.

SQUIDs can be used as nearly ideal current controlled voltage sources with two possible values: zero voltage or gap voltage. Low drift is expected for a D/A converter with Josephson junctions due to the constant helium bath temperature. An integration of up/down counter and D/A converter with a sufficiently large number of digits yields the maximum possible slew rate, but requires a large number of Josephson junctions per chip. However, the chip complexity may be reduced strongly at the expense of a limited output signal linearity in modifying the feedback circuit in Figure 1. The up/down counter is replaced by a D-type flip-flop as a stretcher and the D/A converter by a passive low/pass filter. The digital output signal, i.e. the switching probability of the comparator is fed back into the sensor or comparator after passing through the low-pass filter. Since the switching probability is proportional to the output signal, the working points of the sensor and comparator depend weakly on the input signal leading to a small nonlinearity of the output signal. The output range of magnetometers can be increased in a similar way as with the integrating feedback circuit in Figure 1.

The modified feedback circuit has been simulated with the magnetometer and comparator in Figure 2 for a clock frequency of 700 MHz and a switching probability range 25% < P <- 75% corresponding to a signal flux range -~0-<~s-<~o. For T = 24 ~ts, where T is the time constant of the passive first-order low-pass filter, the simulated energy resolution is ~ 12 h close to the intrinsic energy resolution of ~ 8.5 h. The simulated energy resolution increases with decresing time constant, T. The simulated nonlinearity is better than = _+ 5 × 10-4~o. It may be improved in optimizing design parameters and working points or in reducing the switching probability range at the expense of tighter tolerances of the flip-flop output signal. Conclusions

Digital feedback loops for sensors with Josepbson junctions are proposed. Their integration on a chip in Josephson technology seems feasible. Hybrid simulations of carefully designed sensor and feedback loops yield an overall noise dominated by the intrinsic noise of the sensor. The low intrinsic noise of the sampler feedback loop is confirmed by measurements. It is expected that noise and slew rate of the feedback loops can be improved in optimizing circuit parameters and working points.

Acknowledgements The author would like to thank Professor Dr W. Jutzi for his encouragement and for many helpful discussions. The contributions of E. Crocoll and H.-J. Wermund in designing and fabricating the digital feedback loop prototype, and of R. Strebler in maintaining the hybrid computer facilities at the computing centre of the Karlsruhe University are also greatly appreciated. References

Integration

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For very high slew rates the up/down counter and the D/A converter should be integrated on the sensor chip. Binary ripple counters with SQUID flip-flops for A/D conveners have been demonstrated ~'9. New circuit designs may allow very fast up/down counters in Josephson technology. High speed, low glitch D/A converters with Joseph-

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1 Clarke, J., Goubau, W.M. and Ketchen, M.B. Tunnel junction d.c. SQUID: fabrication, operation, and performance J Low Temp Phys (1976) 25 99-144 2 Ketchen, M.B., Goubau, W.M., Clarke, J. and Donaldson, G.B. Superconducting thin-film gradiometer J Appl Phvs (1978) 49

4111-4116

3 Ketehen, M.IB. DC SQUIDs 1980: The state of the art IEEE Trans Magn (19811 MAG---I7387-394

Digital feedback loops for d.c. SQUIDs: D. Drung 4 Tuckerman, D.B. A Josephson ultrahigh-resolution sampling system Appl Phys Left (1980) 36 1008-1010 5 Harris, R.E., Wolf, P. and Moore, D.F. Electronically Adjustable Delay for Josephson Technology IEEE Electron Dev Lett (1982) EDL--3 261-263 6 Kratz, H.A. and Jutzi, W. A direct coupled Josephson sampler concept with a vortex to vortex transition sampling pulse generator IEEE Trans Magn (1985) MAG-21 582-585 7 Wolf, P., Van Zeghbroeck, B.J. and Deutsch, U. A Josephson sampler with 2.1 ps resolution IEEE Tram Magn (1985) MAG-21 226-229 8 Hamilton, C.A. 100 GHz binary counter using SQUID flip flops IEEE Trans Magn (1983)MAG--19 1291-1292 9 Silver, A.H., Phillips, R.R. and Sandell, R.D. High speed non-latching SQUID binary ripple counter IEEE Trans Magn (1985) MAG-21 204--207

10 Drung, D. and Jutzi, W. Hysteretic noise in DC SQUIDs. IEEE Trans Magn (1985) MAG-21 430-433 11 Drung, D. and Jutzi, W. Hysteretic noise simulation of d.c. SQUIDs with input coil, in: Superconducting Quantum Imerference Devices and their Applications Walter de Gruyter & Co, Berlin, FRG (1985) 807-812 12 Drung, D. DigitalesMagnetometer. German patent pending P36 16 865.3 FRG (1986) 13 Beha, H Asymmetric 2-Josephson-Junction interferometer as a logic gate Electronics Lett (1977) 13 216-218; Digitale Speicherzellen mit Josephson-Kontakten VDE-Verlag GmbH, Berlin, FRG (1981) 14 Drung, D Amplifier noise simulation of d.c. SQUIDs with grounded input coil, in: Superconducting Quantum Interference Devices and their Applications Walter de Gruyter & Co, Berlin, FRG (1985) 1023-1028

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