CN100488051C - Analytical method for electromagnetic bearing switch power amplifier - Google Patents

Abstract

Provided is an analysis method used for the electromagnetic bearing switch power amplifier, the invention is a method which can be used to analyze the output current characteristic of the loading coil of the electromagnetic bearing switch power amplifier system. The method equally effects the practical electromagnetic bearing switch power amplifier system, and gets the nonlinear model of the pulse width modulation generator according to the switching function model of the power main circuit, the pulse width modulation generator is provided with harmonic linearization treatment using harmonic linearization principle. The invention realizes the harmonic linearization of the electromagnetic bearing switch power amplifier whose essence is a nonlinear system, the invention can not only simplify the analysis to the electromagnetic bearing switch power amplifier system, but accurately describe the output current characteristics of the electromagnetic bearing switch power amplifier, the invention brings more convenient to the stability analysis of the electromagnetic bearing controlling system.

Description

A kind of analytical method that is used for electromagnetic bearing switch power amplifier

Technical field

The present invention relates to the analytical method of a kind of electromagnetic bearing switch power amplifier system, be used for the electric current Output Characteristic of electromagnetic bearing switch power amplifier.

Background technology

Electromagnetic bearing is a kind of high-tech electromechanical integrated product, advantage such as it has high rotating speed, no friction, long-life, need not to lubricate, pollution-free is with a wide range of applications in fields such as Aero-Space, high-speed machine tool, turbomachinery and centrifuge, vacuum and clean room technology.Power amplifier is as the actuator of magnetic bearings control system, and its performance will directly influence the stability of magnetic bearings control system.

When electromagnetic bearing switch power amplifier was analyzed, art methods all was that the electromagnetic bearing switch power amplifier equivalence is first order inertial loop.With the approximate electromagnetic bearing switch power amplifier of first order inertial loop,, can not reflect the output characteristic of real system though can simplify the analytical model of electromagnetic bearing switch power amplifier.In fact, the output characteristic of electromagnetic bearing switch power amplifier not only is subjected to the influence of frequency input signal, and is subjected to the influence of input signal amplitude.When too high or amplitude was excessive when the frequency of input signal, the output current of electromagnetic bearing switch power amplifier all can not be followed the variation of input signal fully, but shows as the variation that output current lags behind input signal.Therefore, the shortcoming of art methods is not consider the nonlinear characteristic of electromagnetic bearing switch power amplifier, but utilizes the analysis transfer function model method that linear system adopted to analyze this non linear system of electromagnetic bearing switch power amplifier.This approximate processing method can not reflect the output current characteristic of real system fully, has caused the inaccurate result of real system model, finally can influence magnetic bearings control system stability analysis result's confidence level.

Summary of the invention

The objective of the invention is: the deficiency that overcomes art methods, a kind of analytical method of electromagnetic bearing switch power amplifier control system of directly perceived and reflection real system characteristic is provided, adopt the analytical model of the electromagnetic bearing switch power amplifier that this method obtains to approach real system more accurately, and step is simple, and final analysis result is with the high conformity of actual test result.

Technical solution of the present invention is: a kind of analytical method that is used for electromagnetic bearing switch power amplifier is characterized in that comprising the following steps:

(1) actual electromagnetic bearing switching power amplifier system is carried out equivalence

With the equivalence of the signal processing circuit between current controller and the electromagnetic bearing loading coil in the actual electromagnetic bearing close power amplifier system is pwm generator.Actual electromagnetic bearing close power amplifier system is total to four processes by current controller, pwm generator, electromagnetic bearing loading coil and current sensor and forms through after the equivalence.

(2) set up the nonlinear model of pwm generator

The nonlinear model of pwm generator is:

$u_{\mathrm{AB}}\left(t\right)=\left\{\begin{array}{cc}U,& \mathrm{kT}\&le;t<\mathrm{kT}+t_k\\ -U,& \mathrm{kT}+t_k<t\&le;(k+1)T\end{array}\right.$

$t_k=\frac{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="C200710064065E00122.png,C200710064065E00132.png"\; state="\{\{state\}\}">T</figure-callout>}}{2}+\frac{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="C200710064065E00122.png,C200710064065E00132.png"\; state="\{\{state\}\}">T</figure-callout>}}{2}\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)$

$\mathrm{sat}\left(x\right)=\left\{\begin{array}{cc}x,& \left|x\right|<1\\ \&PlusMinus;1,& \left|x\right|\&GreaterEqual;1\end{array}\right.$

U represents the dc bus supply voltage in the formula; T represents the cycle of sawtooth carrier signal; A represents the amplitude of sawtooth carrier signal; t _kThe output voltage u at interval [kT, (k+1) T] the electromagnetic bearing coil two ends, lining of expression _AB(t) switching instant, k=0,1,2 ...

(3) set up the harmonic linearization model of pwm generator

The model of the harmonic linearization of pwm generator is:

$u_{\mathrm{AB}}\left(t\right)\&ap;\frac{U}{A}{u}_r\left(\mathrm{kT}\right)+[a_1\cos \left(\mathrm{\&omega;t}\right)+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="C200710064065E00122.png,C200710064065E00131.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\sin \left(\mathrm{\&omega;t}\right)]$

Wherein: $\mathrm{\&omega;}=\frac{2\mathrm{\&pi;}}{T}$

$a_1=-\frac{2U}{\mathrm{\&pi;}}\sin [\mathrm{\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]$

$b_1=\frac{2U+2U\cos [\mathrm{\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]}{\mathrm{\&pi;}}$

U in the formula _r(kT) input signal of expression pulse-duration modulation signal generator, ω is the angular frequency of first harmonic.

(4) set up the electromagnetic bearing switch power amplifier analytical model, and the electric current output characteristic of electromagnetic bearing switch power amplifier is analyzed.The step of setting up the electromagnetic bearing switch power amplifier analytical model is as follows:

(a) transfer function of current controller is $G_{\mathrm{pi}}\left(s\right)=\frac{K_{\mathrm{pi}}\mathrm{\&tau;s}+1}{\mathrm{\&tau;s}},$ K _PiBe the amplification coefficient of proportional integral current controller proportional parts, τ is the integration time constant of proportional integral current controller;

(b) the harmonic linearization model of pwm generator is provided by above-mentioned steps (3);

(c) transfer function of electromagnetic bearing loading coil is $G_{\mathrm{coil}}\left(s\right)=\frac{1}{\mathrm{Ls}+R},$ L and R are respectively the equivalent inductance and the equivalent resistance of electromagnetic bearing coil;

(d) transfer function of current sensor is G _Cm(s)=K _Ui, K _UiBe the current sensor gain coefficient.

The Mathematical Modeling of current controller, pwm generator, electromagnetic bearing loading coil and 4 links of current sensor has been formed the analytical model of electromagnetic bearing switch power amplifier jointly.

The present invention's advantage compared with prior art is: the present invention considers the nonlinear characteristic of electromagnetic bearing switch power amplifier, and will cause the pwm generator enforcement harmonic linearization of system's nonlinear characteristic to handle, realized the harmonic linear fractional analysis of electromagnetic bearing switch power amplifier.The inventive method has overcome art methods and has adopted linear model to be similar to the shortcoming of actual non linear system, and the analytical model of the electromagnetic bearing switch power amplifier that employing the inventive method obtains can be approached real system more accurately.The invention process step is simple, and final analysis result is with the high conformity of actual test result.

Description of drawings

Fig. 1 is a flow chart of the present invention;

Fig. 2 is an electromagnetic bearing switch power amplifier system architecture diagram of the present invention;

Fig. 3 is the harmonic linearization illustraton of model of pulse-duration modulation signal generator of the present invention;

Fig. 4 is an electromagnetic bearing switch power amplifier analytical model of the present invention;

Fig. 5 is electromagnetic bearing switch power amplifier electric current output characteristic curve figure of the present invention;

Fig. 6 is the experimental curve diagram of electromagnetic bearing switch power amplifier.

Embodiment

In conjunction with a kind of electromagnetic bearing switch power amplifier system introduction particular content of the present invention.

The step that the inventive method is implemented as shown in Figure 1.

(1) actual electromagnetic bearing switching power amplifier system is carried out equivalence

Actual electromagnetic bearing switching power amplifier system is through after the equivalence, and its system architecture diagram mainly is total to four processes by current controller 1, pwm generator 2, electromagnetic bearing loading coil 3 and current sensor 4 and forms as shown in Figure 2; Current sensor 4 detects the actual current signal in the electromagnetic bearing loading coil 3, and the current error signal that this signal and the given signal of electric current are produced inputs to current controller 1; The current error signal after overregulating that current controller 1 produces inputs to pwm generator 2; The output voltage pulse signal of pwm generator 2 directly is added in electromagnetic bearing loading coil two ends, thus the electric current in the control electromagnetic bearing loading coil.

(2) set up the nonlinear model of pwm generator

For the ease of setting up the nonlinear model of pwm generator, the switch function of definition full bridge power main circuit.For the full bridge power main circuit, two power switch pipe alternate conduction on each brachium pontis and shutoff, insertion switch function S ₁And S ₂, be defined as follows respectively:

According to switch function definition, the output voltage u at electromagnetic bearing coil two ends _AB(t) can be expressed as:

u _AB(t)＝S ₁U-S ₂U＝(S ₁-S ₂)U

When usually the full bridge power main circuit being controlled, power switch pipe VT ₁, VT ₄Conducting simultaneously or shutoff simultaneously, VT ₂, VT ₃Also conducting simultaneously or shutoff simultaneously, and power switch pipe VT ₁, VT ₂Then alternate conduction and shutoff.Therefore, switch function S ₁And S ₂Satisfy:

U in the formula _c(t) expression sawtooth carrier signal, u _r(t) input signal of expression pwm generator.

According to the definition of the switch function of full bridge power main circuit, obtain the output voltage u of pulse-duration modulation signal generator _AB(t) and the input voltage u of pulse-duration modulation signal generator _r(t) relational expression:

$u_{\mathrm{AB}}\left(t\right)=\left\{\begin{array}{cc}U,& \mathrm{kT}\&le;t<\mathrm{kT}+t_k\\ -U,& \mathrm{kT}+t_k<t\&le;(k+1)T\end{array}\right.$

$t_k=\frac{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="C200710064065E00122.png,C200710064065E00132.png"\; state="\{\{state\}\}">T</figure-callout>}}{2}+\frac{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="C200710064065E00122.png,C200710064065E00132.png"\; state="\{\{state\}\}">T</figure-callout>}}{2}\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)$

$\mathrm{sat}\left(x\right)=\left\{\begin{array}{cc}x,& \left|x\right|<1\\ \&PlusMinus;1,& \left|x\right|\&GreaterEqual;1\end{array}\right.$

T represents the cycle of sawtooth carrier signal in the formula; A represents the amplitude of sawtooth carrier signal; t _kThe output voltage u at interval [kT, (k+1) T] the electromagnetic bearing coil two ends, lining of expression _AB(t) switching instant.

(3) set up the harmonic linearization model of pwm generator

The input signal of supposing pulse-duration modulation signal generator is u _r(t), its frequency is far below frequency of carrier signal f (f=1/T), with the output signal u of pulse-duration modulation signal generator _AB(t) carry out fourier progression expanding method:

$u_{\mathrm{AB}}\left(t\right)=\frac{a_0}{2}+\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}[a_n\cos \left(\mathrm{n\&omega;t}\right)+b_n\sin \left(\mathrm{n\&omega;t}\right)]$

In the formula: $\mathrm{\&omega;}=\frac{2\mathrm{\&pi;}}{T}$

$a_n=\frac{2}{T}{\&Integral;}_{\mathrm{kT}}^{(k+1)T}{u}_{\mathrm{AB}}\left(t\right)\cos \left(\mathrm{n\&omega;t}\right)\mathrm{dt},n=\mathrm{0,1,2},\&CenterDot;\&CenterDot;\&CenterDot;$

$b_n=\frac{2}{T}{\&Integral;}_{\mathrm{kT}}^{(k+1)T}{u}_{\mathrm{AB}}\left(t\right)\sin \left(\mathrm{n\&omega;t}\right)\mathrm{dt},n=\mathrm{0,1,2},3,\&CenterDot;\&CenterDot;\&CenterDot;$

By with u _AB(t) and t _kBring following formula into:

$a_0=\frac{2}{T}{\&Integral;}_{\mathrm{kT}}^{(k+1)T}{u}_{\mathrm{AB}}\left(t\right)\mathrm{dt}=\frac{2U}{A}{u}_r\left(\mathrm{kT}\right),\left|u_r\left(\mathrm{kT}\right)\right|<A$

$a_n=\frac{2}{T}[{\&Integral;}_{\mathrm{kT}}^{\mathrm{kT}+t_k}U\cos \left(\mathrm{n\&omega;t}\right)\mathrm{dt}+{\&Integral;}_{\mathrm{kT}+t_k}^{(k+1)T}(-U)\cos \left(\mathrm{n\&omega;t}\right)\mathrm{dt}]$

$=\frac{{(-1)}^n}{n}\&CenterDot;\frac{2U}{\mathrm{\&pi;}}\sin [\mathrm{n\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]$ (n＝1，2，3，…)

$b_n=\frac{2}{T}[{\&Integral;}_{\mathrm{kT}}^{\mathrm{kT}+t_k}U\sin \left(\mathrm{n\&omega;t}\right)\mathrm{dt}+{\&Integral;}_{\mathrm{kT}+t_k}^{(k+1)T}(-U)\sin \left(\mathrm{n\&omega;t}\right)\mathrm{dt}]$

$=\frac{2U}{\mathrm{n\&pi;}}\{1-{(-1)}^n\&CenterDot;\cos [\mathrm{n\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]\}$ (n＝1，2，3，…)

If only consider output u _AB(t) DC component and first-harmonic:

$u_{\mathrm{AB}}\left(t\right)\&ap;\frac{U}{A}{u}_r\left(\mathrm{kT}\right)+[a_1\cos \left(\mathrm{\&omega;t}\right)+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="C200710064065E00122.png,C200710064065E00131.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\sin \left(\mathrm{\&omega;t}\right)]$

Wherein: $a_1=-\frac{2U}{\mathrm{\&pi;}}\sin [\mathrm{\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]$

${\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="C200710064065E00122.png,C200710064065E00131.png"\; state="\{\{state\}\}">b</figure-callout>}}_1=\frac{2U+2U\cos [\mathrm{\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]}{\mathrm{\&pi;}}$

Can get the harmonic linearization model of pwm generator as shown in Figure 3 according to above relational expression.

(4) set up the analytical model of electromagnetic bearing switch power amplifier, and the electric current output characteristic of electromagnetic bearing switch power amplifier is analyzed.

According to the system architecture diagram of electromagnetic bearing switch power amplifier and the harmonic linearization model of pwm generator, can obtain as shown in Figure 4 based on MATLAB/Simulink electromagnetic bearing switch power amplifier analytical model.The system parameters of electromagnetic bearing switch power amplifier is as follows among Fig. 4: the equivalent resistance R=2 Ω of electromagnetic bearing coil, the equivalent inductance L=6mH of electromagnetic bearing coil, dc bus supply voltage U=28V, period T=the 0.00005s of sawtooth carrier signal, amplitude A=the 10V of sawtooth carrier signal, the gain coefficient of current sensor are 5V/A.When importing direct current and sine voltage signal, carry out MATLAB emulation by analytical model, just can obtain the electric current curve of output that electromagnetic bearing switch power amplifier is followed direct current and sine voltage signal electromagnetic bearing switch power amplifier shown in Figure 4.Fig. 5 is electromagnetic bearing switch power amplifier electric current output characteristic curve figure of the present invention, wherein Fig. 5 a is that electromagnetic bearing switch power amplifier is when following the d. c. voltage signal of 5V, output current mean value is the current waveform figure of 1A, and Fig. 5 b is an electromagnetic bearing switch power amplifier that to follow amplitude be 1V, when frequency is the sine wave signal of 800Hz, output amplitude is that 0.2A, frequency are the sinusoidal current waveform figure of 800Hz.Fig. 6 is the actual experiment oscillogram of electromagnetic bearing switch power amplifier, wherein Fig. 6 a is that electromagnetic bearing switch power amplifier is when the d. c. voltage signal of input 5V, output current mean value is the electric current experimental waveform figure of 1A, and Fig. 6 b be electromagnetic bearing switch power amplifier in the input amplitude be 1V, when frequency is the sine wave signal of 800Hz, output amplitude is that 0.2A, frequency are the sinusoidal current experimental waveform figure of 800Hz.Current curve by difference comparison diagram 5a, Fig. 6 a and Fig. 5 b, Fig. 6 b, adopt the current waveform of the electromagnetic bearing switch power amplifier electric current output waveform of the inventive method gained and actual experiment gained fine identically as can be seen, thereby verified the feasibility and the validity of the inventive method.

The content that is not described in detail in the specification of the present invention belongs to this area professional and technical personnel's known prior art.

Claims (2)

1, a kind of analytical method that is used for electromagnetic bearing switch power amplifier is characterized in that comprising the following steps:

(1) actual electromagnetic bearing switching power amplifier system is carried out equivalence;

With the equivalence of the signal processing circuit between current controller (1) and the electromagnetic bearing loading coil (3) in the actual electromagnetic bearing close power amplifier system is pwm generator (2), actual electromagnetic bearing close power amplifier system is total to four processes by current controller (1), pwm generator (2), electromagnetic bearing loading coil (3) and current sensor (4) and forms through after the equivalence;

(2) set up the nonlinear model of pwm generator

The nonlinear model of pwm generator is:

$u_{\mathrm{AB}}\left(t\right)=\left\{\begin{array}{cc}U,& \mathrm{kT}\&le;t<\mathrm{kT}+t_k\\ -U,& \mathrm{kT}+t_k<t\&le;(k+1)T\end{array}\right.$

$t_k=\frac{T}{2}+\frac{T}{2}\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)$

$\mathrm{sat}\left(x\right)=\left\{\begin{array}{cc}x,& \left|x\right|<1\\ \&PlusMinus;1,& \left|x\right|\&GreaterEqual;1\end{array}\right.$

U represents the dc bus supply voltage in the formula; T represents the cycle of sawtooth carrier signal; A represents the amplitude of sawtooth carrier signal; t _kThe output voltage u at interval [kT, (k+1) T] the electromagnetic bearing coil two ends, lining of expression _AB(t) switching instant, k=0,1,2,

(3) set up the harmonic linearization model of pwm generator

The model of the harmonic linearization of pwm generator is:

$u_{\mathrm{AB}}\left(t\right)\&ap;\frac{U}{A}{u}_r\left(\mathrm{kT}\right)+[a_1\cos \left(\mathrm{\&omega;t}\right)+b_1\sin \left(\mathrm{\&omega;t}\right)]$

Wherein: $\mathrm{\&omega;}=\frac{2\mathrm{\&pi;}}{T}$

$a_1=-\frac{2U}{\mathrm{\&pi;}}\sin [\mathrm{\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]$

$b_1=\frac{2U+2U\cos [\mathrm{\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]}{\mathrm{\&pi;}}$

U in the formula _r(kT) input signal of expression pulse-duration modulation signal generator, ω is the angular frequency of first harmonic;

(4) set up the electromagnetic bearing switch power amplifier analytical model, Mathematical Modeling by current controller, pwm generator, electromagnetic bearing loading coil and 4 links of current sensor is formed jointly, and the electric current output characteristic of electromagnetic bearing switch power amplifier is analyzed.

2, the analytical method that is used for electromagnetic bearing switch power amplifier according to claim 1, it is characterized in that: described step (4) is set up the electromagnetic bearing switch power amplifier analytical model and is made up of jointly the Mathematical Modeling of current controller, pwm generator, electromagnetic bearing loading coil and 4 links of current sensor, wherein:

(1) transfer function of current controller is $G_{\mathrm{pi}}\left(s\right)=\frac{K_{\mathrm{pi}}\mathrm{\&tau;s}+1}{\mathrm{\&tau;s}},$ K _PiBe the amplification coefficient of proportional integral current controller proportional parts, τ is the integration time constant of proportional integral current controller;

(2) model of the harmonic linearization of pwm generator is:

$u_{\mathrm{AB}}\left(t\right)\&ap;\frac{U}{A}{u}_r\left(\mathrm{kT}\right)+[a_1\cos \left(\mathrm{\&omega;t}\right)+b_1\sin \left(\mathrm{\&omega;t}\right)]$

Wherein: $\mathrm{\&omega;}=\frac{2\mathrm{\&pi;}}{T}$

$a_1=-\frac{2U}{\mathrm{\&pi;}}\sin [\mathrm{\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]$

$b_1=\frac{2U+2U\cos [\mathrm{\&pi;}\&CenterDot;\mathrm{sat}\left(\frac{u_r\left(\mathrm{kT}\right)}{A}\right)]}{\mathrm{\&pi;}}$

U in the formula _r(kT) input signal of expression pulse-duration modulation signal generator, ω is the angular frequency of first harmonic;

(3) transfer function of electromagnetic bearing loading coil is $G_{\mathrm{coil}}\left(s\right)=\frac{1}{\mathrm{Ls}+R},$ L and R are respectively the equivalent inductance and the equivalent resistance of electromagnetic bearing coil;

(4) transfer function of current sensor is G _Cm(s)=K _Ui, K _UiBe the current sensor gain coefficient.

The Mathematical Modeling of current controller, pwm generator, electromagnetic bearing loading coil and 4 links of current sensor has been formed the analytical model of electromagnetic bearing switch power amplifier jointly.

Images