• 検索結果がありません。

Comments on the Anti-Resonance Method to Measure the Circuit Con-stants of a Coil Used as a Sensor of an Induction Magnetometer

N/A
N/A
Protected

Academic year: 2021

シェア "Comments on the Anti-Resonance Method to Measure the Circuit Con-stants of a Coil Used as a Sensor of an Induction Magnetometer"

Copied!
8
0
0

読み込み中.... (全文を見る)

全文

(1)

Comments on the Anti-Resonance Method to

Measure the Circuit Con-stants of a Coil Used

as a Sensor of an Induction Magnetometer

著者

Ueda Hajime, Watanabe Tomiya

雑誌名

Science reports of the Tohoku University. Ser.

5, Geophysics

22

3-4

ページ

129-136

発行年

1975-04

URL

http://hdl.handle.net/10097/44723

(2)

Sci. Rep. TOhoku Univ., Ser. 5, Geophysics, Vol. 22, Nos. 3-4, pp. 129-435, 1975.

Comments on the Anti—Resonance Method

the Circuit Constants of a Coil Used as

of an Induction Magnetometer

Technical Notes

HAJIME

UEDA

Department

of Geophysics

and Astronomy

University

of British

Columbia

Vancouver,

Canada

and

TOMIYA

WATANABE*

Onagawa

Magnetic

Observatory

Faculty

of Science

TOhoku

University

Sendai,

Japan

(Received

January 10, 1975)

to Measure a Sensor

Abstract: In this notes, comments are made on the theory of the antiresonance

method which is often employed to measure circuit constants of sensors of an tion magnetometer. A few remarks on the practical side of this method are also

made. It was found that the anti-resonance method is rather limited for a

ment of self-inductance. An alternative to the anti-resonance method is proposed also

in this regard.

1. Introduction

For a design of an induction magnetometer, it is basic to precisely measure the equivalent circuit constants of the sensor, viz, the d.c. resistance, R, self-inductance, L, and the capacitance, C. It is not difficult to measure the reistance, R. It can be measured, using a voltmeter and a d.c. power source. It is also easy to find a reliable ohmmeter on market. However, it is safer to avoid a meter using an a.c. power source.

To measure L and C is harder. According to our experience, the anti-resonance

method is powerful for these purposes. According to this method, the resonance frequency 27r/ i/LC can be measured precisely, as far as the Q factor of the R-L-C series cricuit is sufficiently large. According to the antiresonance method, it is also possible to measure the capacitance C separately, though less accurate. The self-inductance can be measured more precisely by the step-function method than by the antiresonance method. It is not necessarily easy to measure L and C using an impedance bridge.

The principles of the anti-resonance method is described by Campbell (1969). In this paper, we will first present several new theoretical results to complement the

* On leave from the University of British Columbia

(3)

130 H. UEDA and T. WATANABE

Campbell's description. Secondly, the principle of the step-function method will be

explained. And, finally, we will make a few remarks on the practical side of the anti-resonance method.

2. Theory of the Anti-resonance Method

In the anti-resonance method, the coil, a signal generator and a load resistance , Rg, are connected in series, and the voltage across the load resistance is measured as a function of frequency of the signal. The equivalent circuit of this system is shown in Fig. 1. The voltage across the load resistance, Vg, is given by

V— g Z E ,(1) sd-Rg-FR,

jcuL

Fig. 1. Experimental set-up for t where E and R, are the electro-motiv force a generator and ZS the impedance of the senso.

the output impedance much smaller than the relation holds in good approximation:

Vg = Rg 1

We assume that E changes sinusoidally wi

where w is related to frequency f by

= 27r f

The impedance of the sensor is given as follov . 1 1 (i)C (-R Zs= _ - COIL R g ,g - J E=E0ejwt

t--up for he anti-resonance method.

e force and the output impedance of the signal he sensor. We choose the load resistance and

han the series impedance. Then, the following

(2) idally with time,

E0e-iwe (3) w = f (4) as follows: . 1 (R+ j cvL) 1(5) \

R+

j(wL—

wCI

(4)

CONSTANTS OF A SENSOR OF AN INDUCTION MAGNETOMETER 131

This can be rewritten into the following form, omitting the factor el':

+

Z u21161432)} S — R

it40+1

2-2+

12)u2exp[—i

Tan-1

u (u2—

1132

(.(6)

where

for = reonance frequency (7)

u colco, (8)

and

=

LR C . (9) co„y

The parameter 13 is the inverse of the Q factor of the R-L-C series circuit (Takahashi, 1970). The Q values of three coils which we have examined fall in a range from about 50 to 3,000. Refer to Table 1 for more details. The parameter /3 is also related to the damping coefficient y of the R-L circuit:

y RIL (10)

and

fi = Yiwr. (11)

The anti-resonance frequency coa is obtained, requiring that the impedance Z, be minimum at that frequency:

ua2(wata),)2 =-1/1 +2,82 _132 (12)

The right-hand side of this equation has to be positive in order that an anti-resonance frequency can exist. The condition for that is

13 < (17 2 +1)"2 (13)

The right-hand side of (12) can never be larger than 1 for ,8>0. Therefore, the

anti-resonance frequency can never be higher than the anti-resonance frequency. If /32<1,

u. coakor T 1—

2164. (14)

This shows that the anti-resonance frequency is almost equal to the resonance

frequency. Under the same condition, we also have, at u=ua,

R L

Z ,1(15)

132 RC

The anti-resonance frequency can also be found, performing an x-y plot of Vg and E on an oscilloscope. The electromotive force E can be measured, monitoring the voltage between the output terminals of the signal generator. As seen from the phase factor of (6), the plot will be elliptically polarized in general but will be linear at the following frequency, col:

(5)

132 H. UEDA and T. WATANABE

H 132-<1, coi is almost equal to coa and o.),.. 3. 45° Line Method

It is possible to measure L and C separately, making use of the experimental set-up for the anti-resonance method. To measure C, we use frequencies much higher than the anti-resonance frequency. Then, the sensor impedance is given approximately as follows: R 1 = (17) 1 Zsl P "co C .

This is derived from (6). This is also clear from Fig. 1. For sufficiently higher frequencies, the impedance of the capacitance is much smaller than that due to wL and R. The voltage Vg becomes proporitional to frequency if the amplitude of the electro-motive force is held constant. This can be checked plotting the voltage Vg versus frequency on a log-log paper: The plot will be a straight line inclined 45° upwards to the right if frequency is taken on the abscissa.

A 45° line technique can also be employed for a measurement of L. If there is a range of frequencies <co, such that coL is much larger than R, the plot of Vg versus frequency will be a 45° line downwards to the right, taking frequency on the abscissa. The frequency range such that IZ, 1:-,--7coL is narrower, if R is larger and co, is lower. In each of the three coils listed in Table 1, the frequency range of a 45° line is narrower compared to the case for the capacitance. Consequently, L is determined less accurately than C, by the 45° line method. For frequencies sufficiently lower than the anti-resonance frequency, the sensor impedance is approximately given by

Z, k- R+ jtoL . (18)

Accordingly, the ratio between the mean square value of E and that of Vg becomes a quadratic function of frequency:

(E2) _ R2+ co2L2 . (19)

( Vg2) Rg2

If we plot the quantity of the left-hand side on the ordinate and the square of frequency on the abscissa, the plot will be a straight line and its gradient given by (LIR02.

4. Step-Function Method

The self-inductance can be measured much more easily, using a rectangular wave. Suppose that the electro-motive force from the signal generator changes abruptly from 0 to E0. Then, Vg increases exponentially and attains half the final value, (RgfR) E, in a time T given by

Li

T =R-In 2 . (20)

Therefore, L can be determined, measuring T. Of course, the period of the rectangular wave has to be taken much longer than T.

(6)

CONSTANTS OF A

Table

SENSOR OF AN INDUCTION MAGNETOMETER

1. Physical constants of sensor coils

133 -- ___ Made Characteristics Type Core material Core cross-section Core length Weight Turn number, N D.C. resistance, R Self-inductance, L Capacitance, C

Resonance frequency, co, Resonance frequency, fr Damping coefficient, y i6

Q(=1113)

Univ. of Tokyo

High-p metal core Permalloy sheets laminated Rectangular, 2 cm x 2 cm 1 meter 12 kg 50,000 158 S2 170 H 0.80 nF 2.7 x 103 radisec 430 Hz 0.93 sec-1 3.43x 10-4 2.91 x 103 Univ.. of B.C.

High-p metal core Mumetal sheets laminated Circular, Diameter= 1.25" 6 feet 18 kg 83,000 3.7 kS2 6.4 kH 0.81 nF 4.4 x 102 radisec 70 Hz 0.58 sec-1 1.32x 10-3 7.57 x 102 Victoria Observatory Air-core Circular, Diameter = 1.27 m 175 kg 16,000 5.45 kS2 530 H 6.7 nF 5.3 x 102 radisec 84 Hz 10.3 sec-1 1.94 x 10-2 5.15x 10 •

In this method, it is assumed that no current flows through the capacitance C. The assumption is justifiable, if the following two conditions are satisfied. Firstly, the load resistance Rg be much smaller than the coil resistance R:

a RIRg> 1 , (21)

where Rg should include the output impedance of the signal generator. Secondly,

the time scale T given by (20) is much larger than CRg. The time scale T is equal to the inverse of the damping coefficient y in (10) except for the factor In 2 which is about 0.7. Therefore, the second condition can be rewritten as follows:

lfy CRg (22)

which can be modified further,

aco,2> y2. (23)

Referring to Fig. 1, let II and I, be currents through the R-L path and the capacitance,

respectively. It is easily shown that 1-1 and /2 are governed by the following

simultaneous differential equations: d21.

dtzl + y dt =(.0,24(24) and

di, . di, aw,,2 a dE

dtdt 12 -R dt(25)

Subject to the initial conditions i = 0 ,d

dt 0 and /2 = 0 , at t = 0 (26)

(7)

134 H. UEDA and T. WATANAAE

0 t < 0

E (27)

E,(= const.) t > 0 ,

the differential equations (24) and (25) yield the solutions which are written approxim-ately as follows under the conditions (21) and (23):

, 2 2

e-Ou'r±i19')

(28)

R cu.°, and

—E0(a-- \

"

2)e-(2/7)‘ '

(29)

2

Therefore, the voltage across the load resistance is given by

Vg— Rg(Ii+I2) = R (1—e-Y`d-ae-(°wri/7)t) (30)

The third term inside the bracket on the right-hand side decays much faster than the second term due to the condition (23). Omitting the third term, the voltage Vg is entirely due to the current through the R-L path.

5. Laboratory Experiments

The values of L and C of the University of Tokyo coil in Table 1 were

determined by the 45° line method. The resonance frequency calculated theoretically

using these values of L and C is 430 Hz. On the other hand, the anti-resonance

frequency obtained by measuring the minimum voltage method is 410 Hz which agrees with the above-mentioned value within a difference of about 5%. Incidentally, the values of L and C obtained by an impedance bridge deviate about 10% from those obtained by the 45° line method. The impedance bridge has a built-in signal generator of a frequency of 1 kHz. At this frequency, the impedances due to L and C, viz., coL and licoC are not much different from each other and, therefore, they are likely to influence one another in the measurements. This is presumably the reason why the measurements by the bridge do not agree with those by the 45° line method.

With respect to the Universirty of B.C. coil mentioned in Table 1, C was measured with a satisfactory precision by the 45° line method, although, L could not be measured accurately by this method. However, the linearity against the square of frequency in (19) was satisfactory and the value of L was determined to be about 6,500

Henry. On the other hand, the step-function method gave L=6,400 Henry. The

difference between the two is less than 2%. The resonance frequency in the table

was theoretically calculated, using these measured values for C and L. The

anti-resonance frequency was found to fall between 70 Hz and 75 Hz due to the minimum

voltage method. However, the minimum was much shallower than in the case of the

University of Tokyo coil and consequently, the anti-resonance frequency could not be measured precisely. The reason is presumably that the Q value of the University of

(8)

CONSTANTS OF A SENSOR OF AN INDUCTION MAGNETOMETER 135

In summary, we think that the step-function method is most reliable to measure L. The 45° line method works for a measurement of C. The anti-resonance method measuring the minimum voltage is more reliable if the Q value is higher. It is useful to compare the measured anti-resonance frequency with the resonance frequency calculated based on the measured values of L and C.

Acknowledgements: The authors sincerely thank Dr. R.N. Michkofsky and Mr . D. Sch-reiber at the University of British Columbia for their assistance in carrying out several experiments for this research. The authors are indebted also to Dr . T. Saito, Director of the Onagawa Magnetic Observatory, Tohoku University who has encouraged publication of this paper. The author also appreciate help of Dr. T. Sakurai of the same observatory in preparing the manuscript. The research was supported by the National Research Council of Canada, and also in part by the Ministry of Education of Japan.

References

Campbell, W.H., 1969: Induction loop antennas for geomagnetic field variation measurements,

Technical Report, ERL 123-ESL 6, Environmental Science Services Administration,

Boulder, Colorado, U.S.A.

参照

関連したドキュメント

We show that a discrete fixed point theorem of Eilenberg is equivalent to the restriction of the contraction principle to the class of non-Archimedean bounded metric spaces.. We

As explained above, the main step is to reduce the problem of estimating the prob- ability of δ − layers to estimating the probability of wasted δ − excursions. It is easy to see

Keywords: continuous time random walk, Brownian motion, collision time, skew Young tableaux, tandem queue.. AMS 2000 Subject Classification: Primary:

Related to this, we examine the modular theory for positive projections from a von Neumann algebra onto a Jordan image of another von Neumann alge- bra, and use such projections

“rough” kernels. For further details, we refer the reader to [21]. Here we note one particular application.. Here we consider two important results: the multiplier theorems

To derive a weak formulation of (1.1)–(1.8), we first assume that the functions v, p, θ and c are a classical solution of our problem. 33]) and substitute the Neumann boundary

This paper presents an investigation into the mechanics of this specific problem and develops an analytical approach that accounts for the effects of geometrical and material data on

The time-frequency integrals and the two-dimensional stationary phase method are applied to study the electromagnetic waves radiated by moving modulated sources in dispersive media..