Characterization of sigma–delta modulator with LR-type integrator for superconducting analog-to-digital converter

Physica C 463–465 (2007) 1092–1095 www.elsevier.com/locate/physc

Characterization of sigma–delta modulator with LR-type integrator for superconducting analog-to-digital converter F. Furuta b

a,*

, K. Saitoh a, A. Yoshida b, H. Suzuki

b

a Advanced Research Laboratory, Hitachi, Ltd., 1-280 Higashi-Koigakubo, Kokubunji, Tokyo 185-8601, Japan Superconductivity Research Laboratory, International Superconductivity Technology Center, 1-10-13, Shinonome, Koutou-ku, Tokyo 135-0062, Japan

Accepted 19 January 2007 Available online 2 June 2007

Abstract We studied the signal-to-noise ratios (SNRs) of a superconducting first-order sigma–delta modulator with an LR integrator. Effects of leakage in the LR integrator and thermal noise on SNR were investigated analyzing a transfer function and simulating circuits with thermal noise sources. Leakage resulted in a decrease in the SNR of 1.5 dB and thermal noise in a decrease of 5.5 dB, at a sampling frequency of 20 GHz. We found that decreases in SNR due to leakage and thermal noise in a comparator were independent of the sampling frequency. Thermal noise in an integrating resistor on the other hand became dominant for a decrease in SNR at frequencies above 5 GHz. We also designed a modulator and evaluated its SNRs experimentally. An SNR of 71 dB was obtained at a sampling frequency of 10 GHz. The dependency of SNR on sampling frequency was consistent with the simulated one. The present results indicate that transfer function analysis and circuit simulations with thermal noise can provide practical estimates of SNRs for modulators.  2007 Published by Elsevier B.V. PACS: 85.25.N Keywords: Analog-to-digital converter; SFQ; Niobium

1. Introduction Analog-to-digital (A/D) converters based on single flux quantum (SFQ) circuitry are promising candidates for use in wireless communication base stations [1]. Because SFQ circuitry originates from flux quantization, analog signals can be accurately sampled in flux quantum steps, r0  2.07 · 1015 (Wb). In addition, because SFQ circuitry has the potential for high-speed operation at frequencies of up to 100 GHz, high-speed sampling can readily be achieved [2]. By using these features, we aimed to attain high-performance superconducting A/D converter that exceeded the semiconductor types; e.g., achieving a sig-

*

Corresponding author. Tel.: +81 42 323 1111x2611; fax: +81 42 327 7722. E-mail address: [email protected] (F. Furuta). 0921-4534/$ - see front matter  2007 Published by Elsevier B.V. doi:10.1016/j.physc.2007.01.062

nal-to-noise ratio (SNR) of 86 dB at a bandwidth of 10 MHz. Fig. 1 outlines the framework for the superconducting A/D converter in this study [3]. We adopted a hybrid architecture, i.e., the converter consists of a front-end circuit based on SFQ circuitry and a back-end circuit based on semiconductor devices. To take full advantage of the high-speed operation of SFQ circuitry, we utilized an over-sampling technique; the analog signal is sampled at a hundred times the objective bandwidth. The input analog signal is sampled with a modulator in the front-end circuit. The modulator is a key circuit in the converter since the modulator determines SNR. Various modulation methods have been proposed in theoretical studies, i.e., sigma–delta, band-pass sigma– delta, and delta modulation. The order for the feed back loop also has several variations [4]. However, it is very difficult to implement a higher order for the feed back loop in

F. Furuta et al. / Physica C 463–465 (2007) 1092–1095

Input analog signal

1-bit digital data

Front-end circuit based on SFQ circuitry Over-sampling at 4.2K

n-bit digital data

Back-end circuit based on semiconductor circuits Digital signal processing at room-temperature

Fig. 1. Block diagram of superconducting A/D converter.

superconducting circuitry due to lack of gain in Josephson junctions (JJs). We thus adopted first-order sigma–delta modulation to construct the modulator reliably. In ideal sigma–delta modulation, that is, there is no leakage of an integrator and no thermal noise in the modulator, the theoretical SNR is expressed as   3 ! 9 fS SNR  10 log ð1Þ 16p fB where fS is the sampling frequency and fB is the bandwidth [4]. The sampling frequency of 20 GHz is sufficient for the objective SNR of 86 dB at the bandwidth of 10 MHz. However, there are leakage of the integrator and thermal noises in a practical superconducting modulator and these effects reduce its SNR. In this study we investigated the effects of leakage and thermal noise on SNR and clarified the most critical factor for reducing of SNR. We analyzed the transfer function and simulated circuits with thermal noises to estimate SNRs. We designed the modulator and experimentally evaluated its SNR to verify the experimental results. 2. Characterization of sigma–delta modulator Fig. 2 is a schematic of the first-order sigma–delta modulator we analyzed [5]. The circuit consists of an LR integrator and a balanced comparator. The LR integrator consists of an inductor, LINT, and a resistor, RINT, and the balanced comparator consists of four JJs, J1–J4. Due to the simple structure of the modulator, high-speed operation can easily be achieved based on the SFQ circuitry. We investigated the effect of leakage in the LR integrator and thermal noise on SNR. First, the SNR of the mod-

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ulator was estimated analyzing the transfer function. The effect of leakage in the LR integrator was taken into account in the analysis. The transfer function of the output signal from the modulator, Y(z), is expressed as  1 fCO 1 fCO z 2p X ðzÞ Y ðzÞ ¼ 1 þ 2p fS fS  1     fCO 1 fCO 1 z 1  1  2p þ 1 þ 2p z EðzÞ fS fS ð2Þ where, fCO is a cut-off frequency of the integrator, (1/ 2pRINT/LINT). X(z) is the transfer function of the input signal and E(z) is that of quantization noise [5]. Parameter, z is an operator of ‘‘z-transformation’’. We calculated the SNR with leakage from the equation and plotted the results in Fig. 3. We can see the dependency of SNR on the cut-off frequency, fCO. The sampling frequency was 20 GHz and the bandwidth was 10 MHz. By comparing the theoretical and calculated SNRs, we found that the SNR with leakage slightly decreased up to 5 MHz, but then steeply decreased above 5 MHz. Here, note that 5 MHz is half the bandwidth. Fig. 4 plots the dependency of calculated SNR on sampling frequency. We can see that the decrease in SNR due to leakage was independent of the sampling frequency. To reduce the effect of leakage on SNR, we should set the cut-off frequency, fCO to less than half the bandwidth. In this study, we set fCO to 3.88 MHz, that is, the inductance, LINT and resistance, RINT were set to be 40 pH and 1 mX, respectively. Next, to study the effect of thermal noises in the modulator on SNR, we performed circuit simulations by using WRSPICE[6]. As shown in Fig. 2, random current sources were connected to JJs in the comparator and RINT in the integrator. The root-mean square of noise current, hINi, can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k B Temp IN ¼ ; ð3Þ Rdt 80 1.5 dB

75

IOFFSET

iN_J3

50 Ω

J3 50 Ω Analog signal

iN_R

LINT RINT

iN_J2

RSJ3

iN_J4 J4

RSJ2

RSJ4 RSJ1

J2 iN_J1 J1

Sampling clock signal

Comparator

LR integrator Modulated data signal

Fig. 2. First-order sigma–delta modulator. IC1 = IC4 = 140 lA, IC2 = IC3 = 100 lA, RSJ1 = RSJ4 = 2.7 X, RSJ2 = RSJ3 = 3.8 X, ICOMP = 50 lA.

SNR (dB)

70

ICOMP

Theoretical including leakage

65 60 55

4 MHz

50 105

106

107

108

Cut-off frequency (Hz) Fig. 3. Dependency of SNR on cut-off frequency of LR integrator.

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F. Furuta et al. / Physica C 463–465 (2007) 1092–1095

70

10-3

Transfer function Experimental

10-5

Power (W)

SNR [dB]

80

Theoretical including leakage including thermal noise Total Comparator RINT

60

10-7 10-9 10-11 10-13

50

106 3

4

5

6 7 8 9

10

107

2

Sampling frequency [GHz] Fig. 4. Dependencies of simulated SNRs on sampling frequency.

where kB is the Boltzmann constant (1.38 · 1023 J/K), Temp is the operational temperature (4.2 K). R is the resistance component of shunt resistors (RSJ1–RSJ4) or RINT [7,8]. Parameter dt determines the bandwidth of the noise source, which was set to the time period for circuit simulation (0.1 ps). We investigated the dependency of SNRs on the sampling frequency using this circuit model with or without thermal noise. Simulated SNRs with thermal noises are also depicted in Fig. 4. Compared with the theoretical SNR, we found that the decrease in SNR due to thermal noise in the comparator was 0.5 dB and was independent of the sampling frequency. On the other hand, the decrease in SNR due to the thermal noise of RINT depended on the sampling frequency and became more dominant than that in the comparator above 5 GHz. This is because the resistance, RINT is much less than the shunt resistances (RSJ1–RSJ4). The total effect of thermal noise resulted in a decease in SNR of 5.5 dB at a sampling frequency of 20 GHz. 3. Design and experiment We designed the modulator, including the front-end circuit, and experimentally evaluated its SNR to validate the result of circuit simulation. The decrease in SNR due to leakage from the theoretical SNR was estimated to be 1.5 dB from Fig. 2. The decrease in SNR due to total thermal noise was also estimated to be 4.5 dB at a test frequency of 10 GHz from Fig. 3. Thus, a total discrepancy between the theoretical SNR and prospective SNR under leakage and thermal noise condition was estimated to be 6.0 dB. In the experiment, the analog and sampling clock signals were supplied to the modulator [9]. The data signal from the modulator was distributed by a demultiplexer to four channels and was amplified by stacked-JJ amplifiers to 4 mV in the superconducting front-end circuit. The signals from the front-end circuit were sent to a data-collection system based on room-temperature electronics, in which the signals were stored in a high-speed memory with a

108

109

Frequency (Hz) Fig. 5. Example of experimentally obtained power spectrum at the sampling frequency of 10 GHz. Noise power spectrum obtained by transfer analysis is also plotted.

length of 131,072 points. Finally, the stored data stream was read out by a PC and was calculated by FFT to obtain a power spectrum. Fig. 5 shows an example power spectrum for a data signal we acquired. The frequency of the sampling clock signal was 10 GHz and that of the input analog signal was 4.88 MHz. The SNR was estimated to be 70.9 dB with a band-width of 10 MHz from this spectrum. The theoretical noise power spectrum derived from the transfer function is also depicted in Fig. 5. The noise characteristics of the experimental data agree with the theoretical ones. We found that a slope of 6 dB/oct, noise shaping, terminates at around 4 MHz and flattens at less than 4 MHz. The change in the spectrum can be attributed to the leakage in the LR integrator mentioned in Section 2. The bending point of 4.00 MHz corresponds to the cut-off frequency of the integrator, fCO  3.88 MHz. These results indicate that the modulator was operated perfectly and the spectrum for the data signals from the modulator was correctly evaluated.

90

Thoretical Circuit simulation

80

(including leakage+thermal noise)

Experimental SNR (dB)

2

70

60

50 2

3

4

5

6 7 8 9

2

10

Sampling frequency (GHz) Fig. 6. Dependencies of simulated and experimental SNRs on sampling frequency.

F. Furuta et al. / Physica C 463–465 (2007) 1092–1095

Next, we investigated the dependency of the experimental SNR on sampling frequency. SNRs were evaluated at sampling frequencies of 2.5, 5.0 and 10 GHz. The results are summarized in Fig. 6 with the theoretical SNR, and the simulated one under total thermal noise. The experimental results are in agreement with those simulated. At the sampling frequency of 10 GHz, the experimental SNR, 70.9 dB was consistent with the simulated SNR with thermal noise, 72 dB. From these results, we concluded that a practical SNR could be estimated by analyzing the transfer function and simulating circuits with thermal noise models. For more accurate estimation of SNR, cause of the difference between estimated SNR and experimental one, 1.1 dB should be established. 4. Discussion The decrease in SNR, due to leakage in the integrator and thermal noise in the comparator, is independent of the sampling frequency. This is because these effects on SNR can be reduced by noise shaping of the sigma–delta modulation and protrudes at higher sampling frequencies. On the other hand, the decrease in SNR due to thermal noise in RINT depends on the sampling frequency and becomes dominant at higher frequencies. We speculated that thermal noise in the resistor overlapped the input analog signal at the input stage of the integrator and the effect of this noise on SNR cannot be reduced by noise shaping. To obtain higher SNR in an actual modulator, the effect of thermal noise on SNR should be reduced. In particular, the noise in the low resistance, RINT becomes a dominant factor at higher sampling frequencies. Thus, a solution may be to construct an integrator with only a superconductive loop and remove RINT. Leakage in the integrator should also be minimized by using this structure. 5. Conclusion We investigated the SNRs of a practical superconducting sigma–delta modulator through using circuit simulations and experimental evaluations. The modulator was

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based on first-order sigma–delta modulation with an LR integrator. Transfer-function analysis revealed that leakage in the LR integrator at its cut-off frequency 3.88 MHz, results in a decrease in the actual SNR of 1.5 dB from the theoretical value and this decrease is independent of the sampling frequency. Effects of thermal noises in the comparator and the integration resistor on SNR were investigated by using circuit simulations with noise sources. We found that total thermal noise resulted in a decrease in the SNR of 5.5 dB at a sampling frequency of 20 GHz and this decrease depended on the sampling frequency. In particular, the noise in the resistor became a dominant factor at frequencies above 5 GHz. We also designed the modulator and evaluated its SNRs experimentally. An SNR of 70.9 dB was obtained at a sampling frequency of 10 GHz, which was consistent with that simulated, i.e., 72 dB. From these results, we concluded that the practical SNR can be estimated by using the transfer function analysis and circuit simulations with thermal noise models. Acknowledgement This work was supported by the New Energy and Industrial Technology Development Organization (NEDO) through ISTEC as the Collaborative Research of Superconductor Network Device Project. References [1] E.B. Wikborg, V.K. Semenov, K.K. Likharev, IEEE Trans. Appl. Supercond. 9 (1999) 3615. [2] K.K. Likharev, V.K. Semenov, IEEE Trans. Appl. Supercond. 1 (1999). [3] F.FurutaK. Saitoh, IEEE Trans. Appl. Supercond. 15 (2005) 445. [4] S. Norsworthy, R. Shreier, G. Temes (Eds.), Delta–Sigma Data Converters: Theory, Design, and Simulation IEEE Press (1997) 1–16. [5] J.X. Przybysz, D.L. Miller, E.H. Naviasky, J.H. Kang, IEEE Trans. Appl. Supercond. 3 (1993) 2732. [6] Circuit simulator WRSPICE. . [7] M. Jeffery, P.Y. Xie, S.R. Whiteley, T. Van Duzer, IEEE Trans. Appl. Supercond. 9 (1999) 4095. [8] K. Saitoh, Y. Soutome, T. Fukazawa, K. Takagi, Supercond. Sci. Technol. 15 (2002) 280. [9] F. Furuta, K. Saitoh, Supercond. Sci. Technol. 19 (2006) S366.