Resonant frequency detection and adjustment apparatus and method for differential transformer resonant bridge

By combining a differential transformer resonant bridge device and a fast-tuning capacitor bridge, and utilizing analog-to-digital conversion and digital control, high-precision detection and adjustment of the resonant frequency in inertial sensors are achieved, solving the detection and adjustment problems in existing technologies and improving the accuracy and reliability of the sensors.

CN120934360BActive Publication Date: 2025-12-05CHANGCHUN INST OF OPTICS FINE MECHANICS & PHYSICS CHINESE ACAD OF SCI
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Patent Information

Application Number
CN202511421281.2
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2025-09-30
Publication Date
2025-12-05
Estimated Expiration
2045-09-30

AI Technical Summary

Technical Problem

Existing technologies make it difficult to conveniently, quickly, and accurately detect and adjust the resonant frequency in inertial sensors, resulting in reduced sensor accuracy. Furthermore, the tuning capacitor is difficult to operate after being integrated into the whole machine and cannot be precisely adjusted.

Method used

A differential transformer resonant bridge device is used. The peak-to-peak value of the system link output voltage is read through the analog-to-digital conversion circuit. The resonant frequency is adjusted by combining it with a fast-tuning capacitor bridge. The signal acquisition and demodulation are controlled by a digital control circuit to achieve high-precision resonant frequency detection and adjustment.

Benefits of technology

It achieves high-precision and rapid resonant frequency detection and adjustment, avoids repetitive operations on the front-end circuit, ensures the safety and reliability of the sensor, and improves the sensor's performance.

✦ Generated by Eureka AI based on patent content.

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Abstract

The present application relates to the technical field of electronic circuit, especially to a differential transformer resonant bridge resonant frequency detection and adjustment device and method, a fast tuning capacitor bridge is composed of multiple tuning capacitors, by adjusting the capacitance value of the tuning capacitor in the access circuit, the change of the resonant frequency is completed; in addition, the voltage data is collected by using an analog-to-digital conversion circuit, and the minimum value of the output voltage peak-to-peak value is directly observed by the host computer to judge the resonant state. The device and method provided by the present application can detect the resonant frequency, and adjust the tuning range according to the demand, the tuning capacitor step precision can be adjusted according to the demand, the tuning precision is high, the efficiency of adjusting the resonant frequency is improved; at the same time, the fast tuning capacitor bridge is a separate adjustment unit, which effectively avoids the repeated welding and unwelding of the front-end circuit when changing the tuning capacitor during the tuning process, ensures the safety and reliability of the front-end bridge, and better meets the engineering application requirements.
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Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of electronic circuits, and particularly relates to a differential transformer resonant bridge resonant frequency detection and adjustment device and method. BACKGROUND

[0002] The differential variable gap capacitance displacement sensor has been used in the front-end electronics of the inertial sensor for gravitational wave detection, wherein the resonant bridge noise is a main influencing factor of the sensor performance and directly affects the sensor accuracy. When the resonant bridge works at the resonant frequency, the bridge thermal noise is minimum, and at this time, the sensor accuracy is highest. When the transformer parameters, cable length, tuning capacitor, stray capacitance and the like change, the resonant frequency will also deviate from the working frequency, resulting in that the bridge thermal noise becomes larger and the sensor accuracy is reduced. The convenient, fast and accurate resonant frequency judgment and resonant frequency adjustment method can effectively improve the sensor performance.

[0003] Currently, there are mainly two resonant frequency judgment methods. One is to observe the resonant frequency through a spectrum analyzer, and the frequency at which the noise minimum value is obtained is the current resonant frequency. This method measures the front-end electronics noise rather than the complete system link noise, and the judgment accuracy is limited by the resolution of the spectrum analyzer. The other is to change the excitation signal frequency, and observe the output voltage value corresponding to the excitation signal at different frequencies. The maximum output voltage value is the current resonant frequency. This method has higher requirements for the front-end bandwidth, and requires that the gain is flat in a wide frequency band.

[0004] When the resonant frequency deviates from the working frequency, the tuning capacitor of the resonant bridge is adjusted to adjust the resonant frequency to the expected state. In the existing scheme, the tuning capacitor is in the front-end electronics circuit. When the resonant frequency deviates, the final tuning capacitor value is confirmed by repeatedly adjusting the tuning capacitor of the front-end circuit. However, in this scheme, once the whole machine is integrated and the system state is fixed, the operation is difficult, the circuit is not easy to change, and based on the above limitations, even accurate resonant frequency adjustment cannot be performed. SUMMARY

[0005] Therefore, the present application aims to provide a differential transformer resonant bridge resonant frequency detection and adjustment device and method. When the resonant state is reached at the working frequency, the circuit noise is minimum, the peak-to-peak value of the output voltage of the complete system link is read through an analog-to-digital conversion circuit to judge the resonant frequency. When the resonant frequency deviates, the resonant frequency is adjusted through the resonant bridge with the tuning capacitor, and the resonant frequency adjustment is no longer limited, which can ensure the tuning accuracy and efficiency.

[0006] To achieve the above purpose, the technical scheme of the present application is as follows:

[0007] The application discloses a differential transformer resonant bridge resonant frequency detection and adjustment device, which comprises an input circuit, a resonant bridge, a signal conversion circuit, a demodulation circuit, an analog-digital conversion circuit and an upper computer.

[0008] Further, in each bridge arm, each tuning capacitor has a corresponding switch, and by adjusting the switch, the corresponding tuning capacitor is controlled to be connected to the bridge arm, and then the number of the tuning capacitors connected to the fast tuning capacitor bridge is adjusted, so that the capacitance value of the tuning capacitors connected to the fast tuning capacitor bridge is changed.

[0009] Further, the resonant bridge further comprises a differential transformer; the differential transformer comprises two primary windings and one secondary winding, the two bridge arms are connected to the two primary windings correspondingly, and the secondary winding is connected to the signal conversion circuit.

[0010] Further, the input circuit comprises a test mass, the excitation signal is applied to the test mass through an electrode plate, and the capacitance signal between the test mass and an upper electrode plate and the capacitance signal between the test mass and a lower electrode plate are respectively converted into current signals and transmitted to the two bridge arms of the fast tuning capacitor bridge.

[0011] Further, the signal conversion circuit is further used for amplifying the voltage signal.

[0012] Further, the device further comprises a filtering module connected to the signal conversion circuit and used for filtering out the out-band noise in the voltage signal output by the signal conversion circuit.

[0013] Further, the device further comprises a digital control circuit connected to the analog-digital conversion circuit, the signal conversion circuit, the demodulation circuit and the upper computer, wherein the digital control circuit controls the signal acquisition and conversion of the analog-digital conversion circuit; the digital control circuit controls the amplification gain of the voltage signal of the signal conversion circuit; the digital control circuit controls the demodulation of the demodulation circuit, so that the phase of the demodulation signal of the demodulation circuit is aligned with the phase of the input signal; and the digital control circuit controls the waveform and frequency of the excitation signal.

[0014] A method for detecting and adjusting the resonant frequency of a differential transformer resonant bridge includes:

[0015] S1: Construct a differential transformer resonant bridge resonant frequency detection and adjustment device as provided in this invention;

[0016] S2: Input an excitation signal of a given frequency into the input circuit and control the demodulation circuit to align the phase of the demodulated signal with the demodulated input signal. At the same time, observe the peak-to-peak value of the voltage in the time domain in the host computer.

[0017] S3: Adjust the capacitance value of the tuning capacitor connected to the fast-tuning capacitor bridge in fixed steps, and observe the change of the peak-to-peak value in the time domain in the host computer until the peak-to-peak value in the time domain reaches its minimum. At this time, the resonant frequency adjustment is completed and the resonant bridge reaches the expected resonant state.

[0018] Furthermore, step S3 includes:

[0019] S31: Reduce the number of connected tuning capacitors by a fixed step size and observe the change of the time domain peak-to-peak value in real time: if the time domain peak-to-peak value decreases, reduce the number of connected tuning capacitors by a fixed step size until the time domain peak-to-peak value reaches its minimum; otherwise, proceed to step S32.

[0020] S32: Increase the number of connected tuning capacitors in fixed steps and observe the change of peak-to-peak value in the time domain in real time: if the peak-to-peak value in the time domain decreases, repeat step S32; otherwise, execute step S31 until the peak-to-peak value in the time domain reaches its minimum.

[0021] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0022] This invention provides a device and method for detecting and adjusting the resonant frequency of a differential transformer resonant bridge. It uses a high-resolution analog-to-digital converter to acquire voltage data and, via a host computer, directly observes the minimum peak-to-peak value of the output voltage to determine the resonant state. The criteria are clear and intuitive, requiring no change to the excitation signal frequency. The circuit gain is stable, and the influence of the accuracy of the testing instrument on the resonant frequency detection is avoided. Regarding resonant frequency adjustment, the fast-tuning capacitor bridge is replaceable, the tuning range can be adjusted as needed, the tuning capacitor can be adjusted in precise steps, and the tuning accuracy is high, improving the efficiency of resonant frequency adjustment. As a separate adjustment unit, the resonant bridge effectively avoids the repeated soldering and desoldering of the front-end circuit during tuning capacitor changes, ensuring the safety and reliability of the front-end bridge and better meeting engineering application requirements. Attached Figure Description

[0023] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments and descriptions of the invention are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0024] Figure 1 A schematic diagram of the circuit structure of the differential transformer resonant bridge resonant frequency detection and adjustment device described in the embodiment of the present invention;

[0025] Figure 2 This is a schematic flowchart of the differential transformer resonant bridge resonant frequency detection and adjustment method described in an embodiment of the present invention. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof.

[0027] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0028] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on this invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, features defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0029] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art will understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0030] The invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0031] like Figure 1 As shown in the embodiment of the present invention, the differential transformer resonant bridge resonant frequency detection and adjustment device includes an input circuit, a resonant bridge, a signal conversion circuit, a demodulation circuit, an analog-to-digital conversion circuit, and a host computer. The input circuit converts the excitation signal into an input signal. The resonant bridge includes a fast-tuning capacitor bridge, which receives the input signal and transmits it to the signal conversion circuit via a differential transformer. The fast-tuning capacitor bridge includes two arms, each arm containing multiple parallel tuning capacitors. The resonant frequency is changed by adjusting the capacitance values ​​of the tuning capacitors connected in the fast-tuning capacitor bridge. The signal conversion circuit converts the current signal output by the resonant bridge into a voltage signal. The demodulation circuit demodulates the output signal of the signal conversion circuit. The analog-to-digital conversion circuit converts the analog signal output by the demodulation circuit into a digital signal, outputting a voltage value. The host computer receives the voltage value output by the analog-to-digital conversion circuit and determines whether the resonant bridge is in a resonant state based on the peak-to-peak value of the voltage value in the time domain.

[0032] In this embodiment of the invention, the signal conversion circuit consists of a transimpedance amplifier and an AC amplifier circuit. The transimpedance amplifier converts the current signal into a voltage signal, and the AC amplifier circuit amplifies the voltage signal. The analog-to-digital conversion circuit is implemented using an analog-to-digital converter (A / D converter). The signal conversion circuit, demodulation circuit, and analog-to-digital conversion circuit in this embodiment of the invention all employ existing circuitry.

[0033] In some embodiments, each tuning capacitor has a corresponding switch. By adjusting the switch, the corresponding tuning capacitor is connected to its respective bridge arm, thereby adjusting the number of tuning capacitors connected to the fast-tuning capacitor bridge. Furthermore, each tuning capacitor and its corresponding switch are connected in series. When the switch is closed, the corresponding tuning capacitor is connected to the circuit, thus enabling the toggle switch to adjust the number of tuning capacitors connected to the fast-tuning capacitor bridge. The number of tuning capacitors in the two bridge arms and the capacitance value of each tuning capacitor are adaptively adjusted according to requirements and actual conditions.

[0034] In this embodiment of the invention, each bridge arm includes four parallel tuning capacitors and four switches connected in series with the four tuning capacitors. Specifically, the first bridge arm includes tuning capacitors C1, C2, C3, and C4 connected in parallel, and also includes switches K1, K2, K3, and K4 connected in series with tuning capacitor C1, K2, K3, and K4; the second bridge arm includes tuning capacitors C5, C6, C7, and C8 connected in parallel, and also includes switches K5, K6, K7, and K8 connected in series with tuning capacitor C8. The capacitance value of each tuning capacitor is determined by the tuning precision. For example, if a tuning precision of 1pF is required, tuning capacitors of 1pF, 2pF, 3pF, and 5pF can be selected; if a tuning precision of 5pF is required, tuning capacitors of 5pF, 10pF, 10pF, and 20pF can be selected. Tuning capacitors can be combined to achieve the required capacitance value according to the step precision, and more groups and combinations can be added as needed.

[0035] In some embodiments, the differential transformer includes two primary windings and one secondary winding, with two bridge arms respectively connected to the two primary windings, and the secondary winding connected to a signal conversion circuit.

[0036] In this embodiment of the invention, the primary windings L1 and L2 of the differential transformer are connected to the two bridge arms respectively, and the secondary winding L3 is connected to the signal conversion circuit. Because the resonant bridge has extremely high sensitivity, it is also susceptible to external interference (such as temperature drift, power fluctuations, and electromagnetic noise). The differential output characteristics of the differential transformer can suppress these interferences and perfectly match the balance characteristics of the bridge. Therefore, this invention combines the differential transformer with a fast-tuning capacitor bridge to significantly suppress interference, ultimately enabling the resonant bridge to achieve high-precision and high-stability physical quantity measurement.

[0037] In some embodiments, the input circuit includes a test mass (TM). An excitation signal is applied to the test mass through electrode plates, and the tolerance signals between the test mass and the upper electrode plate and between the test mass and the lower electrode plate are converted into current signals, which are then transmitted to the two arms of the fast-tuning capacitor bridge, respectively. When the resonant bridge is in a resonant state, the thermal noise of the resonant bridge reaches its minimum value, at which point the sensor performance is optimal.

[0038] In some embodiments, the apparatus provided by the present invention further includes a digital control circuit, which is connected to an analog-to-digital converter (ADC), a signal conversion circuit, a demodulation circuit, and a host computer. Specifically: the digital control circuit controls the ADC to perform signal acquisition and conversion; the digital control circuit controls the amplification gain of the voltage signal by the signal conversion circuit; the digital control circuit controls the demodulation of the demodulation circuit, aligning the phase of the demodulated signal with the phase of the input signal; and the digital control circuit controls the waveform and frequency of the excitation signal. In this embodiment, the digital control circuit sends control signals for signal acquisition and conversion to the ADC, thereby controlling the ADC to perform signal acquisition and conversion; the digital control circuit sends gain control signals to the signal conversion circuit, thereby controlling the amplification gain of the voltage signal by the signal conversion circuit; and the digital control circuit sends demodulation control signals to the demodulation circuit, thereby controlling the start and stop of demodulation by the demodulation circuit, thereby aligning the phase of the demodulated signal with the phase of the input signal.

[0039] This invention also provides a method for detecting and adjusting the resonant frequency of a differential transformer resonant bridge, combined with... Figure 1 and Figure 2 The methods include:

[0040] S1: Construct a differential transformer resonant bridge resonant frequency detection and adjustment device as provided in this invention.

[0041] In this embodiment of the invention, step S1 includes: confirming the expected operating frequency of the resonant bridge, estimating the required tuning capacitor size for the resonant state based on the known inductance value of the differential transformer and the cable length, confirming the tuning accuracy, and welding the fast tuning capacitor bridge. Specifically, the resonant frequency is estimated using the following formula:

[0042] ;

[0043] Where f represents the resonant frequency, L represents the inductance of the differential transformer, and C represents the capacitance at the resonant state, including the capacitance on the fast-tuning capacitor bridge, the capacitance of the cable connecting the electrode plate and the fast-tuning capacitor bridge, and other stray capacitances.

[0044] S2: Input an excitation signal of a given frequency to the input circuit and control the demodulation circuit to align the phase of the demodulated signal with the demodulated input signal. At the same time, observe the peak-to-peak value of the voltage in the time domain in the host computer.

[0045] S3: Adjust the capacitance value of the tuning capacitor connected to the fast-tuning capacitor bridge in fixed steps, and observe the change of the peak-to-peak value in the time domain in the host computer until the peak-to-peak value in the time domain reaches its minimum. At this time, the resonant frequency adjustment is completed and the resonant bridge reaches the expected resonant state.

[0046] The fixed step size is selected based on application requirements. A higher fixed step size results in higher resonant frequency adjustment accuracy. The capacitance value of the tuning capacitor is selected according to the resonant frequency accuracy requirements, and the minimum capacitance value represents the step accuracy. For example, if the minimum capacitance value is 1pF, connecting it to the circuit increases the tuning capacitor by 1pF, and disconnecting it decreases the tuning capacitor by 1pF. In this case, the minimum step accuracy is 1pF.

[0047] In some embodiments, step S3 includes:

[0048] S31: Reduce the number of connected tuning capacitors by a fixed step size and observe the change of the time domain peak-to-peak value in real time: if the time domain peak-to-peak value decreases, reduce the number of connected tuning capacitors by a fixed step size until the time domain peak-to-peak value reaches its minimum; otherwise, proceed to step S32.

[0049] S32: Increase the number of connected tuning capacitors in fixed steps and observe the change of peak-to-peak value in the time domain in real time: if the peak-to-peak value in the time domain decreases, repeat step S32; otherwise, execute step S31 until the peak-to-peak value in the time domain reaches its minimum.

[0050] In step S3, the capacitors on both bridge arms can be adjusted simultaneously with the same fixed step size, which can ensure the symmetry of the resonant bridge while adjusting the resonant frequency. Alternatively, the capacitors on each bridge arm can be adjusted individually to adjust the symmetry of the resonant bridge.

[0051] It should be understood that the various forms of processes shown above can be used to reorder, add, or delete steps. For example, the steps described in this invention disclosure can be executed in parallel, sequentially, or in different orders, as long as the desired result of the technical solution disclosed in this invention can be achieved, and this is not limited herein.

[0052] The specific embodiments described above do not constitute a limitation on the scope of protection of this invention. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this invention should be included within the scope of protection of this invention.

Claims

1. A differential transformer resonant bridge resonant frequency detection and adjustment apparatus, characterized by, The device comprises an input circuit, a resonant bridge, a signal conversion circuit, a demodulation circuit, an analog-to-digital conversion circuit and a host computer. The input circuit is used to convert an excitation signal into an input signal; the resonant bridge comprises a fast tuning capacitor bridge, which receives the input signal, and then transmits the signal to the signal conversion circuit through a differential transformer; the signal conversion circuit is used to convert a current signal output by the resonant bridge into a voltage signal; the demodulation circuit is used to demodulate an output signal of the signal conversion circuit; the analog-to-digital conversion circuit is used to convert an analog signal output by the demodulation circuit into a digital signal and output a voltage value; and the host computer is used to receive the voltage value output by the analog-to-digital conversion circuit and determine whether the resonant bridge is in a resonant state according to a time-domain peak-to-peak value of the voltage value. The fast tuning capacitor bridge comprises two bridge arms, each of which comprises a plurality of parallel tuning capacitors; by adjusting the capacitance values of the tuning capacitors connected to the fast tuning capacitor bridge, the resonant frequency can be changed.

2. The differential transformer resonant-bridge resonant-frequency detection and adjustment apparatus according to claim 1, wherein In each bridge arm, each tuning capacitor has a corresponding switch, and by adjusting the switch, the corresponding tuning capacitor can be controlled to be connected to the bridge arm, thereby adjusting the number of tuning capacitors connected to the fast tuning capacitor bridge and changing the capacitance values of the tuning capacitors connected to the fast tuning capacitor bridge.

3. The differential transformer resonant-bridge resonant-frequency detection and adjustment apparatus of claim 1, wherein, The resonant bridge further comprises a differential transformer. The differential transformer comprises two primary windings and one secondary winding, and the two bridge arms are respectively connected to the two primary windings, and the secondary winding is connected to the signal conversion circuit.

4. The differential transformer resonant-bridge resonant-frequency detection and adjustment apparatus of claim 1, wherein, The input circuit comprises a test mass, and the excitation signal is applied to the test mass through an electrode plate, and the capacitance signals between the test mass and an upper electrode plate and between the test mass and a lower electrode plate are respectively converted into current signals and transmitted to the two bridge arms of the fast tuning capacitor bridge.

5. The differential transformer resonant bridge resonant frequency detection and adjustment apparatus of claim 1, wherein, The signal conversion circuit is further used to amplify the voltage signal.

6. The differential transformer resonant bridge resonant frequency detection and adjustment apparatus of claim 1, wherein, The device further comprises a filtering module connected to the signal conversion circuit and used to filter out out-of-band noise in the voltage signal output by the signal conversion circuit.

7. The differential transformer resonant-bridge resonant-frequency detection and adjustment apparatus according to claim 6, wherein The device further comprises a digital control circuit connected to the analog-to-digital conversion circuit, the signal conversion circuit, the demodulation circuit and the host computer. The digital control circuit controls the analog-to-digital conversion circuit to collect and convert signals; The digital control circuit controls the amplification gain of the signal conversion circuit on the voltage signal; The digital control circuit controls the demodulation of the demodulation circuit to align the phase of the demodulation signal with the phase of the input signal; The digital control circuit controls the waveform and frequency of the excitation signal.

8. A method of differential transformer resonant bridge resonant frequency detection and adjustment, comprising: The method comprises the following steps: S1: building the differential transformer resonant bridge resonant frequency detection and adjustment device according to any one of claims 1-7; S2: inputting an excitation signal with a given frequency to the input circuit, controlling the demodulation circuit to align the phase of the demodulation signal with the phase of the demodulation input signal, and observing the time-domain peak-to-peak value of the voltage value in the host computer. S3: adjusting the capacitance value of the tuning capacitor accessed in the fast tuning capacitor bridge with a fixed step, and observing the change of the time-domain peak-to-peak value in the host computer until the time-domain peak-to-peak value reaches the minimum, at which time the resonance frequency adjustment is completed and the resonance bridge reaches the expected resonance state.

9. The differential transformer resonant-bridge resonant-frequency detection and adjustment method according to claim 8, characterized in that, Step S3 comprises: S31: reducing the number of accessed tuning capacitors with a fixed step, and observing the change of the time-domain peak-to-peak value in real time: if the time-domain peak-to-peak value decreases, reducing the number of accessed tuning capacitors with a fixed step until the time-domain peak-to-peak value reaches the minimum; otherwise, performing step S32; S32: increasing the number of accessed tuning capacitors with a fixed step, and observing the change of the time-domain peak-to-peak value in real time: if the time-domain peak-to-peak value decreases, repeating step S32, otherwise, performing step S31, until the time-domain peak-to-peak value reaches the minimum.

Citation Information

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