322 IEEE T’RANSACTIONS ON INSTRUMENTATION AND MEASUKEMENT, VOL. 44, NO. 2. APRIL 1995 Resonant Frequencies and Q Factors of Dielectric Parallelepipeds C. W. Trueman, S. R. Mishra, C. L. Larose, and R. K. Mongia Abstract-This paper describes the measurement and compu- tation of the resonant frequencies and the associated Q factors of low-loss ceramic parallelepipeds with relative permittivity 37.84 and 79.46. Each sharp resonance peak is measured separately with a fine frequency step. The calculation uses FDTD to compute the initial portion of the backscatteredfield as a function of time, due to a Gaussian pulse plane wave. The transient response is extended in time to zero value by Prony’s method. The measured and computed resonant frequencies and Q factors are presented for two parallelepiped orientations and are in good agreement. I. INTRODUCTION HIS paper investigates the resonant behavior of paral- T lelepiped resonators in free space by measurement and b j computation. The accurate calculation of the resonant frequencies and Q factors of resonators in free space [I] underlies the more complex problem of the calculation of these parameters for a resonator in a stripline environment [2]. The resonators studied in this paper are parallelepipeds with a square cross section of size w x UI and have depth h. Table I gives the dimensions and electrical properties of the resonators. Resonator A has a width-to-thickness ratio of about 2.5, relative permittivity 79.46, and loss tangent 1/3000. Resonators B and C have width-to-thickness ratios of 2.5 and 3.33, respectively, and both have relative permittivity 37.84 and loss tangent 1/6000. 11. RCS MEASUREMENT The radar cross section (RCS) of each dielectric resonator was measured in a 6 x 6 x 6 m anechoic chamber. The resonator was mounted on a Styrofoam column 1.2 m from a pair of horns, for a bistatic angle of about 8”. The amplitude and phase were measured at each frequency-first with the resonator on the support column, then with an 11.9 mm metallic reference sphere on the same support column, and last with nothing on the column. The system is calibrated as follows. The “target response” is calculated as the difference at each frequency between the response with the target on the support column and the response with nothing on the column. The “sphere response” is the difference between the response with the sphere on the support kolumn and the response with the column alone. The calibration factor is obtained at each Manuscript received July I, 1994; revised October 15, 1994. C. W. Trueman is with Concordia University, Montreal, PQ, Canada. S. R. Mishra and C. L. Larose are with David Florida Laboratory, Canadian R. K. Mongia is with the Communications Research Centre, Ottawa, ON, IEEE Log Number 9408688. Space Agency, Ottawa, ON, Canada. Canada. TABLE I Resonator Dimensions Wh Relative Loss Tangent mm Permittivity THE DIMENSIONS AND ELECTRICAL PARAMFTERS OF THE DlELECTRIC RESONATOR^ w h A 7.45 2.98 2.50 79.46 U3000 B 8.77 3.51 2.50 37.84 1/6ooo C 8.60 2.58 3.33 37.84 1/6oOO frequency as the exact scattered field of the sphere, evaluated using the Mie series [3] with the appropriate bistatic angle, divided by the “sphere response.” The scattered field of the target is the “target response” times the calibration factor. Note that for these high-Q resonators, no time gating was used in the measurement. The scattered field was measured initially at 801 frequencies from 4 to 18 GHz. To clearly define each sharp, narrow resonance peak, a separate measurement was made using many frequencies over a narrow band surrounding the peak. Also, for each orientation of the resonator, several equivalent measurements can be made. For example, for the “narrow face vertical” orientation in the insert in Fig. 1, the backscattered field should be the same regardless of which of the four narrow faces the plane wave is incident upon, and so four equivalent measurements can be made. This was done to ensure consistency and repeatability, with good agreement in most cases. 111. FDTD COMPUTATION The finite-difference time-domain (FDTD) method [4], [5], as used here, divides space into cubical cells of side length Ax, with a total space size of N, x Ny x N, cells. To represent a resonator, a region of N,, x X,, x Npz cells at the center of the cell space is filled with dielectric material, hence the ceramic’s permittivity and conductivity are used in the FDTD “update” equations [5]. The remaining cells are filled with free space. The field components that lie on the surface of the resonator must be updated with the average of the dielectric’s permittivity and that of free space [6], [7]. All the surfaces of the resonator are separated from the outer boundary of the cell space by a layer of N, free-space cells. In this work, the Penn State FDTD code has been used [5]. The outer boundary is made to absorb most of the energy incident upon it using the Liao absorbing boundary condition [8]-[ IO]. Resonator A in Table I was represented with a region of NTz = 25 x NTy = 10 x N,, = 25 dielectric cells, for a cell size of Ax = 7.45/25 = 0.298 mm. The free-space layer was made N, = 30 cells thick to eliminate interactions between the resonator and the somewhat-imperfect outer boundary as 0018-9456/95$04,00 0 1995 IEEE
Transcript
322 IEEE T’RANSACTIONS ON INSTRUMENTATION AND MEASUKEMENT, VOL. 44, NO. 2. APRIL 1995
Resonant Frequencies and Q Factors of Dielectric Parallelepipeds
C. W. Trueman, S. R. Mishra, C. L. Larose, and R. K. Mongia
Abstract-This paper describes the measurement and compu- tation of the resonant frequencies and the associated Q factors of low-loss ceramic parallelepipeds with relative permittivity 37.84 and 79.46. Each sharp resonance peak is measured separately with a fine frequency step. The calculation uses FDTD to compute the initial portion of the backscattered field as a function of time, due to a Gaussian pulse plane wave. The transient response is extended in time to zero value by Prony’s method. The measured and computed resonant frequencies and Q factors are presented for two parallelepiped orientations and are in good agreement.
I. INTRODUCTION
HIS paper investigates the resonant behavior of paral- T lelepiped resonators in free space by measurement and b j computation. The accurate calculation of the resonant frequencies and Q factors of resonators in free space [ I ] underlies the more complex problem of the calculation of these parameters for a resonator in a stripline environment [2]. The resonators studied in this paper are parallelepipeds with a square cross section of size w x UI and have depth h. Table I gives the dimensions and electrical properties of the resonators. Resonator A has a width-to-thickness ratio of about 2.5, relative permittivity 79.46, and loss tangent 1/3000. Resonators B and C have width-to-thickness ratios of 2.5 and 3.33, respectively, and both have relative permittivity 37.84 and loss tangent 1/6000.
11. RCS MEASUREMENT The radar cross section (RCS) of each dielectric resonator
was measured in a 6 x 6 x 6 m anechoic chamber. The resonator was mounted on a Styrofoam column 1.2 m from a pair of horns, for a bistatic angle of about 8”. The amplitude and phase were measured at each frequency-first with the resonator on the support column, then with an 11.9 mm metallic reference sphere on the same support column, and last with nothing on the column. The system is calibrated as follows. The “target response” is calculated as the difference at each frequency between the response with the target on the support column and the response with nothing on the column. The “sphere response” is the difference between the response with the sphere on the support kolumn and the response with the column alone. The calibration factor is obtained at each
Manuscript received July I , 1994; revised October 15, 1994. C. W. Trueman is with Concordia University, Montreal, PQ, Canada. S. R. Mishra and C . L. Larose are with David Florida Laboratory, Canadian
R. K. Mongia is with the Communications Research Centre, Ottawa, ON,
IEEE Log Number 9408688.
Space Agency, Ottawa, ON, Canada.
Canada.
TABLE I
Resonator Dimensions Wh Relative Loss Tangent mm Permittivity
THE DIMENSIONS AND ELECTRICAL PARAMFTERS OF THE DlELECTRIC RESONATOR^
w h A 7.45 2.98 2.50 79.46 U3000 B 8.77 3.51 2.50 37.84 1/6ooo C 8.60 2.58 3.33 37.84 1/6oOO
frequency as the exact scattered field of the sphere, evaluated using the Mie series [3] with the appropriate bistatic angle, divided by the “sphere response.” The scattered field of the target is the “target response” times the calibration factor. Note that for these high-Q resonators, no time gating was used in the measurement. The scattered field was measured initially at 801 frequencies from 4 to 18 GHz. To clearly define each sharp, narrow resonance peak, a separate measurement was made using many frequencies over a narrow band surrounding the peak. Also, for each orientation of the resonator, several equivalent measurements can be made. For example, for the “narrow face vertical” orientation in the insert in Fig. 1, the backscattered field should be the same regardless of which of the four narrow faces the plane wave is incident upon, and so four equivalent measurements can be made. This was done to ensure consistency and repeatability, with good agreement in most cases.
111. FDTD COMPUTATION
The finite-difference time-domain (FDTD) method [4], [ 5 ] , as used here, divides space into cubical cells of side length Ax, with a total space size of N , x Ny x N , cells. To represent a resonator, a region of N,, x X,, x Npz cells at the center of the cell space is filled with dielectric material, hence the ceramic’s permittivity and conductivity are used in the FDTD “update” equations [5] . The remaining cells are filled with free space. The field components that lie on the surface of the resonator must be updated with the average of the dielectric’s permittivity and that of free space [6], [7]. All the surfaces of the resonator are separated from the outer boundary of the cell space by a layer of N, free-space cells. In this work, the Penn State FDTD code has been used [ 5 ] . The outer boundary is made to absorb most of the energy incident upon it using the Liao absorbing boundary condition [8]-[ IO].
Resonator A in Table I was represented with a region of NTz = 25 x NTy = 10 x N,, = 25 dielectric cells, for a cell size of Ax = 7.45/25 = 0.298 mm. The free-space layer was made N, = 30 cells thick to eliminate interactions between the resonator and the somewhat-imperfect outer boundary as
0018-9456/95$04,00 0 1995 IEEE
TRUEMAN et ul.: RESONANT FREQUENCIES AND Q FACTORS OF DIELECTRIC PARALLELEPIPEDS 323
: - 0 . 4
d] - 0 . 6
Fig. I . excitation for resonator A, which has f, = 79.46.
The backscattered field as a function of time due to a Gaussian pulse
a consideration in interpreting the results. The cell space size was N , = Nr,+2N, = 85 x Ny = 70 x N, = 85 cells. The conductivity of the dielectric cells was set to correspond to a loss tangent of 113000 at 5 GHz. The excitation was a plane wave of Gaussian pulse time dependence with the width of the pulse set to confine its energy content to frequencies where the cells are smaller than 1\10 wavelength in the dielectric [5] , approximately 11 GHz. The time step was set equal to the Courant limit [ 5 ] . The FDTD program is used to calculate the field components at each time step in the cell space, and then a near- to far-zone transformation is used to find the far-zone backscattered field [ l 11 as a function of time.
Fig. 1 shows the backscattered field from resonator A in the "narrow face vertical" orientation, where the incident electric field vector is parallel to a long edge of the resonator, and the direction of travel is perpendicular to a narrow face. The Gaussian pulse excitation lasts approximately 800 time steps or about 0.5 ns. The sharp resonances of the dielectric parallelepiped cause the backscattered field to "ring down" very slowly, and the figure shows that the FDTD computation would need to be run for about 75 ns or 132000 time steps to trace the transient to zero. This would be very expensive. Here, the FDTD algorithm is used to compute the field for 8192 time steps, or about 4.7 ns. Prony's method [12] is used on the FDTD data from step 800 to 8192 to derive a pole series representation of the backscattered field time function, as described in [6]. The pole series is used to extend the transient until it reaches zero value, to obtain the response shown in Fig. 1. The pole series is evaluated with a time step of Atp = 8 At to obtain the backscattered field to a maximum time of 16384At, = 131072 At, equivalent to running the FDTD code for 131 072 time steps, with a CPU time saving of nearly 90%.
IV. NARROW FACE VERTICAL ORIENTATION
Fig. 2 shows the RCS of resonator A in the narrow face vertical orientation. There is generally good agreement be- tween the measured RCS and that computed by FDTD with
10 - 7 0 i ' " ' ' I ' ' 0 ' I 1. ' I ' ' ' I 8 ' 1 'I ' ' 1 ' 1
Frequency [ GHz I Fig. 2. The RCS of resonator A in the "narrow face vertical" orientation.
Prony extrapolation. The first two resonance peaks agree well in frequency, but are taller in the computation than in the measurement. Resonance peaks 3 and 4, and the notch near 10.5 GHz, show a systematic shift in the computation to lower frequencies than the measurement. This suggests that either the value of the permittivity supplied by the manufacturer was slightly in error, or that, effectively, FDTD models a slightly larger parallelepiped than the size specified in cells.
The measurement shows a sharp, narrow peak at the location marked "A' in Fig. 2. This peak was not found in the computation. Sometimes a sharp peak is missing because there are too few terms in the Prony series. The number of terms was greatly increased, but peak "A" was not found. The calculation was repeated to simulate the bistatic nature of the measurement, with the plane wave incident from -5" and the scattering direction set to 5", roughly approximating the 8' degree bistatic angle in the measurement. Little change was seen near peaks 1, 2, or 3, and no additional peak was found near "A." The fourth resonance was much less sharp in the bistatic computation.
Each resonance peak was measured individually with a fine frequency step. The resonant frequency and Q factor were determined as follows. The resonant frequency f o is the frequency at which the RCS has its largest value 00. The measured data points near a resonance peak do not lie on a smooth curve. A quadratic polynomial was derived as a least-square error best fit to the measured data very near the peak. The largest value of the polynomial was taken as the RCS at resonance, 00, and the corresponding frequency as the resonant frequency, f o . The "quality factor" or Q factor of the resonance peak is Q = f0 / ( f2 - f l ) , where fl and f2 are the lower and upper 3 dB frequencies, respectively, at which the RCS is 3 dB less than 00. To find fi, a second quadratic polynomial was derived to approximate the measured data points near the lower 3 dB point; the quadratic was used to determine the 3 dB frequency. Similarly, a third polynomial was derived to find the upper 3 dB frequency.
Table I1 lists the resonant frequencies and Q factors for the three resonators in the "narrow face vertical" orientation. For
lEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 44, NO. 2, APRIL 1995
Resonator
Resonator A
Resonator B
Resonator C
TABLE I1 RESONANT FREQUENCIES ,\?.ID Q FACTORS FOR THE “NARROW F A C ~ VERTICAL’’ ORIENTATION
Resonance Measured Measured Computed Frequency Q Frequency
resonator A, the measured and computed frequencies of the resonance peaks agree to within 0.66% in the worst case. The CJ factors for the computed RCS are higher for peaks 1, 2, and 4, but lower for peak 3.
v. RESONATORS B AND c Resonators B and C with cr = 37.84 have about half the
permittivity of resonator A with cr = 79.46. Resonator B measures 8.77 x 8.77 x 3.51 mm and was modeled with 2.5 x 25 x 10 cells, with the cell size set to Ax =8.77/25 = 0.351 mm. The computation obtains the RCS to about 14 GHz, where the size of cells is 1/10 of the wavelength in the dielectric. Resonator C has different proportions than A or B, with ratio w / h = 3.33. Resonator C was modeled with 30 x 30 x 9 cells, preserving its width-to-thickness ratio. The cell size was Ax = 8.60/30 =0.287 mm. The computation obtains the RCS to about 17 GHz. The conductivity for both resonators B and C was chosen such that the loss tangent is U6000 at 5 GHz. The FDTD computation was run for 8192 time steps and extrapolated to 131 072 steps, as before.
The RCS for both resonators B and C in the narrow face vertical orientation is generally similar to that shown in Fig. 2, except that resonance peaks 1 and 2 are much broader for resonators B and C. Table I1 shows that the Q factors for resonances 1 and 2, about equal to 30, are much lower than for resonator A, which has Q of 95 or more for these peaks. Conversely, resonance peak 3 is much higher Q for the lower- permittivity resonators, at about 230-280, compared to 36-46 for resonator A. The fourth resonance has a comparable Q factor for both the low- and the high-permittivity resonators.
VI. NARROW FACE HORIZONTAL ORIENTATION
Fig. 3 shows the RCS of resonator A in the “narrow face horizontal” orientation, with the electric field parallel to a short edge of the resonator and the plane wave incident perpendicular to a narrow face. The measurement and the computation agree quite well for resonance peak 1. Table I11 shows that the resonant frequency differs by 0.65% between the measurement and the computation, and that the measured and computed Q factors are both about 60. Peaks 2 and 3 are taller in the computation than in the measurement. Peak
Y
C 0
L, 0 0
-.-I
Ln I m m 0 L U
c 0 U 0 cc
___ measured
Frequency [ GHz I Fig. 3. The RCS of resonator A in the “narrow face horizontal” orientation.
2 has a measured Q of 279, and a computed Q of 432. But peak 3 is sharper in the measurement with Q = 700 than in the computation with Q = 628. Resonance peak 4 agrees well with a 0.40% difference in resonant frequency between the measurement and the computation, and with Q = 260 in the measurement and 220 in the computation. At frequencies above about 9 GHz, the computation differs considerably from the measurement.
Some difficulty was encountered computing the RCS for resonator A in the narrow face vertical orientation. The free space layer had to be increased to 35 cells in thickness to suppress interactions with the resonator. The Prony extrapola- tion initially missed resonance peak 3 entirely when it was run with 210 terms; the number of terms was gradually increased to 260, and peak 3 emerged from the calculated data.
The RCS for resonator B is generally similar to that shown in Fig. 3 for resonator A. Resonance peaks 1 and 2 are broader for the lower-permittivity resonator. The measured Q is 22.5 and 39.8, respectively, for peaks 1 and 2, compared to 58 and 279 for these peaks for resonator A. The measured Q is 241 for peak 3, compared to 700 for resonator A, so the resonance is much less sharp with the lower permittivity. Peak 4 is also much broader for resonator B.
TRUEMAN et al.: RESONANT FREQUENCIES AND Q FACTORS OF DIELECTRIC PARALLELEPIPEDS 325
The RCS for resonator C is different than that for resonators A or B in the “narrow face horizontal” orientation. Recall that the width-to-thickness ratio is 3.33 for resonator C, compared to 2.5 for the other two. For resonator C, peak 1 is so broad that we cannot calculate a meaningful value of the quality factor &. Peak 2 is sharp with Q at 144 in the measurement; peak 3 is broad with Q at 91; and peak 4 has Q of about 110. Note that the measured and computed resonant frequencies agree more poorly for resonator C than for A or B, with a maximum error of 0.81%.
VII. CONCLUSIONS
Tables I1 and 111 summarize the measured and computed resonant frequencies and Q factors for three different orienta- tions of the three resonators. There is good agreement between the measured and computed resonant frequencies: the largest difference is less than- 1%. In general, where the measurement obtains a sharp resonance, the computation does also, although the sharp peak at “A’ in Fig. 2 could not be found. This paper has shown that FDTD can accurately predict the resonances of high-permittivity dielectric parallelepipeds in free space, and therefore should provide a good basis for analyzing their behavior in an MMIC environment.
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