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Appendices Appendix a Values of Constants Symbol Value APPENDICES APPENDIX A VALUES OF CONSTANTS SYMBOL NAME VALUE UNIT -12 €o Permittivity of free space 8.85 x 10 farad/meter (F/m) -6 JL o Permeability of free space 1.26 x 10 henry/meter (Him) e Elementary charge 1.60 F< 10-19 coulomb (C) -24 JL B Bohr magneton 9.27 x 10 joule/tesla (J/T) -15 ~o Flux quantum 2.07 x 10 weber (W) -34 h Planck constant 6.63 x 10 . joule' second (J's) 8 c Speed of light in vacuum 3.00 x 10 meter/sec (m/s) -23 kB Boltzmann constant 1.38 x 10 joule/kelvin (J/K) -8 0 Stefan-Boltzmann constant 5.67 x 10 watt(m2 'K' (w/m 'K') 1\ 23 1 NA Avogadro number 6.02 x 10 mole- (mole -1 ) 633 634 APPENDICES R Universal gas constant * 8.31 joule/mole'kelvin (J/mole'K) * The word "mole" means a gram-mole in both cgs and SI units. The definition has not been changed to kilogram-mole. Thus one mole of a substance comprised of molecules is an amount whose mass in grams is numerically equal to its molecular weight, or if comprised of atoms its atomic weight. APPENDIX B CONVERSION BETWEEN CG5 AND SI UNITS Many scientists who deal with magnetic quantities and measure- ments still use the so-called unrationalized electromagnetic units in the centimeter-gram-second system ("cgs" or "cgs-emu" system). Therefore a conversion table is given here. This in no way consti- tutes a recommendation for the use of these units. Multiply the value in cgs units by the number in the fourth column to obtain the value in 51 units. SYMBOL QUANTITY CGS UNIT FACTOR SI UNIT B flux dens.tty gauss 10-4 tesla (G) (T) -3 H field strength oersted 411 Ie 10 ampere/meter (oe ) (Ajm) -8 ~ flux maxwell 10 weber 2 (G'em ) (W or T'm ) 3 X susceptibility emu/em 411 dimensionless -3 3 Xp mass emu/g 411 x 10 m /kg susceptibilty -6 3 Xm molar .,. emu/mole 411 x 10 m /mole susceptibility * One mole of a substance comprised of molecules is an amount whose mass in grams is numerically equal to its molecular weight, or if comprised of atoms its atomic weight. APPENDICES 635 APPENDIX C CALCULATION OF INDUCTANCE P. Carelli, I. Modena, G.L. Romani Istituto di Elettronica dello Stato Solido - C.N.R. Rome, Italy S.J. Williamson New York University New York, New York, U.S.A. Most biomagnetic research is now done with commercial SQUIDs, and the only component of the field-sensing system that the experi- menter is free to design is the detection coil of the flux trans- former. As explained in Chapter 5, the geometry and size of the coil should be chosen to maximize sensitivity for the biomagnetic source of interest while minimizing sensitivity for interfering "noise" sources. In relatively quiet environments, the intrinsic noi.se of the SQUID or SQUID electronics limits the sensitivity of the field-mesuring system, in which case greater sensitivity is achieved by increasing the radius of the pickup coil and number of turns of wire to increase the total magnetic flux of the signal within it. However as the size of the coil increases there is a corresponding degredation in the lateral resolution, which may be- come intolerable if a field pattern having fine detail is to be measured. Consequently a compromise must be made between sensitivi- ty and spatial resolution. Some analysis to help with this choice has been provided by Williamson and Kaufman (1981b) and by Romani et al. (1982e). An additional consideration when designing a coil for maximum sensitivity is the desire to transfer the greatest possible signal energy from the detection coil to the input coil of the the SQUID. To accomplish this, the detection coil's inductance Ld should match the input coil's inductance L .. The match need not De done very l. precisely because a small mis-match (up to ~20') has a comparative- ly small effect (see Eq. 5.3.2). Thus when designing a detection coil for a particular application, it is necessary to calculate its inductance. This appendix shows how to do that .. C.l. MAGNETOMETER The first example of a d.etection coil is a simple pickup coil 636 APPENDICES of one turn of wire (a "magnetometer"). We recall from Section 2.6.2 that the inductance L of a coil specifies the magnetic flux ~ within it when a current I flows in the wire, as given by ~ .. LI. For a single-turn coil of radius "a" and wire radius "c" the induc- tance is -7 L (C.1 ) o 47TxlO a[ Ln( sa/c) - 1. 75] This gives L in the SI unit of the henry (H) when "a" and "c" are o expressed in meters; and "Ln" indicates the natural logarithm (to the base e = 2.171S ... ), which can be found in standard tables. To illustrate the use of this formula, we take typical values of a = 2 em and b .. 0.1 mm and obtain Lo .. 0.141 ~H. The value of inductance is insensitive to the wire radius "b" since the radius enters log- arithmically in the formula. For instance, if "c" is doubled to the size c .. 0.2 mm, L decreases by only 13'. o For a given coil diameter, the total signal flux increases with increasing number of turns N of wire. However the total in- ductance should not appreciably exBeed that of the SQUID'S input coil. If turns of the same radius are wound close to one another, the total inductance increases by more than a factor of N expected if the individual self inductances were simply to add. TRis is be- cause of the mutual inductance between the various pairs of turns: the flux within a given turn is enhanced by a factor of N owing to the contribution of the other turns. Therefore the total ~nductance L of the closely-wound coil is proportional to N 2: c p L .. N 2 L (C.2 ) cpo Accordingly, if the coil of the preceding example is wound with N P .. 3 turns its inductance becomes L a 1.27 ~H. P It should be noted that winding turns close to one another is not the most advantageous way to increase the inductance of a pick- up coil. If instead a separation is allowed between adjacent turns, the mutual inductance is reduced and therefore more turns can be added while still respecting the condition for inductance matching to the SQUID. The additional turns are an advantage because they increase the total signal flux within the coil. We let "s" denote the separation distance (between wire centers) and for convenience introduce as a parameter the "reduced" separation distance x s/2a, where "a" is the coil's radius. The mutual inductance between two coaxial, single-turn coils of equal radii varies with "x" as shown in Fig. C.l. By conSidering the mutual inductances between all pairs of turns for a multiple-turn coil, it is found that the total inductance L can be expressed as p L N 2-a L (C.3 ) P p 0 APPENDICES 637 IU5+-______________~--------------_+--------------~~--------------~ -6 10 E "- I 0 "- ~ Qj u c 0 t; :> -7 "0 E 10 (5 :> "S ~ "0 Q) u :> "0 n:Q) 10- 8 169+-~------------~------------~--------------_+----------~--~ 10-3 10- 1 10 Reduced separation, x Fig. <':.1. Reduced mutual inductance (mutual inductance divided by the coil radius "a") versus reduced separation distance x = s/2a between two coaxial, single-turn coils of equal radius. For a coil whose turns are closely spaced, a = 0, and so the formu- la reduces to that in Eq. C.2; for a coil with turns far apart, a = I, and the total inductance is just the sum of the individual self inductances of the turns. A general expression for a is N -2 P 2 - !n{2 + L 2[N -n] K[(n+I)X]}/!n(Np ) (C.4 ) n=O p 638 APPENDICES The function K(x) is the coupling coefficient of mutual inductance between two turns of reduced separation "x" (Grover 1962), which is approximately K(X) = - 0.08 - 0.139 Ln(x) (C.s) This expression is valid for values of x that range between 10- 3 and 2X10- 2 , covering the range appropriate for most coils used for biomagnetic measurements. As an example of using Eqs. C.3 - C.S we consider the three- turn coil of radius a = 2 cm discusssed earlier, but now we allow a separation s = 1 mm between the adjacent turns. Thus only two terms appear in the sum of Eq. C.4, and we have x - 0.025. Adding terms gives a(3, 0.025) = 0.23. With this calculated a, Eq. C.3 then gives an inductance of 0.99 #H. This is a substantial reduction from the close-wound value of 1.27 ~H. The decrease of inductance by separating turns allows the addition of one more turn without appreciably exceeding the inductance.of the closely-wound case, as- sumed to match the input coil of the SQUID. A four-turn coil of s = 1 mm and a = 2 cm has a = 0.31 and a total inductance of 1.29 ~H. This is virtually the same inductance as for the three-turn closely wound coil. The extra turn permitted by separating turns thus en- hances the coil's sensitivity by 33\ over that of the closely wound geometry.
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