JP7550403B2 - Resonator characteristic measurement method - Google Patents

Description

The present invention relates to a method for measuring resonator characteristics.

One type of resonator used in high frequency bands such as the millimeter wave band is the waveguide excitation resonator. The waveguide excitation resonator can be used, for example, as an antenna or a waveguide in the high frequency band. Patent Document 1 discloses technology related to a laminated waveguide line. Patent Document 2 discloses technology related to a high frequency conductivity measuring device that can easily measure the conductivity of a conductor plate to be measured.

JP 2010-34826 A JP 2015-227850 A

When designing a resonator that can be applied to a specific device, it is necessary to determine the conductivity and dielectric loss tangent of the resonator in the high frequency band. For example, in the case of a rectangular parallelepiped resonator, it is necessary to determine the conductivity of the wide wall surface, the conductivity of the narrow wall surface, and the dielectric loss tangent. However, with the technology disclosed in Patent Document 2, it is not possible to determine these values at once, which makes it difficult to determine the characteristics of the resonator.

In view of the above problems, the object of the present invention is to provide a method for measuring resonator characteristics that can easily determine the characteristics of a resonator.

A method for measuring resonator characteristics according to one aspect of the present invention includes the steps of:
a first step of evaluating a sensitivity coefficient and an uncertainty with respect to a resonator size and determining a resonator size that reduces the sensitivity coefficient;
a second step of measuring a pass characteristic of a resonator having the resonator dimensions determined in the first step, and calculating a resonant frequency and an unloaded Q value of the resonator;
a third step of calculating a relative dielectric constant of the resonator using the resonance frequency calculated in the second step and actual measurements of the resonator dimensions;
and a fourth step of calculating the conductivity of a wide wall surface, the conductivity of a narrow wall surface, and a dielectric tangent of the resonator using the unloaded Q value calculated in the second step, the relative dielectric constant calculated in the third step, and the actual measured values of the resonator dimensions.

In the above-mentioned resonator characteristic measuring method,
The resonator may be a rectangular parallelepiped resonator, and when the dimension in the width direction of the resonator is a, the dimension in the thickness direction is b, and the dimension in the longitudinal direction is d,
In the first step, a sensitivity coefficient equation between the dielectric constant and the dimension a, a sensitivity coefficient equation between the dielectric constant and the resonant frequency, a sensitivity coefficient equation between the dimension d and the dimension a, and a sensitivity coefficient equation between the dimension d and the resonant frequency are prepared;
Assuming a standard uncertainty of the unloaded Q value, a standard uncertainty of the dimensions a, b, and d, a standard uncertainty of the resonant frequency, and a standard uncertainty of the dielectric constant;
The sensitivity coefficient may be calculated when the dimensions a, b, and d are changed.

In the first step, the spacing between adjacent resonant frequencies may be set so that the error in the unloaded Q value is equal to or less than a predetermined value, and then the resonator dimensions may be determined so that the sensitivity coefficient is small.

The interval between adjacent resonant frequencies is set as follows:
Assuming a parallel circuit of two or more resonators,
Given the resonant frequency and the unloaded Q value Qu, calculate the pass characteristic.
Calculate the unloaded Q value Qum from the pass characteristic;
This may be implemented by setting the interval between adjacent resonant frequencies so that the difference between Qu and Qum is equal to or smaller than a predetermined value.

In the second step, the transmission characteristics of a resonator may be measured, the resonator comprising first and second conductor substrates arranged to face each other and metal vias arranged between the first and second conductor substrates, and the space surrounded by the first and second conductor substrates and the plurality of metal vias functions as a waveguide.

In the third step,
The relative dielectric constant _εr is an unknown quantity, and the resonator dimensions a, b, d, resonance frequency f _mnl , resonance mode orders m, l, and relative permeability _μr are known quantities,
The dielectric constant of the resonator may be calculated using the measured values of the resonator dimensions and the following linear equation (wherein the resonator is a rectangular parallelepiped resonator, a is the dimension in the width direction of the resonator, b is the dimension in the thickness direction, d is the dimension in the longitudinal direction, n=0, and _x1 , _x2 , _x3 , and _x4 are defined as follows):

In the fourth step,
The electrical conductivity σ _f of the wide wall surface of the resonator, the electrical conductivity σ _r,t of the narrow wall surface, and the dielectric loss tangent tan δ are unknown quantities, and the resonator dimensions a, b, d, magnetic permeability μ, wave number k, impedance η of free space, angular frequency ω, unloaded Q value Qu, and resonance mode order l are known quantities,
The conductivity of the wide wall surface, the conductivity of the narrow wall surface, and the dielectric tangent of the resonator may be calculated using the following linear equations (wherein the resonator is a rectangular parallelepiped resonator, a is the dimension in the width direction of the resonator, b is the dimension in the thickness direction, and d is the dimension in the longitudinal direction, ^a1 ₌ ( ^l2a3d + ^ad3 ) x [2π2 ^/ {(kad) ^3bη }] x (ωμ/2) ^1/2 , _a2 = ( ^2l2a3b + ^2bd3 ) x [2π2 ^/ {(kad) ^3bη }] x (ωμ/2) _{1/2, and x5} ^{, x6} _, and _x7 are defined as follows):

The present invention provides a method for measuring resonator characteristics that can easily determine the characteristics of a resonator.

FIG. 1 is a perspective view illustrating an example of a resonator according to an embodiment. 1 is a perspective view illustrating a configuration example of a resonator according to an embodiment; FIG. 2 is a top view showing a configuration example of a resonator according to an embodiment. 1 is a flowchart for explaining a resonator characteristics measuring method according to an embodiment. 5 is a flowchart for explaining details of the operation of step S1 in FIG. 4; 5 is a graph showing an example of a measurement result of the pass characteristic in step S2 of FIG. 4. 5 is a graph for explaining a method of calculating a Q value in step S2 of FIG. 4.

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.
First, a resonator to which the resonator characteristics measuring method according to the present embodiment is applied will be described. Fig. 1 is a perspective view showing an example of a resonator according to the present embodiment. As shown in Fig. 1, the resonator 1 is a rectangular parallelepiped resonator, and is a structure having a length a in the x-axis direction, a length b in the y-axis direction, and a length d in the z-axis direction.

Coupling slots 11 and 12 are formed on the xz plane of resonator 1. Specifically, coupling slot 11 is provided on the negative side of resonator 1 in the z-axis direction, and coupling slot 12 is provided on the positive side of resonator 1 in the z-axis direction. The size of coupling slots 11 and 12 is length l in the x-axis direction and length w in the z-axis direction. For example, the inside of resonator 1 is hollow, and high frequency waves introduced from coupling slot 11 propagate inside resonator 1 and are then extracted from coupling slot 12.

2 and 3 are perspective and top views showing an example of the configuration of a resonator according to this embodiment. The resonator 2 shown in FIGS. 2 and 3 is a waveguide-excited SIW (Substrate Integrated Waveguide) resonator, and is configured using two conductor substrates 31 and 32 and multiple metal vias (through conductors) 33. The conductor substrate 31 is disposed on the upper side (positive side in the y-axis direction), and the conductor substrate 32 is disposed on the lower side (negative side in the y-axis direction). Between the conductor substrates 31 and 32, multiple metal vias 33 are disposed to form a waveguide. In other words, in the resonator 2 shown in FIGS. 2 and 3, the space surrounded by the conductor substrates 31 and 32 and multiple metal vias 33 functions as a waveguide. Note that the space sandwiched between the two conductor substrates 31 and 32 may be hollow, or a dielectric material may be disposed therein.

Coupling slots 21 and 22 are formed on the surface of the resonator 2 facing the conductor substrate 32. Specifically, the coupling slot 21 is provided on the negative side of the z-axis direction of the resonator 2, and the coupling slot 22 is provided on the positive side of the z-axis direction of the resonator 2. The waveguide 35 is connected to the coupling slot 21, and the waveguide 36 is connected to the coupling slot 22. The negative sides of the waveguides 35 and 36 in the y-axis direction are connected to a network analyzer (not shown). For example, a high frequency generated by the network analyzer is introduced into the resonator 2 via the waveguide 35 and the coupling slot 21. After passing through the resonator 2, the high frequency introduced into the resonator 2 is transmitted to the network analyzer via the coupling slot 22 and the waveguide 36. In step S2 of FIG. 4 described below, the resonator 2 having such a configuration is used to measure the transmission characteristics of the resonator.

Next, a resonator characteristics measurement method according to the present embodiment will be described. FIG. 4 is a flowchart for explaining the resonator characteristics measurement method according to the present embodiment. The resonator characteristics measurement method according to the present embodiment includes the following first to fourth steps.

The first step is to evaluate the sensitivity coefficient and uncertainty with respect to the resonator dimensions, and to determine the resonator dimensions that reduce the sensitivity coefficient (step S1 in FIG. 4).
The second step is to measure the pass characteristic of a resonator having the resonator dimensions determined in the first step, and to calculate the resonant frequency and the unloaded Q value of the resonator (step S2 in FIG. 4).
The third step is to calculate the relative dielectric constant of the resonator using the resonance frequency calculated in the second step and the actual measured values of the resonator dimensions (step S3 in FIG. 4).
The fourth step is a step of calculating the conductivity of the wide wall surface, the conductivity of the narrow wall surface, and the dielectric loss tangent of the resonator using the unloaded Q value calculated in the second step, the relative dielectric constant calculated in the third step, and the actual measured values of the resonator dimensions (step S4 in FIG. 4).

The resonator characteristic measuring method according to this embodiment will be described in detail below with reference to the flowchart shown in FIG. 4. The following describes a method for measuring the characteristics of the resonator 1 having the structure shown in FIG. 1.

In step S1 of FIG. 4, the sensitivity coefficient and uncertainty with respect to the resonator dimensions are evaluated (step S1-1), and the resonator dimensions that reduce the sensitivity coefficient are determined (step S1-2). Specifically, hypothetical conditions are defined, and the sensitivity coefficient and uncertainty for the resonator 1 having the structure shown in FIG. 1 are evaluated.

Figure 5 is a flowchart for explaining the details of the operation of step S1 in Figure 4. As shown in Figure 5, in step S1, first, a sensitivity coefficient formula is created (step S11).

The following formula is a general formula for the combined uncertainty, where u _fi is the combined uncertainty of _fi , and u _xj is the uncertainty of _xj . _{∂ fi} /∂ _xj is the sensitivity coefficient. In addition, a, b, and d are the dimensions of the resonator 1 shown in Figure 1, ε _r is the relative dielectric constant, σ _f is the conductivity of the wide wall surface, σ _r,t is the conductivity of the narrow wall surface, tan δ is the dielectric tangent, and Q _u is the unloaded Q value.

The following formula is the general formula for the sensitivity coefficient:

In step S11, the sensitivity coefficient equation for each parameter is created using the general equation for the combined uncertainty and the general equation for the sensitivity coefficient described above.

Specifically, the sensitivity coefficient equation for the relative dielectric constant _εr and the dimension a is as follows:

The sensitivity coefficient equation for the relative dielectric constant _εr and the resonant frequency f is as follows:

In addition, the sensitivity coefficient equation for the dimensions d and a is as follows:

The sensitivity coefficient equation for the dimension d and the resonance frequency f is as follows:

Next, the standard uncertainty of the measured values is assumed to obtain the uncertainties (step S12). As an example, the standard uncertainty of the unloaded Q value _Qu is assumed to be 0.3%, the standard uncertainty of the dimensions a, b, and d is assumed to be 0.01%, the standard uncertainty of the resonant frequency f is assumed to be 0.01%, and the standard uncertainty of the relative dielectric constant is assumed to be 0.3%.

Then, the sensitivity coefficient is calculated when each parameter is changed (step S13). As an example, the sensitivity coefficient is calculated by setting the analysis conditions as follows: frequency 57 to 95 GHz, relative dielectric constant 2.18, dielectric loss tangent 0.001, conductivity of the wide wall surface 1.82×10 ⁷ S/m, and conductivity of the narrow wall surface 1.12×10 ⁷ S/m.

After that, the resonator dimensions are determined so that the sensitivity coefficient becomes small (step S1-2 in FIG. 4). For example, the resonator dimension a is set small. The resonator dimension b does not affect the relative dielectric constant _εr and the dimension d, but when it is increased, the uncertainty of the conductivity and dielectric loss tangent of the narrow wall surface decreases, and the uncertainty of the conductivity of the wide wall surface increases. When the resonator dimension d is increased, the uncertainty of all parameters decreases. However, when the resonator dimension d is increased, the number of modes increases, so the difference in resonant frequencies becomes small, and the Q value changes due to the influence of adjacent frequencies.

Below, steps S14 to S19 in FIG. 5 are used to explain how to determine the resonator dimensions that reduce the sensitivity coefficient. In steps S14 to S19, the effect of the spacing between adjacent resonant frequencies on the unloaded Q value is evaluated. That is, when the resonator length d is increased, the difference between the resonant frequencies becomes smaller, and the Q value changes due to the effect of the adjacent resonant frequency, so the effect of the spacing between adjacent resonant frequencies on the unloaded Q value is evaluated.

First, assume a two-resonator parallel circuit (step S14). Next, the resonant frequency and the unloaded Q value Qu are given, and the pass characteristic is calculated (step S15). Then, the unloaded Q value Qum is calculated from the pass characteristic (step S16). After that, the difference between Qu and Qu due to the interval Δf between two or more resonant frequencies is evaluated (step S17). Then, the resonant frequency interval Δf is set so as to satisfy a predetermined error range (step S18). For example, the resonant frequency interval Δf is set so that the error between the unloaded Q values Qu and Qum is 0.1% or less. Under the above conditions, in order to make the error of the unloaded Q value 0.1% or less, the interval Δf between the resonant frequencies is set to satisfy Δf>0.02. In this embodiment, the processes of steps S15 to S18 are repeated so that the error between Qu and Qum satisfies the predetermined error range. After that, the resonator dimensions are determined so that the sensitivity coefficient becomes small (step S19). In this embodiment, in order to evaluate adjacent resonant frequencies, it is sufficient to use at least two resonators in a parallel circuit, and it is also possible to use a parallel circuit with more than two resonators, i.e., a parallel circuit with two or more resonators.

Next, the pass characteristic of the resonator having the resonator dimensions determined in step S1 is measured, and the resonant frequency and unloaded Q value of the resonator are calculated (step S2 in FIG. 4). For example, resonator 2 having the configuration shown in FIG. 2 and FIG. 3 is prepared. At this time, resonator 2 is configured to have the resonator dimensions a, b, and d determined in step S1. Then, the pass characteristic of resonator 2 is measured (step S2-1).

Specifically, a waveguide 35 is connected to the coupling slot 21 of the resonator 2, and a waveguide 36 is connected to the coupling slot 22. A network analyzer (not shown) is also connected to the negative y-axis side of the waveguides 35 and 36. A high frequency wave is then generated by the network analyzer, and introduced into the resonator 2 via the waveguide 35 and the coupling slot 21. After passing through the resonator 2, the high frequency wave introduced into the resonator 2 is transmitted to the network analyzer via the coupling slot 22 and the waveguide 36. The resonator 2 having such a configuration is used to measure the transmission characteristics of the resonator.

Next, the measured pass characteristic of the resonator is used to calculate the resonant frequency and the unloaded Q value of the resonator (step S2-2). Figure 6 is a graph showing an example of the measurement result of the pass characteristic in step S2. As shown in Figure 6, the resonant frequency is the frequency at which the pass amount S21 reaches a peak.

Fig. 7 is a graph for explaining the method of calculating the Q value in step S2. As shown in Fig. 7, if the frequency on the low frequency side is _f1 and the frequency on the high frequency side is _f2 at a value 3 dB lower than the peak value of the resonance frequency _f0 , the loaded Q value _Ql can be calculated using the following formula. In other words, the loaded Q value _Ql is calculated using the pass characteristic of the 3 dB bandwidth.

The insertion loss IL ₀ of the coupling slots 21 and 22 and the waveguides 35 and 36 can be calculated using the following formula.

The unloaded Q value Q _u,m can be expressed by the following equation using the loaded Q value Q _l and the insertion loss IL ₀ .

The unloaded Q value Qu _,m is a value indicating the sharpness of the resonance, and the larger Qu _,m is, the smaller the loss is.

Next, the relative dielectric constant of the resonator is calculated using the resonant frequency calculated in step S2 and the actual measured values of the resonator dimensions (step S3 in FIG. 4). Specifically, the relative dielectric constant of the resonator is calculated using the following method.

As a precondition, the resonator dimensions a, b, d, and the relative dielectric constant _εr are unknown quantities, and the resonant frequency f _mnl , the resonant mode orders m and l, and the magnetic permeability _μr are known quantities.

In the resonator 1 having the configuration shown in Fig. 1, the resonance frequency _fmnl is expressed by the following formula: where c is the speed of light.

The above formula is transformed into the following formula.

Then, x ₁ , x ₂ , x ₃ , and x ₄ are defined as follows.

Then, the above formula can be expressed as follows using x ₁ , x ₂ , x ₃ , and x ₄ .

Here, since n=0, the above formula can be expressed as follows:

Here, the resonator dimensions a, b, and d are found by mechanically measuring the dimensions of the resonator used in step S2. Therefore, _x1 and _x3 are known values. Therefore, since the above formula reduces to simultaneous linear equations, an exact solution can be calculated for N=2, and a least-squares solution can be calculated for N>2. Thus, in step S3, the relative dielectric constant _εr of the resonator can be calculated using the actual measured values of the resonator dimensions and the above linear equation.

Next, the conductivity of the wide wall surface, the conductivity of the narrow wall surface, and the dielectric loss tangent of the resonator are calculated using the unloaded Q value calculated in step S2, the relative dielectric constant calculated in step S3, and the actual measured values of the resonator dimensions (step S4 in FIG. 4). Specifically, the conductivity of the wide wall surface, the conductivity of the narrow wall surface, and the dielectric loss tangent of the resonator are calculated using the following method.

As a prerequisite, the electrical conductivity of the wide wall surface of the resonator _σf , the electrical conductivity of the narrow wall surface σr _,t , and the dielectric tangent tan δ are unknowns. In addition, the resonator dimensions a, b, d, magnetic permeability μ, wave number k, angular frequency ω, unloaded Q value Qu, and resonant mode order (z-axis direction) l are known quantities.

In the resonator 1 having the configuration shown in Fig. 1, the unloaded Q value Q _u,a is expressed by the following formula. From the following formula, it can be said that the unloaded Q value is determined according to the current flowing through the wall surface of the resonator. Note that k is the wave number, η is the impedance of free space, and R _i,j and σ _i,j are the skin resistance and conductivity for the current flowing in the j direction in the i plane.

Moreover, the unloaded Q value _Qu has the following relationship.

Moreover, 1/ _Qc is expressed as follows.

In this case, _x5 and _x6 are defined as follows, ( ^l2a3d + ^ad3 ) ×[2π2 ^/ {(kad) ^3bη }] ×(ωμ/2) ^{1/2 is set to a1, and (2l2a3b +} _2bd3 ^{) ×} [2π2 ^/ {(kad) ^3bη }] ×(ωμ/2) ^1/2 is set to _a2 , so that ₁ / _Qc = _a1x5 ⁺ _{a2x6 .}

Moreover, 1/ _Qd is expressed as follows.

Therefore, 1/Q _u is ultimately expressed by the following linear equation:

Measuring N resonant modes results in N linear equations. In this case, since there are three unknowns, _x5 , _x6 , and _x7 , an exact solution is obtained when N=3, and a least-squares solution is obtained when N>3. Also, errors can be absorbed by measuring four or more resonant modes. Note that _x5 is the square root of the reciprocal of the conductivity _σf of the wide wall surface, _x6 is the square root of the reciprocal of the conductivity σr _,t of the narrow wall surface, and _x7 is the dielectric loss tangent tan δ.

In the resonator characteristics measurement method according to the present embodiment described above, the conductivity of the wide wall surface, the conductivity of the narrow wall surface, and the dielectric loss tangent can be obtained at once for a single resonator, so that the characteristics of the resonator can be easily obtained. In other words, in the resonator characteristics measurement method according to the present embodiment, the conductivity of the wide wall surface, the conductivity of the narrow wall surface, and the dielectric loss tangent can be obtained at once without measuring multiple resonators, so that the characteristics of the resonator can be easily obtained. In addition, in the resonator characteristics measurement method according to the present embodiment, in step S2, the pass characteristics of the resonator are measured using a resonator having the configuration shown in Figures 2 and 3, so that the electrical properties can be evaluated in a form close to a practical form.

The present invention has been described above in accordance with the above embodiment, but the present invention is not limited to the configuration of the above embodiment, and of course includes various modifications, alterations, and combinations that a person skilled in the art could make within the scope of the invention of the claims of this patent application.

1, 2 Resonator 11, 12 Coupling slot 21, 22 Coupling slot 31, 32 Conductor substrate 33 Metal via (through conductor)
35, 36 Waveguide

Claims (7)

a first step of evaluating a sensitivity coefficient and an uncertainty with respect to a resonator size and determining a resonator size that reduces the sensitivity coefficient;
a second step of measuring a pass characteristic of a resonator having the resonator dimensions determined in the first step, and calculating a resonant frequency and an unloaded Q value of the resonator;
a third step of calculating a relative dielectric constant of the resonator using the resonance frequency calculated in the second step and actual measurements of the resonator dimensions;
a fourth step of calculating a conductivity of a wide wall surface, a conductivity of a narrow wall surface, and a dielectric loss tangent of the resonator using the unloaded Q value calculated in the second step, the relative dielectric constant calculated in the third step, and an actual measurement value of the resonator dimension.
Method for measuring resonator characteristics. The resonator is a rectangular parallelepiped resonator. If the dimension of the resonator in the width direction is a, the dimension in the thickness direction is b, and the dimension in the longitudinal direction is d, then:
In the first step, a sensitivity coefficient equation between the dielectric constant and the dimension a, a sensitivity coefficient equation between the dielectric constant and the resonant frequency, a sensitivity coefficient equation between the dimension d and the dimension a, and a sensitivity coefficient equation between the dimension d and the resonant frequency are prepared;
Assuming a standard uncertainty of the unloaded Q value, a standard uncertainty of the dimensions a, b, and d, a standard uncertainty of the resonant frequency, and a standard uncertainty of the dielectric constant;
Calculate the sensitivity coefficient when the dimensions a, b, and d are changed.
The method for measuring resonator characteristics according to claim 1 . The method for measuring resonator characteristics according to claim 1 or 2, wherein in the first step, the spacing between adjacent resonant frequencies is set so that the error in the unloaded Q value is equal to or less than a predetermined value, and then the resonator dimensions are determined so that the sensitivity coefficient is small. The interval between adjacent resonant frequencies is set as follows:
Assuming a parallel circuit of two or more resonators,
Given the resonant frequency and the unloaded Q value Qu, calculate the pass characteristic.
Calculate the unloaded Q value Qum from the pass characteristic;
This is implemented by setting the interval between adjacent resonant frequencies so that the difference between Qu and Qum is equal to or less than a predetermined value.
The method for measuring resonator characteristics according to claim 3 . The resonator characteristics measuring method according to any one of claims 1 to 4, wherein in the second step, the transmission characteristics of a resonator are measured, the resonator comprising first and second conductor substrates arranged to face each other and metal vias arranged between the first conductor substrate and the second conductor substrate, and the space surrounded by the first conductor substrate, the second conductor substrate, and the plurality of metal vias functions as a waveguide. In the third step,
The relative dielectric constant _εr is an unknown quantity, and the resonator dimensions a, b, d, resonance frequency f _mnl , resonance mode orders m, l, and relative permeability _μr are known quantities,
Calculate the relative dielectric constant of the resonator using the measured values of the resonator dimensions and the following linear equation:
The method for measuring resonator characteristics according to any one of claims 1 to 5.
Here, the resonator is a rectangular parallelepiped resonator, a is the dimension in the width direction of the resonator, b is the dimension in the thickness direction, d is the dimension in the longitudinal direction, n=0, and x ₁ , x ₂ , x ₃ , and x ₄ are defined as follows.
In the fourth step,
The electrical conductivity σ _f of the wide wall surface of the resonator, the electrical conductivity σ _r,t of the narrow wall surface, and the dielectric loss tangent tan δ are unknown quantities, and the resonator dimensions a, b, d, magnetic permeability μ, wave number k, impedance η of free space, angular frequency ω, unloaded Q value Qu, and resonance mode order l are known quantities,
Calculate the broad wall conductivity, narrow wall conductivity, and dielectric tangent of the resonator using the following linear equations:
The method for measuring resonator characteristics according to any one of claims 1 to 6.
Here, the resonator is a rectangular parallelepiped resonator, a is the width dimension of the resonator, b is the thickness dimension, and d is the longitudinal dimension, _a1 = ( ^l2a3d + ^{ad3 )} × [2π2 ^/ {(kad) ^3bη }] × (ωμ/2) ^{1/2 ,} _a2 = ( ^2l2a3b + ^2bd3 ) × [2π2 ^/ {(kad) ^3bη }] × (ωμ/2) ^1/2 , and _x5 , _x6 , and _x7 are defined as follows:

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