Magnetic Energy Domain Magnetic Capacitance Magnetic Resistance Magnetic Domain Example: Inductor with Flux Return Magnetic Circuit

Transcription

1 Michael David Bryant 9/8/07 Electromechanics, sensors & actuators Magnetic Energy Domain Magnetic Capacitance Magnetic Resistance Magnetic Domain Example: Inductor with Flux Return Magnetic Circuit Example: Variable Reluctance Actuators / Relay Example: Loudspeaker Example: Loudspeaker with Acoustic Tube Bond Graphs for Magnetostrictive Actuators

2 Michael David Bryant 2 9/8/07 Magnetic Energy Domain Magnetic field intensity: H [Amp m - ] Magnetic induction (or flux density): B = µ H [Webers m -2 = Tesla] Magnetic flux: φ = B ds [Webers = Volt sec] surface Permeability: µ [Henry m - = Volt sec Amp - m - = Ohm m - sec] Faraday's law defines magnetic flow: magnetic flux rate! [Webers sec - = Volt] V = E dl = - B t ds C urve S urface = - n dφ dt = - n! = - dλ dt The minus sign is grouped into definition of a voltage drop: V = n dφ dt = n! = dλ dt =! Ampere's law defines magnetic effort: magnetomotive force M [Amp] M = H dl = J ds C urve S urface + D t ds S urface = I c + Id = I Magnetic power: P = M! [ Watts ]

3 Michael David Bryant 3 9/8/07 Magnetic Capacitance Magnetic power de dt = P = M! "Potential" Energy stored in magnetic field E M = M dφ dt dt = M (φ) dφ Energy variable = Magnetic flux (magnetic displacement): φ Effort (magnetomotive force) dependence: M = M(φ) Flow (magnetic flux rate) obeys kinematics:! = dφ dt M = M(!)! C

4 Michael David Bryant 4 9/8/07 Magnetic Capacitance Define Reluctance R: R = l µ A M = φ R Analogous to Ohm's law for a resistor V = i R R resistance of element to current flow R reluctance of element to flux flow Differences from Ohm's law magnetic circuit instead of electrical circuit potential energy stored in R instead of power dissipation in R Potential energy storage E M = # M(")d" = $ "#d" = " 2 # 2 Generates linear magnetic capacitance C = φ /Μ = /R

5 Michael David Bryant 5 9/8/07 Magnetic Resistance Power dissipation in magnetic circuit from eddy currents time varying magnetic field H(t) in core time varying induction B(t) = µ H(t) time varying electric field E(t) via Faraday's law x E = - B t current density J = σe via Ohm's law resistive power dissipation in core B H magnetic hysteresis: B-H follows different paths, increasing & decreasing shaded area between paths energy (power) loss prevalent in iron Generates magnetic resistance R

6 Michael David Bryant 6 9/8/07 Magnetic Domain Example: Inductor with Flux Return + V - i flux rate! from flux! magnetomotive force M = n i n turn coil Coil: converts current i magnetomotive force M GY:r Reluctance of flux return path: stores magnetic potential energy E M Bond Graph Model V = r! = " - i = r M(! ) =i( ") GY: r M = M(!)! C Equivalent bond graph V L =! i = i (!) I

7 Michael David Bryant 7 9/8/07 Magnetic Circuit! Iron core + i M a M b V - n turn coil M o " 0! air gap M c n turn coil applies magnetomotive force ni = Ma - Mo Iron core routes magnetic flux Break in core: air gap define reference Mo 0

8 Michael David Bryant 8 9/8/07 Bond Graph of Magnetic Circuit C top R top V i GY! C air C bottom R bottom junction for common flux rate " through circuit R top & R bottom magnetic losses C top & C bottom potential energy stored in magnetic field in iron C air potential energy stored in magnetic field in air Potential energy E = E air + E iron " E air since µ iron >> µ air

9 Michael David Bryant 9 9/8/07 C top R top M a 0! 0 M b V i GY!! C air! M o 0 0 M c C bottom R bottom 0's mark magnetomotive force values Ma,, Mo in bond graph; 's distribute power to elements (C' 's & R 's from core) & mark flow! no magnetic losses in air (only Cair)

10 Michael David Bryant 0 9/8/07 Simplify Bond Graph: Eliminate 0 Junctions C top R top! V i GY!!! C air C bottom R bottom 0 junctions non-essential: conceptual aid for constructing bond graph 2 bonds on 0 's same (e, f) on input and output bonds to 0 's simplify bond graph: eliminate 0 's equivalent bond graph

11 Michael David Bryant 9/8/07 Simplify Bond Graph: Merge Neighboring Junctions C top R top V i GY! C air C bottom R bottom All 's have same flow! and are neighbors collapse into single equivalent bond graph

12 Michael David Bryant 2 9/8/07 Leakage Flux from Coil: Bond Graph! Iron core + i V air gap - n turn coil leakage flux in air! Leakage circuit in parallel with iron Leakage &! add about coil 0 junction and energy storage in leakage field C leak C top R top V i GY 0! C air C bottom R bottom

13 Michael David Bryant 3 9/8/07 Example: Variable Reluctance Actuators / Relay Magnetic field energy E = E air + E iron " E air in iron & air gap relay arm moves, reduces potential energy F = "E "x flux flow: magnetic energy storage E air = " 2 #(x) 2 in air gap, R(x) x µ air A Mechanical force F alters x changes reluctance R(x) E air x!! V + - resistor M b M a relay arm M o!! M d M c spring

14 Michael David Bryant 4 9/8/07 Relay Arm 2 Port C: Magnetic & Mechanical Ports Potential energy E = E air + E iron " E air since µ iron >> µ air E air = E air (φ, x) = " 2 #(x) 2 Efforts on kth bond from ek = E qk : F = "E air "x = # 2 2 M = E air φ "$(x) "x = φ R(x) = φ = # 2 2µ air A x µ air A M F C : /! air " ẋ

15 Michael David Bryant 5 9/8/07 Bond Graphs of Relay R C leak C top R top R: Fµ S : V e V(t) GY 0! M F C : /"! air x TF C: /k R bar C bottom R bottom C bar I: J C top R top R: Fµ R M a 0! 0 M b M F C : /" air! x TF C: /k S : V e V(t) GY! C leak! M c 0 I: J 0 0 R bar M o M d C bar C bottom R bottom Siimilar to magnetic circuit Relay Arm replaces Air Gap: C air becomes 2 port capacitance BG elements off 2-port C accounts for relay arm mass, pin friction & spring C leak accounts for leakage flux in air

16 Michael David Bryant 6 9/8/07 Loudspeaker Example speaker cone I + - E(t) voice coil motion x(t) D permanent magnet with flux density B voice coil assembly (N-turns) low reluctance iron routes flux to coil Input voltage E(t) & electric currents I to voice coil Voice coil: N-turns, diameter D, resistive losses Permanent magnet flux B, crosses voice coil radially, returns axially Stiffness and damping in cone & air Displacements x(t): voice coil/cone assembly

17 Michael David Bryant 7 9/8/07 Magnetic Lorentz force on voice coil F = - I B x dl Curve = - IBNπD Voice coil back emf (Curve: N turns, coil circumference) Eback = - t B ds S urface = - B da dt = - BNπD dx dt (Surface A: Outer shell, coil) Generates GY: BNπD

18 Michael David Bryant 8 9/8/07 speaker cone I + - E(t) voice coil motion x(t) D permanent magnet with flux density B voice coil assembly (N-turns) low reluctance iron routes flux to coil R: R coil I : M S e V P GY: BN!D P/ M kx x C: / k R: B

19 Michael David Bryant 9 9/8/07 Example: Loudspeaker with Acoustic Tube ACOUSTIC IMPEDANCE MATCHING TUBE SOUND WAVES v(t) = A dx/dt plunger/cone motions x(t) speaker cone (area A) I + - E(t) permanent magnet voice coil (N-turns & diameter D) Conversion: mechanical power into acoustic power Tube for impedance matching R: R coil I : M I : M air P P / M air P P/ M k x V air air S GY: BN!D e TF: A C: / k air x kx air x R: B C: / k R: B air

20 Michael David Bryant 20 9/8/07 Bass Reflex Speaker System Woofer Tweeter Enclosure with tuned port Crossover network (capacitor)

21 Bond Graph of Bass Reflex Speaker System 2 S e :E(t) E(t) 0 C:C R:R c w q q/c!t!w I:L t I:L w R:R c t!t / L t!w / L w R:b c t GY: r t GY: r w R:b c w pt pw k t x t I:m t I:m w C:k t pt / m t k w x w xt pw / m w xw C:k w TF: A t TF: A w r TF: A w f 0 k e v e ve C:k e I:I a p a p a / I a 0 k a v a C:k a va R:b p p p R:b a p p / I p I:I p

22 Bond Graphs for Magnetostrictive Actuators 22 Magnetostriction strain smn = smn(hi, Tjk) induced by stress Tjk induced by magnetic field Hi present in most ferromagnetic materials iron, nickel, cobalt, rare earths magnetostrictive strains 0-5 transformer hum: Fe core magnetostrictively extends/contracts under AC special rare earth alloy, terfenol D: strains 0-3 to 0-2

23 Magnetostrictive variables: (uniaxial) fields & stresses aligned with z direction 23 magnetic induction B = B(H, T) axial displacement u and strain s = u z applied (axial) magnetic field H & magnetic induction B Constitutive Equations / Linear Magnetostriction B = B(H, T) = do T + µ T H (a) s = s(h, T) = S H T + do H (b) elastic compliance S H, measured with magnetic intensity H = 0 magnetic permeability µ T, measured with stress T = 0 magnetostrictive coefficient do energy coupling coefficient k = d 2 o µ T S H

24 24 Invert H = d s + B µ s (2a) T = Y B s + d B (2b) d = - k2 ( - k2) do, elastic modulus Y B = S H, measured with inductance B = 0 ( - k2) permeability µ s = µ T ( - k2) measured with strain s = 0 "stiffness" term (Y B s) magnetically induced stress (d B) strain induces magnetic field component (d s) Multiport capacitance with displacements (s, B) & efforts (T, H)

25 Magnetostrictive Actuator mm Aluminum Attachment Plunger Rod Nylon Bearing Body Terfenol Rod Top Cover Belleville Washers Coil Aluminum Spool Coil 2 Permanent Magnet Bottom Cover Steel Spacer

26 Bond Graph Structure / Actuator S e V R:R elec # Ni F(!,x) GY:N C r TF:n r 0 OUT N! i #/L air I:L air R mag!! "! ret R:R mag M(!,x) C:/ " ret! R:b F(!,x) x r x I:m p F(!,x) n x n p /m p /m TF:n TF:n n p /m n n C:/k p k p x p x p I:M P P/M R:B b e OUT f OUT 26 R:b n p n p /m n n I:m n Excitation circuit: coil input V coil resistance Relec leakage inductance Lair electric power magnetic power (Ni! ) by coil GY:N Magnetic Circuit magnetic effort: magnetic potential H = Ni magnetic flow: magnetic flux rate! magnetic capacitance C:/Rret, reluctance Rret resistance R:Rmag incorporates eddy current & hysteresis losses

27 Multi-port C 27 S e V R:R elec # Ni F(!,x) GY:N C r TF:n r 0 OUT N! i #/L air I:L air R mag!! "! ret R:R mag M(!,x) C:/ " ret! R:b F(!,x) x r x I:m p F(!,x) n x n p /m p /m TF:n TF:n n p /m n n C:/k p k p x p x p I:M P P/M R:B b e OUT f OUT R:b n p n p /m n n single magnetic port from magnetic circuit I:m n multiple mechanical ports continuous rod incorporates vibration modes

28 Efforts on Multi-port C 28 Ampere's law over magnetic circuit (using eqn (2a)) magnetomotive force on 2 port C M(φ, x) = H l = d x + Rterf φ x : total end displacement φ : flux in rod Rterf reluctance of magnetostrictive rod (n + ) ports /axial vibration displacement modes uj = Uj(z) xj(t) modal forces Fj(φ, x) = nj d φ + Kj xj shape function Uj(z) modal mass I:mj, stiffness C:/Kj, damping R:bj modal junctions & transformers TF:nj couple forces to modes construct modal displacements 0 junction sums modal flows nj p j mj

29 29 final junction applies load to the end mass M R: Bb bushing friction external element(s) "OUT" plunger rod stiffness generates capacitance C:/kp

30 30 Multi-port Capacitance power flows over multiport C = time derivative of the potential energy de dt = M(φ, x)! + (5) j = Fj(φ, x) ẋj = d dt { 2 Rterf φ 2 + d φ x + 2 j = Kj xj 2 } M(φ, x) = E φ Fj(φ, x) = E xj = d x + Rterf φ = nj d φ + Kj xj x = n j xj(t) j =

31 3 State Equations from Bond Graph! = - R elec Lair λ - N! + V(t) (6a) Rmag φ = N λ Lair - Rret φ - M(φ, x) (6b) ṗj = - Fj(φ, x) - bj p j mj - nj kp xp, ( j =, 2,, n) (6c) ẋj = p j mj, ( j =, 2,, n) (6d) ẋp n = nj p j mj j = + j = n+ nj ẋj - M P (6e) Ṗ = - B b M P - e OUT + k p xp (6f)

32 Substitute equations (3) and (4) into (6) & rearrange into state equation form 32! = - [ R elec Lair + N2 Lair Rmag ] λ + N ret + terf - d2/kr Rmag φ + N d Rmag n j = nj xj - N d kp Rmag Kr xp + V(t) (7a)! = N Lair Rmag (7b) ṗj = - nj d φ - Kj xj - b j mj λ - ret + terf - d2/kr Rmag φ - d Rmag n j = nj xj + pj - nj kp xp,( j =, 2,, n) (7c) d kp Rmag Kr x ẋj = mj pj, ( j =, 2,, n) (7d) ẋp = Ṗ Kr n kp + Kr j = - d Rmag nj p j mj n j = - d kp + Kr nj xj + -{ d kp Rmag Kr N Lair Rmag xp} - M λ - ret + terf - d2/kr Rmag Kr kp + Kr = - B b M P - e OUT + k p xp (7f) P (7e) φ

33 Frequency Response 33 measured / actuator: solid line predicted / BG model: dashed line two modes X 0 t P M dt = j = nj xj - xp x - x2 - ( + kp/kr) xp - d Kr φ geometry: design material properties: handbook -00 X/V (m V - ) X/i (m A - ) Magnitude (m V - ) Phase (degrees) Magnitude (m A - ) Phase (degrees) Frequency (Hz) Frequency (Hz)

34 20 Table : Parameter values for the bond graph model of the magnetostrictive actuator SYMBOL DESCRIPTION SOURCE VALUE UNITS R elec solenoid resistance measured 036 Ω L air solenoid leakage inductance x 0-3 H N turns in excitation coil measured 200 R ret reluctance of flux return circuit Appendix 2 6 x 0 8 H - R mag resistance of flux return circuit Appendix 2 2 x 0 5 Ω - R terf reluctance of magnetostrictive rod = l 9 x 0 8 H - µ s A terf l length of magnetostrictive rod measured 0038 m D diameter of magnetostrictive rod measured m A terf cross sectional area of rod = π (D/2) x 0-5 m 2 k energy coupling coefficient Butler (988) 072 do magnetostrictive coefficient Butler (988) 5 x 0-9 m A - Y B elastic modulus-coil open circuited Butler (988) 55 x 0 0 N m -2 µ s permeability - rod clamped Butler (988) 565 x 0-7 V s m - A - ρ terfenol D mass density Butler (988) 925 x 0 3 kg m -3 ω /2π st mode natural frequency of eq (3c) & (3e) 6,000 Hz ω 2 /2π clamped-free rod 2nd mode natural frequency of eq (3c) & (3e) 48,000 clamped-free rod bj modal damping coefficient = 2 ζ j ω j mj 68 kg s - ζ, ζ 2 modal damping ratios lightly damped estimate 005, 0067 mj modal mass eq (3d) 675 x 0-3 kg K st mode stiffness eq (3f) 683 x 0 7 N m - K2 2nd mode stiffness eq (3f) 64 x 0 8 N m - Kr residual stiffness eq (4b) 557 x 0 8 N m - Kp plunger rod stiffness geometry and materials Hz 860 x 0 7 N m - M attached end mass measured 0036 kg Bb damping of guide bearing Appendix kg s -