Real-time measurement method and system for instantaneous input power and impedance characteristic of ultrasonic transducer
By applying discrete Fourier transform technology in ultrasonic transducers, real-time sampling and calculation of input power and impedance, the problems of insufficient power measurement accuracy and real-time, dynamic impedance changes and inaccurate frequency response in the prior art are solved, and the problem of inaccurate power response in the prior art is realized. Higher measurement accuracy and stronger real-time performance ensure the safety and reliability of the equipment.
Patent Information
- Application Number
- CN202510074279.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-17
- Publication Date
- 2025-02-21
- Estimated Expiration
- Not applicable · inactive patent
AI Technical Summary
The existing ultrasonic transducer input power measurement methods have problems such as insufficient power measurement accuracy and real-time performance, ineffective dynamic changes in impedance, inaccurate frequency response, high performance requirements for sampling circuits, and difficult transducer status monitoring.
The discrete Fourier transform method is used to sample and digitize the voltage signal and current signal of the ultrasonic transducer in real time, and calculate the input power value and impedance value through discrete Fourier transform to realize real-time monitoring and accurate measurement.
It significantly improves the accuracy and real-time performance of the input power measurement of ultrasonic transducer, can effectively monitor impedance characteristics, reduces the system's hardware performance requirements, and improves the reliability and safety of the equipment.
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Figure CN119492910A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of ultrasonic transducer working state monitoring, and in particular relates to a method and system for real-time measurement of instantaneous input power and impedance characteristics of an ultrasonic transducer based on discrete Fourier transform. Background Art
[0002] With the widespread application of ultrasonic technology in the medical field, accurate and real-time monitoring of the input power of ultrasonic transducers has become the key to ensuring treatment effects and patient safety. In ultrasonic therapy, the ultrasonic transducer excites the transducer element (usually piezoelectric material) through an electrical signal, causing it to vibrate mechanically, thereby generating ultrasonic signals, which are transmitted to the target area for treatment or diagnosis. Ultrasonic transducers have a specific operating frequency, only respond to excitation signals of a specific frequency, and convert electrical energy into ultrasonic output. Since the operating frequency of the transducer presents a partial capacitive impedance, there is a certain phase difference between its excitation voltage and current. These characteristics make the power measurement of ultrasonic transducers complicated. In the medical field, ultrasonic therapy equipment has very strict requirements on power. Too low power may result in insignificant treatment effects, while too high power may cause harm to patients; therefore, real-time and accurate monitoring of the input power of ultrasonic transducers is crucial.
[0003] Existing power measurement methods mainly rely on two approaches, one is to estimate by combining the voltage signal with the impedance characteristics of the transducer, and the other is to calculate by integrating the voltage and current signals, but both have certain limitations. First, the voltage-impedance combined calculation method cannot fully consider the dynamic changes in impedance encountered by the ultrasonic transducer in actual work. Specifically, the ultrasonic transducer will produce impedance changes in different environmental media and working conditions, and this change makes it difficult for traditional estimation methods to adapt. Secondly, the integral calculation method estimates the input power by integrating the voltage and current signals over time. However, this method will also calculate the electrical signals at other frequencies into the input power, including harmonics, ringing, noise and other electrical powers with low ultrasonic power output response, thereby affecting the accuracy of subsequent output power prediction. For example, the Chinese patent application with publication number CN117452054A proposes a voltage, current, frequency, phase impedance detection device and detection method for an ultrasonic knife transducer, which uses the integration of voltage and current to calculate the apparent power and active power, and can calculate the overall electrical power, but cannot separate the power of a specific frequency, and cannot further combine the frequency specificity of the ultrasonic transducer to estimate the output ultrasonic power. For another example, a Chinese patent application with publication number CN215534823U proposes a signal acquisition circuit for an ultrasonic transducer, which has similar defects and only detects the amplitude and phase difference of the voltage signal and the current signal, without separating the frequency of the electric power.
[0004] In summary, the existing ultrasonic transducer input power measurement method mainly has the following technical problems: 1. Insufficient power measurement accuracy and real-time performance. In the medical field, ultrasonic therapy equipment has extremely strict power requirements. Too low power may lead to poor treatment effects, and too high power may cause harm to patients. Existing power estimation methods cannot provide sufficiently accurate and real-time power monitoring, making it difficult to achieve fine control and safety assurance.
[0005] 2. Dynamic impedance change problem. Existing power measurement methods mainly estimate power by combining voltage signals with transducer impedance characteristics. Such methods cannot effectively solve the problem of dynamic impedance changes encountered by ultrasonic transducers in actual work; the impedance of the transducer will change in different environmental media and working conditions, resulting in deficiencies in power measurement accuracy and real-time performance of existing methods.
[0006] 3. Frequency response problem. Ultrasonic transducers have a specific operating frequency and only respond to excitation signals of a specific frequency. Traditional time-integrated power measurement methods for voltage and current signals cannot accurately separate and identify the frequency components, especially when there are interferences from harmonics, noise or other non-specific frequencies. These methods will mistakenly include signals of irrelevant frequencies in the input power calculation, thereby affecting the accuracy of subsequent output power measurement.
[0007] 4. The sampling circuit performance requirements are high. The traditional time-integrated power measurement method of voltage and current signals usually requires the analog-to-digital converter sampling frequency to be much higher than the ultrasonic frequency itself, which makes the traditional method have high hardware requirements, which not only increases the cost of the system, but also may lead to increased system complexity and power consumption.
[0008] 5. Transducer status monitoring. The impedance of the transducer will change under different working conditions. Real-time measurement of the impedance characteristics of the ultrasonic transducer can be used to evaluate whether the transducer is damaged, aged, or in a dangerous empty state, ensuring the stability, reliability and safety of the system under various conditions and avoiding misoperation or equipment failure. Summary of the invention
[0009] In view of the above, the present invention provides a real-time measurement method and system for the instantaneous input power and impedance characteristics of an ultrasonic transducer based on discrete Fourier transform, which can solve the interference of problems such as frequency response mismatch and impedance instability that traditional methods cannot cope with, and at the same time realize real-time impedance monitoring of the ultrasonic transducer, thereby greatly improving the accuracy and real-time performance of power measurement, ensuring the treatment effect and patient safety.
[0010] A real-time measurement method for the instantaneous input power and impedance characteristics of an ultrasonic transducer based on discrete Fourier transform, specifically: firstly, the voltage signal and current signal of the ultrasonic transducer are sampled in real time, then the sampled voltage and current values are converted from analog to digital, and finally the input power value and impedance value of the ultrasonic transducer are calculated by discrete Fourier transform using the digitized voltage and current values.
[0011] Furthermore, in the real-time sampling process, it is required that the voltage signal and the current signal themselves produce the same phase offset or no phase offset, the sampling process needs to filter out noise and interference above the analog-to-digital conversion frequency, and the sampled voltage and current values must be kept within the input range required by the analog-to-digital conversion.
[0012] Furthermore, the conversion of the voltage and current values from analog to digital quantities requires synchronous analog-to-digital conversion, and the conversion rate is at least twice the operating frequency of the ultrasonic transducer (to ensure accuracy, it is recommended to be more than 10 times).
[0013] Furthermore, the specific process of calculating the ultrasonic transducer input power value and impedance value is as follows: (1) Each set of digitized voltage and current values received is stored, and step (2) is executed after a total of N sets of data are collected, where N = a × Fs / Fu, a is an arbitrary positive integer, Fs is the analog-to-digital conversion frequency, and Fu is the operating frequency of the ultrasonic transducer; (2) Use the following formulas to perform discrete Fourier transform on the voltage and current values respectively:
[0014] Where: x[n] is the voltage or current value in the nth group of data collected, X[k] is the kth frequency component of the corresponding frequency domain signal, j is an imaginary unit, and e is a natural constant; (3) According to the above formula, the frequency components corresponding to the voltage and current values when k=N×Fu / Fs are denoted as U and I respectively; (4) The input power value P and impedance value Z of the ultrasonic transducer are calculated based on the frequency components U and I, and Z = U / I.
[0015] Furthermore, in step (4), the input power value P is calculated by the following formula: P = Re [U × I * ] Where: Re[ ] represents the real part of the complex number, I * is the complex conjugate of I.
[0016] Furthermore, in step (4), the input power value P is calculated by the following formula: P=|U|×|I|×cos(φ U -φ I ) Among them: |U| and φ U are the amplitude and phase of U, |I| and φ I are the amplitude and phase of I respectively.
[0017] A real-time measurement system for instantaneous input power and impedance characteristics of an ultrasonic transducer based on discrete Fourier transform, comprising: A voltage and current sampling module is used to sample the voltage signal and current signal of the ultrasonic transducer in real time; An analog-to-digital conversion module is used to convert the voltage and current values sampled by the voltage and current sampling module from analog quantities to digital quantities; The signal processing module uses the digitized voltage value and current value to calculate the input power value and impedance value of the ultrasonic transducer through discrete Fourier transform.
[0018] Furthermore, the signal processing module is implemented by an embedded microprocessor such as a single chip microcomputer, a DSP (Digital Signal Processor) or an FPGA (Field Programmable Gate Array).
[0019] Furthermore, the specific process of the signal processing module calculating the ultrasonic transducer input power value and impedance value is as follows: (1) Each set of digitized voltage and current values received is stored, and step (2) is executed after a total of N sets of data are collected, where N = a × Fs / Fu, a is an arbitrary positive integer, Fs is the analog-to-digital conversion frequency, and Fu is the operating frequency of the ultrasonic transducer; (2) Use the following formulas to perform discrete Fourier transform on the voltage and current values respectively:
[0020] Where: x[n] is the voltage or current value in the nth group of data collected, X[k] is the kth frequency component of the corresponding frequency domain signal, j is an imaginary unit, and e is a natural constant; (3) According to the above formula, the frequency components corresponding to the voltage and current values when k=N×Fu / Fs are denoted as U and I respectively; (4) Calculate the input power value P and impedance value Z of the ultrasonic transducer based on the frequency components U and I, that is, P = Re [U × I * ], Z = U / I, where Re [] represents the real part of the complex number, I * is the complex conjugate of I.
[0021] Based on the above technical solution, the present invention has the following beneficial technical effects: 1. Improve the accuracy and real-time performance of power measurement. The present invention adopts the discrete Fourier transform (DFT) method, which can effectively separate the various frequency components in the voltage and current signals and accurately calculate the input power at a specific frequency. Compared with the traditional method, the DFT method can eliminate the influence of factors such as frequency response mismatch and impedance instability on power measurement, significantly improving the measurement accuracy and real-time performance. The input power combined with the electromechanical coupling coefficient to estimate the output power of the ultrasonic transducer at the characteristic operating frequency will be more accurate, ensuring the precise control of the output power during ultrasonic treatment, thereby ensuring the safety of patients.
[0022] 2. Real-time monitoring of the impedance characteristics of the ultrasonic transducer. The present invention can simultaneously calculate the impedance characteristics of the ultrasonic transducer at the operating frequency, helping users to evaluate in real time whether the equipment is damaged, aged, or in an idle state, thereby timely discovering equipment failures, avoiding equipment damage or safety risks, and significantly improving the reliability and safety of ultrasonic therapy equipment.
[0023] 3. Strong anti-interference ability. Since the present invention uses DFT technology to effectively decompose the signal by frequency, thereby suppressing interference from other frequencies (such as harmonics, noise, etc.), even if the excitation signal has high-order harmonic components or nonlinear distortion, or uses a non-sinusoidal excitation signal (such as the commonly used square wave pulse, etc.), the present invention can accurately extract the effective signal of the operating frequency to ensure the reliability and stability of the measurement results. This frequency separation capability can significantly improve the measurement accuracy in complex environments.
[0024] 4. Real-time and rapid response. The present invention simplifies the discrete Fourier transform method and only calculates the frequency components at the operating frequency point of the ultrasonic transducer, thereby improving the calculation speed, ensuring real-time monitoring of the ultrasonic treatment equipment, and making rapid responses to abnormal power output and working conditions.
[0025] 5. The system hardware requirements are relatively low. Traditional power measurement methods usually require high-speed ADC (analog-to-digital conversion) sampling, which places high demands on hardware performance. The present invention uses discrete Fourier transform to perform frequency domain analysis on the signal, so that the ADC sampling frequency requirement is reduced to more than twice the ultrasonic operating frequency. The system reduces the hardware requirements for the ADC, which can effectively reduce costs and circuit complexity.
[0026] 6. Wide range of applications. The present invention is not only applicable to ultrasonic therapeutic equipment, but also can be widely used in industrial fields such as ultrasonic welding, ultrasonic cleaning, and ultrasonic flaw detection. By providing accurate power monitoring and real-time status feedback for various ultrasonic systems, the present invention has important application value in various ultrasonic equipment and can provide more stable and reliable monitoring and control solutions for the industrial and medical fields.
[0027] Therefore, the present invention is mainly used in the field of ultrasonic transducer operation monitoring and control, and is particularly suitable for ultrasonic working state monitoring in medical treatment equipment. It can accurately measure the input electrical power in real time. The input electrical power combined with the electromechanical coupling coefficient can estimate the output power of the ultrasonic transducer. The present invention can be widely used in medical equipment related to ultrasonic treatment, and can ensure the accuracy of the output power during the treatment process to avoid the adverse effects of too low or too high output on the patient; by real-time monitoring of the impedance characteristics of the transducer, it can timely detect whether the transducer is damaged, aged, or in a dangerous empty state, thereby effectively avoiding equipment failure or safety risks. In addition, the present invention can also be used in industrial fields such as ultrasonic welding and ultrasonic cutting. It is one of the core technologies in ultrasonic systems and has important medical and industrial application value. BRIEF DESCRIPTION OF THE DRAWINGS
[0028] Figure 1 It is a structural schematic diagram of the instantaneous input power and impedance characteristic real-time measurement system of the ultrasonic transducer of the present invention.
[0029] Figure 2 It is a schematic diagram of the signal processing flow of the signal processing module in the measurement system of the present invention. DETAILED DESCRIPTION
[0030] In order to describe the present invention more specifically, the technical solution of the present invention is described in detail below in conjunction with the accompanying drawings and specific implementation methods.
[0031] like Figure 1 As shown, the instantaneous input power and impedance characteristic real-time measurement system of ultrasonic transducer based on discrete Fourier transform of the present invention comprises a current and voltage sampling module 1, an analog-to-digital conversion module 2 and a signal processing module 3, wherein: The current and voltage sampling module 1 collects the excitation voltage signal and the corresponding current signal on the ultrasonic transducer, and ensures that the signal amplitude meets the processing requirements of the subsequent analog-to-digital conversion module. Specifically, the voltage signal can be sampled as a differential signal through a voltage divider resistor, and then converted into a single-ended signal through a differential-to-single-ended circuit, and output to the analog-to-digital conversion module 2; the current signal can be sampled using a sampling resistor or a current transformer.
[0032] During the sampling process, the voltage signal and the current signal themselves need to have the same phase shift or no phase shift; in addition, in order to improve the signal quality, a filtering circuit can be added to the signal path to filter out high-frequency signals (noise and clutter) that are higher than the conversion rate of the analog-to-digital conversion module to reduce noise interference; at the same time, ensure that the amplitude of the analog signal obtained by sampling is within the input range of the analog-to-digital conversion module 2.
[0033] The analog-to-digital conversion module 2 converts the current and voltage analog signals sampled by the current and voltage sampling module 1 into digital signals. It is necessary to perform synchronous analog-to-digital conversion on the current and voltage signals, and the conversion rate Fs of the analog-to-digital conversion module 2 must be higher than twice the operating frequency Fu of the ultrasonic transducer; in order to achieve better accuracy, Fs should be set to more than 10 times Fu. Synchronous analog-to-digital conversion can be achieved by providing the same sampling clock signal to two independent ADC chips, or by using two ADC chips to directly achieve synchronous sampling; in this embodiment, the operating frequency of the ultrasonic transducer is 500kHz, and we use a dual-channel synchronous sampling ADC chip to perform analog-to-digital conversion on the current and voltage analog signals at 10Msps.
[0034] The signal processing module 3 can be any computing unit with data processing capability, such as a single chip microcomputer, DSP or FPGA and other embedded microprocessors, which is responsible for executing the following Figure 2 The signal processing flow shown uses discrete Fourier transform to calculate the input power and impedance characteristics of the ultrasonic transducer at the operating frequency. It is divided into three steps including data acquisition and storage S1, single-frequency discrete Fourier transform S2, power calculation S3 and impedance calculation S4.
[0035] In the data acquisition and storage step S1, the current digital signal i and the voltage digital signal u converted by the analog-to-digital conversion module 2 are stored, and a total of N data points are stored for calculation in subsequent steps, where N=a×Fs / Fu, a can be any positive integer, representing the number of complete waveforms of the ultrasonic working frequency stored, and the discrete current signal sequence i[n] and the discrete voltage signal sequence u[n] in the memory are recorded; in this embodiment, a=20, that is, 20 complete waveforms of 500kHz are sampled, then N=20×10Msps / 500kHz=400, that is, 400 data points are stored.
[0036] In the single-frequency discrete Fourier transform step S2, the discrete current signal sequence i[n] and the discrete voltage signal sequence u[n] in the data acquisition storage S1 are subjected to single-frequency discrete Fourier transform, and the calculation formula is:
[0037] Where: X[k] is the kth frequency component of the frequency domain signal, and x[n] is the nth sampling value of the time domain signal.
[0038] The measurement of ultrasonic output power only requires the excitation signal power and impedance characteristics of the ultrasonic transducer at the operating frequency. Therefore, it is only necessary to calculate the frequency component at k=N×Fu / Fs. The calculated current frequency domain value and voltage frequency domain value are denoted as I and U respectively. In this embodiment, k=20.
[0039] In the power calculation step S3, the input power of the ultrasonic transducer at the operating frequency is calculated according to the following formula: P = Re [U × I * ] Where: I * is the conjugate complex number of the current frequency domain value I, and Re[ ] represents the real part of the complex number.
[0040] The power can also be calculated based on the amplitude and phase of voltage and current, that is: P=|U|×|I|×cos(φ U -φ I ) Where: |U| and |I| are the amplitudes of voltage and current respectively, φ U and φ I are the phases of voltage and current respectively.
[0041] In the impedance calculation step S4, the impedance Z=U / I of the ultrasonic transducer at the operating frequency is calculated.
[0042] In summary, the core technical points and innovations of the present invention are mainly reflected in the following points: 1. Instantaneous input power measurement method based on discrete Fourier transform. The core innovation of this invention is to use DFT to perform frequency domain analysis on the current and voltage signals of the ultrasonic transducer, so as to calculate the input power at the working frequency in real time; through frequency domain decomposition, the signal components at a specific frequency can be accurately extracted, solving the inaccuracy of the traditional integration method under multi-frequency interference.
[0043] 2. Real-time impedance monitoring. The present invention can not only measure the input power in real time, but also monitor the impedance characteristics of the ultrasonic transducer in real time by calculating the amplitude and phase of the frequency domain signal; impedance monitoring can help detect damage, aging or dangerous empty state of the equipment, greatly improving the safety and reliability of the equipment.
[0044] 3. Single-frequency discrete Fourier transform. The working characteristics of the ultrasonic transducer make it respond only to the input power at the working frequency. Therefore, the present invention simplifies the complexity of the Fourier transform by calculating the frequency domain component at a single frequency point. Only a single-frequency discrete Fourier transform is required, thereby effectively reducing the amount of calculation and improving processing efficiency.
[0045] 4. Synchronous analog-to-digital conversion module. The present invention adopts a synchronous analog-to-digital conversion module to ensure that the phases of current and voltage signals are not offset during the sampling process, thereby avoiding errors caused by asynchronous sampling.
[0046] 5. Flexibility in hardware implementation. The current and voltage sampling module of the present invention can be implemented using a variety of sampling circuits, and the signal processing method can be implemented through a variety of hardware platforms (such as single-chip microcomputers, DSPs, FPGAs, etc.), with good hardware compatibility, and supports the selection of the most appropriate hardware platform for implementation according to different application scenarios.
[0047] The above description of the embodiments is to facilitate the understanding and application of the present invention by those skilled in the art. It is obvious that those skilled in the art can easily make various modifications to the above embodiments and apply the general principles described herein to other embodiments without creative work. Therefore, the present invention is not limited to the above embodiments. Improvements and modifications made by those skilled in the art to the present invention based on the disclosure of the present invention should be within the protection scope of the present invention.
Claims
1. A real-time measurement method for instantaneous input power and impedance characteristics of an ultrasonic transducer based on discrete Fourier transform, characterized in that: First, the voltage signal and current signal of the ultrasonic transducer are sampled in real time, and then the sampled voltage and current values are converted from analog to digital. Finally, the digitized voltage and current values are used to calculate the input power value and impedance value of the ultrasonic transducer through discrete Fourier transform.
2. The method for real-time measurement of instantaneous input power and impedance characteristics of an ultrasonic transducer according to claim 1, characterized in that: During the real-time sampling process, it is required that the voltage signal and the current signal themselves produce the same phase offset or no phase offset. The sampling process needs to filter out noise and interference above the analog-to-digital conversion frequency, and the sampled voltage and current values must be kept within the input range required by the analog-to-digital conversion.
3. The method for real-time measurement of instantaneous input power and impedance characteristics of an ultrasonic transducer according to claim 1, characterized in that: Converting the voltage and current values from analog to digital quantities requires synchronous analog-to-digital conversion, and the conversion rate must be at least twice the operating frequency of the ultrasonic transducer.
4. The method for real-time measurement of instantaneous input power and impedance characteristics of an ultrasonic transducer according to claim 1, characterized in that: The specific process of calculating the input power value and impedance value of the ultrasonic transducer is as follows: (1) Each set of digitized voltage and current values received is stored, and step (2) is executed after a total of N sets of data are collected, where N = a × Fs / Fu, a is an arbitrary positive integer, Fs is the analog-to-digital conversion frequency, and Fu is the operating frequency of the ultrasonic transducer; (2) Use the following formulas to perform discrete Fourier transform on the voltage and current values respectively: ; Where: x[n] is the voltage or current value in the nth group of data collected, X[k] is the kth frequency component of the corresponding frequency domain signal, j is an imaginary unit, and e is a natural constant; (3) According to the above formula, the frequency components corresponding to the voltage and current values when k=N×Fu / Fs are denoted as U and I respectively; (4) The input power value P and impedance value Z of the ultrasonic transducer are calculated based on the frequency components U and I, and Z = U / I.
5. The method for real-time measurement of instantaneous input power and impedance characteristics of an ultrasonic transducer according to claim 4, characterized in that: In step (4), the input power value P is calculated by the following formula: P=Re[U×I * ] Where: Re[ ] represents the real part of the complex number, I * is the complex conjugate of I.
6. The method for real-time measurement of instantaneous input power and impedance characteristics of an ultrasonic transducer according to claim 4, characterized in that: In step (4), the input power value P is calculated by the following formula: P=|U|×|I|×cos(φ U -f I ) Among them: |U| and φ U are the amplitude and phase of U, |I| and φ I are the amplitude and phase of I respectively.
7. A real-time measurement system for instantaneous input power and impedance characteristics of ultrasonic transducers based on discrete Fourier transform, characterized in that: include: A voltage and current sampling module is used to sample the voltage signal and current signal of the ultrasonic transducer in real time; An analog-to-digital conversion module is used to convert the voltage and current values sampled by the voltage and current sampling module from analog quantities to digital quantities; The signal processing module uses the digitized voltage value and current value to calculate the input power value and impedance value of the ultrasonic transducer through discrete Fourier transform.
8. The real-time measurement system for instantaneous input power and impedance characteristics of an ultrasonic transducer according to claim 7, characterized in that: The signal processing module is implemented by a single chip microcomputer, a DSP or a FPGA.
9. The instantaneous input power and impedance characteristic real-time measurement system of ultrasonic transducer according to claim 7, characterized in that: The specific process of the signal processing module calculating the ultrasonic transducer input power value and impedance value is as follows: (1) Each set of digitized voltage and current values received is stored, and step (2) is executed after a total of N sets of data are collected, where N = a × Fs / Fu, a is an arbitrary positive integer, Fs is the analog-to-digital conversion frequency, and Fu is the operating frequency of the ultrasonic transducer; (2) Use the following formulas to perform discrete Fourier transform on the voltage and current values respectively: ; Where: x[n] is the voltage or current value in the nth group of data collected, X[k] is the kth frequency component of the corresponding frequency domain signal, j is an imaginary unit, and e is a natural constant; (3) According to the above formula, the frequency components corresponding to the voltage and current values when k=N×Fu / Fs are denoted as U and I respectively; (4) Calculate the input power value P and impedance value Z of the ultrasonic transducer based on the frequency components U and I, that is, P = Re [U × I * ], Z = U / I, where Re [] represents the real part of the complex number, I * is the complex conjugate of I.
Citation Information
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