Thermal decomposition method for dynamic pH regulation and control of PEG derivative modified superparamagnetic ferrite nanoparticles
By monitoring the pH value in real time through FTIR and selecting appropriate group regulators, PEG derivative-modified superparamagnetic ferrite nanoparticles were prepared by dynamically controlling the high-temperature thermal decomposition method, which solved the problem of difficult-to-control pH environment and improved the stability and magnetic properties of the particles.
Patent Information
- Application Number
- CN202511136478.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-14
- Publication Date
- 2025-09-16
AI Technical Summary
The existing high-temperature thermal decomposition method makes it difficult to achieve dynamic and precise control of the pH environment in the preparation of PEG derivative-modified superparamagnetic ferrite nanoparticles, resulting in inconsistency between particle size and magnetic properties. In addition, traditional methods cannot accurately measure pH values in high-temperature organic solvents.
Fourier transform infrared spectroscopy (FTIR) was used to monitor the pH value in real time. By selecting regulators with different groups, the pH value was dynamically controlled during the nucleation, growth and modification stages. Combined with temperature regulation, dynamic pH regulation of PEG derivative-modified superparamagnetic ferrite nanoparticles was achieved.
It effectively reduces the standard deviation of nanoparticle size, improves the stability and magnetic saturation of the particles, enhances the magnetization intensity and biocompatibility, and solves the problem of dynamic fluctuation of pH value during high-temperature thermal decomposition.
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Figure CN120646918A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of nanocomposite material production, and in particular to a thermal decomposition method of polyethylene glycol (PEG) derivative-modified superparamagnetic ferrite nanoparticles using dynamic pH regulation. Background Art
[0002] High-temperature thermal decomposition is an existing technology used to prepare PEG derivative-modified superparamagnetic ferrite nanoparticles. These PEG derivative-modified superparamagnetic ferrite nanoparticles exhibit low defectivity and high crystallinity. The superparamagnetism of PEG derivative-modified superparamagnetic ferrite nanoparticles is primarily manifested in their magnetic saturation, which refers to the maximum magnetization a magnetic material can achieve in an applied magnetic field. It is a key characteristic of ferromagnetic materials. High magnetic saturation increases magnetization, thereby enhancing the energy exchange efficiency between the spin system and the lattice.
[0003] The range of magnetic saturation is influenced by multiple factors, including particle size, morphology, crystallinity, and surface modification. Smaller particles (e.g., 5-10 nm) may exhibit lower magnetic saturation due to a higher proportion of surface atoms. Particles with higher crystallinity exhibit higher magnetic saturation due to fewer lattice defects and a more ordered magnetic moment arrangement. Surface modifiers (e.g., dextran or polyvinyl alcohol) may slightly reduce magnetic saturation but can improve particle stability and biocompatibility. In the preparation of PEG-derivative-modified superparamagnetic ferrite nanoparticles by high-temperature thermal decomposition, dynamic and precise control of the pH environment is a key challenge in regulating particle size, morphology, and magnetic properties. There are the following technical problems in achieving dynamic and precise control of the pH environment during high-temperature thermal decomposition: 1. High-temperature thermal decomposition is usually carried out in organic solvents. In non-polar solvent environments such as oleic acid and oleylamine at temperatures greater than 200°C, traditional glass electrodes cannot withstand high temperatures and cannot accurately measure pH values in organic solvents; 2. The precursor releases acidic products during high-temperature decomposition, resulting in dynamic fluctuations in the system pH, and the buffer system in the organic solvent is difficult to respond quickly; 3. Surfactants are both pH regulators and ligands, and high-temperature decomposition products cause pH changes; 4. In mass-production reactors, the pH difference between the high-temperature and low-temperature regions can reach 0.5~1.0, causing inconsistent particle growth rates.
[0004] When preparing PEG derivative-modified superparamagnetic ferrite nanoparticles by controlling the pH value through high-temperature thermal decomposition, the influence of the temperature and pH environment at different reaction stages on the regulation of the precursor decomposition rate and the control of the nucleation-growth balance must be considered to ensure that the biocompatibility of the PEG derivative-modified superparamagnetic ferrite nanoparticles matches the pH value. In order to improve the production capacity of PEG derivative-modified superparamagnetic ferrite nanoparticles while also improving the modification efficiency, it is necessary to provide a dynamic pH-controlled thermal decomposition method for PEG derivative-modified superparamagnetic ferrite nanoparticles that can meet the dynamic acquisition requirements of the pH value of the high-temperature organic phase, establish a pH correction method based on dynamic pH and temperature coupling, and form fusion control information for controlling the rate of regulator addition, which is used to control the preparation process of PEG derivative-modified superparamagnetic ferrite nanoparticles. Summary of the Invention
[0005] To address the problems raised in the above-mentioned background technology, the present invention provides a method for dynamically controlling the thermal decomposition of PEG derivative-modified superparamagnetic ferrite nanoparticles by pH. The method dynamically detects the pH value during the high-temperature thermal decomposition process of preparing PEG derivative-modified superparamagnetic ferrite nanoparticles by using online Fourier transform infrared and Raman spectroscopy. Based on the temperature, real-time pH value, and target pH value of the nucleation, growth, and modification stages, regulators with different groups are selected to dynamically control the pH value at different stages. To achieve the above-mentioned objectives, the present invention provides a method for dynamically controlling the thermal decomposition of PEG derivative-modified superparamagnetic ferrite nanoparticles by pH, comprising: S1, determine the temperature window and pH threshold window of each stage, where the temperature window of the nucleation stage is , the pH threshold window is , the temperature window during the growth phase is , the pH threshold window is , the temperature window of the modification stage is , the pH threshold window is ; S2, determine the target pH value of each stage. The target pH value of the nucleation stage is , the pH target during the growth phase is , the pH target in the modification stage is ; S3, dynamic monitoring of pH during the nucleation phase , inject the precursor and NH group regulator, when the real-time temperature , the unit is ℃, satisfy , using Fourier transform infrared to monitor the real-time peak shift of NH ,according to and Determining the pH value for the nucleation phase correction , when the temperature meets , go to S4; S4, dynamic monitoring of pH during the growth phase , inject OH group regulator, when the real-time temperature satisfy Real-time peak shift monitoring of OH by Fourier transform infrared ,according to and Determining pH values for growth phase correction , when the temperature meets , go to S5; S5, dynamic monitoring of pH value during the modification phase , inject CH group regulator, when the real-time temperature satisfy Real-time peak shift monitoring of CH by Fourier transform infrared ,according to and Determining pH values for growth phase correction , when the temperature meets ,Finish.
[0006] As a further description of the above technical solution: S3, in real-time temperature satisfy ,according to and Determining the pH value for the nucleation phase correction ,in is the temperature-peak shift coupling coefficient of the injected regulator during the nucleation phase, which is determined by the thermal expansion characteristics of the injected regulator and is expressed in °C / nm. Temperature-peak coupling offset for injection of modifiers during the growth phase, in nm.
[0007] As a further description of the above technical solution: the S3 satisfy: (1) in: is the linear coefficient of NH peak shift and pH change, in nm / pH. is the compensation coefficient of NH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. The reference temperature for the nucleation stage is the target pH value for the nucleation stage. The corresponding standard temperature is in °C. satisfy: (2) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies .
[0008] As a further description of the above technical solution: S3 also includes The step of determining the addition rate of the nucleation stage regulator, the addition rate of the nucleation stage regulator satisfy: (3) in, is the proportional coefficient of the nucleation stage, is the integral coefficient of the nucleation stage, is the differential coefficient in the nucleation stage.
[0009] As a further description of the above technical solution: S4, in real-time temperature satisfy ,according to and Determining pH values for growth phase correction ,in The temperature-peak shift coupling coefficient of the injected regulator during the growth phase is determined by the thermal expansion characteristics of the injected regulator and is expressed in °C / nm. Temperature-peak coupling offset for injection of modifiers during the growth phase, in nm.
[0010] As a further description of the above technical solution: the S4 satisfy: (4) in: is the linear coefficient of OH peak shift and pH change, with the unit of nm / pH, is the compensation coefficient of OH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. The reference temperature for the growth phase is the target pH value for the growth phase. The corresponding standard temperature is in °C. satisfy: (5) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies .
[0011] As a further description of the above technical solution: S4 also includes according to Step of determining the rate of addition of a growth phase regulator, the rate of addition of the growth phase regulator satisfy: (6) in, is the growth stage proportional coefficient, is the integral coefficient of the growth stage, is the differential coefficient in the growth stage.
[0012] As a further description of the above technical solution: S5, in real-time temperature satisfy ,according to and Determine pH value for modification phase ,in is the temperature-peak shift coupling coefficient of the injected modifier during the modification phase, which is determined by the thermal expansion characteristics of the injected modifier and is expressed in °C / nm. The temperature-peak coupling shift for the injection of modulators during the modification phase, in nm.
[0013] As a further description of the above technical solution: the S5 satisfy: (7) in: is the linear coefficient of CH peak shift and pH change, in nm / pH. is the compensation coefficient of CH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. The reference temperature for the modification phase is the target pH value for the modification phase. The corresponding standard temperature, satisfy: (8) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies .
[0014] As a further description of the above technical solution: S5 also includes the following steps: The step of determining the rate of addition of the modification phase regulator, the rate of addition of the modification phase regulator satisfy: (9) in, is the proportional coefficient of the modification stage, is the integral coefficient of the modification stage, is the differential coefficient of the modification stage.
[0015] The present invention provides a method for the thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH regulation. In the nucleation stage, the Fe³⁺ / Fe²⁺ hydrolysis equilibrium is monitored in real time by Fourier transform infrared spectroscopy (FTIR). The NH group regulator is selected according to the peak shift. The pH value is dynamically controlled to control the rate and number of crystal nuclei, effectively reducing the standard deviation of particle size. During the growth phase, the temperature and pH are coordinated to capture the changes in the vibration frequency of the Fe-O bond through Fourier transform infrared spectroscopy (FTIR). The OH group regulator is selected according to the peak shift. Dynamic addition rate, inhibiting the Ostwald ripening effect, inhibiting the crystal growth rate while improving the product lattice integrity. In the modification stage, the CH group regulator is selected according to the peak shift Dynamically controlling the pH value can match the thiol or carboxyl active sites at the PEG end and improve the colloidal stability. BRIEF DESCRIPTION OF THE DRAWINGS
[0016] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for describing the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.
[0017] Figure 1 Flow chart of the thermal decomposition method for dynamically controlling PEG derivatives to modify superparamagnetic ferrite nanoparticles proposed in this invention DETAILED DESCRIPTION The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention. Specific implementation method 1 As attached Figure 1 As shown, the dynamic pH regulation of PEG derivative modified superparamagnetic ferrite nanoparticles thermal decomposition method includes: S1, determine the temperature window and pH threshold window of each stage, where the temperature window of the nucleation stage is , the pH threshold window is , the temperature window during the growth phase is , the pH threshold window is , the temperature window of the modification stage is , the pH threshold window is ; S2, determine the target pH value of each stage. The target pH value of the nucleation stage is , , the pH target during the growth phase is , , the pH target in the modification stage is , ; S3, dynamic monitoring of pH during the nucleation phase , inject the precursor ferric acetylacetonate and NH group regulator oleylamine, when the real-time temperature, The unit is ℃, satisfying The peak shift of 3330cm⁻¹ of NH group modifier oleylamine was monitored in real time by Fourier transform infrared. and Determining the pH value for the nucleation phase correction , the S3 satisfy: As a further description of the above technical solution: S3, (10) in: is the linear coefficient of NH peak shift and pH change, in nm / pH. is the compensation coefficient of NH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. ℃ is the reference temperature The corresponding standard temperature, satisfy: (11) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies , when the temperature meets , go to S4; S4, dynamic monitoring of pH during the growth phase , inject OH group regulator, when the real-time temperature satisfy Real-time monitoring of OH peak shift by Fourier transform infrared ,according to ℃ and Determining pH values for growth phase correction , the S4 satisfy: (12) in: is the linear coefficient of OH peak shift and pH change, with the unit of nm / pH, is the compensation coefficient of OH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. ℃ is the reference temperature The corresponding standard temperature, satisfy: (13) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies , when the temperature meets , go to S5; S5, dynamic monitoring of pH value during the modification phase , inject CH group regulator, when the real-time temperature satisfy Real-time monitoring of CH peak shift by Fourier transform infrared ,according to ℃ and Determining pH values for growth phase correction , the S5 satisfy: (14) in: is the linear coefficient of CH peak shift and pH change, in nm / pH. is the compensation coefficient of CH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. ℃ is the reference temperature The corresponding standard temperature, satisfy: (15) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies , when the temperature meets ,Finish. Specific implementation method 2 As attached Figure 1 As shown, the dynamic pH regulation of PEG derivative modified superparamagnetic ferrite nanoparticles thermal decomposition method includes: S1, determine the temperature window and pH threshold window of each stage, where the temperature window of the nucleation stage is , the pH threshold window is , the temperature window during the growth phase is , the pH threshold window is , the temperature window of the modification stage is , the pH threshold window is ; S2, determine the target pH value of each stage. The target pH value of the nucleation stage is , , the pH target during the growth phase is , , the pH target in the modification stage is , ; S3, dynamic monitoring of pH during the nucleation phase , injecting precursors and NH group modifiers, at real-time temperature satisfy ,according to and Determining the pH value for the nucleation phase correction ,in is the temperature-peak shift coupling coefficient of the injected regulator during the nucleation phase, which is determined by the thermal expansion characteristics of the injected regulator and is expressed in °C / nm. is the temperature-peak coupling offset of the injected regulator in the nucleation stage, in nm, then ,when When satisfied ,according to and Determining the pH value for the nucleation phase correction , the S3 satisfy: (16) in: is the linear coefficient of NH peak shift and pH change, in nm / pH. is the compensation coefficient of NH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. ℃ is the reference temperature The corresponding standard temperature, satisfy: (17) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies , when the temperature, when the temperature meets , go to S4; S4, dynamic monitoring of pH during the growth phase , inject OH group regulator, when the real-time temperature satisfy ,according to and Determining pH values for growth phase correction ,in The temperature-peak shift coupling coefficient of the injected regulator during the growth phase is determined by the thermal expansion characteristics of the injected regulator and is expressed in °C / nm. is the temperature-peak coupling offset of the injection regulator during the growth phase, in nm, then ,when When satisfied ,according to and Determining pH values for growth phase correction , the S4 satisfy: (18) in: is the linear coefficient of OH peak shift and pH change, with the unit of nm / pH, is the compensation coefficient of OH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. ℃ is the reference temperature The corresponding standard temperature, satisfy: (19) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies , when the temperature meets , go to S5; S5, dynamically monitors the pH value during the modification phase, injects CH group regulator, and satisfy ,according to and Determine pH value for modification phase ,in is the temperature-peak shift coupling coefficient of the injected modifier during the modification phase, which is determined by the thermal expansion characteristics of the injected modifier and is expressed in °C / nm. is the temperature-peak coupling offset of the modulator injected during the modification phase, in nm. ,when When satisfied ,according to and Determining pH values for growth phase correction , the S5 satisfy: (20) in: is the linear coefficient of CH peak shift and pH change, in nm / pH. is the compensation coefficient of CH peak offset and pH change, which is related to the ionic strength of the regulator and is expressed in nm / ℃. ℃ is the reference temperature The corresponding standard temperature, satisfy: (twenty one) Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies , when the temperature meets ,Finish.
[0020] Although the embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to these embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the appended claims and their equivalents.
[0021] Those skilled in the art will appreciate that the units and algorithm steps of each example described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professional and technical personnel can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the present invention.
[0022] In the embodiments provided by the present invention, it should be understood that the systems / terminal devices and methods shown can be implemented in other ways. For example, the system / terminal device embodiments described above are merely illustrative. For example, the division of the modules or units is merely a logical functional division. In actual implementation, there may be other division methods, such as multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. In addition, the mutual coupling or direct coupling or communication connection shown or discussed can be through some interfaces, and the indirect coupling or communication connection of the system or unit can be electrical, mechanical or other forms.
[0023] The units described as separate components may or may not be physically separate, and the components described as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the units may be selected according to actual needs to achieve the purpose of the solution of this embodiment.
[0024] The embodiments described above are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. These modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present invention, and should all be included in the scope of protection of the present invention.
Claims
1. Dynamic pH regulation of PEG derivative-modified superparamagnetic ferrite nanoparticles thermal decomposition method, characterized in that: The following steps are involved: S1, determine the temperature window and pH threshold window of each stage, where the temperature window of the nucleation stage is , the pH threshold window is , the temperature window during the growth phase is , the pH threshold window is , the temperature window of the modification stage is , the pH threshold window is ; S2, determine the target pH value of each stage. The target pH value of the nucleation stage is , the pH target during the growth phase is , the pH target in the modification stage is ; S3, dynamic monitoring of pH during the nucleation phase , inject the precursor and NH group regulator, when the real-time temperature ,satisfy , using Fourier transform infrared to monitor the real-time peak shift of NH ,according to and Determining the pH value for the nucleation phase correction , when the temperature meets , go to S4; S4, dynamic monitoring of pH during the growth phase , inject OH group regulator, when the real-time temperature satisfy Real-time peak shift monitoring of OH by Fourier transform infrared ,according to and Determining pH values for growth phase correction , when the temperature meets , go to S5; S5, dynamic monitoring of pH value during the modification phase , inject CH group regulator, when the real-time temperature satisfy Real-time peak shift monitoring of CH by Fourier transform infrared ,according to and Determining pH values for growth phase correction , when the temperature meets ,Finish.
2. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH control according to claim 1, characterized in that: The S3, real-time temperature ,satisfy To satisfy ,according to and Determining the pH value for the nucleation phase correction ,in The temperature-peak shift coupling coefficient for the injection of modulators during the nucleation phase, Temperature-peak coupling offset for injection of modifiers during the growth phase.
3. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH control according to claim 1, characterized in that: The S3 satisfy: in: is the linear coefficient of NH peak shift and pH change, in nm / pH. is the compensation coefficient of NH peak offset and pH change, which is related to the ionic strength of the regulator. The reference temperature for the nucleation stage is the target pH value for the nucleation stage. The corresponding standard temperature, satisfy: in, Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies .
4. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH control according to claim 3, characterized in that: The S3 also includes the following The step of determining the addition rate of the nucleation stage regulator, the addition rate of the nucleation stage regulator satisfy: in, is the proportional coefficient of the nucleation stage, is the integral coefficient of the nucleation stage, is the differential coefficient in the nucleation stage.
5. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH control according to claim 1, characterized in that: The S4, real-time temperature satisfy To satisfy ,according to and Determining pH values for growth phase correction ,in Temperature-peak shift coupling coefficient for injection of modifiers during the growth phase, Temperature-peak coupling offset for injection of modifiers during the growth phase.
6. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH control according to claim 5, characterized in that: The S4 also includes the following Step of determining the rate of addition of a growth phase regulator, the rate of addition of the growth phase regulator satisfy: in, is the growth stage proportional coefficient, is the integral coefficient of the growth stage, is the differential coefficient in the growth stage.
7. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH control according to claim 1, characterized in that: The S4 satisfy: in: is the linear coefficient of OH peak shift and pH change, with the unit of nm / pH, is the compensation coefficient of OH peak offset and pH change, which is related to the ionic strength of the regulator. The reference temperature for the growth phase is the target pH value for the growth phase. The corresponding standard temperature, satisfy: in: Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies .
8. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH regulation according to claim 1, characterized in that: The S5, real-time temperature satisfy To satisfy ,according to and Determine pH value for modification phase ,in is the temperature-peak shift coupling coefficient of the injected regulator during the modification phase, which is determined by the thermal expansion characteristics of the injected regulator. Temperature-peak coupling offset for injection of modulators for the modification phase.
9. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives under dynamic pH control according to claim 1, characterized in that: The S5 satisfy: in: is the linear coefficient of CH peak shift and pH change, is the compensation coefficient of CH peak offset and pH change, which is related to the ionic strength of the regulator. The reference temperature for the modification phase is the target pH value for the modification phase. The corresponding standard temperature, satisfy: in, Dynamically modify the weight factor to meet , adjusted according to the system response speed, The target pH approximation coefficient satisfies .
10. The method for thermal decomposition of superparamagnetic ferrite nanoparticles modified with PEG derivatives by dynamic pH regulation according to claim 7, characterized in that: The S5 also includes the following The step of determining the rate of addition of the modification phase regulator, the rate of addition of the modification phase regulator satisfy: in, is the proportional coefficient of the modification stage, is the integral coefficient of the modification stage, is the differential coefficient of the modification stage.