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Analysis and design of a magnetic bearing Soukup, Vladimir 1988

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j A N A L Y S I S A N D D E S I G N O F A M A G N E T I C B E A R I N G Vladimir Soukup B. Sc. (Electrical Engineering) Czech Technical University A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F E L E C T R I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A June 1988 © Vladimir Soukup , 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract Magnetic bearings have recently begun to be employed in rotating machinery for vi-bration reduction, elimination of oil lubrication problems and prevention of failures. This thesis presents an analysis and design of an experimental model of a magnetic suspension system. The magnetic bearing, its control circuit and the supported ob-ject are modeled. Formulas are developed for the position and current stiffness of the bearing and the analogy with a mechanical system is shown. The transfer function is obtained for the control and experimental results are presented for the double pole one axis magnetic support system. i i Table of Contents Abstract i i List of Figures v 1 I N T R O D U C T I O N 1 2 D Y N A M I C M O D E L O F M A G N E T I C B E A R I N G S 6 2.1 Introduction 6 2.2 Lifting Force of an Electromagnet 6 2.3 Stiffness of Magnetic Bearings 9 2.4 Mechanical Analogy of Magnetic Bearing 11 2.5 Mathematical Model of Magnetic Bearing 13 3 S O L U T I O N O F T H E D I F F E R E N T I A L E Q U A T I O N 15 3.1 Introduction 15 3.1.1 Critical Points of a System 16 3.1.2 Transformation of the 2nd Order to the l " Order Differential Equations 17 4 L I N E A R I Z E D M O D E L O F M A G N E T I C B E A R I N G S 21 4.1 Introduction 21 4.2 Linear Approximation of an Electromagnetic Force 22 4.3 Linearized Model and the Transfer Function 23 iii 5 I M P L E M E N T A T I O N O F A M A G N E T I C B E A R I N G 26 5.1 Introduction 26 5.2 Velocity Transducer 27 5.3 Position Transducers 28 5.4 Power Drivers 31 5.4.1 Voltage Driver 31 5.4.2 Current Feedback 33 5.4.3 Current Driver 34 6 M E A S U R E M E N T S 36 6.1 Introduction 36 6.2 Double Ended and Single Ended Models 36 6.3 Electric Time Constant of the Electromagnet 38 6.4 Position Constant of the Hall Effect Sensor 39 6.5 Velocity Constant 39 6.6 Measurement of The Position Stiffness 41 .7 C O N C L U S I O N S 45 Bibliography 4 7 A P P E N D I X 49 iv Lis t of Figures 1.1 Model for a Magnetic Suspension System 2 1.2 Radial Suspension of a Rotor 3 2.3 Coordinate System of the Double Ended Suspension System 7 2.4 The Electromagnet 7 2.5 Model of an Analogous Mechanical System 11 2.6 Characteristic Roots of a Second Order System in the Complex Plane . 13 4.7 Graphical Representation of Transfer Function Poles in the s plane . . . 24 5.8 Velocity Transducer 27 5.9 Resonant Circuit Measurement Method 29 5.10 Phase-Locked Loop 29 6.11 Voltage vs. Current in a Series RL Circuit 38 6.12 Measurement of the Velocity Constant 40 v Chapter 1 I N T R O D U C T I O N The idea of suspending a mechanical object in a magnetic field is not new. It spans nearly 150 years back to the year 1842 when S. Earnshaw [1] developed a theorem which states that a system using permanent magnets or electromagnets without control of current is inherently unstable. His work was later followed by W. Braunbeck [2] and others. Although Earnshaw was right, there are now magnetic suspension systems that use both permanent magnets and electromagnets with controlled currents together in that the permanent magnets are employed to carry all static loads whereas the electromagnets are used just for control purposes to stabilize the system. These systems are known as VZP (for " Virtually Zero Power "), because such an arrangement results in minimal power requirements since they approach zero power consumption as a limit, regardless of how much mass is suspended. This type of suspension system is especially convenient in applications where space, weight and power are limited. Now a magnetically suspended 1100 kg shaft capable of spinning at 10,000 rpm or higher is becoming a commercial reality. Already such shafts turning at various speeds are being used in grinding and polishing machinery, vacuum pumps, compressors, tur-bines, generators and centrifuges. Electronically controlled magnetic bearings offer two major advantages: there is no mechanical wear and no frictional losses or lubrication requirements. In order to fully position a rotating shaft, a magnetic force must be applied along five axes: two perpendicular axes at each shaft end and a fifth axis parallel to the 1 Chapter 1. INTRODUCTION 2 Figure 1.1: Model for a Magnetic Suspension System shaft's rotational axis. A complete model [3} for such a suspension system resembles a set of springs and dampers where each spring represents one electromagnet and is shown in Fig.1.1. This thesis deals with analysis and design of just one axis control (axial suspension). Once the control circuitry is developed and its characteristics found satisfactory, it will be used in the remaining axes as well. The electronic system is intended to control the position of the rotor by acting on the current in the electromagnets on the basis of the signal from the position and velocity sensors. The signal from the position sensor is compared with the difference signal, which defines the rotor's nominal position. If the reference signal is zero, the nominal position is in the centre of stator. By acting upon the reference signal, it is possible to shift the nominal position of the shaft by up to half the air gap. The error signal is proportional to the difference between the nominal position and actual position of the Chapter 1. INTRODUCTION 3 Electromagnet Figure 1.2: Radial Suspension of a Rotor rotor in any given time. This signal is transmitted to the analog signal processing part which produces a control signal to the power amplifier. The ratio of the output signal to the error is chosen so as to maintain the rotor as precisely as possible at its nominal position and to return it rapidly to the nominal position, with a well damped movement, in the event of any disturbance. The servo system defines the stiffness and damping of the magnetic suspension. Practical radial suspension [4], as shown in Fig.1.2., is used on each end of the shaft. In this arrangement there is no mechanical contact between rotor and stator. Rotor size has little effect on the signal processing of the control circuit. Only the power amplifier design depends on bearing capacity. The time constants of typical electromagnets might be as large as several hundred milliseconds and yet the magnet-amplifier combination must act as a closed loop con-trol system with a bandwidth of at least 10 kHz [5]. A fairly substantial reserve voltage Chapter 1. INTRODUCTION 4 to force rapid current change in the electromagnet, in order to overcome the inductive voltage, is therefore an essential feature in the design of DC power amplifiers for con-trolled DC electromagnets. This requirement can lead to large power dissipation and low efficiences in quiescent operating conditions. Switching amplifiers are much more efficient than linear amplifiers and only con-verters of this type are considered in this thesis. The switching amplifiers can cover a wide range of power ratings from small applications, such as spindles for revolving mirrors where very high rotating speed and high speed stability is required, to large turbine generators where magnetic bearings do not require regular maintenance. High frequency switching amplifiers (choppers), using pulse-width modulation to control a duty-ratio of the switching element and thus electric current for the electromagnets, provide an effective and economical solution. This is described in more detail in Chap-ter 5. In order to modify the force-distance characteristic so that the current in the elec-tromagnet, and thus the force of attraction, decreases as the gap decreases (and vice versa), some form of feedback control must be used. From physical laws governing the equation of an electromagnet it becomes obvious that the system is highly nonlinear. To analyze this system and to explore possible linearization techniques in the case of a vertical arrangement of the magnetic bearing is the purpose of this thesis. The chapters in the thesis are arranged to form logical blocks. The general case of an electromagnet is discussed in Chapter 2, covering the derivation of equations for the lifting force and position and current stiffnesses. The analogy to a mechanical system is also briefly described. The differential equation governing the non-compensated, un-stable magnetic system is also derived here. Chapter 3 is focused more on the nonlinear equation and finding the type of singularity, by using a linearized form of the equation. Chapter 1. INTRODUCTION 5 Chapter 4 continues with the linearized equation and a transfer function of the uncom-pensated system is found. Chapter 5 describes the practical realization of the magnetic bearing and gives an evaluation of different types of control. Some experimental results are given which are supplemented by material in Appendix. Chapter 6 discusses the measurement techniques, used to calibrate the actual model for the calculations and computer simulation. Overall conclusions are presented in chapter 7. Chapter 2 D Y N A M I C M O D E L O F M A G N E T I C B E A R I N G S 2.1 I n t r o d u c t i o n In this chapter equations for the lifting force of an electromagnet and stiffnesses of the double-ended axial magnetic suspension will be deduced. A simplified view of such an arrangement and its coordinate system is introduced in Fig.2.3. Several assumptions will be made and their correctness will be evaluated in the conclusions of this thesis. 2.2 Lifting Force of an Electromagnet An equation for the lifting force of a single electromagnet [9] can be obtained from the work done in the magnetic system to change the field energy as F = —2lli (2-1) where L is an inductance and x is an air gap of the electromagnet. For further analysis a few assumptions are made: • magnetic reluctance of the iron core of the electromagnet is negligible in compar-ision to the reluctance of the air gap • flux levels are well below saturation limits • fringing and leakage effects are neglected 6 Chapter 2. DYNAMIC MODEL OF MAGNETIC BEARINGS © Figure 2.3: Coordinate System of the Double Ended Suspension System For an electromagnet such as shown in Fig.2.4 the inductance is expressed as L = Rm COIL N rc/R/v.s' CROSS- SECTION A, A 2 Figure 2.4: The Electromagnet Chapter 2. DYNAMIC MODEL OF MAGNETIC BEARINGS where the magnetic reluctance R„ _d/2 + x , d/2 + x {d/2 + x).l 1 •n-m — - I = ( 1 VQA! fj,0A2 MO M i Ai and using 1 1 Ai + A2 -r + -r Ay Ai A\Ai equation (2.3) can be written as _ jd/2 + x) Ax + A2 Ho MM Symbols used : H [A/m]... magnetic field intensity j4i,j42[m2]... cross-sectional area B [T]... magnetic flux density Ho [H/m]... permeability of free space $ [Wb]... magnetic flux L [H]... inductance Rm [l/H]... magnetic reluctance N... number of turns of the coil x [m]... air gap now the equation (2,4) can be substituted into (2.2), yielding I ^ »oN2 ( MA2 \ {d/2 + xyAr + A2' and — = 2MQ7V2 AJA2 dx {d/2 + xy[A1 + A2' The magnetic flux can be derived from the magnetic equation Rm$ = Ni Chapter 2. DYNAMIC MODEL OF MAGNETIC BEARINGS 9 as Mil and finally the force As Ho, N, Ai, A2, are constants, equation (2.9) can be simplified to i2 F l = K(d/2 + x)> (2-10) and similarly F2 = if -(d/2 - x) 2 1 ^ = ^ M^rrr) (2.11) where Ai-\- A2 From the analysis done so far, it is obvious that the lifting force of an electromagnet depends on two variables: the current i and the air gap x, and that the equation is not a linear function. 2.3 Stiffness of Magnetic Bearings Since there are two independent parameters that can change, position x and current i, equation (2.10) can be used to define a position and current stiffness as P O S I T I O N S T I F F N E S S : 2 K ^ ^ = - 2 K W T x ) - t = c o n s t - ( 2 - 1 2 ) C U R R E N T S T E F F N E S S : Ki = -—• — 2K —— , x — const. (2.13) 01 (a/2 + x)2 v ' Chapter 2. DYNAMIC MODEL OF MAGNETIC BEARINGS 10 Partial derivatives dx, di are used rather then dx, di since the force F is a function of two independent variables x and i. The stiffness terminology was used here since the situation is analogous to a spring. However, for the electromagnet the effective stiffness is negative; this reflects the phys-ical behaviour in which a positive displacement of the shaft -fx, from the top elec-tromagnet, decreases the attractive force. Conversely, an actual spring would apply a force tending to restore the original position of the mass. The consequence of the negative spring stiffness causes the system to be essentially unstable. In the compensated system a closed loop control is used to stabilize the position by changing the current and the effective stiffness. Chapter 2. DYNAMIC MODEM OF MAGNETIC BEARINGS 11 2.4 M e c h a n i c a l A n a l o g y o f M a g n e t i c B e a r i n g There is an analogy between an electromagnet and the mass-spring-damper mechanical system [17]. The case of a mechanical system is shown in Fig.2.5. Fig.2.5: (a) simple mass-spring-damper mechanical system (b) corresponding network The end of the spring and the damper have positions denoted'as the reference position. If a force fext is applied to the mass m it will result in a displacement x. This displace-ment must be balanced by an extension of the spring forcing the mass to the original position. Fig.2.5(a) can be drawn into a network in Fig.2.5(b). According to Newton's law, sum of the forces at each node must add to zero. There is only one node here and the equation is where x, x and x are the displacement, velocity and acceleration of the mass m, B represents the damping and k is elastance or stiffness, which provides a restoring force represented by the spring. Assuming thatm, B and k are constants and all initial conditions are zero, the Laplace transform can be used yielding Fext{s) = s2X{s)m + sX{s)B + X{s)k and the transfer function between Fext(s) and the resulting displacement X(s) is fext{t) = mi + Bx + kx Fext{s) = m(s2 + s 1 m m Chapter 2. DYNAMIC MODEL OF MAGNETIC BEARINGS 12 The expression s* + a l + A = 0 (2.14) m m is called the characteristic equation of the system from Fig.2.5. Its roots reflect the behaviour of the system. Eq.2.14 can be compared to s2 + 2Zwns + w2n = 0 (2.15) yielding bJr, B 2mu>r ( 2.16) Location of 5 roots in eq.2.14 corresponds to the stability and dynamic characteristics of the system. In case of a single electromagnet with constant current any displacement of the mass from the balanced position will result in further move either towards or from the magnet. Thus the electromagnet with a constant current demonstrates a negative stiffness k. This instability is also apparent from the location of roots of eq.2.14. (Fig 2.6.), due to the positive real root, in the s-plane. Lm I UNSTABLE. Re Figure 2.6: Characteristic Roots of a Second Order System in the s - Plane To achieve a stable magnetic suspension the current in the electromagnet must be controlled, which will effectively change the negative stifness k to a positive value and remove the positive root from the s-plane. Chapter 2. DYNAMIC MODEL OF MAGNETIC BEARINGS 13 2.5 Mathematical Model of Magnetic Bearing To obtain a mathematical model of the magnetic bearing an equation of motion is used which, for any mechanical model, is Newton's law Application of this law involves defining convenient coordinates to account for the body's motion, i.e. position, velocity and acceleration. A simplified arrangement of the vertical magnetic bearing is shown in Fig. 2.3. Here the total gap d is equal to the top plus bottom gap (in the state of equilibrium d/2 and d/2). Fx, F2 are attractive forces of the top and bottom electromagnet. The equation of motion can be written as F = ma (2.17) where JP[N]. . . vector sum of all forces applied to the body in the system a[ms - 2]... acceleration ^[ms -2]... gravity constant m[kg]... mass of the body d2x mlm =m9 + F2-F1 + Fl ext (2.18) or by using d2x dt2 = x as m'x — mg + K i (2.19) {d/2 - x) {d/2 + x) ext Chapter 2. DYNAMIC MODEL OF MAGNETIC BEARINGS 14 the term mg can be lumped together with F'txt as Fext and the equation can be written as " - K _ K *'l F e x t X ~ m {d/2 - x) 2 m {d/2 + i ) 2 + m ^• 2 0^ The coordinate system is symmetrical around d/2 (as was shown in Fig.2.3), where d/2 is the origin of the new coordinate system. Thus x can vary in the range from -d/2 to d/2, where d is the total air gap of both electromagnets. Chapter 3 S O L U T I O N O F T H E D I F F E R E N T I A L E Q U A T I O N 1.1 Introduction A function f is linear if: / ( x i + x j ) = / ( s i ) + / ( x 2 ) (3.21) and for any real number a f{ax) = af{x) (3.22) Any other function is nonlinear. Equation (2 .20) , obtained from the equation of motion, evidently does not meet the above conditions. It is a nonlinear second order differential equation with time invariant coefficients. Since frequency response techniques and root locus diagrams are not applicable to nonlinear systems, there is a need for a graphical tool to allow nonlinear behavior to be displayed. The phase plane diagram, which plots velocity x(t) vs. displacement x(t), is a convenient technique. Although it is applicable only to second order processes, it can be used for higher order systems, which can be approximated by a second order equation. The variation of velocity x(t) vs. displacement x(t) for a specific initial condition is called a trajectory. A set of trajectories for several initial conditions is called a phase portrait. 15 Chapter 3. SOLUTION OF THE DIFFERENTIAL EQUATION 16 Let E represent a nonlinear second order dynamic system [13], which has two state variables x\, x2. The vector x = (x\, x2)T is an dement of the real two-dimensional state space X or briefly x 6 X. In the case where system E is time invariant and receives no input, x(t) is deter-mined by t, x(0) and E. Thus: £(*,Xo) = x ( « ) and x(t) is called a particular solution for the system E. The trajectory through any point x 6 X is denoted 7r(x) and is defined as 7r(x) = [x(t) | x(0)] = x, -oo < t < oo The positive semi-trajectory 7T+ through x is defined 7T+(x) = [x(t) | X(0)] = X, 0 < t < OO A point x 6 X is called a critical point of the system E if E ( t , x ) = x for every t €E R1 3.1.1 C r i t i c a l Po in t s of a Sys tem Critical (singular) points of a system are points of dynamic equilibrium. They corre-spond to positions of rest for the system and may be stable or unstable. The significant feature of a critical point is that all derivatives are zero. In other words, the derivatives of the state variables are zero. Chapter 3. SOLUTION OF THE DIFFERENTIAL EQUATION 17 A critical point of the system £ is: (a) stable if, given a circular region of radius 6 > 0 around the critical point, there exists another circular region of radius e concentric with the 6 region, where e > 6, such that every positive half trajectory starting in the 6 region remains within the e region. A solution x(t), originating at x0(t), t0 is stable with respect to the critical point x if: (1) x(t) is defined for all t satisfying t0 < t < oo (2) if | x 0 — x |< 6, for some positive constant <5, then there exists another positive constant e such that | x(t) — x |< e, Vr G (£o,°°) (b) asymptotically stable if, it is stable and if in addition, every half trajectory satisfying the conditions in (a), reaches the critical point in the limit as t —• oo. A solution is asymptotically stable to the critical point x if it is stable and in addition lim | x(t) — x |= 0 Unlike linear systems, which have only one type of behaviour everywhere in the phase plane, nonlinear systems can have many different types of behaviour in different regions. If a linear approximation can be done in each region, then the knowledge of the whole system can be obtained. 3.1.2 Transformation of the 2 n d Order to the 1" Order Differential Equa-tions If the nonlinear differential equation cannot be easily integrated, then a linearized form Chapter 3. SOLUTION OF THE DIFFERENTIAL EQUATION 18 of the equation can be obtained by means of Taylor or MacLaurin expansion. Once the linear form of the second order differential equation is available it can be rewritten into a form of two coupled differential equations of the first order. Starting with the original nonlinear differential equation (2.20) and assuming Fext-0 .. K x — — m „-2 (3.23) (d/2-x) 2 (d/2 + x)2 The expansion with respect to x will be made about point x = 0 (i.e. MacLaurin expansion). The current can be set to t'i = i2 = i (const.) and the linearized differential equation will be derived: x(x,i) « £(0,i) + 0 x x(0, i) = K t5 dx K i2 m m (d/2)' = 0 dx{0,i) _ 2K i2 2K i2 dx m (d/2)* m (d/2)' 32Ki2 md3 Thus the linearized equation will be: i(x,i) 32Ki2 md3 -x This equation can be written as a set of coupled differential equations and the character of the critical point determined according to the following theorem: Let E be a second order linear system described by the equation: X = AX A critical point occurs at X = 0 Chapter 3. SOLUTION OF THE DIFFERENTIAL EQUATION 19 and the type of singularity depends on the eigenvalues of the matrix A accordingly: • real and negative A l 5 A2 indicate stable node • real and positive A l 5 A2 indicate unstable node • real A l 5 A2 of opposite sign indicate saddle point • complex A x, A2 with a negative real part indicate stable focus • complex Ai, A2 with a positive real part indicate unstable focus • imaginary Ai, A2 of opposite sign indicate centre Using the linearized second order equation and making the following substitution : x — Xi X\ = x2 yields %2 Z2Ki2 md3 (3.24) x2 = (3.25) These two equations can be expressed in a matrix form: X = AX (3.26) where (3.27) A = an al2 \ ° 2 1 «22 J (3.28) Chapter 3. SOLUTION OF THE DIFFERENTIAL EQUATION 20 (3.29) Example Equations (3.24), (3.25) can be evaluated for one practical situation: K = 15.006xl0-6 m = 0.683 kg d = 0.003 m (i.e. top and bottom air gap 1.5 mm) H = «2 = 2 A Then the A matrix will be ( 0 1 A = ^ 72670 0 with eigenvalues +269 and -269. Here, the eigenvalues are real and of opposite sign which indicates a saddle point. Chapter 4 L I N E A R I Z E D M O D E L O F M A G N E T I C B E A R I N G S 4.1 Introduction A great majority of physical systems are linear within some range of the variables. However, all systems ultimately become nonlinear as the variables are increased without limit. A system is defined as linear in terms of the system excitation and response. In general, a necessary condition of linearity was given in the previous chapter by eq.(3.21) and (3.22). The physical meaning is as follows: when the system at rest is subjected to an excitation xi{t), it provides a response yi(t). Furthermore, when the system is subjected to an excitation x2(t), it provides a corresponding response y2(t). For a linear system,it is necessary that the excita-tion xi(t) + x2(t) results in a response y^t) + y2[t). This is called the principle of superposition. Furthermore, it is necessary that the magnitude scale factor is preserved in the linear system. Again, consider a system with an input x{t) which results in an output y{t). Then it is necessary that the response of a linear system to a constant multiple a of an input x{i) is equal to the response to the input multiplied by the same constant so that the output is equal to ay{t). This is called a property of homogenity. 21 Chapter 4. LINEARIZED MODEL OF MAGNETIC BEARINGS 22 4.2 Linear Approximation of an Electromagnetic Force In the case of the magnetic bearing, it was shown in Chapter 2, that the equations describing the system are highly nonlinear. Since, for a nonlinear system the principle of superposition does not hold, the Laplace transform cannot be used and a transfer function cannot be defined. It is very difficult to obtain some useful information about the system and to control it by using classical control methods. The force-distance characteristic (Appen. p.54) shows that the measured charac-teristic resembles a straight line in the range of interest (i.e. <0, 3>mm). Since the magnetic bearing is designed to operate at a fixed point (d/2 = 1.5mm), the system is clearly a good candidate for linearization at that point. The linear model then repre-sents behaviour of the system to a small signal or perturbation from the equilibrium point. A nonlinear equation can be expressed in a Taylor expansion about a point x0 or sim-ilarly in a McLaurin expansion about the point x0 = 0. As the equation of the lifting force is a function of two variables (x,i), the linearized equation will be: the linearization can be illustrated by placing a tangent plane onto the nonlinear surface f(x,i) in the point i 0 , t'o. The lifting force of an electromagnet can be linearized about point (x0,i0) as above eq.(4.33). The forces F x , F2 were found in Chapter 1 to be will be: F{x,i) « F ( io , t 0 ) + dF{x0, t'o) 6\F(x 0,t 0) . L x _) 1 L t dx di (4.33) K (4.34) F 2 = K ' 2 (d /2-x) 2 (4.35) Chapter 4. LINEARIZED MODEL OF MAGNETIC BEARINGS 23 The linearized form of the forces Fi, F2 about an equilibrium point x 0 = 0 and i 1 0 (resp.1'20) Note that +2 and +i represent increments of the displacement and current. 4.3 Linearized Model and the Transfer Function Now, the equation of motion m i = F2 - Fi + Fext can be written, using linearized forces as (4.38) A negative sign in (-i) means a decrease of current in the top coil (F{) as an opposite of the increase (+i) of current in the bottom coil (F2). It is assumed that | + t | = | — t | in both coils. In a steady state the shaft will be in a stable position between the two electromagnets. The forces Fi, F2 will provide a resulting steady state force (due to bias currents i'i0 and t 2o) which will compensate weight of the shaft (mg is part of Fext): (d/2)2 (d/2) The resulting equation using the stiffnesses (eq. 2.12, 2.13) will simplify to mi = —Kxx — Kit (4.39) Chapter 4. LINEARIZED MODEL OF MAGNETIC BEARINGS 24 Try, -te Figure 4.7: Graphical Representation of Transfer Function Poles in the s plane where tf.fO.iWa,) = 2Kj^ + 2K- l™ (d/2Y (d/2)* Ki{0,i10,i20)=2K- + 2K-(«f/2)» (d/2)' Now, the magnetic bearing can be approached as a linear system with small per-turbations of current i, as an input and the air gap x, as an output. In order to express the transfer function (i.e. output x to input i), the equation of motion must be transformed from the time domain to the 5 domain by means of the Laplace trans-formation. Suppose that the initial conditions are all zero and there is no damping; then the equation (4.38) can be transformed into the s domain by means of Laplace transformation as s2X{s)m = -KxX{s) -KJis) (4-40) Both Kx and K, are negative so the mass simply accelerates to the surface of the Chapter 4. LINEARIZED MODEL OF MAGNETIC BEARINGS 25 electromagnet. Thus no electromagnet by itself can operate as a bearing. Solving the equation of motion for the ratio X(s)/I(s) gives X(s) -Ki/m 1(a) a 2 + Kx/m (4.41) where Kx and K , are negative values. Thus the roots of the system equation are +y and —\[^- . Due to the the positive root the system is unstable. This is the case of a negative spring compared to a genuine spring (with no damping) where its roots would be both imaginary +jyf^ and located on the imaginary axis symmetrically to the origin. In that case there would be undamped oscillations with a natural frequency of [K~X V m And in a case with added damping the poles would be complex conjugates placed in the left half plane, so that the oscillations would cease in a finite time. The graphical representation is shown in Fig.4.7. Chapter 5 I M P L E M E N T A T I O N O F A M A G N E T I C B E A R I N G 5.1 Introduction Based on the theoretical analysis in Chapters 1, 2, 3 and the resulting linearized model in Chapter 4, several variants of the one axis magnetic suspension control circuits were built and tested for stability and reliability. An evolution of the control circuitry followed practical experience and gained insight into the problems. After experimenting with different sensors and power drivers a simple, stable and reliable controller was constructed and is presented in this thesis. Three different ways of controlling the electromagnets were investigated and built. The comparison of their functions and the evaluation of results are discussed later in the conclusions. The most critical part of any system is its interfacing with the physical environment i.e. the input sensors. The system developed here uses both position and velocity transducers and the quality of signals produced by them is very critical for the whole system. Experiments with substitution of the velocity signal by differentiating the position signal were also performed. The resulting lead forward compensator could not be practically used here, mainly because of the position signal quality. Any noise in the position signal is emphasized after differentiation. For this reason both velocity and position transducers are normally required in practice. A procedure for designing a lead compensator from a root-locus diagram is presented in the Appendix. 26 Chapter 5. Implementation of a Magnetic Bearing 27 This compensator was tested experimentally, but the results confirmed the need for a separate velocity signal. This chapter presents the evalution of a practical design for the velocity and position sensors and discusses the design of a practical power circuit. 5.2 Velocity Transducer The original velocity transducer consisted of an air coil with a permanent magnet connected with the shaft as shown in Fig.5.8(a). The resulting velocity signal was very COIL 1 1 P p N »S COIL (b) Figure 5.8: Velocity Transducer small and sensitive to any external magnetic disturbance. A new transducer was built and it is shown in the same figure in (b). In that new design the coil is magnetically shielded by a ferromagnetic case which forms part of the magnetic circuit. The magnetic field is formed by a permanent magnet in the middle of the sensing coil. This way the resulting output voltage is proportional to axial velocity while practically insensitive to a radial movement. Since the gap between moving plate and the coil is very small, no noise is induced and the signal is of a very good quality. Chapter 5. Implementation of a Magnetic Bearing 28 The voltage induced in the coil is d$ v = N-j-dt where N represents number of turns of the coil and ^ is a change of magnetic flux, by change of the magnetic reluctance of the magnetic circuit. The output voltage of the transducer can be easily increased either by increasing number of turns N or by adding a small amplifier close to the transducer: (as was done here). 5.3 Position Transducers Several methods were used to sense the displacement of the shaft, with different results. Resonant C i r c u i t A circuit consisting of a series arrangement of a coil L and a capacitor C was used in this method. A frequency generator was used to produce a sinusoidal signal with a frequency set on the side of the resonant curve. A change of inductance L then results in a change of the output voltage across the resonant circuit LC. The high frequency output signal is then processed by a peak detector to provide a DC voltage. This method did not provide a stable, reliable signal. One reason was that the frequency generator was not very stable and any change in the frequency or the amplitude added an unwanted offset which required manual correction. Also since the peak detector provides an "envelope" of the detected signal, any outside disturbance is automatically included in the resulting signal. Chapter 5. Implementation of a Magnetic Bearing 29 tt f V Figure 5.9: Resonant Circuit Measurement Method Frequency Detector In this circuit the sensing coil L forms a part of an oscillator [6] of frequency 2TTVLC Thus as the inductance is changed by an axial displacement of the shaft the resulting change in frequency can be detected by a phase detector. A convenient solution [7], [8];provides a Phase-Locked Loop (PLL) circuit, which operates as following: a phase LPF. CONTROL VOLTAGE Figure 5.10: Phase-Locked Loop detector compares two input frequencies and the output is a measure of their difference. If they differ in frequency, it gives a periodic output at the difference frequency. If UN Chapter 5. Implementation of a Magnetic Bearing 30 does not equal /vco» the phase-error signal, after being filtered and amplified, causes the VCO frequency to deviate in the direction of fjN- The VCO will quickly "lock" to ///vr, maintaining a fixed phase relationship with the input signal. The filtered output of the phase detector is a DC signal, and the control input to the VCO is a measure of the input frequency. A phase-lock loop is shown simplified in Fig.5.10, After initial difficulties with the low pass filter, the PLL worked well, but yet the long-time stability was not sufficient. The resulting DC output signal provided a volt-age change of about 0.2V for the full gap range. This signal, compared to a reference voltage, results in an error signal which can be further amplified. Hall Effect Sensor This signal is of a good quality with long term stability and requires minimal additional processing. This technique was therefore used in all further experiments. The disk, connected to the shaft and forming a part of the velocity transducer, was also used to produce position information. This was done by placing a Hall effect sensor on the top of the permanent magnet measuring the flux density B of the resulting magnetic field. The only drawback, which is the temperature sensitivity, can be solved by placing the sensor further from the heat source. It could be also improved by employing an electronic compensation technique, but in this case it was not necassary. All the circuits listed in the Appendix use this type of sensor. Although the air gap - output voltage characteristic is nonlinear, as shown on p.54, it can be considered linear over the range of interest (air gap 0 - 3mm). Chapter 5. Implementation of a Magnetic Bearing 31 5.4 P o w e r D r i v e r s Once the signals are detected and processed, they form a control signal for the power drivers. Two types of power drivers [9], can be used for the magnetic bearing. A class A chopper (one quadrant) is shown on p. 66 . This circuit can decrease the output current, if desired, by effectively applying zero voltage across the electromagnet (i.e. using only the +V.+I quadrant).During that (off) time the current continues to flow through the freewheeling diode. A class D chopper (two quadrant) is shown on p.67. Its advantage is that it can control the output current even further, by applying full negative supply voltage across the electromagnet, while the output current freewheels through the power supply. This could produce improved dynamic performance at increased cost. However, in the dou-bly excited system analyzed in this thesis, a fast rise in force in either direction is possible even if only one quadrant controllers are used. Therefore, for simplification, this was the only type of circuit considered here. The resulting system performed satis-factorily, but the investigation of two quadrant controllers can be carried in the future. 5.4.1 V o l t a g e D r i v e r The first control circuit that was designed and built was simple, yet fully operational and gave us the "feel" of a magnetic suspension. The name voltage driver is used here to refer to a voltage output of the amplifier which is the result of input signals, position and velocity. The block diagram is shown in the Appendix on p.58. The relationship between the voltage applied across the electromagnet V(s) and the resulting current I(s) is expressed, using lumped parameters R and L as: V(s) = sLI{s) + RI{s) = I{s){sL + R] Chapter 5. Implementation of a Magnetic Bearing 32 and V(s) I(s) = where r = j- is known as the electric time constant. The meaning of the above equation is that the current I(s) is not linearly proportional to the voltage V(s), but it is lagging. Furthermore T depends on the air gap of the electromagnet. Since the linearized model is already being used, r can be measured for the specific air gap and considered constant within a small range. The method of an air gap measurement is discussed in Chapter 6 and, for an air gap of 1.5mm, r equals 15.6ms. The transfer function of the closed loop system is derived by consequent simplifications analogous to those, used on pp.62 - 65 for a current driver. Use of that procedure yields the transfer function i KPK, xjfi s 2m •+• sK,Kv + KpK,H — Kz where K, ^(Ka + K,!)—!— ST •+- 1 and Kx = Kxi + Kz2 From Chapters 2 and 3 K, and Ks are known to be the current and position stiff-nesses and are calculated in A43 for a fixed gap 1.5mm and several values of current. The electric time constant can be expressed from 1 _ 1 sr + 1 ~ r(s + -J) as a pole s=-* in s - plane of value -64.10. Its effect can be seen in the root locus where being close to the origin it constitues a dominant pole pushing the root locus branch to the right half plane. Thus with no velocity feedback the system is unstable for any value of position gain Kp. Chapter 5. Implementation of a Magnetic Bearing 33 For an illustration several root locus plots are shown on pp.85 - 88. A region of stability is investigated for different values of the velocity gain Kv. Only a narrow range of Kp demonstrates a stable state. This is in full agreement with the practical experience, where the system had to be frequently adjusted. Since this arrangement could not be kept stable over a longer time, it was really of no practical use and it was abandoned. 5.4.2 Current Feedback The situation with the unwanted dominant pole, due to the electric time constant, can be improved by introducing current feedback. As an example let 57 + 1 to be a transfer function of the electromagnet (i.e. I0ut/Vin), resulting thus in a pole s=-^. Let Keur be a gain of an amplifier connected in cascade with the electromagnet. Then by closing a negative unity feedback (see A6) the new transfer function will be G= HT + l _ '•••cur 1 + 7rTl *T+l+Kcur and the new pole will be at 1 + Kcur s = T Thus by using current feedback the unwanted significant pole can be moved further from the origin into the left half plane and become less significant for the dynam-ics of the system. The model shown in the block diagram on p.59 is very similar to the voltage driver diagram. The current feedback can be implemented either only on one electromagnet (p.73) or on both ends (pp.74 - 75), where both currents are monitored independently. Chapter 5. Implementation of a Magnetic Bearing 34 The transfer function has again the same general form as with the voltage driver, only the Kt is different, since it cointains the local current feedback loop: ^ " " • t r + l  1 + * c „ r ^ 5.4.3 Current Driver It becomes obvious that by increasing the current loop gain without limit the original time constant s = approaches -infinity. This practically means that the electric time constant becomes negligible and can be omitted. Thus the current driver is actually a transconductance amplifier with the transfer function hut . —— = const. Yin By employing a current source the system will become much easier to stabilize and will remain stable both over a wider range of control parameters Kp, Kv and system parameters (which may drift over an extended period of time). A block diagram of such a system and its simplification process is shown in detail on pp.60 - 65. The system used here will have, under normal operating conditions, a zero input (x,„) and is referred to as a "zero tracking regulator". For a measurement of dynamic characteristics it is convenient to observe the system response to a standard arbitrary input, such as an unit impulse, unit step, sinusoidal or square wave input which may represent both a driving function or a disturbance. As an example a sudden load being applied to the magneticaly suspended rotor will result in a precendented reaction. This disturbance can correspond either to a position disturbance x,„ or to a force disturbance Fext. Thus by introducing the transfer function as — or is pos-* m f a t sible. The correspondence between those two functions becomes clear from the block diagram on pp.64 - 65. The only difference is a scaling factor. Chapter 5. Implementation of a Magnetic Bearing 35 The equations i KPK, (5.42) s*m + sK,Kv + KPK,H - Kx or x Kp K, [ m (5.43) in s2 + sK.KJm + (tfptf.tf - Kz)/m can be compared to the standard form of a second order differential function : where C is a constant and Kx is the position stiffness calculated from eq.2.12 and listed on pp.98 - 99. An examination of the coefficients in the denominators provides an important insight into the change of an unstable (negative) spring situation to a real spring situation with a positive (restoring) force type. By increasing the position gain K p , the stiffness of the bearing can be increased (theoretically to infinity). Practically this is not possible be-cause of nonideal position and velocity signals. Also the damping of the system £ depends on the velocity gain Kv. The boundary of the theoretical stiffness and practical limitations due to a nonideal physical world will now be investigated. C (5.44) a1 + 2£w ns + ul 2mwn Chapter 6 M E A S U R E M E N T S 6.1 I n t r o d u c t i o n In order to calibrate the model of one axis double ended vertical bearing, several mea-surements had to be done. A brief description of each measurement technique, together with the results obtained, are presented in this chapter. The model used for the measurements was the current driver system with: top and bottom gaps .d/2 - 1.5 [mm] mass of the shaft m = 0.683 [kg] gravity constant g = 9.81 [m/s2] The other parameters were being changed for different measurements and they are specified in the appropriate place in the text. 6.2 D o u b l e E n d e d a n d S i n g l e E n d e d M o d e l s All the analysis of the magnetic bearing so far was concerned with the "double ended'' model, mainly for reasons of practical use and also because the equations obtained can be easily changed to describe the single ended model (i.e. using only the top electromagnet) by setting t 2 = 0 and thus (=> F2 =0). Although the single ended model is of little use for the practical bearing, it can be utilized for a convenient calibration of the model. As mentioned earlier, a number 36 Chapter 6. MEASUREMENTS 37 of simplifications were used (those concerned with fringing, leakage, saturation and magnetic reluctance of iron ). In order to obtain a practical model for further analysis, it must be checked that the model does not stray too far from reality. A theoretical constant K was calculated in section 2.2. This can be checked ex-perimentally as follows. The current in the bottom coil is set to zero and the current required by the top electromagnet to maintain a particular gap is then measured. The gap, which will be used in the constructed bearing is the most important. The equation of motion will simplify to m i = mg — F x = 0 (6.45) so that Fi = mg K —-, — = TOO (d/2 + z)» (where d/2 is the gap) and since the values of mass m, gap and current can be easily obtained. A corrected value of the constant K can be thus determined from the equation: t2 In the case of the model used with Kp = 150 and Kv = 0.6, the current z'i was found to be 1.19 [A], corresponding to a force of 6.73 [N]. For further values of the airgap see A51. Thus a new constant of the electromagnet was found to be: K = 10.646 * 10"6 Chapter 6. MEASUREMENTS 38 This corresponds to a 29% decrease from the originally calculated value 15.005* 10 6 . All the calculations in this thesis are based on this new, corrected value. 6.3 Electric Time Constant of the Electromagnet A series combination of L and R is used here to represent the electromagnet. Thus by solving the equation for current as a function of applied unit step voltage we obtain « = Imax{l - C ' ) (6.47) where I max — D represents a steady state value of a DC current. This equation can be evaluated for t = T, where r is the electric time constant as denned earlier in section 5.4.1 to yield: t = / m ox ( l - e~r) = / m a i ( l - e"1) = 0.632/maI 0 T T N ^ ^ Figure 6.11: Voltage vs. Current in a Series RL Circuit Now, using a low frequency square wave generator and a power amplifier, the output Chapter 6. MEASUREMENTS 39 voltage is applied across the electromagnet (fixed airgap) and the resulting current is displayed (by means of a current transducer) on an oscilloscope. In the practical case /m a i=3.5 [A] and 63.2% of which was 2.21 [A]. The time corresponding to this current level was found to be: r = 15.6[ms] 6.4 P o s i t i o n C o n s t a n t o f t h e H a l l E f f e c t S e n s o r The characteristic, i.e. the output voltage as a function of the air gap, of the Hall effect sensor was measured and is shown in A l . It can be seen that the function, in the range of <0, 3> mm, can be substituted with a straight line of appropriate slope. The slope, and thus the transfer function, was found to be: H = 110[V/m] 6.5 V e l o c i t y C o n s t a n t Similarly, the velocity sensor transducer had to be calibrated. The appropriate part of the block diagram, together with x = x(t) and x = x(t) is shown in Fig.6.12. Position and velocity response for a simulated impulse input and Kp = 150, H = 110 and Kv = 0.6 (underdamped case), were displayed on an oscilloscope simultaneously. The fol-lowing can be written x = Xmaxsinujt u = — = 2nf T where w [rad/s] is radian frequency, T [s] is the period and f [Hz is cyclic frequency. Chapter 6. MEASUREMENTS X 1 x /5-0 4 ®* K ? - ^ - T 2.7 110 Figure 6.12: Measurement of the Velocity Constant and for velocity dx X = — = 0jXmaxCOSUt = XnaxCOSUt at The following data were obtained: T = 28 [ms], so that: u> = 224[ra<i/s] X^ = 8 V « 0.485mm (i.e. 8/150/110) yielding : KPabs = 16495[F/m] Similarly: Xmax = 12 V and Xmax = w l m 0 1 = 224 * 0.485 * 10~3 = 0.1086 [m/s] 12/0.1086 = 110.5 [V/m/s] and 110.5/2.7 = 40.93 so that: KVab, = 40.93\V/m/s} Chapter 6. MEASUREMENTS 41 6.6 Measurement of The Position Stiffness As outlined briefly in 5.4.3 (page 34), a technique simulating an unit impulse response [10],[11],[12] was used here in order to obtain the stiffeness values for the closed loop suspension system. An analogous system to the magnetic bearing is the mechanical mass-spring-damper system, as discussed earlier, where the input R(s) and output X(s) are related through the transfer function T(s) r(5) = fw = + + <6"8> The poles of the above equation are * = -fw„ ± j uny/l - (2 If the unit impulse 6(t) is used as an input then the response of the system X(s) will be equal to T(s) X(s) = T(s) since the Laplace transform £6(t) = 1 The transient response in the time domain can be obtained by using inverse Laplace transform of X(s) (= T(s)): x(t) = L-lX{a) = —e-^sinu.t = 4 e - ^ ' « n ( W r i V T q j ) t where u>„ . . . natural radian frequency ojd ... damped radian frequency £ . . . damping ratio a . . . damping coefficient r . . . time constant of the system Chapter 6. MEASUREMENTS 42 and the basic equations: = un^l i2 - 1 a = (wn _ 1 a £ = sin'16 0 = tan'1^ As £ decreases, the poles approach the imaginary axis and the system becomes increas-ingly oscillatory. It is common to take several performance measures from the transient response to a step input. The swiftness of the response is measured by the rise time Tr and the peak time T p . For underdamped systems with an overshoot, the 0-100 % rise time is a useful index. If the system is overdamped then the peak time is not defined and the 10-90 % rise time is normally used. The similarity with which the actual response matches the step input is measured by the percent overshoot, P.O., and the settling time, T s . The settling time, T s , is defined as the time required for the system to settle withtin a certain percentage of the input amplitude. For a second order system with a closed-loop damping constant £, the response remains within 2 % after four time constants. In practice a demand for a fast response yet with an overshoot of less then 5 % can be matched by the minimum damping ratio of 0.707 with a resulting overshoot of 4.3 %. Chapter 6. MEASUREMENTS 43 The stiffness of the closed loop compensated system was obtained from the tran-sient response to a simulated unit impulse (mechanical impulse). The natural radian frequency of the undamped system un was obtained from the os-cilloscope, while position gain Kp and velocity gain Kv were set so that the system produced a sustained periodic oscillation. The damped radian frequency uj was ob-tained in a similar manner for different settings of Kp and Kv. Pictures were taken from the screen of a storage oscilloscope and they are shown together with computer simulated responses in Appendix pp.78 -83. parameters measured values calculated values K„ 100 0.6 222 227 0.209 12900 100 0,7 209 227 0.209 12900 100 2.7 122 227 0.844 12900 150 0.9 326 336 0.242 48600 150 1.5 290 336 0.505 48600 150 2.7 140 336 0.909 48600 150 5.0 97 336 0.957 48600 £ = l - ( ^ ) 2 Kcomp = ' un • mu\ where Keomp is the stiffness of the compensated system. Chapter 6. MEASUREMENTS 44 As can be seen from the table the natural radian frequency w„ and the effective stiffness K depend on the position gain K p , whereas the damping ratio £ and thus the damped radian frequency depend on the velocity gain Kt,. An expression for a steady state error (i.e. in this case it would be the steady state displacement of the shaft if a constant force is applied) can be calculated, using the steady-state or the final-value theorem: limx(t) = UmsX(s) the situation corresponds to a unit step being applied limsX(s) = lim s s{s* + 2&n + W 2 ) in case of the bearing it would be 1 m KVK FH—KX KPKSH m Chapter 7 C O N C L U S I O N S The goal of this thesis was to analyse and design a practical one-axis double-ended magnetic bearing. A complete analysis of the vertical arrangement was done, forming a good base for further investigation. This included: equation of motion, equation for the lifting force of an electromagnet, general expressions for stiffnesses, linearized form of the force and finally a model for small signals (perturbations). From the analogy with a mechanical system, the equation for the position stiffness of the closed loop suspension system was derived and measured, using a simulation of the unit impulse response. The results were found to be in good agreement with the calculated values. The formula for a steady-state error, in the case of a unit step input (i.e. a case of a suddenly applied load) was derived from the final value theorem. A good insight into the system properties was obtained by using the root-locus plots and an example af a complete procedure for synthesizing any type of cascade compensator is presented in the Appendix. A high importance was placed on the position transducer by analyzing its role in the ultimate bearing stiffnesses. This is still a vast area to be investigated. By comparing different types of transducers the Hall-effect transducer was found to be a very convenient type for its good quality output signal with no further processing necessary. The key goal, which was to actually build a circuit that will perform the control function, was achieved. The identical circuitry can be now made for those remaining 45 Chapter 7. CONCLUSIONS 46 four axes to completely suspend the rotor. Further investigation may still be necessary in the multi-axes suspension case, if a possible cross coupling would occure. The future of magnetic bearings is still open which is obvious from a number of papers being published. Some other types of position transducer candidates would be capacitance sensors and optoelectric sensors. Further explorations of this topic promises to pose an interesting challenge. 47 Bibliography [l] Earnshaw, S., On the nature of the Molecular Forces which regulate the Constitu-tion of Luminoferous Ether, Trans. Camb. Phil. Soc, 7, pp. 97-112 (1842) [2] Braunbeck,W., Free Suspension of Bodies in Electric and Magnetic Fields, Zeitschrift fur Physik, 112, 11, pp. 753-63 (1939) [3] Habermann, W., Liard Guy L., Practical Magnetic Bearings, Societe de Mecanique Magnetique, France; IEEE Spectrum, September (1979) [4] ACTIDYNE - Application of Active Magnetic Bearings to Industrial Rotating Ma-chinery, Societe de Mecanique Magne'tique, France (5] Jayawant, B. V., Electromagnetic Levitation and Suspension Techniques, Edward Arnold Publishers (1981) [6] Fogiel, M . , The Electronic Problem Solver, Research and Education Association, New York N.Y. (1982) (7] Horowitz, P., Hill, W., The Art of Electronics, Cambridge University Press (1983) [8] Geiger, Dana F., Phaselock Loops for DC Motor Speed Control, Wiley ic Sons, Inc. (1981) [9] Dewan, S. B., Slemon, G.R., Power Semiconductor Drives, Wiley &: Sons, Inc. (1984) (10] Dorf, R.C., Modern Control Systems, Addison-Wesley (1982) 48 [11] Di Stefano J.J., Feedback and Control Systems, McGraw-Hill, 1976 [12] Franklin Gene F., Powell David J., Feedback Control of Dynamic Systems, Addison-Wesley, 1986 [13] Leigh J. R., Essentials of Nonlinear Control Theory, Peter Peregrinus Ltd., Lon-don, U.K., 1983 [14] SIGNETICS - Linear Circuits Catalog, 1986 [15] UNITRODE - Linear Circuits Catalog, 1986 [16] Haberman H. et al., IEEE Spectrum, Sept. 1979, p.26 [17] D'Azzo John J., Houpis Constantine H., Linear Control System Analysis and De-sign, McGraw-Hill, 3rd Edition, 1988 49 APPENDIX 50 List of Appendix 1 Output Voltage of the Hall Effect Sensor vs. Air Gap 54 2 Output Current vs. Input Voltage of the Current Driver using 3843 Integrated Circuit 55 3 Force of the Electromagnet acting on the Shaft vs. Displacement (il = i2 = 5 A const.) 56 4 Force vs. Displacement (calculated, 1st order approximation and 3rd order approximation) 57 5 Block Diagram for Small Signal with linearized Force (Voltage Driver) . 58 6 Block Diagram for Small Signal with Linearized Force (Current Feed-back) 59 7 Basic Block Diagram of a Magnetic Bearing (Current Driver) 60 8 Block Diagram with Linearized Force (Current Driver) 61 9 Block Diagram for Small Signal with Linearized Force (Current Driver) 62 10 Simplification 1 (Current Driver) 63 11 Simplification 2 (Current driver), Simplification 3 (Current Driver) . . . 64 12 Simplification 4 (Current Driver) 65 13 Class A Chopper (one quadrant switching amplifier) 66 14 Class D Chopper (two quadrant switching amplifier) 67 15 A Saw Wave Generator for the Comparator Circuit 68 16 Comparator Circuit as a Generator of Control Signals for the D Class Chopper 69 17 D Class Chopper (two quadrant switching amplifier) 70 51 18 Analog Part of the Control Circuit for the Magnetic Bearing (position and velocity signals) 71 19 Simple A Class Chopper (Voltage Driver) 72 20 Simple A Class Chopper (Current Feedback) 73 21 A Class Chopper (Current Feedback) 74 22 The Complete Control Circuit (Current Driver) 75 23 Block Diagrams of Integrated Circuits 5561 and 3842 76 24 Impulse Response of a Second Order Linear System (for different values off l 77 25 Impulse Response of the Single Axis Magnetic Bearing 78 26 Impulse Response of the Single Axis Magnetic Bearing 79 27 Impulse Response of the Single Axis Magnetic Bearing 80 28 Impulse Response of the Single Axis Magnetic Bearing 81 29 Impulse Response of the Single Axis Magnetic Bearing 82 30 Impulse Response of the Single Axis Magnetic Bearing 83 31 Measured Stiffnesses of the Compensated Magnetic Bearing 84 32 Root Locus of the Stabilized System using a Voltage Driver 85 33 Root Locus of the Stabilized System using a Voltage Driver 86 34 Root Locus of the Stabilized System using a Voltage Driver 87 35 Root Locus of the Stabilized System using a Voltage Driver 88 36 Root Locus of the Stabilized System using a Current Feedback 89 37 Root Locus of the Stabilized System using a Current Feedback 90 38 Root Locus of the Unstable System (no velocity feedback) 91 39 Root Locus of the System (without the velocity feeback) using a Lead Cascade Compensator 92 40 RC Network Synthesis for a Feedback Amplifier 93 52 41 Root Locus of the Stabilized System using a Current Driver 96 42 Root Locus of the Stabilized System using a Current Driver 97 43 Calculated Values of Forces and Stiffnesses for the Uncompensated Sys-tem 98 53 Symbols used in the Appendix: K t ...velocity gain (adjustable) K p . . .position gain (adjustable) K c u r . . . current loop gain (adjustable) K e . . .system constant (-0.321 A/V) K,; . . . converter constant (+0.5, -0.5) H . . . Hall sensor constant (110 V/m) K , . . . current stiffness of the uncompensated system (calculated) K, . . . position stiffness of the uncompensated system (calculated) T . . .electric time constant of the electromagnet (15.6 ms for 1.5 mm air gap) m . . . mass of the shaft (0.683 kg) g . . .gravity constant (9.81 m/s2) Fext ... force representing a disturbance Output Voltage of the Hall Effect Sensor vs. Air Gap 54 Output Current vs. Input Voltage of the Current Driver using 3843 Integrated Circuit 55 Vo l tage - C u r r e n t C h a r a c t e r i s t i c of 3843 < 0 I ' 1 1 i i i I 0 2 4 6 8 10 12 14 INPUT VOLTAGE V[V] 56 Force of the Electromagnet acting on the Shaft vs. Displacement (il = i2 = 5 A const.) F o r c e v s . D i s p l a c e m e n t (ca lcu la ted , 1st o r d e r a p p r o x i m a t i o n a n d 3rd order a p p r o x i m a t i o n ) 57 58 Block Diagram for Small Signal with linearized Force (Voltage Driver) 59 Block Diagram for Small Signal with Linearized Force (Current Feedback) - + or 0-60 61 Block Diagram for Small Signal with Linearized Force (Current Driver) 4 <3 Simplification 2 (Current driver) 64 Kr, +KxZ H k's - kct / s J ( k.^ + K-cxkszkiz Simplification 3 (Current Driver) Simplification 4 (Current Driver) 65 66 Class A Chopper (one quadrant switching amplifier) [9] 67 Class D Chopper (two quadrant switching amplifier) [9] A Saw Wave Generator for the Comparator Circuit (A16) 68 Comparator Circuit as a Generator of Control Signals for the D-Class Chopper (A17) 69 Analog Part of the Control Circuit for the Magnetic Bearing (position and velocity signals) 71 Simple A Class Chopper (Voltage Driver) 72 Simple A Class Chopper (Current Feedback) 73 A Class Chopper (Current Feedback) 22 k The Complete Control Circuit (Current Driver) 75 Block Diagrams of Integrated Circuits 5561 and 3842 NE/SE5561 [14] REF VOLTAGE CURRENT __». SENSE u — RT.CT SAWTOOTH GENERATOR yS^-O OUTPUT UC3842 [15] A • Dli.fi Pin N u m b e r B • SO 14 Pin N u m b e r . I I I - T L T L T L rtQURE 1 TWCH.OOR CURRENT-UOOE CONTROC SYSTEM POS. : KP = 100 2V/div VEL. : Kv = 0.6 5V/div TIME : 50ms/div POS. : KP = 100 2V/div VEL. : Kv = 0.7 5V/div TIME : 50ms/div, 20ms/div POS. : KP = 100 2V/div VEL. : Kv = 2.7 5V/div TIME : 50ms/div POS. : KP = 150 2V/div VEL. : Kv = 0.9 5V/div TIME : 50ms/div, 10»ms/div POS. : KP = 150 2V/div VEL. : Kv = 1.5 5V/div TIME : 50ms/div, lOms/div POS. : KP = 150 2V/div VEL. : Kv = 2.7 5V/div TIME : 50ms/div 84 Measured Stiffnesses of the Compensated Magnetic Bearing In the compensated magnetic suspension system with closed loop the equation is derived from the block diagram in the Appendix on p.58. The simplification process is shown on pp.58 - 65. The result is the following expression: and compared to it gives 2 , KSKV KPKSH — Kx S + 5 h — m m s2 + 2s£w n + w2 KtKv m 2 KpKgH - Kx K m m The following values were calculated: Parameters Calculated Measured Difference K„ K K % 100 0.6 10870 12900 18.7 100 0.7 10870 12900 18.7 100 2.7 10870 12900 18.7 150 0.9 42110 48600 15.4 150 1.5 42110 48600 15.4 150 2.7 42110 48600 15.4 150 5.0 42110 48600 15.4 Root Locus of the Stabilized System using a Voltage Driver Root Locus of the Stabilized System using a Voltage Driver 86 Root Locus of the Stabilized System using a Voltage D 87 river CO CD U l m I n a 9 s 3000 2000 1000 -VOLTAGE DRIVER  Kp stable range <13.2 - 583> o o o c (71 tr P a-0 -1000 N Q . C/3 CO 3 c 5' -2000 £ OP ro -300 3 00 -100 0 100 200 < n i-i Real s oo oo 2000 CURRENT FEEDBACK Kcr = 50 1000 0 -1000 -200 Kv = 1U KP stable range <13.5 297> / / / / • . * » [ ; \ i I i I i \ 000 -2000 -1000 0 1000 Real_s fa o o o o c co O ro co T <D a. co CO •rt-(t> (3 co 5' (TO p» o c •-1 <-l <T> 13 >n ft) CL, c r p> rt oo co mm 2000 1000 0 1000 -2000 -300 3-l 000 -3000 CURRENT FEEDBACK Kcr = 50 NO VELOCITY FEEDBACK unstable systen -2000 -1000 j V \ \ \ 1000 Real s 92 Root Locus of the System (without the velocity feeback) using a Lead Cascade Compensator CD t o 93 2.7 R C Network Synthesis for Feedback Amplifier RC networks can be used for synthesis of various feedback amplifier functions, using the transmitance function. Appendix A44 - A49 contains a table of the most common RC configurations 1 . We assume the following for the operational amplifier: A0 = oo op en-loop gain B W = oo bandwidth IB = 0 bias current RJN — oo input impedance Ro — 0 output impedance The transimpedance ZT = ^ of a circuit is defined as the input voltage divided by the output current when the output is shorted. This is the type of transfer function which is needed since the currents are summed at the op-amp input node (in the inverting arrangement of an op-amp the + input is grounded). As an example a lead type of network is design here to represent a transfer function: s + 3000 Various combinations could be chosen for Zn, ZT2, but let's choose Z T 1 = 10(s + 800) and Z T 2 = s + 3000 Reproduced from F.R.Bradley and R.McCoy, "Driftless DC Amplifier", Electronics, April 1952. This table was developed by S.Godet of the Reeves Instrument Corporation, New York. 94 Referring to the table in A44, equation (Jl-3), the procedure is following: Zn = 10(s + 800) = 10 * 800(0.00125s + 1) so that A = 8000 and T = 0.00125 [s]. Let C, = 1 /xF (a practical value picked) then = 2 5 0 0 ( n l And similarly ZT2 = s + 3000 = 3OO0(0.00033s + 1) so that A = 3000 and T = 0.00033 [s]. Again, let pick a value of C 2= 10 nF then R2= = 66700 [n] If the calculated value is not available, the procedure car be repeated for a different value of C. At DC (i.e. s = 0), the gain is: A — 1 0 * 8 0 0 _ 2 667 A » ~ 3000 — Z - D D / R. R, I 'VVWr—f-^ VNAA , -AAA/VV— i - | 2> RC Network Synthesis for a Feedback Amplifier £ .2 V -> « " a. x n as n II as O II n as (J n n II os n es &s II II n II OS + o ft; ee n n 05 + as n - O c + 5 V 5 I n a g 5 606 400 200 0 -200 -400 -60 5 CURRENT DRIVER Kv = KP stable 10 nge <41, 1 - infinity) \ m 1 1 1 1 1 1 1 1 / i i i i i 00 -400 -300 -200 -100 0 100 200 300 Real s o o o o c co ro CO P c r is Cu CO ••<! co ro 3 CO 5 ' OP o c "1 ro _ ro CO I ft a 9 5 666 466 266 6 -266 -466 CURRENT DRIVER -60 Kv = 1 KP stable range <41. 1 - infinity) m _. 1 l 1 l 1 l 1 1 1 i 1 i 1 i 66 -468 -360 -288 -106 6 166 266 368 Real s o o f o o a cn fD CO N zn fD 0 cn_ 5' m O c »-» 3 rtt CO 98 <f/2[mm] WeightlN] I [A] i[.A] F1[N] 0. 25 6 .70 0.20 0.20 6 .73E+00 0. 50 6 .70 0.40 0.40 6 .73E+00 0. 75 6 .70 0.59 0.59 6 .73E+00 1. 00 6 .70 0.79 0.79 6 .73E+00 1. 25 6 .70 0.99 0.99 6 .73E+00 1. 50 6 .70 1.19 1.19 6 .73E+00 1. 75 6 .70 1.39 1.39 6 .73E+00 2. 00 6 .70 1.59 1.59 6 .73E+00 2. 25 6 .70 1.78 1.78 6 .73E+00 2. 50 6 .70 1.98 1.9 8 6 .73E+00 2. 75 6 .70 2.18 2.18 6 .73E+00 3. 00 6 .70 2.38 2.38 6 .73E+00 /21mm] <f/2Imm] 11 A) KxtN/m] Kx/m Ki[N/A] Ki/m 1. 50 1. 50 0 . 5 1 .94E+02 2 •84E+02 1.16E+00 1 .70E+00 1. 50 1 . 50 1 . 0 7 .75E+02 1 .14E+03 2.33E+00 3 .41E+00 1. 50 1. 50 1 . 5 1 .74E+03 2 .55E+03 3.49E+00 5 •11E+00 1. 50 1. 50 2. 0 3 .10E+03 4 .54E+03 4.65E+00 6 .81E+00 1. 50 1. 50 2. 5 4 .85E+03 7 •10E+03 5.82E+00 8 .52E+00 1 . 50 1. 50 3. 0 6 .98E+03 1 .02E+04 6.98E+00 1 .02E+01 1 . 50 1 . 50 3. 5 9 .50E+03 1 •39E+04 8.14E+00 1 .19E+01 1 . 50 1 . 50 4. 0 1 .24E+04 1 .82E404 9.31E+00 1 .36E+01 1 . 50 1 . 50 4. 5 1 •57E+04 2 .30E+04 1.05E+01 1 .53E+01 1. 50 1 . 50 5. 0 1 .94E+04 2 .84E+04 1.16E+01 1 .70E+01 1 . 50 1 . 50 5. 5 2 .35E+04 3 .43E+04 1.28E+01 1 .87E+01 1. 50 1 . 50 6. 0 2 .79E+04 4 .09E+04 1.40E+01 2 .04E+01 1. 50 1. 50 6. 5 3 •28E+04 4 .80E+04 1.51E+01 2 .21E+01 1. 50 1. 50 7. 0 3 .80E+04 5 .56E+04 1.63E+01 2 .38E+01 1. 50 1. 50 7. 5 4 .36E+04 6 .39E+04 1.74E+01 2 .55E+01 1. 50 1. 50 8. 0 4 .96E+04 7 .27E+04 1.86E+01 2 .72E+01 1. 50 1. 50 8. 5 5 .60E+04 8 .20E+04 1.98E+01 2 .90E+01 1. 50 1. 50 9. 0 6 .28E+04 9 •20E+04 2.09E+01 3 .07E+01 1. 50 1. 50 9. 5 7 .00E+04 1 .02E+05 2.21E+01 3 .24E+01 1 . 50 1 . 50 10. 0 7 .75E+04 1 .14E+05 2.33E+01 3 .41E+01 99 d/2 [mm] d/2fmm] i l l A] i2[A] FUN] F2INJ KxIN/m] Kx/m 1. 50 1. 50 2. 33 2.00 2 .53E+01 1 .86E+01 5.B5E+04 8 .56E+04 1.50 1. 50 1. 00 1.00 4 .65E400 4 .65E+00 1.24E+04 1 .82E+04 1.50 1.50 1. 50 1.50 1 .05E+01 1 •05E+01 2.79E+04 4 . 09E+04 1.50 1.50 2. 00 2.00 1 .86E+01 1 .86E+01 4.96E+04 7 .27E+04 1.50 1.50 2. 50 2.50 2 .91E+01 2 .91E+01 7.75E+04 1 .14E+05 1.50 1.50 3. 00 3.00 4 .19E+01 4 .19E+01 1.12E+05 1 •63E+05 1.50 1.50 3. 50 3.50 5 .70E+01 5 •70E+01 1.52E+05 2 .23E+05 1.50 1.50 4. 00 4.00 7 •44E+01 7 .44E+01 1.99E+05 2 .91E+05 1.50 1.50 A. 50 4.50 9 .42E+01 9 .42E+01 2.51E+05 3 .6BE+05 1.50 1.50 5. 00 5.00 1 .16E+02 1 .16E+02 3.10E+05 4 .54E+05 1.50 1.50 5. 50 5.50 1 .41E+02 1 .41E+02 3.75E+05 5 .50E+05 1.50 1.50 6. 00 6.00 1 •68E+02 1 .68E+02 4.47E+05 6 .54E+05 1.50 1.50 6. 50 6.50 1 .97E+02 1 .97E+02 5.24E+05 7 .68E+05 1.50 1.50 7. 00 7.00 2 .28E+02 2 .28E+02 6.08E+05 8 .90E+05 1.50 1.50 7. 50 7.50 2 .62E+02 2 .62E402 6.98E+05 1 .02E+06 1.50 1.50 8. 00 8.00 2 .98E+02 2 .98E+02 7.94E405 1 .16E+06 1.50 1.50 8. 50 8.50 3 .36E+02 3 .36E+02 8.96E+05 1 •31E+06 1.50 1.50 9. 00 9.00 3 •77E+02 3 .77E+02 1.01E+06 1 .47E+06 1.50 1.50 9. 50 9.50 4 .20E+02 4 .20E+02 1.12E+06 1 .64E+06 1.50 1.50 10. 00 10.00 4 .65E+02 4 .65E+02 1.24E+06 1 .82E+06 <*/2lmm) <f/2[mm] 11IA] 12(A] FUN] F2IN] Ki[N/A] Ki/m 1. 50 1. 50 2. 33 2. 00 2 .53E+01 1. 86E+01 4 .03E+01 5 .90E+01 1. 50 1. 50 1. 00 1. 00 4 .65E+00 4. 65E+00 1 .86E+01 2 .72E+01 1. 50 1. 50 1. 50 1. 50 1 .05E+01 1. 05E+01 2 .79E+01 4 •09E+01 1. 50 1. 50 2. 00 2. 00 1 .86E+01 1. 86E+01 3 •72E+01 5 .45E+01 1. 50 1. 50 2. 50 2. 50 2 .91E+01 2. 91E+01 4 .65E+01 6 .81E+01 1. 50 1. 50 3. 00 3. 00 4 .19E+01 4. 19E+01 5 .58E+01 8 .17E+01 1. 50 1. 50 3. 50 3. 50 5 . 70E + 01 5. 70E+01 6 .51E+01 9 .54E+01 1. 50 1. 50 4. 00 4. 00 7 .44E*01 7. 44E+01 7 .44E+01 1 •09E+02 1. 50 1. 50 4. 50 4. 50 9 .42E+01 9. 42E+01 e .38E+01 1 .23E+02 1. 50 1. 50 5. 00 5. 00 1 .16E+02 1. 16E+02 9 .31E+01 1 .36E+02 1. 50 1. 50 5. 50 5. 50 1 .41E+02 1. 41E+02 1 .02E+02 1 .50E+02 1. 50 1. 50 6. 00 6. 00 1 •68E+02 1. 68E+02 1 .12E+02 1 .63E+02 1. 50 1. 50 6. 50 6. 50 1 .97E+02 1. 97E+02 1 .21E+02 1 .77E+02 1. 50 1. 50 7. 00 7. 00 2 .28E+02 2. 28E+02 1 .30E+02 1 .91E+02 1. 50 1. 50 7. 50 7. 50 2 .62E+02 2. 62E+02 1 .40E+02 2 .04E+02 1. 50 1. 50 8. 00 8. 00 2 .98E+02 2. 98E+02 1 .49E+02 2 .18E+02 1. 50 1. 50 8. 50 8. 50 3 .36E+02 3. 36E+02 1 .58E+D2 2 .32E+02 1. 50 1. 50 9. 00 9. 00 3 .77E+02 3. 77E+02 1 .68E+02 2 .45E*02 1. 50 1. 50 9. 50 9. 50 4 •20E+02 4. 20E+02 1 .77E+02 2 .59E+02 1. 50 1. 50 10. 00 10. 00 4 .65E+02 4 . 65E+02 1 .66E+02 2 .72E+02 

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