superconductive quantum fluxmeter—superconductive quantum magnetometer with external feedback

ELSEVIER PII: SOOll-2275(97)00077-S

Cryogenics 37 (1997) 511-516

C? 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved

001 l-2275/97/$17.00

Superconductive quantum superconductive quantum with external feedback Vladimir Zrubec and Anton Urban

fluxmeter- magnetometer

Institute of Measurement Science, Slovak Academy of Sciences, Dtibravskd cesta 9, 842 19 Bratislava, Slovak Republic

Received 25 November 1996; revised 19 March 1997

In this article transfer and noise parameters of superconductive quantum magnetometers with external feedback functioning as superconductive quantum fluxmeters (SQFs) are analysed. From the derived relationships it follows that they are capable of measuring magnetic flux acting on superconductive coils with unknown geometrical structure, and unknown inductance and size parameters. The data on high input impedance of fluxmeters of this type are specified. Relationships are derived referring to the connection between inductive (typically (0.1 t IO) H) or resistive ( - (0.1 + 100) fi) nature of the input impedance of electronic unit (EU) SQFs and transfer parameters of their complete feedback loop. Simple methods permitting utilization of SQFs for the measurement of inductance of superconductive coils in situ in cryogenic environments in the range - (1O-gs IO-‘) H are presented. 0 1997 Elsevier Science Ltd.

Keywords: SQM with external feedback; superconductive quantum fluxmeter; inductance of superconductive coils

Superconductive quantum magnetometers (SQMs) with external negative feedback have become more important in connection with the problem of cross-talk between channels of multichannel systems for the measurement of biomagnetic fields’. The cross-talk develop when the SQUID sen- sors are connected with an electronic unit (EU) with internal negative feedback. They are caused by mutual inductance and non-zero current in antenna coils of individual channels, induced by the measured magnetic field. The principle of elimination of cross-talk by the use of a SQM in the external feedback mode essentially consists of the high input impedance of its EU and ensuing low current in the antennae of individual channels. The transmitting and noise properties of these SQMs have been analysed in more detail in previous articles ‘3’ Unlike SQMs with internal . feedback, these magnetometers have other, as yet little known, specific properties, predestinating them especially for the measurement of magnetic flux as well as for the measurement of inductance of superconductive coils. In essence they are superconductive quantum fluxmeters

(SQFs).

Theoretical analysis

The analysis of noise properties of a SQM with external feedback has been presented in a previous article2. It follows from the described experiments, that the noise ratios

are well accounted for by the equivalent network with two sources of an input equivalent noise signal. The block diagram of the fluxmeter is shown in Figure la and its equivalent substitute circuit is shown in Figure lb. It consists of the source of magnetic flux noise with the instantaneous value $” and the source of current noise with the instantaneous value ?” while these two sources are not correlated. In all the other time-dependent values, unless stated other- wise, instantaneous values are considered. The antenna is represented by a circular superconductive coil with inductance L,. The measured magnetic field affects it with magnetic flux & The inductance Li, is the efficient inductance of the input circuit of the SQUID sensor, consisting of the input coil connected to it through the mutual inductance M,,, (unmarked in the diagram). The feedback current branch consists of the circuit with a serial resistance Rf and a coup- ling transformer with inductances L, and Lf of the primary and secondary coil. The mutual inductance between them is M. Provided that R,>>q,Lp, where LO,, is the circular frequency of the highest frequency component of the assumed spectrum of low-frequency signals, the feedback current if in the entire band is proportional to the output voltage u of EU SQFs.

We assume that the transfer function of the direct path of the EU SQF is the invariable 9 = u/ii, where ii is the current flowing in its input circuit. The inductance Li, is measured in the condition with disconnected feedback loop on the clamps of the SQUID coil at a low signal ( iiMi,<<~o,

Cryogenics 1997 Volume 37, Number 9 511

Superconductive quantum fluxmeter: V. Zrubec and A. Urban

Ri

r---- ___---_-_.-----

1

1 l”

JT

EU SQF

(u,) and its spectral density S,,, will be marked analogi- cally, while it holds that

U” 2

&“=‘k (4)

If the signal component u, in Equation (3) is separated from the noise component u, we will obtain following relationships:

where

I u=qi,

(W

Figure 1 (a) Simplified scheme of SQF composed of EU SQF and antenna coil with inductance Ls. (b) Equivalent network of SQF with sources of equivalent input noise signals T, and &

&, = 2.07 x 10 - I5 Wb), wherein the unidirectional component of the SQUID operating point must be in the position corresponding to the operation mode. It follows from the circuit equations of Figure lb that for the output signal u = qii in connection with the feedback loop it holds that

With a high amplification of the disconnected feedback loop Aa = (+s/$ff(, where +f = ifM, i.e. provided that

Aa= qM

&XL, + Lf + Lid B-1

then Equation (1) is simplified to

u = [+s + 4” + i-,(L, + L,)] 2

We assume that the noise signal has the characteristics of white noise. The effective value of the equivalent noise flux and the mean square value of the equivalent noise magnetic flux are denoted by (i,) and (&), respectively. Their power spectral densities are Si and S,, respectively, where Si = (i,)*/Af and S, = (4J2/Af, with Af being the noise frequency width of the transmission channel. The effective value of the noise component of the output noise signal

A,,=$[VWb-‘1 (W

We considered the correlative independence between equivalent noise sources and the principles of summing the noise signals of white noise nature. From Equation (5) for the signal component of the output voltage it holds that the system behaves as a converter of the magnetic flux & to the voltage u,. From this direct proportionality between the output voltage and magnetic flux it also follows that to the output noise signal the equivalent input noise magnetic flux & with mean square value (&J and spectral density S, = S,,/A%a can be added. From Equation (5) it follows that

s@e = s* + si(Ls + Lf) (6) (A,,) = Gdf)“*

For the quantitative analysis of the noise properties the parameters of the sources of the input equivalent noise have to be known. Equation (5) for the noise component u, represents the equation of two unknowns. It follows from this that spectral densities of the sources of equivalent input noise flux given by a SQF can be detected by two measure- ments of the output noise signal at two different inductance values on the input of the connected coil. It is rec- ommended to choose one large coil and the other as small as possible. The effective value of the output noise signal after the connection of the coil with high inductance L,,>L, will be denoted as (u,]) and its spectral density as S,, . The small connected inductance (short-circuited input) L,, for the effective value of the output noise is denoted as (u,,) and the spectral density as S,,.

In ordinary magnetometers with external feedback the condition Lf << Lie, following from the optimization of the properties of a magnetometer designed for the measurement of intensity of magnetic field, is complied with. Such a condition is also favourable for a SQF designed for the measurement of the magnetic flux. In this case, following the connection of the low inductance L,, it holds that i”(L,, + L,)<<&. The parameters of the sources of equivalent input noise signal are calculated from the measured spectral densities S,, and S,, of the output noise signal by the use of the following relatively simple relationships:

S, = LJAZ,, Si = su~I(Au&,)~ (7) (L,O + Lf e Lio)v tLslBLio)

512 Cryogenics 1997 Volume 37, Number 9


From Equation (5) it follows that the dependence of the spectral density of the output noise of SQFs on the inductance L, can then be expressed by the relationship

S’i2 = (L, + Lf)2 IL2

U”

t s U” + s,,, ~~

Cl -) (8)

Measurement of the magnetic flux and the inductance of superconductive coils

Magnetic flux

In an ordinarily understood magnetometer with external feedback, the antenna coil is ‘part and parcel’ of the magnetometer. Its geometrical parameters and inductance are known. Usually, its geometry is simple, so that its directional properties can be explicitly defined. The transfer invariable between the measured magnetic field (B,H) and the output signal of the magnetometer can either be computed or experimentally determined in a known magnetic field. Typically, by the measurement of an unknown magnetic field, the information on its layout and size are obtained. For a magnetometer optimization the criteria given previously2 hold.

The situation is different in the measurement of a magnetic flux. The magnetic flux 4 is a quantity dependent on the direction and size of the magnetic inductance vector L? and at the same time on the vector s of the total area of coil turns:

(9)

If the coil is of the usual simple shape, the magnetic flux can easily be computed from the data on magnetic induction measured by a standard SQM. However, in physical experiments it is sometimes a problem to determine the magnetic flux influencing a coil with unknown inductance and unknown (and virtually immeasurable, e.g. small ball of wire) geometry of the coils. The relationship (Equation (5)) between the input magnetic flux & and the output voltage u, of the SQF is remarkable for the fact that there occurs neither the inductance of the scanning coil attached to the input EU SQF nor any of the parameters from Equ- ation (9). With the known transmitting variable A,,

(Equation (5a)) we can measure the magnetic flux influencing it without knowing these parameters. It is essential to know the limits within which this notion holds.

(a) The noise properties of SQFs in relation to the size of the measured magnetic flux & are characterized by the signal-to-noise ratio:

Psln = Ze) where (&) is the mean square value of the input equivalent noise signal (Equation (6)). The lowest measurable will be considered the magnetic flux &,,, in which it holds that psln = 1. It follows from Equation (6) that its value is given by:

&,,, = (Se + S,(L, + L,)2)“2Af”2 (11)

As we can see, the noise of a SQF depends on the inductance L, of the coil attached to the input. Therefore, the lowest measurable magnetic flux &,, will alternate in the interval from values of approximately (S~Af)‘” (for L, << Li,) UP to values of Ll(SiA~“2 (for L,>>Li,).

(b) In the calculation of the simplified Equation (3) it was assumed that the transfer function q is frequency independent (q = qo, has a dimension of a real resistance) and at the same time Equation (2) holds. However, this condition is never fully met (qJR,#m). It follows from Figure la that under this condition between the measured magnetic flux and the signal component (i = 0, & = 0) of the input current ii, the following relationship holds:

!i!L=L +L_ +I,+%Lt iis ’ lo Rf e’

(12)

where L,, is the equivalent inductance of the closed loop of the input circuit. Li” = L,, - L, represents the equivalent input inductance between the input clamps l-l (Figure la). For its value it follows from the above mentioned relationship that

Li, = L, + Lf + + f

(13)

Since it is necessary to meet the requirement of Equation (2) for the very reason of ensuring the undisturbed transfer of the input signal to the output of the fluxmeter (linearization of the triangular transfer characteristics of the SQUID) in the band of the regular function of the SQF, it always holds that MqJR,>Li, + Lf. Therefore, from the previous equation the following always holds for an EU SQF:

Li,>>Lj, (13a)

Under usual conditions, Li, of the EU SQF surpasses the value of the input inductance of the EU SQM in the mode with internal feedback (its value is Li, - L,,) by several orders of magnitude. The accompanying phenomenon in the SQF is a markedly lower current in the antenna circuit, which is used for the above-mentioned suppression of inter- channel cross-talk in multichannel magnetometers and gradiometers’.

The situation is more complex when the transfer path q

of the EU SQF contains elements with frequency dependent amplitude and phase characteristics. Such an internal circuit structure of the SQF is shown in Figure 2. Generally, it can be illustrated by a frequency independent amplifier with amplification q,, (it also comprises an additional phase detector in case of an RF SQM with an auxiliary low-frequency modulation signa13) in series with a frequency dependent second stage. Hitherto the assumptions corre- sponded to the state when it did not exist or had the character of PI circuit, but the topical band of the input signal was situated in the frequency independent area, below the cut-off frequencyf, = 1/(2~rR&,) (Figure 2a) of the transfer characteristics. However, the predominant number of SQMs with internal and external feedback have in the frequency dependent stage an integrator rotating the phase of the harmonic signal by - 90’ already at low frequencies (Figure 2b). The main sense of this integrator circuit is to reduce the total gain of open feedback loop below the unit



cl

I

(b) Figure 2 (a) Simplified scheme of a typical direct transfer path q of EU SQF with PI frequency characteristics. (b) Transmission path with the ideal integrator

level, above a certain frequency limit where the phase condition cannot be kept in the necessary range (parasite capacitances and inductances, undefined phase properties of active transfer elements). In this way it is possible to prevent the oscillations in the feedback mode in the higher frequency range; however, the EU SQF transfer parameters worsen in the operating frequency range (non-zero time constant of the feedback loop, decrease of the value of maximum measurable magnetic flux of harmonic behaviour at higher frequencies).

From the phasor solution of the substitute circuit of SQFs it follows that in such a case for the transfer function q and for the input impedance Zi, of SQFs it holds that

so&JR ‘=(l +jo/o,)

Zi, = jo(L, + Lf) + jwM 90R, 1

R,R (1 + jolt,)

(144

For the frequency area below the cut-off frequency of the transfer characteristics (Figure 2~) at w/o, << 1, where the transfer function is a frequency-independent invariable q = q&lR, the input impedance is of inductive character. In accordance with Equation (13) it generally holds that

Zi” = jw(L, + Lf + MqJ$lRfR) = jwLi, w/w, << 1, w/w,, 4 1

(15)

For the frequency above the bend of transfer characteristics

(w/o,>l), where the transfer function is of integrator character, the result is obtained in the form

Zi” = jw(L, + Lf) + MqJRfRCp w/w,>> 1

(16)

In this case, the input impedance contains a frequency-independent series resistance component Ri, = Mq,,/R,RC, [II]. Usually, it considerably exceeds the inductive reactance of the component Li, + Lf. The input of a SQF with the transfer function of integrator type presents for the coil L, a high resistance load (typically tenths to units of ohms). This remarkable result can be explained also in the following way. In the frequency range o>>w,,, the output voltage u is phase-shifted by about - 90” compared with the input current ii. The phase properties of this element do not change when inserting it in the feedback loop. At the same time in the closed feedback loop mode the measured magnetic flux & and the output signal u are practically in phase (in the frequency range 0 I o << Ed,,, where the open loop gain is great). However, it means (if & and u are in phase and there is the phase shift of about - 90” between u and ii in the range w>>w,) that the phase shift - 90” is also between the magnetic flux +s and the current ii (considering the condition that ii--O) flowing through the antenna loop. When the magnetic flux acts on the inductive loop connected to the electric circuit, such a state can set in only in the case if a serial resistance component is dominant in the whole circuit. This interpretation is in full compliance with Equation (16).

Measurement of inductance

If the source of the total magnetic flux & in a coil on the input of EU SQFs is the current iL conducted to it from the auxiliary current source (Figure 3), it holds that

c#~~ = iLLs resp. L, = c#& (17)

Since the value (ps is directly set by the output signal of a SQF, at the known current iL, the inductance L, can be determined from this relationship. The SQF can thus be used as a measuring system for the inductance of superconductive coils.

For the signal-to-noise ratio in this case it holds that

‘S/n = (S, + Si( L, + Lf)2)“2Af”2 (18)

From the analysis of this relationship for two extreme cases of inductance values L,, the information on the required level of the useful signal iL or & from the point of view of the noise can be acquired:

Figure3 Connection of SQF for measuring inductance L, of superconductive coil using auxiliary current signal iL



-for very high (L, + Lf>s&%s~‘2):

-for very low (L, + Lf -cc s&s;‘2):

i,_ = p,,J S,Af, ‘121Ls

4~s = ~s,n&AfI”~

values

values

of

of

inductance

(19a)

inductance

(19b)

In standard cases it is usually enough for the measurement to go along with the signal-to-noise ratio psln in the interval 10 to 100.

Discussion

A practical illustration of the real conditions is given in the following example: an SQM with an internal feedback with standard rf readout was adapted to a SQF by a SQUID, which, in its original version, had a spectral energy sensitivity of - lo- 28 J Hz-’ and a bandwidth of 20 kHz (at low signal). After the change of the feedback to an external one, its parameters had values shown in Table 1.

It is necessary to note that the transfer parameters of the SQF described in this example were initially optimized for its use for magnetometric purposes, i.e. from the point of view of maximum sensitivity for measuring magnetic induction. From the optimization conditions2 it followed that it will be achieved with an antenna coil with inductance L, = 2 x 10 - 6 H. The required amplification qO of the transfer path (Figure 2a) at the given circuit parameters (Table Z) was determined so that at its introductory value in the bandwidth over 20 kHz an absolute stability (IA*/ < 1) of the circuit was secured. The input inductance calculated from Equation ( 15) for the frequency band below f0 = 1 Hz has a value Li” - 85 mH (unlike the value of 2 x 10 - 6 H in the initial SQM with internal feedback). The 1% error margin in measuring inductance or magnetic flux will not be surpassed, provided the attached coil has an inductance L, 5 0.85 mH.

From Equation (2) it follows that the amplification of the feedback loop depends on the inductance L,. If it were assumed that the system is to be designed, e.g. for values L, 2 1 x low3 H, the above-mentioned condition of absolute stability (IA@1 5 1 for f 2 20 kHz), will be met even at the amplification of qO - 7.7 x lo9 V A-’ (Equations (2) and (14a)). The input inductance will then increase to - 20 H and the measurement range will widen as far as the

Table 1 Parameters of illustrative example

M=0.2x 10-6H R=64k0 sy2 = 3 x IO-” Wb Hz - “*

Lf= 0.28 x 10m6H R, = 1.5 MU. S’” = 1 5 x JO-” A’Hz - I/;

L,,=~xIO-~H C, = 0.1 pF CfO = L,, = 0.085 H ( f < 1 Hz)

Rf = 1.8 x 103a 9,=3xlO’VA-’ $ =20Hat 7f”x 109VA-:f=c 1

fi

area suitable even for the measurement of power superconductive magnets. If a fluxmeter is designed for a wide range of inductance values of attached coils, it is expedient to alternate the amplification of any of the partial transmission elements of the feedback loop in jump-switched ranges. It will allow us to keep the total gain in every range on an acquired level but under the level critical for stability. In this respect it is necessary to point out the fact that in the measurement of high inductances at an appropriate signal- to-noise ratio on the output of a SQF, the magnetic flux & might reach a high value (Equation (19a)). In the above- mentioned example, at L, - 10-l H and psln = 100, the output signal should have the value of - 8 x lo”&,. The SQF must be adapted to this level from the viewpoint of maximum admissible amplitude of the measured signals. At the same time the frequency has to be chosen with regard to the maximum slew rate of a SQF.

It follows from Equation (16) that in the frequency range above the frequency f0 the input impedance of EU SQFs will be of resistance character. By calculation we find out that for the above mentioned example the input resistance Ri, has a value - 0.5 fl or - 120 R after the increase in amplification qO to the value of 7.2 x lo9 V A-’ for the topical range of coils L, > 1 mH. In the frequency band w, in which the condition wL, < O.OlRi” is met, the relative error of determined inductance value and the error in the measured magnetic flux & will not exceed the value of 1%. In the given example, it means that the measurement of coils with inductance up to the order of lo-’ H can be carried out with signals even in the frequency range of the order of 10’ Hz. The significance of this lies in the fact that in this frequency range the level of geomagnetic disturb- ances is negligible4 and the measured object does not usually require special magnetic shielding5. Moreover, in measuring, the efficient selective filtering of the output signal of a SQF is possible. However, the conditions following from the maximum slew rate of a given SQF have to be complied with for both the noise and useful signals.

In the circuit structure of SQMs, an ‘ideal’ integrator in which Rp-+” (Figure 2b) is often used. From Equation (16) it follows that in such a case the resistance component of the input impedance would be dominant from ‘zero’ frequencies (fO+O). However, in integrator circuits with real operational amplifiers, the decrease in frequency is tied with the increase in amplification of the harmonic signal only to a certain extent. Below it the circuit loses the character of an integrator and if the decrease in frequency continues the amplification remains constant. This state can be explained by the existence of an equivalent ‘internal resistance’ R,, (attached in parallel to C,) of the operational amplifier (usually R,>>R,). Therefore, the transfer path of every active integrator of the first order has actually the character of PI regulator with constant amplification and negligible phase shift of the harmonic signal in the frequency range below o,, = l/R&,. Therefore, in the band of very low frequencies the input impedance of a SQF even in the case of an ‘ideal’ integrator has inductive character.

The untraditional method of measuring inductance of superconductive coils attached to the input of EU SQFs follows from Equation (5), valid for the noise component. With known transfer and noise parameters L, can be determined from the spectral density of the output noise signal. It is not suitable for the measurement of low inductances (L, - L, + Li,), since in this range the dependence of the output noise voltage of SQFs on L, is very flat*.



The inductance is determined by spectral density of the output noise signal according to the equation

PW

For the inductance L, > lO(&, + L,) a simplified relationship can be used (error < 10%):

L, = S;'2/(S;'2A,,) Gob)

It is even more important in this measurement than in the measurement of inductance with auxiliary current signal i,_ that the noise signal should not contain the components of external disturbing magnetic fields. It can be achieved by the use of the above-mentioned superconductive and ferromagnetic shielding of the measured coil (or shielding of the measuring space). The other possibility is to measure the noise in the band lying out of the spectrum of disturbing magnetic fields. But even in this case it holds that the frequency and amplitude of disturbing signals must not exceed the limit given by the maximum slew rate of the SQF. The advantage is that by this method the inductance of an unknown coil attached to the input of EU SQFs can be determined without any auxiliary signals. A more universal method, however, is a measuring method employing an auxiliary current signal iL (Equation (17)), which is more accurate. This was, for example, the method employed in measuring the inductance of a set of mechanically remote- switched superconductive coils (within the range (7.2 x 10 - 9 to 14.7 x 10 - 6, H] placed in a small superconductive shielding cylinder in experiments described previously*. The advantage of both above-mentioned methods consists in the fact that they determine a real inductance of a coil in situ in a cryogenic medium, influenced, for example, by contingent vicinity of superconductive or ferromagnetic walls of the shielding. If, at the measurement of magnetic flux (or at the measurement of directional diagram of magnetic flux) of an unknown coil by a SQF the value and the direction of a vector of magnetic induction of the acting magnetic field is known, its effective area or a ‘directional diagram’ of the effective area can be determined.

Conclusion

It follows from the derived relationships that superconductive quantum magnetometers with external feedback are virtually superconductive quantum fluxmeters with a transfer constant independent of parameters of a coil attached to the input of their EU SQF. Therefore, they allow measurement of magnetic flux influencing the coil without the necessity of knowing its inductance, geometrical parameters and the magnitude of the active magnetic field. This property is connected with high input (Figure la, input l- 1) impedance of EU SQFs. Under real conditions its inductance can reach the values of units of henries and higher. At the same time the SQF facilitate a simple measurement within a wide range of values (typically 10 - 9 to 10-l H) of the inductance of superconductive coils ‘in situ’ in cryogenic environments. In this article there were derived relationships which, under defined conditions, allow expedient utilization of the described properties in experimental practice.

Acknowledgements

The work reported in this paper was carried out within the framework of the research of utilization of Josephson’s effect in measurement, subsidized by the Grant Agency VEGA of the Ministry of Education and the Slovak Acad- emy of Sciences, Project 2/1159/96.

References

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superconductive quantum fluxmeter—superconductive quantum magnetometer with external feedback

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