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A method for the evaluation of microwave dielectric and magnetic parameters using rectangular cavity perturbation technique
0038-1098/89 $3.00 + .00 Pergamon Press plc
Solid State Communications, Vol. 70, No. 8, pp. 847-850, 1989. Printed in Great Britain.
A M E T H O D FOR THE EVALUATION OF MICROWAVE D I E L E C T R I C AND MAGNETIC PARAMETERS USING R E C T A N G U L A R CAVITY P E R T U R B A T I O N T E C H N I Q U E V.R.K. Murthy and R. Raman Department of Physics, Indian Institute of Technology, Madras - 600 036, India
(Recetved 15 June 1988; in final form 6 February 1989 by T. Tsuzukt) To derive the expression for complex dielectric and magnetic susceptibihties, an easy and alternate approach is proposed. This analysis is based on the assumption that the cavity is equivalent to an isolated lumped resonant circuit. The results are identical with standard expressions. Results of a few samples are also tabulated. INTRODUCTION
L. -
CAVITY perturbation methods have been widely used to characterize the dielectric, magnetic and superconducting materials, at microwave frequencies [1-4]. The dispersive and dissipative terms of the materials are directly related to the change in the resonant frequency and quality factor of the cavity from its respective empty cavity values. In order to understand the overall properties of a material, the solution of Maxwell's equations in the particular configuration must be understood. The standard perturbation analysis of this problem is found elsewhere [5, 6]. As the resonant cavity is microwave analogue to an isolated lumped resonant circuit [7], the present paper goes on to arrive at the standard expressions by calculating the value of capacitance and inductance for a rectangular cavity excited to ~ts fundamental mode.
From the solution of Maxwell's electromagnetic equations, for a rectangular with broad &mention a, narrow dimension b and length l, along X, Y and Z directions respectively, the orientation of respective electric and magnetic field components can be deduced as E~., H~ and/4: (where as E=, Ex and H v are zero). In this case the assumption is that the cavity is a parallel resonant circuit with a capacitance C~ corresponding to average value of E). and parallel combination of two inductances L~ and L, corresponding to the average value of respective magnetic fields He and H=. From the low frequency lumped circuit elements analogy, C) -
coal b
-
(2)
i~oN2alb L~: =
a2 q- 12
(3)
where l = 2g/2 for TE10~ mode and 'N' has usual meaning similar to the number of turns in a solenoid inductor. If N is replaced by 1/Tt, the above assumptions result to, f0 -
1
2rc~>
(4)
where f0 is the resonant frequency of the cavity. In the case of TEI0o mode l = n2g/2 and C). becomes nC( and L~: becomes L'~,./n, so that the resonant condition 1
2rc~
IS always satisfied, where C,~ and L~, are equal to C,. in equation (1) and L~, in equation (3) respectively.
(1)
]~o N 2 ]b L~:
l
From equation (2) the resultant inductance is written as,
f-
GENERALTHEORY
#oNZab
a
847
DIELECTRIC CONSTANT e' AND LOSS e" A sample of complex dielectric susceptibility Xe = X/ -jX~" (equals to e' - j g " ) is kept at the maximum electric field location of the cavity. A thin sample is taken such that the electric field will be uniform throughout the volume of the sample. In order to simplify the problem, the sample is assumed to be cylindrical or rectangular with umform cross sectional area S and length is greater than or equal to b so that it will occupy the entire narrow dimension of the cavity. After the introduction of the sample the empty resonant frequency and Q factor alter, due to the change in the overall capacitance and conductance of
848
EVALUATION OF MICROWAVE PARAMETERS
the cavity, without perturbing the inductance. The new value of capacitance is given as, c,
=
~o(la -- S) eoe*S + b b
(5)
Vol. 70, No. 8
cavity with and without samples. Here also taking f0-fl [ 1 2Qt
1 ] 200
2V~ v~ x~".
-
(13)
The distribution of the electric field for TEl0, mode can be derived from the solution of Maxwell's electromagnetic equation as,
Equations (11) and (13) are of the standard form for the expressions for dielectric parameters as reported in [61.
Ey = E m a x S i n ( ~ ) s i n ( 2 X\ z ~2g ]"
(6)
MAGNETIC PERMEABILITY #' AND LOSS/~"
(7)
In this case the sample is introduced at maximum magnetic field location. The sample is aligned along maximum H x for which H~ and Ey are least perturbed. The samples of complex magnetic susceptibility Xm = X,~ -- jX/,," having uniform cross sectional area 'S' changes Lx to Lxl as
From equation (6), the expression for power is P = Pma*sin2
sin2 \ 2,
"
From equation (7) it is clear that the average electric power is one fourth of the maximum power. Since, the sample is introduced at maximum field position, the effective interaction is four times that of its average value. Hence, the sample area S has to be replaced by 4S. If the sample is kept at some other position say zero electric field location, then S will be zero times the cross sectional area and correspondingly there would not be any change in the characteristics of the cavity. Now equation (5) is changed as,
C, -
e,o(la - 4S) + 4~e*__.~S b b
(8)
Equation (8) can be simplified by comparing with equation (1) as, C, = Co 1 + - ~ c Xe
(9)
where V~ = Sb and V~ = lab are the volume of the sample and cavity respectively and Co = Cy. The complex resonant frequencies [8] with and without samples are f~ and f0 and are given by Ct oc 1/f~2 and Co oc lifoz. Taking the real part of equation (9)
fg - f ? 4V#e' -(10) f? For very small values of V~,fo = fl and hence equation (10) can be simplified as,
i.to(lb -- 4S) 4~/u*S + - ~2a 7r2a
Lx~ -
(14)
Here also, replacement of S by 4S is inducted
Ll
#o Ib + 4SXm + -~
(15)
l L0
n2[a l] #0 ~ + a-b
(16)
where L 1 and L0 are the respective inductances of the cavity. The real part of the inductance Re (L) is proportional to (1/f 2) and the ratio Im (L)/Re (L) is equal to (l/Q), wherefis resonant frequency and Q is the quality factor of the resonant cavity. On simplification of equations (15) and (16) 4a 2 ~ X,, g I
=
g 0
1+
(a 2 + 12) + 4
(12Vs~
Xm
.
(17)
For V~ >> V~, f0 -~ ft, the term 4(VJV~)Xm can be neglected. 4a 2 V~ L0
-
+ t2 .
(18)
af 2V~ Xe' (11) f, where Afis the shift in the resonant frequency denoted by (f0 - f~)- Similarly separation of imaginary terms yield to,
Equation (18) is identical to the equation (9) for the case of capacitative term, and separation of real and imaginary terms results to the similar expressions as,
I1 Q,
and
1 ]f02 4V~ Q0_]f2 = ~ X~"
X~=KV
1 V~( ~ l f ) ~
(19)
(12)
L_
where Qj and Qo are the respective quality factors of
S~r -- g ~
2QI
200
Vol. 70, No. 8
EVALUATION OF MICROWAVE PARAMETERS
849
Table 1. Dielectric constant e" values of a few samples using three different expressions and the corresponding dielectric loss e" given by equation (12) Material
Diameter in cm
Frequency f m MHz
Teflon Plexiglass Quartz Rexolite No. 422 Stycast
0.285 0.240 0.196 0.290
9111 9109 9101 9093
0.264
8765
e' from equation (10)
e' from equation (l l)
2.06 2.56 3.74 2.43
e' from equation (21)
2.06 2.56 3.73 2.43
12.2
2.06 2.56 3.74 2.43
11.96
12.2
Ref. e'
e"
2.08* 2.59* 3.78* 2.5t
0.0008 0.017 0.0004 0.002 0.025
* R.A. von Hippel, Dielectrtc Materials and its Applications, Chapman Hall Ltd., London (1954). t Data provided by the manufacturer, C. Lec plastic Inc., 1201 Chestnut Street, Philadelphia. 19197, USA. where K = (2a2)/(a2 + 12). Details of the derivation of equations (19) and (20) are given in Appendix. Equations (19) and (20) are exactly identical to the results of Artman's [5] perturbation analysis.
Professor T. Fujimura of Tohoku University, Japan, for his instructive comments and suggestions, especially for letting us become familiar with (17) and its derivation. This work was improved by his instructions greatly.
EXPERIMENTAL PARTS
REFERENCES
The experimental arrangement for the dielectric parameter studies consists of X-13 Klystron whose reflector voltage is modulated. The isolator, attenuator, shde screw tuners, Majic Tree and Crystal detector forming a standard test bench follows the Klystron. The rectangular cavity of length 11.75 cm and mode TE~05 with the resonant frequency 9157 MHz is connected to the bench. The sample is kept at the centre of cavity where the maximum electric field clearly exists for odd mode of excitation. The frequency is measured by H.P. frequency counter. For recording the cavity response, a personal computer is interfaced to the detector.
1. 2. 3. 4. 5. 6. 7. 8.
L.J. Buranov & I.F. Schegolev, Prib. Tek. Eksp. 2, 171 (1971). H. Kobayashi & Ogawa Selya, Japan J. Appl. Phys. 10, 345 (1971). N.P. Ong, J. Appl. Phys. 48, 2935 (1977). B. Viswanathan, R. Raman, N.S. Raman & V.R.K. Murthy, Solid State Commun. 66, 409 (1988). J.O. Artman & P.E. Tannenwald, J. Appl. Phys. 26, 1124 (1955). B. Lax & K.J. Button, Microwave Ferrttes and Ferrimagnetws, McGraw-Hill, New York (1962). J.C. Slater, Rev. Mod. Phys. 18, 474 (1946). R.F. Soohoo, Microwave Magnetics, Harper and Row, New York, 1985.
CONCLUSIONS Thus an alternate and simple approach has been discussed for the evaluation of material parameters at microwave frequencies using rectangular cavity technique. Moreover, from the above discussed analysis, the expression for dielectric constant e' of a low loss sample in TE~0n cavity of length I can be written as,
(
[(C') 2 e L --
-~
+
a212 ~ l n E a 2 + l 2 12-+n2a2J_]
Equation (18) in the above text is written as
[
L~ = L 0 1 +
2K~xo]_]
(I)
where 2a 2
K = a 2 + 12 and the complex inductances are
1
=
APPENDIX
4alS (21)
where C' is the free space velocity of electromagnetic waves and the other terms bear the usual meaning. Results of a few samples are tabulated in Table 1 using equation (10), (11), (12) and (21) along with the available standard data.
Acknowledgement - T h e authors would hke to thank
L, = L', - jL~' and
Lo = L~ -- jL'o' separating the real part of equation (I), one can write as
L; -- L~ -
2KV~
(L~X/,, -- L'o'X/,,')
(II)
850
E V A L U A T I O N OF M I C R O W A V E P A R A M E T E R S
After approximating L~ - L6 L~
-
LoX/,; to
zero, eq. (II) becomes
2KV~ - X,~. V~
from equation (III), for X,~ in equation (IV), one can write equation (IV) as
(III) L~' - Lo' -
2KV~
~
/
L'oX/: + ---7 L, L~ -
Lo
In terms of frequenciesfo andf~, equation (III) can be written as
on further simphfication,
fo2 - f? A2
L~ \ L--]~
--
2KV~ - Y~. v~
_
Further by taking fo ~- f~, Af f,
--
Lg
-
2KV,
(L~Xd; + L~X/.)
K
Xm = 2KV~
L-~
tt
2,<
V~ Xm'
Here also by taking f0 "- f~, equation (IV) reduces to the standard result as in equation (20) given in the text: i.e.
(IV) K ~, Xm' =
substRuting
L--~o,1
Lo
fo2 ( L~' L'o') 2KVs f2 \ L---~ L'o - ~ X/,'.
KV~ --X,~ v~
hence the equation (19) in the text. Separating imaginary terms of equation (I), L~' -
Vol. 70, No. 8
I'
2Q2
'1
2Q0
As the main form of the expressions are ldentmal for both C and L, similar analysis of equation (9) will lead to the final expressions of d~electnc parameters.
Solid State Communications, Vol. 70, No. 8, pp. 847-850, 1989. Printed in Great Britain.
A M E T H O D FOR THE EVALUATION OF MICROWAVE D I E L E C T R I C AND MAGNETIC PARAMETERS USING R E C T A N G U L A R CAVITY P E R T U R B A T I O N T E C H N I Q U E V.R.K. Murthy and R. Raman Department of Physics, Indian Institute of Technology, Madras - 600 036, India
(Recetved 15 June 1988; in final form 6 February 1989 by T. Tsuzukt) To derive the expression for complex dielectric and magnetic susceptibihties, an easy and alternate approach is proposed. This analysis is based on the assumption that the cavity is equivalent to an isolated lumped resonant circuit. The results are identical with standard expressions. Results of a few samples are also tabulated. INTRODUCTION
L. -
CAVITY perturbation methods have been widely used to characterize the dielectric, magnetic and superconducting materials, at microwave frequencies [1-4]. The dispersive and dissipative terms of the materials are directly related to the change in the resonant frequency and quality factor of the cavity from its respective empty cavity values. In order to understand the overall properties of a material, the solution of Maxwell's equations in the particular configuration must be understood. The standard perturbation analysis of this problem is found elsewhere [5, 6]. As the resonant cavity is microwave analogue to an isolated lumped resonant circuit [7], the present paper goes on to arrive at the standard expressions by calculating the value of capacitance and inductance for a rectangular cavity excited to ~ts fundamental mode.
From the solution of Maxwell's electromagnetic equations, for a rectangular with broad &mention a, narrow dimension b and length l, along X, Y and Z directions respectively, the orientation of respective electric and magnetic field components can be deduced as E~., H~ and/4: (where as E=, Ex and H v are zero). In this case the assumption is that the cavity is a parallel resonant circuit with a capacitance C~ corresponding to average value of E). and parallel combination of two inductances L~ and L, corresponding to the average value of respective magnetic fields He and H=. From the low frequency lumped circuit elements analogy, C) -
coal b
-
(2)
i~oN2alb L~: =
a2 q- 12
(3)
where l = 2g/2 for TE10~ mode and 'N' has usual meaning similar to the number of turns in a solenoid inductor. If N is replaced by 1/Tt, the above assumptions result to, f0 -
1
2rc~>
(4)
where f0 is the resonant frequency of the cavity. In the case of TEI0o mode l = n2g/2 and C). becomes nC( and L~: becomes L'~,./n, so that the resonant condition 1
2rc~
IS always satisfied, where C,~ and L~, are equal to C,. in equation (1) and L~, in equation (3) respectively.
(1)
]~o N 2 ]b L~:
l
From equation (2) the resultant inductance is written as,
f-
GENERALTHEORY
#oNZab
a
847
DIELECTRIC CONSTANT e' AND LOSS e" A sample of complex dielectric susceptibility Xe = X/ -jX~" (equals to e' - j g " ) is kept at the maximum electric field location of the cavity. A thin sample is taken such that the electric field will be uniform throughout the volume of the sample. In order to simplify the problem, the sample is assumed to be cylindrical or rectangular with umform cross sectional area S and length is greater than or equal to b so that it will occupy the entire narrow dimension of the cavity. After the introduction of the sample the empty resonant frequency and Q factor alter, due to the change in the overall capacitance and conductance of
848
EVALUATION OF MICROWAVE PARAMETERS
the cavity, without perturbing the inductance. The new value of capacitance is given as, c,
=
~o(la -- S) eoe*S + b b
(5)
Vol. 70, No. 8
cavity with and without samples. Here also taking f0-fl [ 1 2Qt
1 ] 200
2V~ v~ x~".
-
(13)
The distribution of the electric field for TEl0, mode can be derived from the solution of Maxwell's electromagnetic equation as,
Equations (11) and (13) are of the standard form for the expressions for dielectric parameters as reported in [61.
Ey = E m a x S i n ( ~ ) s i n ( 2 X\ z ~2g ]"
(6)
MAGNETIC PERMEABILITY #' AND LOSS/~"
(7)
In this case the sample is introduced at maximum magnetic field location. The sample is aligned along maximum H x for which H~ and Ey are least perturbed. The samples of complex magnetic susceptibility Xm = X,~ -- jX/,," having uniform cross sectional area 'S' changes Lx to Lxl as
From equation (6), the expression for power is P = Pma*sin2
sin2 \ 2,
"
From equation (7) it is clear that the average electric power is one fourth of the maximum power. Since, the sample is introduced at maximum field position, the effective interaction is four times that of its average value. Hence, the sample area S has to be replaced by 4S. If the sample is kept at some other position say zero electric field location, then S will be zero times the cross sectional area and correspondingly there would not be any change in the characteristics of the cavity. Now equation (5) is changed as,
C, -
e,o(la - 4S) + 4~e*__.~S b b
(8)
Equation (8) can be simplified by comparing with equation (1) as, C, = Co 1 + - ~ c Xe
(9)
where V~ = Sb and V~ = lab are the volume of the sample and cavity respectively and Co = Cy. The complex resonant frequencies [8] with and without samples are f~ and f0 and are given by Ct oc 1/f~2 and Co oc lifoz. Taking the real part of equation (9)
fg - f ? 4V#e' -(10) f? For very small values of V~,fo = fl and hence equation (10) can be simplified as,
i.to(lb -- 4S) 4~/u*S + - ~2a 7r2a
Lx~ -
(14)
Here also, replacement of S by 4S is inducted
Ll
#o Ib + 4SXm + -~
(15)
l L0
n2[a l] #0 ~ + a-b
(16)
where L 1 and L0 are the respective inductances of the cavity. The real part of the inductance Re (L) is proportional to (1/f 2) and the ratio Im (L)/Re (L) is equal to (l/Q), wherefis resonant frequency and Q is the quality factor of the resonant cavity. On simplification of equations (15) and (16) 4a 2 ~ X,, g I
=
g 0
1+
(a 2 + 12) + 4
(12Vs~
Xm
.
(17)
For V~ >> V~, f0 -~ ft, the term 4(VJV~)Xm can be neglected. 4a 2 V~ L0
-
+ t2 .
(18)
af 2V~ Xe' (11) f, where Afis the shift in the resonant frequency denoted by (f0 - f~)- Similarly separation of imaginary terms yield to,
Equation (18) is identical to the equation (9) for the case of capacitative term, and separation of real and imaginary terms results to the similar expressions as,
I1 Q,
and
1 ]f02 4V~ Q0_]f2 = ~ X~"
X~=KV
1 V~( ~ l f ) ~
(19)
(12)
L_
where Qj and Qo are the respective quality factors of
S~r -- g ~
2QI
200
Vol. 70, No. 8
EVALUATION OF MICROWAVE PARAMETERS
849
Table 1. Dielectric constant e" values of a few samples using three different expressions and the corresponding dielectric loss e" given by equation (12) Material
Diameter in cm
Frequency f m MHz
Teflon Plexiglass Quartz Rexolite No. 422 Stycast
0.285 0.240 0.196 0.290
9111 9109 9101 9093
0.264
8765
e' from equation (10)
e' from equation (l l)
2.06 2.56 3.74 2.43
e' from equation (21)
2.06 2.56 3.73 2.43
12.2
2.06 2.56 3.74 2.43
11.96
12.2
Ref. e'
e"
2.08* 2.59* 3.78* 2.5t
0.0008 0.017 0.0004 0.002 0.025
* R.A. von Hippel, Dielectrtc Materials and its Applications, Chapman Hall Ltd., London (1954). t Data provided by the manufacturer, C. Lec plastic Inc., 1201 Chestnut Street, Philadelphia. 19197, USA. where K = (2a2)/(a2 + 12). Details of the derivation of equations (19) and (20) are given in Appendix. Equations (19) and (20) are exactly identical to the results of Artman's [5] perturbation analysis.
Professor T. Fujimura of Tohoku University, Japan, for his instructive comments and suggestions, especially for letting us become familiar with (17) and its derivation. This work was improved by his instructions greatly.
EXPERIMENTAL PARTS
REFERENCES
The experimental arrangement for the dielectric parameter studies consists of X-13 Klystron whose reflector voltage is modulated. The isolator, attenuator, shde screw tuners, Majic Tree and Crystal detector forming a standard test bench follows the Klystron. The rectangular cavity of length 11.75 cm and mode TE~05 with the resonant frequency 9157 MHz is connected to the bench. The sample is kept at the centre of cavity where the maximum electric field clearly exists for odd mode of excitation. The frequency is measured by H.P. frequency counter. For recording the cavity response, a personal computer is interfaced to the detector.
1. 2. 3. 4. 5. 6. 7. 8.
L.J. Buranov & I.F. Schegolev, Prib. Tek. Eksp. 2, 171 (1971). H. Kobayashi & Ogawa Selya, Japan J. Appl. Phys. 10, 345 (1971). N.P. Ong, J. Appl. Phys. 48, 2935 (1977). B. Viswanathan, R. Raman, N.S. Raman & V.R.K. Murthy, Solid State Commun. 66, 409 (1988). J.O. Artman & P.E. Tannenwald, J. Appl. Phys. 26, 1124 (1955). B. Lax & K.J. Button, Microwave Ferrttes and Ferrimagnetws, McGraw-Hill, New York (1962). J.C. Slater, Rev. Mod. Phys. 18, 474 (1946). R.F. Soohoo, Microwave Magnetics, Harper and Row, New York, 1985.
CONCLUSIONS Thus an alternate and simple approach has been discussed for the evaluation of material parameters at microwave frequencies using rectangular cavity technique. Moreover, from the above discussed analysis, the expression for dielectric constant e' of a low loss sample in TE~0n cavity of length I can be written as,
(
[(C') 2 e L --
-~
+
a212 ~ l n E a 2 + l 2 12-+n2a2J_]
Equation (18) in the above text is written as
[
L~ = L 0 1 +
2K~xo]_]
(I)
where 2a 2
K = a 2 + 12 and the complex inductances are
1
=
APPENDIX
4alS (21)
where C' is the free space velocity of electromagnetic waves and the other terms bear the usual meaning. Results of a few samples are tabulated in Table 1 using equation (10), (11), (12) and (21) along with the available standard data.
Acknowledgement - T h e authors would hke to thank
L, = L', - jL~' and
Lo = L~ -- jL'o' separating the real part of equation (I), one can write as
L; -- L~ -
2KV~
(L~X/,, -- L'o'X/,,')
(II)
850
E V A L U A T I O N OF M I C R O W A V E P A R A M E T E R S
After approximating L~ - L6 L~
-
LoX/,; to
zero, eq. (II) becomes
2KV~ - X,~. V~
from equation (III), for X,~ in equation (IV), one can write equation (IV) as
(III) L~' - Lo' -
2KV~
~
/
L'oX/: + ---7 L, L~ -
Lo
In terms of frequenciesfo andf~, equation (III) can be written as
on further simphfication,
fo2 - f? A2
L~ \ L--]~
--
2KV~ - Y~. v~
_
Further by taking fo ~- f~, Af f,
--
Lg
-
2KV,
(L~Xd; + L~X/.)
K
Xm = 2KV~
L-~
tt
2,<
V~ Xm'
Here also by taking f0 "- f~, equation (IV) reduces to the standard result as in equation (20) given in the text: i.e.
(IV) K ~, Xm' =
substRuting
L--~o,1
Lo
fo2 ( L~' L'o') 2KVs f2 \ L---~ L'o - ~ X/,'.
KV~ --X,~ v~
hence the equation (19) in the text. Separating imaginary terms of equation (I), L~' -
Vol. 70, No. 8
I'
2Q2
'1
2Q0
As the main form of the expressions are ldentmal for both C and L, similar analysis of equation (9) will lead to the final expressions of d~electnc parameters.