Indian Journal of Engineering & Materials Sciences VoL6, february 1999, pp. 34-42

A novel approach based on impedance spectroscopy for measurement of magnetic penneability of ceramics

Rajesh K Katare', Lakshman Pandey', R K Dwivedib, Om Parkashb• & Devendra Kumarc

• Department of Physics, Rani Durgavati University, Jabalpur 482 001, India

b School of Materials Science and Technology, Institute of Technology, Banaras Hindu University, Varanasi 221005, India

c Department of Ceramic Engineering Institute of Technology Banaras Hindu University, Varanasi 221005, India

Received 24 April /997; accepted /5 April /998

A novel technique, based on impedance spectroscopy, is presented for measurement of permeability of magnetic ceramics. The necessary immittance spectra for equivalent circuit models involving resistive and inductive elements to represent the magnetic behaviour are simulated. A direct comparison of experimental immittance plots with these yields the. most appropriate model representing the magnetic behaviour of the material and in tum its permeability. The method is illustrated using YIG sample.

The magnetic susceptibility of ceramics are generally measured using methods of Gouy, Faraday, change in flux, vibrating sample magnetometer or NMR I. In the Gouy and Faraday methods the sample is positioned in a magnetic field gradient and the value of magnetic susceptibility is obtained by measuring the force exerted on the sample by the field gradient. Specially designed pole-caps are required. In the change-in-flux method the resonance frequency of a LC circuit is first measured. The sh.ift in this frequency produced, when the sample is introduced into the coil, directly yields the value of susceptibility. In the vibrating sample magnetometer set-up the sample situated in an uniform magnetic field is vibrated in a sinusoidal fashion and the susceptibility is measured from the electrical signal it generates in near-by pick up coils. NMR provides a convenient method for determining paramagnetic susceptibilities of substances in dilute solutions also. For any of these methods specific equipment and specially designed magnets are required I . A convenient method of measurement of low field magnetic susceptibility has been by means of ac mutual inductance bridge.2

-5 The value of the

susceptibility of a sample is measured by the change in the mutual inductance of a coil when the sample is placed in it. In order to reduce the cost, several modifications have been proposed such as simulation

* For correspondence

of the mutual inductance electronically6-8 and designs using only operational amplifier9

.

The dielecttic measurements lO in ceramics, on the otherhand, are carried out using a suitable ac bridge such as General Radio bridge. Therefore, usually two set-ups, one for dielectric and the other for magnetic measurements are required for the same material. Due to the easy availability of the impedance analyzer, such as HP 4192 LF A, Solartron etc. It is becoming a routine practice in various laboratories to measure the real and imaginary parts of the impedance, Z* = Z' - j

Z ", j = ~, of the system and carry out a detailed complex impedance analysis to determine various polarization processes present in the materiaI I0

-15

. In this -method, experimentally measured values of the complex impedance are plotted as a function of frequency or on a complex plane (i.e. Z" versus Z '). These plots show special features depending upon the relative contributions from the grain, grain-boundaries and electrode polarization processes present in the ceramic 10,13. For the case where these processes have widely separated time constants distinct semicircular arcs are obtained and when they have time constants close to each other depressed looking semicircular arcs or distorted arcs are obtained 13 . A suitable model is chosen to represent the material-electrode system and the values of the components are obtained from the intercepts of the arcs on the Z '-axis, peaks in the plots and the values of the frequency when Z" shows a

KAT ARE ei ttl.: MEASUREMENT OF MAGNETIC PERMEABILITY OF CERAMICS 35

maximum. The choice of a suitable equivalent circuit is a difficult process as many equivalent circuit models can give rise to the same simulated behaviour. A general practice is to repr.esent one polarization process by one parallel R.C combination. Thus a simple model for and electronic ceramic could be a combination of three parallel RC circuits connected in series, representing grain, grain-boundary and contact electrode processes. The capacitances appearing in the model are direct measures of the permittivity. A description of this Complex Impedance Method of so­called Impedance Spectroscopy, has been r,iven by Mac Donald lO

, Jonscherl8 and Pandey et al.13• 4

Just as the information about the permittivity and polarization processes are obtained from the capacitive

elements .present in the models representing the ac impedance behaviour, information about the permeability and the magnetization process could be extracted by observing the presence of inductive elements. The value of inductive elements thus obtained when mUltiplied by a suitable geometrical factor would in tum yield the value of the permeability. Surprisingly, no work on these lines using impedance spectroscopy seems to have been reported so far to the best of our knowledge. In the present paper, the impedance spectroscopic approach for obtaining the value of the magnetic permeability of ceramics has been described . The necessary models involving resistive and inductive elements and their simulated immittance patterns has been given. A~ a demonstration, the experimental measurement of the permeability ofyittrium iron garnet (YJG) sample using the proposed technique has also been discussed.

Models and Their Simulated Spectra When ' a magnetic material is kept in a solenoidal

coil connected to an ac source, an in-phase loss current flows together with the magnetization current. The total impedance can be written as Z* = Z' - jZ" = jmLo}J*, where"Lo is the inductance of the coil without the sample ancf JJ* = j.l - jj.l' is the complex relative permeability of the material 17

. The existence of the in­phase loss current can be equivalently represented by a resistance connected across the coil as shown in Fig. 1 a. The impedance, Z* = Z' - jZ" , of such a paralIel RL circuit is given by :

Z' = (mL)2 R R2 + (mL)2

... (I)

--6-.

t 0..

~o. .!J

(b)

_w

00 0.2 0.4 o.e o.e ".'~ lo/l,­

(d)

(c;)

Fig. la-A parallel combination of RI and [I ' (b-e)-Nonnalized Z, Y, f.J and Mrnag plots.

'2 Z"= _ mLR

R2 + (mL)2 .. . (2)

It can be see that Z' and Z" satisfy the following relation

... (3)

which is the equation of a circle with centre at (RJ2, 0) on a Z" versus Z' plot. Also as Z' tends to R when m~C1) and m, L, R are all positive quantities, the Z"

versus Z' plot is a semicircular arc in the negative half side (Z' and -Z" quadrant) with intercept of the arc at Z' = R at the high frequency end. Also the maximum value of Iz" 1 occurs at mLIR = I and Z' = RJ2 (see Fig. 1 b). Therefore, if experimentalIy measured values of Z" are plotted as a function of Z' at various frequencies and a semicircular arc is obtained in the negative. quadrant then it indicates that the material is magnetic and can be represented by a paralIei RL

circuit. The values of Rand L can be obtained from the intercept on the Z '-axis and the value of frequency when Z " peaks.

The magnetic behaviour of the system can be described bJ the so-called inter-related immittance functions 1o

•1

•

Impedance Z* = Z' - j Z" Admittance 1'* = Y' + jY" = (Z* ) · 1

36 [NDlAN 1. ENG. MATER. SCI., FEBRUARY 1999

R I . b·I·· I • e atJve pennea I Ity JI = -.-- Z jwLO

... (4)

Magnetic modulus M:ag = (f-I*r' In analogy with the dielectric modulus NJ*=( c*r'

we have defined the magnetic modulus M:ag as equal

to (f-I*r'. The four immittance functions are inversely related and therefore can be conveniently used to highlight low or high frequency responses. Figs I a-e show the parallel RL circuit model and the simulated Z, Y, JI and M lrulig plots. Values of Rand L are indicated by R, and L, respectively.

Another useful equivalent circuit is a series combination of R and L shown In Fig. 2 a and has impedance value, given by :

Z'=R,

Z" = -wL, ... (5)

... (6)

The Z, Y, JI and Mmag plots are given in Figs 2b-e. A material showing such a behaviour can be represented by a series RL combination. The c( nponent values used in this model can be expressed inenns of those used in the parallel combination.

If the electronic ceramic system has two magnetization processes the equivalent circuit model would contain two inductive elements. An electronic ceramic having twG magetization processes can, In

Z'/Rl~ y/)! R,~

0 1·0 0 ·2 ·4 .. ·8 1.0

fn ·2 r% ..

·4 ~

':' ~ · 6

Cb) (e )

t

~'U o ·6 "' ..... -' .....

S · 4 .. .J Co> '" 0

.:::1 , E ·2 :E

0 0 ' .0 0 0 · 2 ·4 ·6 ·8 1.0

I4'Lo/L,-- ,", 'mag a L, /Lo ~

Cd) (~)

Fig. 2a-A series combination of RI and [I ' (b-e) : Normalized Z, Y, Ii and Mmag plots.

general, be modelled by two parallel RL circuits connected in series as shown in Fig. 3a. The Z' and Z" are given by :

Z' = (WL, )2 R, + R,2 + (wL, y

(WL 2 Y R2

Ri ~ (WL 2)2 ... (6)

~-~ Ll L2

(0)

Fig. 3a-A parallel RILl circuit connected in seri es with another parallel R2 L2 circuit.

~ o. • .r OJ.

;::.. NO·6

RZ L2/R2 -r,:l, ~ : 1,5,10,100

(b)

Fig. 3 b--Z "/(RI + R2) versus Z '/(R I + Rz) for R21RI = I L2 I R2 - -= 1,5,10.100. L, I R,

v'x (R,+R2)-

00 0.4 0.8 1.2 1.6 2.0 2.4

'NO.4 a: ~ 0.8 a: ; 1.2 t

w

R2 ~ lr,:1, Ll/Rl :1,5,10,100

(c )

and

Fig. 3c-Y" (RI + R2) versus Y' (RI + R2) for RziRI I and

L, I R, = I 5 10 100. L, I R, ",

... -' 0.6 +

Fig. 3d-jl' LoI(LI + L2) versus jI LoI(LI + L2) for RziRI = I and

L, I R, = I 10 \00 L, I R, " .

KATARE et al.: MEASUREMENT OF MAGNETIC PERMEABILITY OF CERAMICS 37

I ~ 2.0

t- 1-..J

!!. \.2 J' -0;0.

o

'l 0.

OQ!-l-;Of. .. "---'!-O.-;-& -

M' e ~SI).eL,+L2)~

(.)

Fig. 3~M'mag (LI + L2)ILo versus M mag (LI + L2)ILo for R21RI = I

and L, I R, = I 5 10 100. LI I RI ",

~ :", ~~/R; ",5.10,100

(f)

Fig. 3f-Z "/(RI + R2) versus Z '/(RI + R2) for R21RI

~ I R2 --= 1,5,10,100. LI I RI

;., +

a: . .. , j ,

v'.(R,+R2)-2 , , 5

1 f

Fig. 3g-1" (R I + R2) versus l' (RI + Rz) for R21R1 L, I R, _ --= 1,) ,10.100 /'1 I RI

I 1.0

-N O·& ..J

P'lO /(L ,1> L2 ) -

( n )

0.3,

0.3.

Fig. 3h-,LI' /.oI(L } + L2) versus )' LoI(I'1 -I- Id for H2IRI =0.3, I., I H, -- = 1.5.10.100. f' l f RI

2·0 .. ..J 1.6 +

~ ' .2 . ]> 0.& "-. r 0-, :I[

0L-~-f~L--L~~­o 0.4 0.& 1·2 ' .6 2·0

.. 0 IE + o. IF

.:;::.. 0.' N ,

0.6

(M:..all/Lol ' ( Ll+L2)

(d

R L2/ R2 l?,- "0'. '1:i'iR,: 1,5,10,100

cj )

Fig. 3j--Z "/(R} + R2) versus Z 'f(R} + R2) for R2/R} L2 I R2 --= 1,5,10,100. LI / RI

_ 00 .. IE .. 0 .5

IE .:::. 1.0 . ~

~ 1.!

I 20

y'.(A,+A2>-0.5 1.0 1.5 2.0

A2 ' ] fA, r, ··or. ' l/ L, ", 5. '0,100

c. }

Fig. Jk- l " (/?) -I- H2) versus I' (RI -I- Hz) for RzlR}

L, f R, - - = 1.5.1 0. 1 00. LI / RI

?- 0 6 . S 0. 4 .:::.. ! 0 ..

0 2 0 4 0.6 0.6 1.0

I"Lo /(L, +L, }-(I)

0.0 1,

0.0 1.

Fig. 31- 11' L(/(L} -I- 1'2 ) vas li s II f.j(l' 1 -I- f.2) for RlfR}=O.OI. f., f H, _ _ -- - 1.1 O.2(J.)O.1 00. f., f 1<,

38 INDIAN 1. ENG. MATER. SCI., FEBRUARY 1999

Other immittance functions are obtained using Eq. (I). Figs 3b-m show the immittance plots for various values of R2/RJ and (L2/R2)/(LJIRJ) where L2/R2 and LJIRJ are the \ime constants. Figs 3 b, f and j show Z" ';s. Z' plots for different values . of R2/RJ and (L2/R2)/(LJ/RJ). When R2 :::: R J and L2 /R2 » LJ/RJ two distinct semicircular arcs are observed. When RJ/R2 =

3 and (L2/R2)/(L/RJ) » I the Z piot contains a small

arc and a big arc. When RJ/R2 » 1 and (L2/R2)/(L\/RJ) » 1 the Z" versus Z' plot would contain one big arc and a small arc but the tiny arc being masked by the big arc. The Z" versus Z' plot is almost a single semicircular ar.c. The Y plot, shown in Fig. 3k for this case, is a vertical straight line. These two plots have appearances similar to those for a single parallel RL c ircuit (see Fig. I b, c.) The J..i and MIru!.g plots (Fig. 31, m), however, show two distinct arcs indicating the presence of two different magnetization processes . Therefore, if Z' and Z" are measured and experimental Z" vs Z' plot is a single semicircular arc in the negative quadrant (Z' positive, Z" hegative) then one would be prompted to assume the equivalent ci rcuit to be a parallel RL combination indicating the presence of only one process. But the above discussion indicates that there might still be two processes where RJ »R2• Therefore, J..i and Mmag plots also must be looked at together with Z and Y plots for deciding the equivalent circuit model. Usually Z or Yand J..i or MIru!.g

suffice. Another circuit model which has been found to be

very useful contains a parallel R\LJ combination connected in series to resistance R2 and inductor L2 as shown in Fig. 4 a. The Z' and Z" are given by :

1--.JW\I\r--'lJH'- •

Ie)

Fig. 4a--A parallel RI LI combination connected in series with resistances R2 and inductor L2.

-<- . , I,~.,,,,,Z,S.1J) ~)

. Z ' Z ' I L, / R, 0 2 5 0 Flg. 4b---c 'IR2 versus IR2 for R2IRI=0.1, L I R- = .1,1, , ,I .

~ • . ,,~/ . " ,5,10 lf1 1 1

(e)

I ' 1

Fig. 4c--Y" R2 versus Y'R2 for R21RI =0.1 , L, I R, =0.1,5,10. LJ R,

t 1 0 .• .......

,,'1.01 I l, .l2) - ­

Id)

ie)

Fig. 4e---M'mag L21Lo versus M mag LziLo for R21RI 0. 1 and (mL )2 R.

Z'= J J +R 2 ( )2 2 RJ + mL J

. .. (8) L, 1 R, =0.1,1,5,10,30. LI 1 RI

KA TARE et al. : MEASUREMENT OF MAGNETIC PERMEABILITY OF CERAMICS 39

0. ~.I

(f)

Fig. 4f-Z "IR2 versus Z 'IR2 for R/ RI = I,

~--0.01,0.1,0.3,0.5,1 .0,10.0. ~I RI

F· 4 L 1 R Ig. g---Y" R2 versus r R2 for R21RI = I _ 2 __ 2 =0.0115. , L, 1 R, "

1 s ~ 0. • ..

.::::. Q4 ~ .~ 0.2

~ 0.1 .... .s 0.4

_f 0.2 :I:

A l,/At ~. I. L."/lr., • -()1. I, S

I I I

I ,

Table I-Values of the immittance functions Z, Y, J.I, and M~ as (j)~0 and ~oo

Models OJ Z

Z' Z" Parallel RI LI combination [Fig. I (a) ] 0 0 0

00 RI 0 Series RI LI combination [Fig. 2 (a) ] 0 RI 0

00 RI -00

Parallel RI LI in series 0 0 0

with another parallel

R2 L2 [Fig. 3 (a) ] 00 RI+R2 0

Parallel RI LI in series 0 R2 0

with another series R2 L2 [Fig. 4 (a) ] 00 RI+R2 - 00

Z" { OJLI RI2 1 = 2 +OJL2 RI2 + (OJL;)

... (9)

The other three immittance functions, viz., Y, f..I and Mmag are obtained using Eq. (4). The behaviour of the immittance values for different ratios R21RI and

Y I:!:. Mmas r r' jJ )..I." Mmag M'mag

I1RI -00 LIllo 0 LolLI 0 I1RI 0 0 0 Lolli 00

I /RI 0 LIllo 00 0 0 0 0 LIllo 0 Lolli 0

I ( 1.1 ) 2 - 00 ...!.. (L , +1'2) 0 -.!:L 0 RIl.l +1.2 1.0 I. , +1.2

I ( 1.2 r + Ii; 1.1 +1.2

I 0 0 0 1.,,( II, r 00 R, +R2 1,\ [(I + R2

1'0 ( /12 r + 1.2 II , + 112

I /R2 0 ...!... (!~ , + 1. 2) 00 0 0 1.0

0 0 L21Lo 0 LoIL2 0

(L21R2)/(LI/R,) are shown in Figs 4b-i. Many other circuit models are possible and are given elsewhere'8 . A preliminary report was presented earlier'9. The limiting values of the immittances as ~O and OJ ~oo for all the four models discussed are summarized in Table I .

40 INDIAN 1. ENG MATER. SCI., FEBRUARY 1999

Measurement on Yig Sample and Analysis of Data A solenoidal coil of 32 mm length, 15 mm dia and

containing 40 turns of 24 SWG copper wire was connected to the HP 4192 LF A Impedance Analyzer and real and imaginary parts of the impedance, Z I and Z ", were meas.ured as a function of frequency. Then a 4 mm thick and about 15 mm dia YIG pellet was inserted in the coil coaxially and Z I and Z" were again measured.

Fig. 5 b shows the Z and Mmag plots for the empty coil in the frequency range 100 Hz to 10kHz at room temperature. The closed circles show the experimental values. The continuous line in Fig. 5 a is a vertical straight line passing through the experimental points . The smooth curve in Fig. 5b is a semicirc le passing through the experimental points with its centre on the real axis. Fig. 6 shows the immittance values when the sample is housed in the coil.

A comparison of the Z" vs Z I and M'mag vs M lIlag

plots of Fig. 5 for the empty coil with the plots simu lated in the previous section indicates that these plots are similar to those simulated for series RL circuit and shown in Fig. 2. This indicates that such an impedance behaviour can be well represented by a

z'(n) 0·1 0·2 0 ·3

0·'

0·3 ~

Fig. 5a--Z I. versus Z I plot for the empty coil. [(e) experimental values, (-) vertical straight line through the experimental points)] .

.103

100~--------~-----------,

80

20 100 60 80 100 120 1100 160 180. 10'

("'~/Lo)

Fig. 5b--M'mag /Lo versus M mag /Lo for the empty coil. [(e) experimental values, (-) a semicircle through the experimental points with its centre at (86x 103, 0)].

series RL circuit with R = 0.17 nand L = 5.8 pH obtained from the intercepts. This is close to the value of L estimated from the expression L (uH) = a2n2/(9a + lOb) where a and b are respectively the radius and length in inches and n is the number of turns in the coil2o

.

The Z and p plots for the case when the YIG sample is inside the coil are shown in Figs 6 and 7. A quick look at the simulated patterns given in the last section indicates that these experimental pl ots resemble with those of Fig. 4 b, d and thus an equivalent c ircui t of Fig. 4 a, i.e., one parallel RILl connected in series wi th

Z'( n.)

2 1 4 5 6 7 e

2

q 3 - 1kHz t 4 ~-l N • .-.5kHl

5

6 10kHZ-.! 7

Fig. 6a--Z I . versus Z I plot for the co il having Y IG pellet inside it.

[(e) experimental values, (- ) calculated values ---corresponding to the model shown in Fig. 4 (a) with RI = 5.8 0 , LI = 5.9x I0-4 H. R2 = 0.2 0 and L2 = 7. 7x I 0.5 H obtained from CNLS fitting)]_

6

5

c:! 4

N 3

2

0 4

109'01 ( HZ)

2

-c:! 3 --'

N 4

5

6

7

Fig. 6b--Z I versus log F and Z I . versus log F plots for the co il

having YIG pellet. [(e) experimental values, ---calculated val ues corresponding to the model shown in Fig. 4(a) with RI = 5.8 O. '- I = 5.9xI0-4 H, R2 = 0.2 0 and L2 = 7.7x I0·5 H obtained from C LS fitting)].

KA TARE et ar : MEASUREMENT OF MAGNETIC PERMEABILITY OF CERAMICS 41

-w

~~~--~--~-7~--~--~6--~~8

Ii Lox 10~

Fig. 7-"" Lo versus'" Lo plot for the coil having YIG pellet inside it. [(e) experimental values, calculated values corresponding to the model shown in Fig. 4 (a) with RI = 5.8 n, LI = 5.9x I0-4 H, R2 =

0.2 n and L2 = 7.7x I0·5 H obtained from CNLS fitting)].

R2 and L2 represents the data well where L2/ LI ~ 0.1, RI ~ 0.2 n, (LI + L2)/Lo ~ 7.2. A complex non-linear

LS) fi · 10 21 22 • h I least squares (CN Ittmg " usmg t ese va ues as initial guesses yielded the values as RI = 5.8 n, LI = 5.9x 10- 4 H, R2 = 0.2 nand L2 = 7.7x 10 - 5 H subjec.t to 20% error. The smooth curves drawn in Fig. 6a, b and Fig. 7 correspond to these fitted values.

The above analysis indicates that the coil without the YIG sample behaved like a series RL circuit but when the Y]G sample was inserted in the coil it behaved like one parallel RILl connected to another series R2L2• This can be explained as follows : when the YIG sample is inserted in the coil the portion of the coil covering the Y]G can be considered to be a separate inductor with . a value equal to say, L I ,

connected in series with another coil of inductance L2 corresponding to the remaining portion of the coil. Since LI contains YIG, its inductance would be equal to the inductance of that portion of the empty coil multiplied by the relative permeability of YIG. The thickness of the YIG is such that it covers about 5 turns of the coil. This portion of the empty coil would have an inductance of 0.6 ,uH. The complex Impedance Analys is carried out above leads to a value LI = 5.9x I 0---4 H for this when the YIG sample was inside it The ratio of these is 983 and must be the value of the permeability of YIG sample. A lso, the value of the inductance corresponding to the remaining portion of the coil comes out to be L2 = 7.7x 10-5 H which is larger than the i'nd uctance of the empty coi l. This is not unreasonable as the increased concentrati on of flux lines would occur upto some distance away from the sample. It is worth mentioning that the Z or ,u plots for the YIG and coil system contain a clear arc and corresponds to a parallel RILl connected in series

to another R2 and L2 where Rio LI, R2 and L2 are lumped elements. It means that LI and L2 are independent of frequency which in tum, indicates that the value 983 obtained for relative permeabilitY is the low frequency value. This also agrees with the fact that measurements were carried out below 10kHz only. By a suitable modification of the coil assembly the high frequency value of the permeability might be proved. The value 983 of relative permeability of YIG agrees well with the value 1000 obtained using other methods23

.

As discussed above the RILl combination in the equivalent circuit model representing the YTG coil system represents the portion of the coi) covering the sample whereas R2L2 repre!)ents the remaining portion. The value 0.25 for R2 is close to the value of resistance obtained when no sample was inserted. However, RI is 5.8 n which is about 20 times larger than the resistance associated with the whole coil. This resistance m<ry be attributed to the resistance associated with the domain reorientation in the YTG

I 24 - 26 samp e .

Conclusions The simulated immittance spectra of equivalent

circuit models involving resistive and inductive elements have been presented here and that may be used to represent magnetization process in materials. The experimental results of complex impedance measurements on a YIG sample placed in a coil are analyzed. The complex impedance analysis carried out for the data on YIG sample y ields the value of relative permeability of YIG. It is proposed that this method may be widely used for permeability measurement for magnetic materials using the same impedance analyzer installed for dielectric measurements. This would, in turn, lead to a very compact and cost effective laboratory arrangement also.

Acknowledgements The necessary CNLS programme, available as

IMPSPEC,BAS with the authors, was developed by one of the authors (LP) using the facilities provided at the Computer Centre, Rani Durgavati University,

Jabalpur. The author is thankful to Dr. KC. Deomurari and R.K. Anand for their help.

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42 INDIAN 1. ENG. MATER. SCI., FEBRUARY 1999

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