CN101033775A - Determination method of magnetic bearing dynamic electric current stiffness based on effect of eddy current - Google Patents

Abstract

A method for determining the dynamic current stiffness of a magnetic bearing based on the eddy current effect. According to the physical characteristics and geometric dimensions of the laminated core material of the magnetic bearing, the comprehensive magnetic permeability model of the core lamination based on the eddy current effect is obtained, and the dynamic item caused by the eddy current effect in the dynamic current stiffness is determined, and the magnetic circuit method or finite element method is used. The dynamic current stiffness is obtained by combining the static current stiffness with the dynamic item caused by the eddy current effect after calculating the static current stiffness of the magnetic bearing. The invention is simple and practical, has clear physical meaning, can be applied to the performance analysis of the magnetic bearing control system, and can also guide the design of the magnetic bearing.

Description

A kind of magnetic bearing dynamic current rigidity based on eddy current effect is determined method

Technical field

The present invention relates to a kind of magnetic bearing dynamic current rigidity and determine method, can be used for the stability analysis of the magnetic bearing and the rotor-support-foundation system of the control system analysis of magnetic bearing, particularly magnetically levitated flywheel and magnetic suspension control torque gyroscope based on eddy current effect.

Background technique.

Magnetic suspension bearing is according to the magnetic force presentation mode, can be divided into the passive magnetic suspension bearing and (provide magnetic force by permanent magnet, also claim passive magnetic suspension bearing), Active Magnetic Suspending Bearing (provide magnetic force by electromagnet, also claim active magnetic bearing) and hybrid magnetic suspension bearing (providing magnetic force) by permanent magnet and electromagnet.Because the stable region of passive magnetic suspension bearing is very little, and the hybrid magnetic bearing of permanent magnet bias utilizes permanent magnet to replace the quiescent biasing magnetic field that is produced by field current in the magnetic bearing, has the loss that reduces power amplifier, reduce the Number of ampere turns of electromagnet, dwindle the volume of magnetic suspension bearing, improve advantages such as bearing load carrying capacity, so hybrid magnetic bearing has obtained using widely in the high-speed motion occasion of magnetic suspension motor, high speed flywheel energy-storage system equimagnetic suspension support.The develop rapidly of space technology has promoted deepening continuously of magnetic bearing research, has particularly adopted the high-speed rotor system of magnetic bearing supporting, makes and has realized low vibration, no friction, highi degree of accuracy and long lifetime when astrovehicles such as satellite, space station carry out attitude control.

Current stiffness is one of important parameter in active magnetic bearings and the hybrid magnetic bearing control system, and the magnetic bearing iron core all adopts lamination usually, and this structure can greatly reduce because the eddy current loss that the high-frequency PWM switch causes.Definite method of present relevant current stiffness has theoretical two kinds of (Magnetic Circuit Method and finite element method) and the test identifications that calculate, no matter be theoretical calculation or test identification, because employed model is not all considered the influence of eddy current effect, thereby calculating only is quiescent current rigidity.But in the control procedure of magnetic bearing, the power that magnetic bearing produces changes along with the variation of coil current, and coil current has High Frequency Dynamic, adopt the designed control system of quiescent current rigidity to tend to make the dynamic characteristic variation of system, particularly along with the rising of magnetic suspension rotor rotating speed, various high frequency mode can appear in high speed rotor, control at these mode, if unsettled phenomenon will appear in the control system that designs according to quiescent current rigidity, cause rotor unstability damage whole system when serious, thereby there is bad dynamic performance in the method for existing definite magnetic bearing current stiffness after enforcement, easily cause the unsettled defective of system.

Summary of the invention

The technical problem that the present invention solves is: overcome the deficiency that existing magnetic bearing current stiffness is determined method, propose a kind of magnetic bearing dynamic current rigidity based on eddy current effect and determine method.

Technical solution of the present invention is: a kind of magnetic bearing dynamic current rigidity based on eddy current effect is determined method, may further comprise the steps:

(1) according to the laminated core physical characteristics of materials and the physical dimension of magnetic bearing, ask for comprehensive permeability model based on the core-lamination stack of eddy current effect, determine the dynamic item that causes by eddy current effect in the dynamic current rigidity T=4 σ μ b wherein ²/ π ², the multiple parameter of s representative.

(2) ask for the quiescent current stiffness K of magnetic bearing _i

(3) quiescent current rigidity and the dynamic item combination that is caused by eddy current effect are drawn dynamic current rigidity $K_i\left(s\right)=K_i\·\frac{1}{1+\mathrm{Ts}}.$

Principle of the present invention is: by Fundamental Theory of Electrical Engineering as can be known: the Maxwell's equations group of electromagnetic field is:

$\mathrm{rotE}=-\frac{\∂B}{\∂t}=-\frac{\mathrm{dB}}{\mathrm{dH}}\frac{\∂H}{\∂t}...\left(1\right)$

rotH＝J (2)

divB＝0 (3)

J＝σE (4)

B＝μH (5)

In the formula: E is an electric field strength, and B is a magnetic induction intensity, and H is a magnetic intensity, and J is a current density, and σ is a conductivity, and μ is a permeability.

In order to obtain the representation of permeability, at first try to achieve the magnetic induction intensity in the laminated core, the one dimension eddy current model of Jian Liing as shown in Figure 1 for this reason, in lamination shown in Figure 1, b is half of laminate thickness, supposes magnetic field only by the Z direction, be y function (b≤y≤+ b), but also be the sine function of time, consequent eddy current is the x direction.

Get by formula (1) and (4):

$\frac{{\∂J}_x}{\∂y}={\mathrm{\σ\μ}}_0{\mathrm{\μ}}_r\frac{{\∂H}_z}{\∂t}...\left(6\right)$

In the formula: J _xBe the current density of x direction, μ ₀Be airborne permeability, μ _rBe relative permeability.

Get by formula (5):

$\frac{\∂H_z}{\∂y}=J_x...\left(7\right)$

In the formula: H _zBe the magnetic intensity on the z direction

Formula (7) differential is got:

$\frac{{\&PartialD;}^2{H}_z}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}={\mathrm{\&sigma;\&mu;}}_0{\mathrm{\&mu;}}_r\frac{{\&PartialD;H}_z}{\&PartialD;t}...\left(8\right)$

Because H _zBe the quantity field of sinusoidal variations, so j ω can be replaced Formula (6)～(8) become:

$\frac{\mathrm{dJ}}{\mathrm{dy}}=\mathrm{j\&omega;\&sigma;}{\mathrm{\&mu;}}_0{\mathrm{\&mu;}}_{r}H={\mathrm{\&alpha;}}^{2}H...\left(9\right)$

$\frac{\mathrm{dH}}{\mathrm{dy}}=J...\left(10\right)$

$\frac{d^2\mathrm{<figure-callout\; id="2"\; label="H\; d\; y"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">H</figure-callout>}}{d{y}^{}}={\mathrm{\&alpha;}}^{2}H...\left(11\right)$

α wherein ²=j ω σ μ ₀μ _r

Solving equation (11) can get

$H=\frac{\mathrm{\&alpha;}{\mathrm{\&phi;}}_s}{2{\mathrm{\&mu;}}_0{\mathrm{\&mu;}}_r}\frac{\cosh \mathrm{\&alpha;y}}{\sinh \mathrm{\&alpha;b}}...\left(12\right)$

So

$\stackrel{\&OverBar;}{b}={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20071006335900112.png"\; state="\{\{state\}\}">b</figure-callout>}}_0\frac{\tanh \mathrm{\&alpha;b}}{\mathrm{\&alpha;b}}...\left(13\right)$

In the formula: b is the magnetic induction intensity in the lamination, b ₀Be the magnetic induction intensity of laminate surface, φ _sBe the magnetic flux in the lamination.

Because the existence of eddy current can be regarded magnetic circuit as a small electronic appliances in the lamination, makes the impedance of hot-wire coil that variation take place.It seems that from the external world variation of this impedance can be regarded the variation of permeability unshakable in one's determination as, define the comprehensive permeability μ of core-lamination stack so _eFor:

${\mathrm{\&mu;}}_e=\frac{b_0}{H}=\frac{{\mathrm{\&phi;}}_{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">s</figure-callout>}}}{2\mathrm{bH}}={\mathrm{\&mu;}}_0{\mathrm{\&mu;}}_r\frac{\tanh \mathrm{\&alpha;b}}{\mathrm{\&alpha;b}}...\left(14\right)$

Make j ω=s, μ ₀μ _r=μ can get:

${\mathrm{\&mu;}}_e\left(s\right)=\mathrm{\&mu;}\left[\frac{\tanh \left(\sqrt{\mathrm{s\&sigma;\&mu;}}b\right)}{\sqrt{\mathrm{s\&sigma;\&mu;}}b}\right]...\left(15\right)$

According to mathematical formulae:

$\tanh \frac{\mathrm{\&pi;<figure-callout\; id="2"\; label="x"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">x</figure-callout>}}{2}=\frac{4x}{\mathrm{\&pi;}}\underset{k=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{1}{{(2k-1)}^2+x^2}...\left(16\right)$

Formula (15) but abbreviation be:

${\mathrm{\&mu;}}_e\left(s\right)=\frac{8\mathrm{\&mu;}}{{\mathrm{\&pi;}}^2}[\frac{1}{1+s\left(\frac{4\mathrm{\&sigma;\&mu;}{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{{\mathrm{\&pi;}}^2}\right)}+\frac{1}{9+s\left(\frac{4\mathrm{\&sigma;\&mu;}{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{{\mathrm{\&pi;}}^2}\right)}+\frac{1}{25+s\left(\frac{4\mathrm{\&sigma;\&mu;}{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{{\mathrm{\&pi;}}^2}\right)}+\mathrm{\&Lambda;}]...\left(17\right)$

Ignore the influence of higher order term, formula (17) can be write as:

${\mathrm{\&mu;}}_e\left(s\right)=\frac{8\mathrm{\&mu;}}{{\mathrm{\&pi;}}^2}\left[\frac{1}{1+s\left(\frac{4\mathrm{\&sigma;\&mu;}{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{{\mathrm{\&pi;}}^2}\right)}\right]\&ap;\frac{\mathrm{\&mu;}}{1+s\left(\frac{4\mathrm{\&sigma;\&mu;}{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20071006335900102.png,A20071006335900121.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{{\mathrm{\&pi;}}^2}\right)}...\left(18\right)$

Make T=4 σ μ b ²/ π ²:

${\mathrm{\&mu;}}_e\left(s\right)=\frac{\mathrm{\&mu;}}{1+\mathrm{sT}}...\left(19\right)$

Formula (19) is the expression-form of dynamic current rigidity dynamic item for only having considered the single order transfer function form of dominant pole influence.

For any magnetic bearing structure, all exist equivalent magnetic circuit, and, all can be reduced to magnetic circuit shown in Figure 2 by the series parallel connection of magnetic resistance in the magnetic circuit for equivalent magnetic circuit arbitrarily, the magnetic flux in the air gap is φ=μ as shown in Figure 2 ₀H ₀A under the situation of not considering leakage field, can think that process magnetic flux unshakable in one's determination equals the magnetic flux in the air gap, i.e. φ=μ ₀H ₀A=μ _eH _eSo A is φ (s)=μ ₀(s) H ₀A=μ _e(s) H _eTherefore A can think μ ₀(s) have and μ _e(s) Yi Yang form.

${\mathrm{\&mu;}}_0\left(S\right)=\frac{{\mathrm{\&mu;}}_0}{1+\mathrm{Ts}}...\left(20\right)$

According to principle of virtual displacement

$F(x,i)=\frac{B{(x,i)}^{2}A}{{\mathrm{\&mu;}}_0}...\left(21\right)$

Wherein x is a gap length, and i is an exciting current, and F is a magnetic force, and A is that magnetic pole section is long-pending.

When i is constant, i.e. i=i ₀, μ so ₀(t)=μ ₀, ${\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="A20071006335900112.png"\; state="\{\{state\}\}">H</figure-callout>}}_0=\frac{{\mathrm{Ni}}_0}{x},$ As can be known $F={\left(\frac{{\mathrm{\&mu;}}_0{\mathrm{Ni}}_0}{x}\right)}^{2}A/{\mathrm{\&mu;}}_0,$ Differentiate gets the time domain representation of current stiffness to i:

$K_i\left(t\right)=\frac{2{\mathrm{\&mu;}}_0\left(t\right)N^2{i}_{0}A}{x^2}...\left(22\right)$

After pull-type conversion, as can be known

$K_i\left(s\right)=\frac{2{\mathrm{\&mu;}}_0{N}^2{i}_{0}A}{x^2}\&CenterDot;\frac{1}{1+\mathrm{Ts}}=K_i\&CenterDot;\frac{1}{1+\mathrm{Ts}}...\left(23\right)$

In the formula: N is a coil turn

This shows that dynamic current rigidity is quiescent current rigidity and the product of the dynamic item that is caused by eddy current effect.

The present invention's advantage compared with prior art is: the present invention has considered the influence of eddy current effect, set up comprehensive permeability frequency model, the model explicit physical meaning, be easy to realize, the current stiffness of Que Dinging is dynamic amount thus, has important directive significance for the analysis and the design of control system.

Description of drawings

Fig. 1 is a magnetic bearing laminated core schematic representation;

Fig. 2 simplifies magnetic circuit model for magnetic bearing;

Fig. 3 determines method flow diagram for the magnetic bearing dynamic current rigidity based on eddy current effect of the present invention;

The dynamic current rigidity amplitude-versus-frequency curve of Fig. 4 for obtaining according to the inventive method;

Fig. 5 a and Fig. 5 b are for to adopt quiescent current rigidity and dynamic current rigidity magnetic bearing to be controlled the rotor stability comparison diagram that obtains respectively, the rotor stability figure that obtains when wherein Fig. 5 a is for employing quiescent current rigidity, Fig. 5 b is the rotor stability figure that obtains when adopting dynamic current rigidity.

Embodiment

As shown in Figure 3, determine method, can be divided into and find the solution quiescent current rigidity, combine by quiescent current rigidity and dynamic item then, so the various in the past methods of finding the solution quiescent current rigidity all can be used based on the dynamic current rigidity of eddy current effect.But since the complicated magnetic circuit of any magnetic bearing all abbreviation be magnetic circuit model shown in Figure 2, so present embodiment just finds the solution with magnetic circuit model shown in Figure 2, wherein coil turn is N=70, lamination thickness is d=0.0002m, the lamination permeability is μ=20000 μ ₀, conductivity=0.22e ⁷, air length of magnetic path x=0.0001m, area of core section A ₁=1.2e ^-4m ², the cross-section of air gap amasss A ₂=1.2e ^-4m ², bias current i ₀=1A.

The first step is asked for comprehensive permeability.

Know according to formula (19): T=4 σ μ b ²/ π ²=0.000224

Therefore, the comprehensive permeability of laminated core is: ${\mathrm{\&mu;}}_e\left(s\right)=\frac{\mathrm{\&mu;}}{1+\mathrm{Ts}}=\frac{20000{\mathrm{\&mu;}}_0}{1+0.000224s}$

Know that according to formula (20) the air comprehensive permeability is: ${\mathrm{\&mu;}}_0\left(s\right)=\frac{{\mathrm{\&mu;}}_0}{1+\mathrm{Ts}}=\frac{{\mathrm{\&mu;}}_0}{1+0.000224s}$

Thereby know that dynamic current rigidity dynamic item is: $\frac{1}{1+\mathrm{Ts}}=\frac{1}{1+0.000224s}$

Second goes on foot, and asks the quiescent current rigidity of magnetic bearing.

Know that according to formula (21) magnetic force is: $F={\left(\frac{{\mathrm{\&mu;}}_0{\mathrm{Ni}}_0}{x}\right)}^{2}A/{\mathrm{\&mu;}}_0$

By (22) Shi Kede: $K_i\left(t\right)=\frac{2{\mathrm{\&mu;}}_0\left(t\right)N^2{i}_{0}A}{x^2}$

Because i (t)=i ₀So, $K_i=\frac{2{\mathrm{\&mu;}}_0{N}^2{i}_{0}A}{x^2}=147.7805$

In the 3rd step, ask for dynamic current rigidity.

Know according to formula (23), $K_i\left(s\right)=K_i\&CenterDot;\frac{1}{1+\mathrm{Ts}}=\frac{147.7805}{1+0.000224s}$

Fig. 4 is the amplitude frequency curve of the dynamic current rigidity asked for, as we know from the figure, dynamic current rigidity is typical first order inertial loop, phase lag is 1.3 ° when 100Hz, phase place has lagged behind 34 ° when 3KHz, and amplitude fading is to-1.62db, has had bigger influence for the Stability Control of high speed rotor mode, need take corresponding corrective action, as the phase place anticipatory control.

The stability of rotor is relevant with the integral stiffness of magnetic bearing, integral stiffness is relevant with current stiffness and displacement rigidity, wherein displacement rigidity determines that in the magnetic bearing physical dimension back just is field planting, therefore under the identical control parameter, current stiffness greatly then integral stiffness is big, the little then integral stiffness of current stiffness is little, and after integral stiffness was crossed and hanged down, rotor then lost suspension stability.Fig. 5 is for to adopt quiescent current rigidity and dynamic current rigidity magnetic bearing to be controlled the rotor stability comparison diagram that obtains respectively, Fig. 5 a is the rotor stability figure that obtains when adopting quiescent current rigidity, and Fig. 5 b is the rotor stability figure that obtains when adopting dynamic current rigidity.As can be seen from Figure 5, because the different current stiffnesses that adopt have designed different controller parameters, in magnetic bearings control bandwidth 3KHz scope, owing to do not consider the dynamic of current stiffness, according to the quiescent current stiffness K _i=340N/A CONTROLLER DESIGN parameter causes integral stiffness low excessively at HFS from Fig. 5 a as can be seen, therefore is subjected to effect of non-linear, causes rotor the low frequency unstability to occur.And according to dynamic current rigidity $K_i\left(s\right)=\frac{340}{1+0.000224s}N/A$ The controller parameter of design has all kept high integral stiffness in magnetic bearings control bandwidth 3KHz, so the rotor suspension is stable.

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

Claims (4)

1, a kind of magnetic bearing dynamic current rigidity based on eddy current effect is determined method, it is characterized in that: may further comprise the steps:

(1) according to the laminated core physical characteristics of materials and the physical dimension of magnetic bearing, ask for comprehensive permeability model based on the core-lamination stack of eddy current effect, determine the dynamic item that causes by eddy current effect in the dynamic current rigidity

(2) ask for the quiescent current stiffness K of magnetic bearing _i

(3) quiescent current rigidity and the dynamic item combination that is caused by eddy current effect are drawn dynamic current rigidity $K_i\left(s\right)=K_i\&CenterDot;\frac{1}{1+\mathrm{Ts}}.$

2, a kind of magnetic bearing dynamic current rigidity based on eddy current effect according to claim 1 is determined method, and it is characterized in that: the comprehensive permeability model in the described step (1) has ${\mathrm{\&mu;}}_e\left(s\right)=\frac{\mathrm{\&mu;}}{1+\mathrm{Ts}}$ Form, wherein μ is the static permeability of laminated core material, T=4 σ μ b ²/ π ², σ is the conductivity of laminate, b is 1/2nd laminated core thickness.

3, a kind of magnetic bearing dynamic current rigidity based on eddy current effect according to claim 1 is determined method, it is characterized in that: the quiescent current stiffness K in the described step (2) _iObtain by Magnetic Circuit Method or finite element method.

4, a kind of magnetic bearing dynamic current rigidity based on eddy current effect according to claim 1 is determined method, it is characterized in that: the laminated core material in the described step (1) is 1J50 or 1J79 or electrical steel plate or silicon steel thin belt or 1J85 or 1J22 or amorphous.

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