CN115020954A - Taylor distribution power divider based on Wilkinson form and design method thereof - Google Patents
Taylor distribution power divider based on Wilkinson form and design method thereof Download PDFInfo
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Abstract
The invention discloses a Taylor distribution power divider based on a Wilkinson form and a design method thereof, wherein the design method comprises the following steps: selecting a Taylor weight according to the power gain intensity of the antenna side lobe, and calculating the power distribution of each port by combining the output port number of the Taylor distribution power divider; taking the reduction of the topological stage as a design target, and adopting a one-to-two power divider in a Wilkinson form to carry out topological design to obtain a topological structure of the Taylor distribution power divider; considering the voltage standing wave ratio of the input and output ports and the requirements of amplitude and phase consistency, carrying out simulation design on one-to-two power dividers in the Taylor distribution power divider one by one; and calculating the phase consistency of the Taylor distributed power divider, and adjusting the wiring length between each level in the topological structure one by one according to the magnitude of the absolute value of the phase until the phase consistency requirement is met. By adopting the technical scheme, the loss of the power divider is reduced, the design flow of the power divider network is simplified, and the design efficiency is improved.
Description
Technical Field
The invention relates to the technical field of antenna design, in particular to a Taylor distribution power divider based on a Wilkinson form and a design method thereof.
Background
The early warning and monitoring radar is generally provided with a phased array antenna for pitching electrical scanning and azimuth mechanical scanning, and in order to effectively suppress ground and sea clutter and active interference, an azimuth directional diagram of the antenna has a low sidelobe characteristic. Under the condition of less active channels, power distribution can be carried out on each unit through the power distribution network, amplitude weighting is achieved, and finally a directional diagram with low side lobes is formed.
The power division network is a feeder line part in an antenna system, and distributes or synthesizes signal power, so that different units of the antenna have different transmitting or receiving amplitudes, and an antenna directional diagram with low side lobes is formed. The design of the power distribution network directly influences the side lobe level of the antenna, and particularly when the power distribution network has low side lobe requirements, the power distribution network needs to achieve low voltage standing wave ratio and high amplitude phase consistency. The power divider based on the Taylor distribution has the characteristics of easy realization of the distribution ratio and high amplitude and phase consistency, and is widely used.
The wilkinson power divider has the characteristics of small standing wave, high isolation degree and high amplitude-phase consistency, and is commonly used in a topological structure (power dividing network) of the power divider, but in the prior art, the wilkinson power divider has the following problems when used in large-scale and multi-stage cascade due to the complex model: (1) because of the large number of ports and large scale, the number of topological stages is large, and the loss of the power divider is large; (2) the design idea is designed step by step, namely firstly, the design of a one-to-two power divider is carried out, then, the design of a one-to-four power divider is carried out, then, the design of a one-to-eight power divider is carried out, and the analogy is repeated, so that the design of the power divider is completed, the design workload is large, and the debugging workload is also large.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a Wilkinson-form-based Taylor distribution power divider and a design method thereof, which reduce the number of topological stages under the condition that the output ports of the Taylor distribution power divider have the same number, and further reduce the loss of the power divider; when the one-to-two power divider in the Taylor distribution power divider meets the design index, the voltage standing wave ratio and the amplitude consistency do not need to be adjusted and optimized, the design flow of the power divider network is obviously simplified, and the design efficiency is improved.
The technical scheme is as follows: the invention provides a design method of a Taylor distribution power divider based on a Wilkinson form, which comprises the following steps: selecting a Taylor weight according to the power gain intensity of the antenna side lobe, and calculating the power distribution of each port by combining the output port number of the Taylor distribution power divider; according to the number of output ports of the Taylor distribution power divider, taking reduction of topological series as a design target, and adopting a one-to-two power divider in a Wilkinson form to carry out topological design to obtain a topological structure of the Taylor distribution power divider; considering the voltage standing wave ratio of the input and output ports and the requirements of amplitude and phase consistency, carrying out simulation design on one-to-two power dividers in the Taylor distribution power divider one by one; and calculating the phase consistency of the Taylor distribution power divider, and adjusting the wiring length between each level in the topological structure one by one according to the absolute value of the phase until the phase consistency requirement is met.
Specifically, on the basis of the power gain intensity of the antenna side lobe, 7dB is subtracted to obtain the taylor weight.
Specifically, the number of output ports of the taylor distributed power divider is 32, and the number of topological stages of the taylor distributed power divider is 6.
Specifically, the voltage standing wave ratio of the input/output port is required to be less than 1.15, the amplitude consistency is required to be less than 0.05dB, and the phase consistency is required to be less than 0.5 °.
Specifically, the simulation design is performed on the one-to-two power divider in the taylor distribution power divider one by one, and the following requirements are met: the power capacity of the isolation resistor in each one-to-two power divider meets the following conditions: power capacity > 2 (P) General assembly -S Isolation of ) Wherein P is General assembly Representing the active power, S, of a one-to-two power divider Isolation of Representing the apparent power of the isolation resistor of the one-to-two power divider; the distance between the output ports of the one-to-two power divider in the sixth stage of the topological structure of the Taylor distribution power divider is equal to the distance between the antenna units; the line spacing of the connecting lines in the topological structure of the Taylor distribution power divider is larger than 2.5 times of the line width.
Specifically, according to a simulation result of one-to-two power dividers in the taylor distribution power divider, which is subjected to simulation design one by one, a central frequency point is selected, and phase consistency is calculated.
Specifically, the routing lengths between every two stages are adjusted one by one according to the sequence of the topological stages of the Taylor distributed power divider from large to small, the routing is shortened when the absolute value of the phase is small, and the routing is lengthened when the absolute value of the phase is large; in the process of adjusting the length of the wiring between the two stages, when the smaller stage meets the requirement of phase consistency, the adjustment between the two stages is completed.
Specifically, when the central frequency point meets the phase consistency requirement and the low-frequency and high-frequency phase consistency is inconsistent with the intermediate-frequency phase consistency, the length difference of the routing between the two stages of power dividers is shortened.
The invention also provides a taylor distribution power divider based on the wilkinson form, the topological structure of which is obtained by carrying out topological design according to the one-to-two power divider adopting the wilkinson form recorded in the invention, and the method comprises the following steps: the topological structure of the Taylor distributed power divider comprises 6 stages, a left half-side structure and a right half-side structure are symmetrical, in the half-side structure, a first stage comprises 1 one-to-two power divider, and two output ports are respectively connected with a second stage and a fifth stage; the second stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the third stage and the fifth stage; the third stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the fourth stage and the fifth stage; the fourth stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the fifth stage and the sixth stage; the fifth stage comprises 4 one-to-two power dividers, wherein two output ports of two one-to-two power dividers are both connected with the 6 th stage, one output port of the other two one-to-two power dividers is connected with the 6 th stage, and the other output port is used as the output port of the taylor distribution power divider.
Specifically, if the Taylor distribution power divider has resonance, a metal block is arranged at a position where a wire is not arranged in a cavity of the Taylor distribution power divider, the distance between the position of the metal block and the wire is larger than 3 times of the line width, the resonance frequency point reaching the low frequency is smaller than 0.97 times of the working low-frequency, and the resonance frequency point reaching the high frequency is larger than 1.03 times of the working high-frequency.
Has the advantages that: compared with the prior art, the invention has the following remarkable advantages: the loss of the power divider is reduced, the design flow of the power divider network is simplified, and the design efficiency is improved.
Drawings
Fig. 1 is a schematic flow chart of a design method of a taylor distributed power divider according to the present invention;
FIG. 2 is a schematic structural diagram of a Wilkinson power divider;
fig. 3 is a structural diagram of the left half side of the taylor distributed power divider provided by the present invention.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
Fig. 1 is a schematic flow chart of a design method of a taylor distributed power divider according to the present invention.
In the embodiment of the invention, on the basis of the power gain intensity of the antenna side lobe, 7dB is subtracted to obtain the Taylor weight.
In specific implementation, when the phase difference between a difference branch and a sum branch of an antenna is an integral multiple of 180 degrees, phase consistency is called, and the amplitude of a sum channel signal is normalized, amplitude consistency is called, and when the sum channel meets the requirement of amplitude consistency, a radar angle servo system can complete stable tracking of a target angle.
In specific implementation, according to the number of antenna elements and the power gain strength requirement of the side lobe (taking-30 dB as an example), considering the coupling effect between ports and the reserved margin required by the side lobe, a taylor weight can be selected as the power gain strength of the side lobe minus 7dB, so that a relatively wide design space can be obtained, the characteristic of the low side lobe is ensured, and meanwhile, when the power of each output port of the taylor distribution power divider is calculated according to taylor distribution, the power ratio is relatively small, and does not exceed or only slightly exceeds the power distribution ratio of 2:1 (the power distribution ratio is greater than 0.71 or less than 1.41), the amplitude and phase consistency of the side lobe is ensured, that is, the performance of the side lobe is ensured, therefore, on the basis of the power gain strength of the side lobe of the antenna, 7dB is subtracted to obtain the taylor weight, which is a relatively reasonable and comprehensive taylor weight calculation mode.
In a specific implementation, the characteristic of the unequal power distribution of the taylor distribution is that the smaller the taylor weight is, the larger the maximum power ratio is when the same topological series and the number of output ports are determined, and the smaller the number of ports is, the larger the maximum power ratio is when the same topological series and the number of taylor weight are selected. Therefore, the Taylor weight value obtained by subtracting 7dB on the basis of the power gain intensity of the antenna side lobe can be determined, and the amplitude and phase consistency of the side lobe can be ensured while the characteristic of the low side lobe is ensured.
Taking the port number of 32 and the weight of-37 dB Taylor distribution as an example, the power distribution is shown in Table 1.
Table 1:
and 2, according to the number of output ports of the Taylor distribution power divider, performing topological design by adopting a one-to-two power divider in a Wilkinson mode by taking the reduction of the topological series as a design target to obtain a topological structure of the Taylor distribution power divider.
Fig. 2 is a schematic structural diagram of a wilkinson (wilkinson) power divider. The return loss performance of three ports of the Wilkinson power divider is good, and meanwhile, the signal isolation degree between the two output ports is also good. Normalizing the impedance of each port to obtain a characteristic impedance Z on the transmission line 1 = 2, the electrical length of the transmission line is 90 °
In the embodiment of the present invention, the number of output ports of the taylor distributed power divider is 32, and the number of topological stages of the taylor distributed power divider is 6.
Fig. 3 is a structural diagram of the left half side of the taylor distributed power divider according to the present invention.
The invention also provides a Wilkinson-form-based Taylor distribution power divider, and the topological structure of the Wilkinson-form-based Taylor distribution power divider is obtained by performing topological design on a one-to-two power divider adopting a Wilkinson form according to the design method of the Taylor distribution power divider provided by the invention.
In the embodiment of the invention, the topological structure of the Taylor distribution power divider comprises 6 stages, a left half-side structure and a right half-side structure are symmetrical, in the half-side structure, a first stage comprises 1 one-to-two power divider, and two output ports are respectively connected with a second stage and a fifth stage; the second stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the third stage and the fifth stage; the third stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the fourth stage and the fifth stage; the fourth stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the fifth stage and the sixth stage; the fifth stage includes 4 one-to-two power dividers, where two output ports of two one-to-two power dividers are both connected to the 6 th stage, one output port of the other two one-to-two power dividers is connected to the 6 th stage, and the other output port is used as an output port of the taylor distribution power divider.
In a specific implementation, the one-to-two power divider is connected to other topological stages, specifically, is connected to an input port of the one-to-two power divider in other topological stages, for example, the first stage includes 1 one-to-two power divider, two output ports are respectively connected to the second stage and the fifth stage, and two output ports of one-to-two power divider in the first stage are respectively connected to an input port of one-to-two power divider in the second stage and the fifth stage.
In the specific implementation, when the Wilkinson power division form is adopted, the one-to-N power divider is composed of N-1 one-to-two cascades. Because the Taylor distribution is symmetrical left and right, only the half side of the Taylor distribution power divider needs to be designed in a topological mode, and the structure of the right half side is the same as that of the left half side.
In the implementation, the topology design is performed according to two principles, namely, reducing the number of topology stages as much as possible to reduce the loss of the power divider, and not generating the power division ratio larger than 2:1 (the power division ratio is larger than 0.71 or smaller than 1.41). The topological stage number of the traditional taylor distribution power divider with 32 output ports is 7 or more, and in order to further reduce the topological stage number to reduce the loss of the power divider, the limitation that the power distribution ratio is larger than 2:1 is not generated, so that the amplitude distribution and the phase consistency of the ports of the power divider are influenced, and the secondary lobe performance of an antenna is further influenced. In order to reduce the influence on the amplitude distribution and phase consistency of the ports of the taylor power divider, an improved topology design is selected for the port with smaller distributed power, and a schematic diagram of an improved topology structure is shown in fig. 3.
In a specific implementation, the power ratios of the 7-level topology are compared to the 6-level topology, as shown in table 2.
Table 2:
in a specific implementation, by comparing the 6-stage topology with the 7-stage topology, it can be known that the power division ratio of the port with higher allocated power is not changed, and the power division ratio of the port with lower allocated power is slightly larger. In order to solve the problem, only 2 one-to-two power dividers with slightly larger power divider ratio need to be subjected to performance index control during one-to-two simulation design, so that the overall amplitude distribution and phase consistency of a one-to-sixteen Taylor power divider cannot be influenced, and the power divider with the 6-level topological structure has the advantage of remarkably reducing the loss.
And 3, considering the voltage standing wave ratio of the input and output ports and the requirements of amplitude and phase consistency, and performing simulation design on the one-to-two power dividers in the Taylor distribution power divider one by one.
In the embodiment of the invention, the requirement of the voltage standing wave ratio of the input/output port is less than 1.15, the requirement of the amplitude consistency is less than 0.05dB, and the requirement of the phase consistency is less than 0.5 degrees.
In specific implementation, on the premise of meeting the above design index requirements, the standing-wave ratio and the amplitude consistency do not need to be optimized, and only phase consistency optimization is needed, so that a certain margin can be reserved according to the design index requirements, and the performance of the designed Taylor distribution power divider is excellent under the above design index requirements.
In the embodiment of the invention, the power capacity of the isolation resistor in each one-to-two power divider meets the following conditions: power capacity > 2 (P) General assembly -S Isolation of ) Wherein P is General assembly Representing the active power, S, of a one-to-two power divider Isolation of Representing the apparent power of the isolation resistor of the one-to-two power divider; the distance between the output ports of the one-to-two power divider in the sixth stage of the topological structure of the Taylor distribution power divider is equal to the distance between the antenna units when the output ports of the one-to-two power divider are integrated with the antenna; the line spacing of the connecting lines in the topological structure of the Taylor distribution power divider is larger than 2.5 times of the line width.
In the specific implementation, the power capacity of the one-to-two isolation resistor, the distance between the output ports and the line distance are numerically constrained, so that the overall amplitude distribution and phase consistency of the Taylor power divider and the characteristic of low side lobe are ensured.
In particular implementations, CAD model stitching may occur after the design is simulated. CAD stitching is a structural implementation of topological design. The length of the wiring is reduced as much as possible to reduce power division loss, and the line spacing needs to be larger than 2.5 times of the line width. The CAD model splicing steps are as follows: after 15 one-to-two simulation designs are finished, respectively exporting dxf files and marking; opening 15 dxf files in the CAD to generate DWG files; arranging one-to-two DWG files corresponding to the sixth level in the topological design drawing according to the port spacing; connecting a one-to-two DWG file corresponding to a fifth level in the topological design drawing to a sixth level, wherein the connecting line uses 50 ohm wiring; sequentially connecting the fourth stage, the third stage, the second stage and the first stage into two DWG files, wherein the connecting line uses 50 ohm wiring; selecting the position of a first-stage one-to-two main port according to the structural requirement; completing model splicing to generate a DWG file; and importing the merged DWG file into HFSS, and performing parameter setting and simulation.
And 4, calculating the phase consistency of the Taylor distribution power divider, and adjusting the wiring length between each level in the topological structure one by one according to the magnitude of the absolute value of the phase until the phase consistency requirement is met.
In the embodiment of the invention, according to the simulation result of one-to-two power dividers in the Taylor distribution power divider, which is subjected to simulation design one by one, the central frequency point is selected, and the phase consistency is calculated.
In the embodiment of the invention, the wiring lengths between every two stages are adjusted one by one according to the sequence of the topological stages of the Taylor distributed power divider from large to small, the wiring is shortened when the absolute value of the phase is small, and the wiring is lengthened when the absolute value of the phase is large; in the process of adjusting the length of the wiring between the two stages, when the smaller stage meets the requirement of phase consistency, the adjustment between the two stages is completed.
In the specific implementation, the phase absolute value is obtained by comparing with a standard absolute value, the standard absolute value can be set correspondingly according to the actual application scene, and in the invention, the phase absolute value can be obtained by calculation based on the topological structure, the phase consistency requirement and the like of the Taylor distribution power divider.
In the embodiment of the invention, when the central frequency point meets the phase consistency requirement and the low-frequency and high-frequency phase consistency is inconsistent with the intermediate-frequency phase consistency, the length difference of the routing between the two stages of power dividers is shortened.
In the specific implementation, on the premise that the 15 one-to-two power dividers meet the design index requirements of voltage standing wave ratio, amplitude consistency and phase consistency, the one-to-sixteen voltage standing wave ratio and the amplitude consistency are not required to be optimized, and only the phase consistency is required to be optimized. Because the cavity structure size of the one-in-sixteen power divider is large, resonance can occur inevitably, and the metal block needs to be added in the cavity and the size and the position of the metal block are optimized to perform resonance elimination.
In a specific implementation, when the absolute value of the phase is smaller, it means that the trace of the output port of the one-to-two power divider is longer, and when the absolute value of the phase is larger, it means that the trace of the output port of the one-to-two power divider is shorter. When the central frequency point meets the phase consistency requirement and the low-frequency and high-frequency phase consistency is inconsistent with the intermediate-frequency phase consistency, the difference value of the routing length between the output ports of the power divider exceeds the microwave period of the antenna, and the relation between the corresponding regulating quantity and the phase consistency can be calculated by software such as ADS or TXLine.
In a specific implementation, in adjusting the trace length between two levels, when the smaller level meets the phase consistency requirement, the adjustment between the two levels is completed. For example, when the trace length between the sixth-level power divider and the fifth-level power divider is adjusted, the trace length between the fourth-level power divider and the fifth-level power divider is adjusted until the fifth-level power divider meets the requirement for phase consistency, and then the trace length between the fourth-level power divider and the fifth-level power divider is adjusted until the fourth-level power divider meets the requirement for phase consistency, and so on, the adjustment of the taylor distribution power divider is completed.
In the embodiment of the invention, after the simulation design and the CAD model are spliced, for the Taylor distribution power divider, if the Taylor distribution power divider has resonance, in order to remove the resonance, a metal block is arranged at a position where no wire is arranged in a cavity of the Taylor distribution power divider, the distance between the position of the metal block and the wire is more than 3 times of the line width, the resonance frequency point of the low frequency is less than 0.97 times of the working low frequency, and the resonance frequency point of the high frequency is more than 1.03 times of the working high frequency.
In specific implementation, when the Taylor distribution power divider designed according to the design method provided by the invention works at a relative bandwidth of 15%, the voltage standing wave ratio of a port of a simulation result is less than 1.25, the amplitude consistency is less than 0.2dB (the amplitude difference between output ports is compared with the theoretical amplitude difference), the phase consistency of the port is less than +/-2 degrees, and the isolation of the port is greater than 18 dB. The design process of the power distribution network is simplified, the design efficiency is improved, and meanwhile, the performance is excellent.
Claims (10)
1. A design method of a Taylor distribution power divider based on a Wilkinson form is characterized by comprising the following steps:
selecting a Taylor weight according to the power gain intensity of the antenna side lobe, and calculating the power distribution of each port by combining the output port number of the Taylor distribution power divider;
according to the number of output ports of the Taylor distribution power divider, taking reduction of topological series as a design target, and adopting a one-to-two power divider in a Wilkinson form to carry out topological design to obtain a topological structure of the Taylor distribution power divider;
considering the voltage standing wave ratio of the input and output ports and the requirements of amplitude and phase consistency, carrying out simulation design on one-to-two power dividers in the Taylor distribution power divider one by one;
and calculating the phase consistency of the Taylor distributed power divider, and adjusting the wiring length between each level in the topological structure one by one according to the magnitude of the absolute value of the phase until the phase consistency requirement is met.
2. The design method of a Wilkinson-form-based Taylor distribution power divider as claimed in claim 1, wherein the selecting Taylor weights according to the power gain intensity of the antenna side lobes comprises:
and subtracting 7dB on the basis of the power gain intensity of the antenna side lobe to obtain a Taylor weight.
3. The Wilkinson-form-based Taylor distributed power divider design method according to claim 2, wherein the number of output ports of the Taylor distributed power divider is 32, and the number of topological stages of the Taylor distributed power divider is 6.
4. The Wilkinson-form-based Taylor distributed power divider design method according to claim 3, wherein the voltage standing wave ratio of the input and output ports is required to be less than 1.15, the amplitude consistency is required to be less than 0.05dB, and the phase consistency is required to be less than 0.5 °.
5. The design method of the Wilkinson-form-based Taylor distribution power divider as claimed in claim 4, wherein the simulation design is performed on one-to-two power dividers in the Taylor distribution power divider one by one, and the following requirements are met:
the power capacity of the isolation resistor in each one-to-two power divider meets the following conditions: power ofCapacity > 2 (P) General assembly -S Isolation of ) Wherein P is General assembly Representing the active power, S, of a one-to-two power divider Isolation of Representing the apparent power of the isolation resistor of the one-to-two power divider; the distance between the output ports of the one-to-two power divider in the sixth stage of the topological structure of the Taylor distribution power divider is equal to the distance between the antenna units; the line spacing of the connecting lines in the topological structure of the Taylor distribution power divider is larger than 2.5 times of the line width.
6. The Wilkinson-form-based Taylor distribution power divider design method according to claim 5, wherein the calculating the phase consistency of the Taylor distribution power divider comprises:
and selecting a central frequency point according to a simulation result of one-to-two power divider in the Taylor distribution power divider by one simulation design, and calculating phase consistency.
7. The Wilkinson-form-based Taylor distribution power divider design method according to claim 6, wherein the adjusting the trace length between each stage in the topology one by one according to the magnitude of the absolute value of the phase comprises:
according to the sequence of the topological stages of the Taylor distribution power divider from large to small, the wiring length between every two stages is adjusted one by one, the wiring is shortened when the absolute value of the phase is small, and the wiring is increased when the absolute value of the phase is large; in the process of adjusting the length of the wiring between the two stages, when the smaller stage meets the requirement of phase consistency, the adjustment between the two stages is completed.
8. The Wilkinson-form-based Taylor distribution power divider design method of claim 7, wherein the adjusting the trace length between each level in the topology one by one comprises:
when the central frequency point meets the phase consistency requirement and the low-frequency and high-frequency phase consistency is inconsistent with the intermediate-frequency phase consistency, the length difference of the routing between the two stages of power dividers is shortened.
9. A taylor distributed power divider based on wilkinson form, characterized in that the topology is designed according to the one-to-two power divider using wilkinson form as recited in claim 1, and the taylor distributed power divider includes:
the topological structure of the Taylor distributed power divider comprises 6 stages, a left half-side structure and a right half-side structure are symmetrical, in the half-side structure, a first stage comprises 1 one-to-two power divider, and two output ports are respectively connected with a second stage and a fifth stage; the second stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the third stage and the fifth stage; the third stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the fourth stage and the fifth stage; the fourth stage comprises 1 one-to-two power divider, and two output ports are respectively connected with the fifth stage and the sixth stage; the fifth stage comprises 4 one-to-two power dividers, wherein two output ports of two one-to-two power dividers are both connected with the 6 th stage, one output port of the other two one-to-two power dividers is connected with the 6 th stage, and the other output port is used as the output port of the taylor distribution power divider.
10. The Wilkinson-form-based Taylor distribution power divider of claim 9, wherein if the Taylor distribution power divider has resonance, a metal block is disposed at a position where no wire is routed in a cavity of the Taylor distribution power divider, a distance between the position of the metal block and the wire is greater than 3 times of a line width, a resonance frequency point of a low frequency is less than 0.97 times of a low frequency of operation, and a resonance frequency point of a high frequency is greater than 1.03 times of a high frequency of operation.
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