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The nonlinearity of the high-Tc rf SQUID system
PHYSICA® ELSEVIER
Physica C 341-348 (2000) 2695-2696 www.elsevier.nl/locate/physc
T h e N o n l i n e a r i t y o f the High-Te r f S Q U I D S y s t e m D.EHe a, Y.Zhang b, A.I.Braginskib, Y.D.Dai a and S.Z.Wang ~ aMesoscopic Physics National Laboratory and Department of Physics, Peking University, Beijing 100871, China blnstitut fur Schicht-und Ionentechnik (ISI), Forschungszentrum Juelich, D-52415 Juelich, Germany The nonlinearity of SQUID system can be described by the total harmonic distortion (THD). For our YBCO High-To rf SQUID system operated in a flux-locked loop without modulation, we measured the THD by analyzing the error signal between input and output, by this way, the influence caused by the harmonics of input signal could be decreased. For input signals at frequency up to 250Hz and amplitude up to 100dP0(250nT), the THD was below 10ppm. We also observed some anomalous behaviors of the THD, such as, the abnormal curve of THD versus frequency for the input signal at amplitude of about 3000. A qualitative explanation for these phenomena was given in this paper. 1. I N T R O D U C T I O N
2. THE M E A S U R E M E N T OF THD When SQUID operates in flux-locked loop, its output is linear with the applied external flux. In some applications of SQUID, The SQUID needs to work in unshielded environmental where the line interference is very big. If the linearity of a SQUID system is not good enough, the output signal will be distorted. So the nonlinearity of a SQUID system must be measured and analyzed. The nonlinearity of SQUID can be described by the total harmonic distortion (THD) [1]. When a sinusoidal signal flux of frequency f is applied to the SQUID, the THD of the output signal can be defined by the following formula: THD --4A 2 + As2£<<" ' ' / A , (1) Where Al is the amplitude of fimdamental frequency f; A2, A3 ...is the amplitude of harmonics of frequency 2f, 3f... which can be measured by a fourier transform spectrum analyzer. In the measurement of THD, the harmonics of input signal should be small enough to ensure the accuracy of the measurement. D.G.Nichols et.al.[2] purified the input signal by using low-pass and bandpass filter and suppressed the fundamental signal of output by a notch filter. D.Dmng[1] suppressed the fundamental signal by subtracting the input signal and the output signal using a resistance trimmer. This method could also decrease the influence of the harmonics of the input signal and didn't add other nonlinearity. We chose D.Drung's method to measure the THD of our YBCO High-To rf SQUID system.
Our High-To rf SQUID system includes three components: readout electronics, tank circuit and rf SQUID sensor. The readout electronics is a 900MHz circuit without modulation. The tank circuit is a YBCO coplanar resonator with high Q-value and the resonant frequency is about 900MHz[3]. The rf SQUID sensor is made of YBCO step-edge junction working at liquid-nitrogen temperature 77K. The inner loop size of the rf SQUID is 10x500 ~tm2. The white flux noise is 10 ~tdPo/qHz and the field sensitivity is 26 tT/~/Hz. When the feedback resistor is 3Kf), the dynamic range is about 200 ~0 and the bandwidth is over 300KHz. The input signal was generated by a PhaseLocking Amplifier of Model SR830 DSP with the harmonic distortion of below 5× 10-4. We analyzed the signal by a fourier transform spectrum analyzer of model HP3562 with the dynamic range of 90dB. For the output signal with high THD, the spectrum analyzer can directly analyze it. For the small THD of output signal, we analyzed the error signal of input and output, thus the fundamental signal of output could be suppressed. We got the error signal by a resistance compensation network showed in Fig. 1. In Fig. 1, The resistance of R could be adjusted to make sure the output voltage of SQUID was approximately equal to the voltage from sine wave generator. We adjusted the resistance trimmer to cancel the fundamental signal of output.
0921-4534/00/$ - see front matter ¢3 2000 Elsevier Science B.V. All rights reserved. Pll S0921-4534(00)01483-0
2696
D.E He et al./Physica C 341-348 (2000) 2695-2696
output
1OK
of
rf SQUID 200 lsin¢ wave I input generator ] IOK "R Figure 1. The resistance compensation network According to our analyses, the error of this method is kD, where D is the harmonic distortion of input signal and k is the compensation coefficient of harmonics of input. We estimated the relative error of our measurement was about 10%. 10 -3
j
1 0 .4
l 0KHz /+ 4KHz 4.._+,,,+-~
"°% 240Hz /
o -iI-- lO-S
~ 1 0 "6
"
~
10 Flux (~0)
~
H
z
100
(a)
region with amplitude between 8~0 and 30~0, where the THD decreased with the increasing of amplitude, which was also observed by D.Drung for his Low-To dc SQUID system [1]. The curve of THD vs frequency in Fig. 2(b) also appeared anomalous. We will give a qualitative explanation for this. There is a follower after the output in our rf SQUID system. We think the follower may cause the anomalous behavior of THD. According to the analyses of D.G.Nichols et.al [2], when a one-pole integrator is used in the flux-locked loop, the third harmonic distortion caused by the nonlinearity of transfer function is proportional to (Aco)2, where A is the amplitude of the input signal and 0~ is its frequency, and the initial phase is equal to the input signal. For a follower, the third harmonic is approximately proportional to A0), but its initial phase is reversal to the input signal. Thus, the total harmonic distortion can be written as following: THD = X/[(aA¢o) 2 - (13Am)] 2 + "t2 (2) Where a, 13 is constant and ~ is the second and fourth harmonic distortion. We used formula (2) to fit the data of Fig. 2b and found the fitted curve (the dashed line in Fig. 2(b)) was consistent with the measured points. 3. D I S C U S S I O N
I 0 -4'
1O.S I I1 0 -6'
0
.....
Yi~0
....
liSb0'
Frequency (Hz)
(b) Figure 2 (a) For different frequency, the THD vs the amplitude of input signal. (b) THD vs frequency with the amplitude of 30~0. The dashed is the fitted curve. We measured the THD vs input signal amplitude for different frequency. The result was showed in Fig. 2(a). For the frequencies of 4KHz and 10KHz, the output was analyzed directly; for the frequencies below 400Hz, the error signal of input and output was analyzed. Considering the 50Hz line signal and its odd harmonics was the main interference in unshielded environment, we especially measured the THD at frequencies of 45Hz, 140Hz and 240Hz. For the curve of 400Hz, there is an anomalous
From Fig. 2(a), for input signals at frequency up to 250Hz and amplitude up to 100~o, the THD of our High-To rf SQUID system was below 10ppm. If the 50Hz line interference is below 100nT in unshielded environment, the distortion caused by the nonlinearity is below lpT. This is good enough for MCG and NDE measurement in unshielded environment. The nonlinearity will neither influenced the balance of electrical gradiometer, since its balance is normally below 10000. REFERENCE [1]. D.Drung, H.Matz and H.Koch, Rev. Sci. Instrum. 66, 3008 (1995) [2]. D.G.Nichols,E.Dantsker, R.Kleiner, M.Muck, and J.Clarke, J. Appl. Phys. 80, 6032 (1996) [3]. Y.Zhang, W.Zander, J.Schubert, ERuders, H.Soltner, M.Banzet, N.Wolters, X.H.Zeng and A.I.Braginski, Appl. Phyas. Lett. 71,704 (1997)
Physica C 341-348 (2000) 2695-2696 www.elsevier.nl/locate/physc
T h e N o n l i n e a r i t y o f the High-Te r f S Q U I D S y s t e m D.EHe a, Y.Zhang b, A.I.Braginskib, Y.D.Dai a and S.Z.Wang ~ aMesoscopic Physics National Laboratory and Department of Physics, Peking University, Beijing 100871, China blnstitut fur Schicht-und Ionentechnik (ISI), Forschungszentrum Juelich, D-52415 Juelich, Germany The nonlinearity of SQUID system can be described by the total harmonic distortion (THD). For our YBCO High-To rf SQUID system operated in a flux-locked loop without modulation, we measured the THD by analyzing the error signal between input and output, by this way, the influence caused by the harmonics of input signal could be decreased. For input signals at frequency up to 250Hz and amplitude up to 100dP0(250nT), the THD was below 10ppm. We also observed some anomalous behaviors of the THD, such as, the abnormal curve of THD versus frequency for the input signal at amplitude of about 3000. A qualitative explanation for these phenomena was given in this paper. 1. I N T R O D U C T I O N
2. THE M E A S U R E M E N T OF THD When SQUID operates in flux-locked loop, its output is linear with the applied external flux. In some applications of SQUID, The SQUID needs to work in unshielded environmental where the line interference is very big. If the linearity of a SQUID system is not good enough, the output signal will be distorted. So the nonlinearity of a SQUID system must be measured and analyzed. The nonlinearity of SQUID can be described by the total harmonic distortion (THD) [1]. When a sinusoidal signal flux of frequency f is applied to the SQUID, the THD of the output signal can be defined by the following formula: THD --4A 2 + As2£<<" ' ' / A , (1) Where Al is the amplitude of fimdamental frequency f; A2, A3 ...is the amplitude of harmonics of frequency 2f, 3f... which can be measured by a fourier transform spectrum analyzer. In the measurement of THD, the harmonics of input signal should be small enough to ensure the accuracy of the measurement. D.G.Nichols et.al.[2] purified the input signal by using low-pass and bandpass filter and suppressed the fundamental signal of output by a notch filter. D.Dmng[1] suppressed the fundamental signal by subtracting the input signal and the output signal using a resistance trimmer. This method could also decrease the influence of the harmonics of the input signal and didn't add other nonlinearity. We chose D.Drung's method to measure the THD of our YBCO High-To rf SQUID system.
Our High-To rf SQUID system includes three components: readout electronics, tank circuit and rf SQUID sensor. The readout electronics is a 900MHz circuit without modulation. The tank circuit is a YBCO coplanar resonator with high Q-value and the resonant frequency is about 900MHz[3]. The rf SQUID sensor is made of YBCO step-edge junction working at liquid-nitrogen temperature 77K. The inner loop size of the rf SQUID is 10x500 ~tm2. The white flux noise is 10 ~tdPo/qHz and the field sensitivity is 26 tT/~/Hz. When the feedback resistor is 3Kf), the dynamic range is about 200 ~0 and the bandwidth is over 300KHz. The input signal was generated by a PhaseLocking Amplifier of Model SR830 DSP with the harmonic distortion of below 5× 10-4. We analyzed the signal by a fourier transform spectrum analyzer of model HP3562 with the dynamic range of 90dB. For the output signal with high THD, the spectrum analyzer can directly analyze it. For the small THD of output signal, we analyzed the error signal of input and output, thus the fundamental signal of output could be suppressed. We got the error signal by a resistance compensation network showed in Fig. 1. In Fig. 1, The resistance of R could be adjusted to make sure the output voltage of SQUID was approximately equal to the voltage from sine wave generator. We adjusted the resistance trimmer to cancel the fundamental signal of output.
0921-4534/00/$ - see front matter ¢3 2000 Elsevier Science B.V. All rights reserved. Pll S0921-4534(00)01483-0
2696
D.E He et al./Physica C 341-348 (2000) 2695-2696
output
1OK
of
rf SQUID 200 lsin¢ wave I input generator ] IOK "R Figure 1. The resistance compensation network According to our analyses, the error of this method is kD, where D is the harmonic distortion of input signal and k is the compensation coefficient of harmonics of input. We estimated the relative error of our measurement was about 10%. 10 -3
j
1 0 .4
l 0KHz /+ 4KHz 4.._+,,,+-~
"°% 240Hz /
o -iI-- lO-S
~ 1 0 "6
"
~
10 Flux (~0)
~
H
z
100
(a)
region with amplitude between 8~0 and 30~0, where the THD decreased with the increasing of amplitude, which was also observed by D.Drung for his Low-To dc SQUID system [1]. The curve of THD vs frequency in Fig. 2(b) also appeared anomalous. We will give a qualitative explanation for this. There is a follower after the output in our rf SQUID system. We think the follower may cause the anomalous behavior of THD. According to the analyses of D.G.Nichols et.al [2], when a one-pole integrator is used in the flux-locked loop, the third harmonic distortion caused by the nonlinearity of transfer function is proportional to (Aco)2, where A is the amplitude of the input signal and 0~ is its frequency, and the initial phase is equal to the input signal. For a follower, the third harmonic is approximately proportional to A0), but its initial phase is reversal to the input signal. Thus, the total harmonic distortion can be written as following: THD = X/[(aA¢o) 2 - (13Am)] 2 + "t2 (2) Where a, 13 is constant and ~ is the second and fourth harmonic distortion. We used formula (2) to fit the data of Fig. 2b and found the fitted curve (the dashed line in Fig. 2(b)) was consistent with the measured points. 3. D I S C U S S I O N
I 0 -4'
1O.S I I1 0 -6'
0
.....
Yi~0
....
liSb0'
Frequency (Hz)
(b) Figure 2 (a) For different frequency, the THD vs the amplitude of input signal. (b) THD vs frequency with the amplitude of 30~0. The dashed is the fitted curve. We measured the THD vs input signal amplitude for different frequency. The result was showed in Fig. 2(a). For the frequencies of 4KHz and 10KHz, the output was analyzed directly; for the frequencies below 400Hz, the error signal of input and output was analyzed. Considering the 50Hz line signal and its odd harmonics was the main interference in unshielded environment, we especially measured the THD at frequencies of 45Hz, 140Hz and 240Hz. For the curve of 400Hz, there is an anomalous
From Fig. 2(a), for input signals at frequency up to 250Hz and amplitude up to 100~o, the THD of our High-To rf SQUID system was below 10ppm. If the 50Hz line interference is below 100nT in unshielded environment, the distortion caused by the nonlinearity is below lpT. This is good enough for MCG and NDE measurement in unshielded environment. The nonlinearity will neither influenced the balance of electrical gradiometer, since its balance is normally below 10000. REFERENCE [1]. D.Drung, H.Matz and H.Koch, Rev. Sci. Instrum. 66, 3008 (1995) [2]. D.G.Nichols,E.Dantsker, R.Kleiner, M.Muck, and J.Clarke, J. Appl. Phys. 80, 6032 (1996) [3]. Y.Zhang, W.Zander, J.Schubert, ERuders, H.Soltner, M.Banzet, N.Wolters, X.H.Zeng and A.I.Braginski, Appl. Phyas. Lett. 71,704 (1997)