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Chapter 3 Electromechanical-Energy-Conversion Principles The electromechanical-energy-conversion process takes place through the medium of the electric or magnetic field of the conversion device of which the structures depend on their respective functions. Transducers: microphone, pickup, sensor, loudspeaker Force producing devices: solenoid, relay, electromagnet Continuous energy conversion equipment: motor, generator This chapter is devoted to the principles of electromechanical energy conversion and the analysis of the devices accomplishing this function. Emphasis is placed on the analysis of systems that use magnetic fields as the conversion medium. The concepts and techniques can be applied to a wide range of engineering situations involving electromechanical energy conversion. Based on the energy method, we are to develop expressions for forces and torques in magnetic-field-based electromechanical systems. §3.1 Forces and Torques in Magnetic Field Systems The Lorentz Force Law gives the force on a particle of charge in the presence of electric and magnetic fields. F q ( ) BvEqF × + = (3.1) F : newtons, : coulombs, : volts/meter, q E B : telsas, : meters/second v In a pure electric-field system, qEF = (3.2) In pure magnetic-field systems, ( ) BvqF × = (3.3) Figure 3.1 Right-hand rule for ( ) BvqF × = . For situations where large numbers of charged particles are in motion, ( ) BvEF v × + = ρ (3.4) vJ ρ = (3.5) BJF v × = (3.6) ρ (charge density): coulombs/m 3 , (force density): newtons/m v F 3 , Jv ρ = (current density): amperes/m 2 . 1 Figure 3.2 Single-coil rotor for Example 3.1. Unlike the case in Example 3.1, most electromechanical-energy-conversion devices contain magnetic material. Forces act directly on the magnetic material of these devices which are constructed of rigid, nondeforming structures. The performance of these devices is typically determined by the net force, or torque, acting on the moving component. It is rarely necessary to calculate the details of the internal force distribution. Just as a compass needle tries to align with the earth’s magnetic field, the two sets of fields associated with the rotor and the stator of rotating machinery attempt to align, and torque is associated with their displacement from alignment. In a motor, the stator magnetic field rotates ahead of that of the rotor, pulling on it and performing work. For a generator, the rotor does the work on the stator. 2 The Energy Method Based on the principle of conservation of energy: energy is neither created nor destroyed; it is merely changed in form. Fig. 3.3(a): a magnetic-field-based electromechanical-energy-conversion device. A lossless magnetic-energy-storage system with two terminals The electric terminal has two terminal variables: (voltage), (current). e i The mechanical terminal has two terminal variables: fld f (force), x (position) The loss mechanism is separated from the energy-storage mechanism. – Electrical losses: ohmic losses… – Mechanical losses: friction, windage… Fig. 3.3(b): a simple force-producing device with a single coil forming the electric terminal, and a movable plunger serving as the mechanical terminal. The interaction between the electric and mechanical terminals, i.e. the electromechanical energy conversion, occurs through the medium of the magnetic stored energy. Figure 3.3 (a) Schematic magnetic-field electromechanical-energy-conversion device; (b) simple force-producing device. : the stored energy in the magnetic field fld W dt dx fei dt Wd fld fld −= (3.7) dt d e λ = (3.8) dxfidWd fldfld − = λ (3.9) Equation (3.9) permits us to solve for the force simply as a function of the flux λ and the mechanical terminal position x . Equations (3.7) and (3.9) form the basis for the energy method. 3 §3.2 Energy Balance Consider the electromechanical systems whose predominant energy-storage mechanism is in magnetic fields. For motor action, we can account for the energy transfer as ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ heat into converted Energy field magneticin stored energyin Increase output energy Mechanical sources electric form inputEnergy (3.10) Note the generator action. The ability to identify a lossless-energy-storage system is the essence of the energy method. This is done mathematically as part of the modeling process. For the lossless magnetic-energy-storage system of Fig. 3.3(a), rearranging (3.9) in form of (3.10) gives fldmechelec dWdWdW + = (3.11) where elec dW id λ = = differential electric energy input = differential mechanical energy output mech fld dW f dx= = differential change in magnetic stored energy fld dW Here is the voltage induced in the electric terminals by the changing magnetic stored energy. It is through this reaction voltage that the external electric circuit supplies power to the coupling magnetic field and hence to the mechanical output terminals. e dteidW = elec (3.12) The basic energy-conversion process is one involving the coupling field and its action and reaction on the electric and mechanical systems. Combining (3.11) and (3.12) results in fldmechelec dWdWdteidW + = = (3.13) §3.3 Energy in Singly-Excited Magnetic Field Systems We are to deal energy-conversion systems: the magnetic circuits have air gaps between the stationary and moving members in which considerable energy is stored in the magnetic field. This field acts as the energy-conversion medium, and its energy is the reservoir between the electric and mechanical system. Fig. 3.4 shows an electromagnetic relay schematically. The predominant energy storage occurs in the air gap, and the properties of the magnetic circuit are determined by the dimensions of the air gap. Figure 3.4 Schematic of an electromagnetic relay. 4 ( ) ixL = λ (3.14) dxfdW fldmech = (3.15) dxfiddW fldfld − = λ (3.16) is uniquely specified by the values of fld W λ and x . Therefore, λ and x are referred to as state variables. Since the magnetic energy storage system is lossless, it is a conservative system. is the same regardless of how fld W λ and x are brought to their final values. See Fig. 3.5 where tow separate paths are shown. Figure 3.5 Integration paths for . fld W () ∫∫ += 2bpath fld 2apath fld00fld , dWdWxW λ (3.17) On path 2a, 0d λ = and fld 0f = . Thus, fld 0dW = on path 2a. On path 2b, 0dx = . Therefore, (3.17) reduces to the integral of id λ over path 2b. () () λλλ λ dxixW ∫ = 0 0 000fld ,, (3.18) For a linear system in which λ is proportional to , (3.18) gives i () ( ) () () ∫∫ = ′ ′ = ′′ = λλ λ λ λ λλλ 0 2 0 fld 2 1 ,, xL d xL dxixW (3.19) : the volume of the magnetic field V ( ) fld 0 B V W H dB dV ′ =⋅ ∫∫ (3.20) If B H µ = , 2 fld 2 V B W µ ⎛⎞ = ⎜⎟ ⎝⎠ ∫ dV (3.21) 5 Figure 3.6 (a) Relay with movable plunger for Example 3.2. (b) Detail showing air-gap configuration with the plunger partially removed. 6 §3.4 Determination of Magnetic Force and Torque form Energy The magnetic stored energy is a state function, determined uniquely by the values of the independent state variables fld W λ and x . ( ) dxfidxdW fldfld , − = λ λ (3.22) () 21 12 1 2 12 , xx FF dF x x dx dx xx ∂∂ =+ ∂∂ (3.23) () dx x W d W xdW x λ λ λ λ ∂ ∂ + ∂ ∂ = fldfld fld , (3.24) Comparing (3.22) with (3.24) gives (3.25) and (3.26): ( ) x xW i λ λ ∂ ∂ = , fld (3.25) ( ) λ λ x xW f ∂ ∂ −= , fld fld (3.26) Once we know as a function of fld W λ and x , (3.25) can be used to solve for (,)ix λ . Equation (3.26) can be used to solve for the mechanical force fld (,) f x λ . The partial derivative is taken while holding the flux linkages λ constant. For linear magnetic systems for which ( )Lxi λ = , the force can be found as () () ( ) dx xdL xL xLx f 2 22 fld 2 2 1 λλ λ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ −= (3.27) ( ) dx xdLi f 2 2 fld = (3.28) 7 Figure 3.7 Example 3.3. (a) Polynomial curve fit of inductance. (b) Force as a function of position x for i = 0.75 A. For a system with a rotating mechanical terminal, the mechanical terminal variables become the angular displacement θ and the torque . fld T ( ) θ λ θ λ dTiddW fldfld , − = (3.29) ( ) λ θ θλ ∂ ∂ −= , fld fld W T (3.30) For linear magnetic systems for which ( )Li λ θ = : () () θ λ θλ L W 2 fld 2 1 , = (3.31) () () ( ) θ θ θ λ θ λ θ λ d dL L L T 2 22 fld 2 1 2 1 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ −= (3.32) (3.33) ( ) θ θ d dLi T 2 2 fld = (3.34) Figure 3.9 Magnetic circuit for Example 3.4. 8 §3.5 Determination of Magnetic Force and Torque from Coenergy Recall that in §3.4, the magnetic stored energy is a state function, determined uniquely by the values of the independent state variables fld W λ and x . ( ) dxfidxdW fldfld , − = λ λ (3.22) () dx x W d W xdW x λ λ λ λ ∂ ∂ + ∂ ∂ = fldfld fld , (3.24) ( ) x xW i λ λ ∂ ∂ = , fld (3.25) ( ) λ λ x xW f ∂ ∂ −= , fld fld (3.26) Coenergy: from which the force can be obtained directly as a function of the current. The selection of energy or coenergy as the state function is purely a matter of convenience. The coenergy is defined as a function of and ),( xiW fld ′ i x such that ( ) ( ) xWixiW ,, fldfld λ λ − = ′ (3.34) ( ) diidid λ λ λ + = (3.35) ( ) ),()(, fld xdWidxiWd fld λ λ − = ′ (3.36) ( ) dxfdixiWd fldfld , + = ′ λ (3.37) From (3.37), the coenergy ),( xiW fld ′ can be seen to be a state function of the two independent variables and i x . () dx x W di i W xiWd i fld x fld ∂ ′ ∂ + ∂ ′ ∂ = ′ , fld (3.38) ( ) x i xiW ∂ ′ ∂ = , fld λ (3.39) ( ) i x xiW f ∂ ′ ∂ = , fld fld (3.40) For any given system, (3.26) and (3.40) will give the same result. 9 By analogy to (3.18) in §3.3, the coenergy can be found as (3.41) () () λλλ λ dxixW ∫ = 0 0 000fld ,, (3.18) () () ∫ ′′ = ′ i idxixiW 0 fld ,, λ (3.41) For linear magnetic systems for which ixL )( = λ , () () 2 fld 2 1 , ixLxiW = ′ (3.42) ( ) dx xdLi f 2 2 fld = (3.43) (3.43) is identical to the expression given by (3.28). For a system with a rotating mechanical displacement, () ( ) idiiW i ′′ = ′ ∫ 0 fld ,, θλθ (3.44) ( ) i iW T θ θ ∂ ′ ∂ = , fld fld (3.45) If the system is magnetically linear, () () 2 fld 2 1 , iLiW θθ = ′ (3.46) ( ) θ θ d dLi T 2 2 fld = (3.47) (3.47) is identical to the expression given by (3.33). In field-theory terms, for soft magnetic materials ∫∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅= ′ V H dVdHBW 0 0 fld (3.48) dV H W v ∫ = ′ 2 2 fld µ (3.49) For permanent-magnet (hard) materials ∫∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅= ′ V H H dVdHBW c 0 fld (3.50) 10 [...]... §3.6 Multiply-Excited Magnetic Field Systems Many electromechanical devices have multiple electrical terminals Measurement systems: torque proportional to two electric signals; power as the product of voltage and current Energy conversion devices: multiply-excited magnetic field system A simple system with two electrical terminals and one mechanical terminal: Fig 3.13 Three independent variables: {θ... the coenergy function is a relatively simple function of displacement The use of a coenergy function of the terminal currents simplifies the determination of torque or force Systems with more than two electrical terminals are handled in analogous fashion 14 Figure 3.15 Multiply-excited magnetic system for Example 3.7 Figure 3.16 Plot of torque components for the multiply-excited system of Example 3.7 . of electric and magnetic fields. F q ( ) BvEqF × + = (3.1) F : newtons, : coulombs, : volts/meter, q E B : telsas, : meters/second v In a pure electric- field. single coil forming the electric terminal, and a movable plunger serving as the mechanical terminal. The interaction between the electric and mechanical