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Submitted on 1 Jan 1978
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MEASUREMENTS OF SQUID EQUIVALENT
CIRCUIT PARAMETERS
G. Ehnholm, S. Islander, P. Ostman, B. Rantala
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplkmenr au no 8, Tome 39, aozir 1978, page ~ 6 - 1 2 0 6
M E A S U R E M E N T S O F S Q U I D E Q U I V A L E N T C I R C U I T P A R A M E T E R S
G.J. Ehnholm, S.T. Islander, P. 'Cfstman and B. Rantala
Low Temperature Laboratory, Helsinki University of Technology, SF-02150 Espoo 15, Finland
RCsum6.- Un circuit equivalent complet pour le SQUID-rf a 6td prSsent6 par Ehnholm. Nous prdsentons des mesures qui vdrifient ce circuit et dsmontrent ses propristss.
Abstract.- A complete equivalent circuit for the rf SQUID has been presentated earlier by Ehnholm. Measurements that verify and demonstrate its properties are presented.
An equivalent circuit for SQUIDs that inclu- des both inductive and resistive components of the output signal has been derived by Ehnholm/l/. Nor- mally the resistive component dominates and the cir- cuit can then be simplified to the one shown in figure 1.
Fig. 1.
On the input side we find a leakage inductance (I-~;~)L~, where ksf is the effective input coupling coefficient and Ls the input coil inductance, and a flux generator with strength k 2 L r i /r. Flux is
s f s p 2 here defined as the time integral of voltage. On the output side is the output resistance ar the
P' voltage generator i r, and the noise voltage gene-
1
rator 4k ar T, which represents the intrinsic noise
B P
of the SQUID. The load impedance Rt, and the noise current and voltage generators of the next amplifier inclucing the noise of the tank circuit, have also been indicated. i and u2 represent changes in peak
2
values from the operating point.
The novel feature of the circuit is the flux generator representing the reverse signal transfer, and the. generator for the intrinsic noise. The mo- del is complete and can be used in all situations.
The circuit parameters are interrelated by the approximate equation/l/
r = k s f h X i T
P s P (1)
where w is the rf frequency. Using ~urkij~rvi's/2/ P
theory for the noise of SQUIDs the value of T be- a
comes approximately
The SQUID circuit has been vindicated in three steps ; first the general form of it has been veri- fied by measuring the input and output impedances. Second, equation(1) has been corroborated by measu- ring the forward signal transfer properties. Final- ly, equation(2) has been tested by measuring the in- trinsic noise.
The input impedance of the SQUID is inductive and has the magnitude/l/ L. In = ~ ~ ( l - k ; ~ + k i ~ b ~ )
,
where the loading parameter b is defined asb = (1+R /r ) - I . It also has resistive part/l/ t P
which we will not treat here. The b parameter is convenient because it is easy to construct from a staircase pattern measured with an open input(i =0)
1
cf. figure 2A.
Fig. 2.
From the figure we find that the small signal resis- tance at the "step" is aboRt. Equating this to the value of ar in parallel with R gives the above ex-
P t
pression for b
.
Lin fot two types of SQUIDs, a thin film device and a toroidal point contact one, was measured as a function of bo, which was varied by placing diffe- rent resistors across the tank circuit. In both ca-
ses L. changed linearly with b as expected, yiel- Ln
ding values for ksf of 0.3 and 0.5, respectively. The output impedance of the SQUID is connected in parallel with Rt. The resulting value Zf has the form/ 1 / Zf = b o ~ t ~ + ( l - b o ) k ~ f L s / ( ~ i n + ~ g ) 7 , where L is the generalized inductance of the signal sour-
g
ce connected across the SQUID input. If a capacitor is used Z
(w
) becomes complex and exhibits a reso-f s
nance at the frequency determined by the capacitor and L in' At this point Z has a large imaginary part, f part, proportional to the Q-value of the input re- sonant circuit.
The expression for Zf was verified experimen- tally in the following way : the rf current fed to the tank circuit of a SQUID with a capacitor at its input was amplitude modulated by a signal, the fre- quency w of which could be swept, and which was also fed to the input of a synchronous detector. The tank circuit voltage was amplitude demodulated and connected to the input of the said detector. Its in-phase and quadrature outputs yielded the real and the imaginary parts of Zf, respectively. Both the shape and the magnitude of the measured Z VS.
w curves followed the :heoretical expression wi- thin the precision of the measurement.
The input and output impedances of the SQUID are independent of r because they depend on the re- serve transfer inductance k2 L r /r and the forward
sf s p
transfer resistance r only through their product. This can be understood from the fact that the si- gnal measuring these entities passes through the SQUID in two directions, first in one and then re- flected back in the other. Equation(1) can be veri- fied by measuring the signal transfer in a single direction, forward or reverse. The forward trans- fer properties are obtained from the staircase pat- tern data combined with the value Ail of input cur- rent that corresponds to a signal change of one flux quantum. Figure 2B shown how r and r are ob-
P tained from the data. Inserting ksf and L from the measurement explained above and r and r from
P the staircase data into equation(]) made the two sides equal to within about 25 %, which is the pre- cision of the measurement and also of the theory. The reverse transfer coefficient has been measured directly by Giffard and Hollenhurst/3/.
Finally, the value of T was checked by measu- ring the intrinsic SQUID noise with a cooled GaAs- FET preamplifier/4/; whose noise was small compared to the intrinsic one. The values of Ta, measured at
different frequencies, were found to follow equation (2) within a 50 % margin, as long as the frequency was below a certain value, determined by the SQUID
ring inductance and the weak link resistance. The presented equivalent circuit is useful when optimizing the properties of SQUID systems. For- merly the reverse signal transfer of the SQUID has simply been omitted ; this gives roughly correct re- sults when the SQUID is used as a low pass amplifier However, if the input is tuned, capacitively or mechanically, a complete SQUID model is essential.
References
/I/ Ehnholm,G.J., J.L.T.P. z(1977) 1
/2/ Kurkijarvi,J., J. Appl. Phys.
