Principles Of Electromechanical Energy Conversion
Principles of Electromechanical EnergyConversion Why do we study this?– Electromechanical energy conversion theory is thecornerstone for the analysis of electromechanical motiondevices.– The theory allows us to express the electromagnetic forceor torque in terms of the device variables such as thecurrents and the displacement of the mechanical system.– Since numerous types of electromechanical devices areused in motion systems, it is desirable to establish methodsof analysis which may be applied to a variety ofelectromechanical devices rather than just electricmachines.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig87
Plan– Establish analytically the relationships which can be usedto express the electromagnetic force or torque.– Develop a general set of formulas which are applicable toall electromechanical systems with a single mechanicalinput.– Detailed analysis of: Elementary electromagnet Elementary single-phase reluctance machine Windings in relative motionActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig88
Lumped Parameters vs. Distributed Parameters If the physical size of a device is small compared tothe wavelength associated with the signalpropagation, the device may be considered lumpedand a lumped (network) model employed.vλ fλ wavelength (distance/cycle)v velocity of wave propagation (distance/second)f signal frequency (Hz) Consider the electrical portion of an audio system:– 20 to 20,000 Hz is the audio range186,000 miles/secondλ 9.3 miles/cycle20,000 cycles/secondActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig89
Conservative Force Field A force field acting on an object is calledconservative if the work done in moving the objectfrom one point to another is independent of the pathjoining the two points.rˆˆˆF F1i F2 j F3krr uurr F dr is independent of path if and only if F 0 or F φCr uurF dr is an exact differentialFdx F2 dy F3dz dφ where φ (x, y,z)1( x2 ,y2 ,z2 ) r uur( x2 ,y2 ,z2 )F dr dφ φ ( x 2 , y2 , z 2 ) φ ( x1 , y1 , z1 ) ( x1 ,y1 ,z1 )( x1 ,y1 ,z1 )Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig90
Energy Balance Relationships Electromechanical System– Comprises Electric system Mechanical system Means whereby the electric and mechanical systems can interact– Interactions can take place through any and allelectromagnetic and electrostatic fields which are commonto both systems, and energy is transferred as a result of thisinteraction.– Both electrostatic and electromagnetic coupling fields mayexist simultaneously and the system may have any numberof electric and mechanical subsystems.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig91
Electromechanical System in Simplified � Neglect electromagnetic radiation– Assume that the electric system operates at a frequencysufficiently low so that the electric system may beconsidered as a lumped-parameter systemWE We WeL WeS Energy DistributionWM Wm WmL WmS– WE total energy supplied by the electric source ( )– WM total energy supplied by the mechanical source ( )Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig92
– WeS energy stored in the electric or magnetic fields whichare not coupled with the mechanical system– WeL heat loss associated with the electric system,excluding the coupling field losses, which occurs due to: the resistance of the current-carrying conductors the energy dissipated in the form of heat owing to hysteresis, eddycurrents, and dielectric losses external to the coupling field– We energy transferred to the coupling field by the electricsystem– WmS energy stored in the moving member and thecompliances of the mechanical system– WmL energy loss of the mechanical system in the form ofheat due to friction– Wm energy transferred to the coupling field by themechanical systemActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig93
WF Wf WfL total energy transferred to thecoupling field– Wf energy stored in the coupling field– WfL energy dissipated in the form of heat due to losseswithin the coupling field (eddy current, hysteresis, ordielectric losses)Wf WfL ( WE WeL WeS ) Conservation of Energy( WM WmL WmS )Wf WfL We WmActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig94
The actual process of converting electric energy tomechanical energy (or vice versa) is independent of:– The loss of energy in either the electric or the mechanicalsystems (WeL and WmL)– The energies stored in the electric or magnetic fields whichare not in common to both systems (WeS)– The energies stored in the mechanical system (WmS) If the losses of the coupling field are neglected, thenthe field is conservative and Wf We W m . Consider two examples of elementaryelectromechanical systems– Magnetic coupling field– Electric field as a means of transferring energyActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig95
v voltage of electric sourcef externally-applied mechanicalforcefe electromagnetic orelectrostatic forcer resistance of the currentcarrying conductorl inductance of a linear(conservative)electromagnetic systemwhich does not couplethe mechanical systemM mass of moveable memberK spring constantD damping coefficientx0 zero force or equilibriumposition of the mechanicalsystem (fe 0, f 0)Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsElectromechanical System with Magnetic FieldElectromechanical System with Electric FieldKevin Craig96
div ri l efdtvoltage equation that describes theelectric systems; ef is the voltage dropdue to the coupling fieldd2 xdxf M 2 D K ( x x 0 ) fedtdtWE ( vi ) dt dx WM ( f )dx fdt dt div ri l efdtWE ( vi ) dtActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsNewton’s Law of MotionSince power is the time rate ofenergy transfer, this is the totalenergy supplied by the electricand mechanical sources di WE r ( i )dt l i dt ( ef i )dt dt WeL WeS We2Kevin Craig97
d2 xdxf M 2 D K ( x x 0 ) fedtdt dx WM ( f )dx fdt dt d x dx WM M 2 dx D dt K ( x x 0 )dx ( f e )dx dt dt 22ΣWmSWf We Wm ( ef i )dt ( f e )dxActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsWmLWmtotal energy transferred tothe coupling fieldKevin Craig98
These equations may be readily extended to includean electromechanical system with any number ofelectrical and mechanical inputs and any number ofcoupling fields. We will consider devices with only one mechanicalinput, but with possibly multiple electric inputs. In allcases, however, the multiple electric inputs have acommon coupling field.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig99
JKj 1k 1Wf Wej WmkJJ W e i dtj 1ejj 1KK Wk 1fj jmk f ek dx kk 1JWf efji jdt f e dxj 1JdWf efji jdt f edxj 1Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsTotal energy supplied to thecoupling fieldTotal energy supplied tothe coupling field from theelectric inputsTotal energy supplied tothe coupling field from themechanical inputsWith one mechanical inputand multiple electric inputs,the energy supplied to thecoupling field, in bothintegral and differential formKevin Craig100
Energy in Coupling Field We need to derive an expression for the energy storedin the coupling field before we can solve for theelectromagnetic force fe. We will neglect all losses associated with the electricor magnetic coupling field, whereupon the field isassumed to be conservative and the energy storedtherein is a function of the state of the electrical andmechanical variables and not the manner in which thevariables reached that state. This assumption is not as restrictive as it might firstappear.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig101
– The ferromagnetic material is selected and arranged inlaminations so as to minimize the hysteresis and eddycurrent losses.– Nearly all of the energy stored in the coupling field isstored in the air gap of the electromechanical device. Air isa conservative medium and all of the energy stored thereincan be returned to the electric or mechanical systems. We will take advantage of the conservative fieldassumption in developing a mathematical expressionfor the field energy. We will fix mathematically theposition of the mechanical system associated with thecoupling field and then excite the electric system withthe displacement of the mechanical system held fixed.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig102
During the excitation of the electric inputs, dx 0,hence, Wm is zero even though electromagnetic andelectrostatic forces occur. Therefore, with the displacement held fixed, theenergy stored in the coupling field during theexcitation of the electric inputs is equal to the energysupplied to the coupling field by the electric inputs. With dx 0, the energy supplied from the electric0Jsystem is:Wf efji jdt f e dxj 1JWf efji jdtj 1Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig103
For a singly excited electromagnetic system:dλef dtWf ( i )dλ with dx 0Wf ( i )dλArea represents energy storedin the field at the instantwhen λ λa and i ia.GraphStored energy and coenergy ina magnetic field of a singlyexcited electromagneticdeviceActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsFor a linear magnetic system:Curve is a straight line and1Wf Wc λi2Wc ( λ )diArea is calledcoenergyλi Wc WfKevin Craig104
The λi relationship need not be linear, it need only besingle-valued, a property which is characteristic to aconservative or lossless field. Also, since the coupling field is conservative, theenergy stored in the field with λ λa and i ia isindependent of the excursion of the electrical andmechanical variables before reaching this state. The displacement x defines completely the influenceof the mechanical system upon the coupling field;however, since λ and i are related, only one is neededin addition to x in order to describe the state of theelectromechanical system.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig105
If i and x are selected as the independent variables, itis convenient to express the field energy and the fluxlinkages as Wf Wf ( i,x )λ λ ( i, x ) λ ( i, x ) λ (i,x )dλ di dx i x λ ( i, x )dλ di with dx 0 ii λ ( ξ, x ) λ ( i, x )Wf ( i )dλ idi ξdξ0 i ξActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsEnergy storedin the field of asingly excitedsystemKevin Craig106
The coenergy in terms of i and x may be evaluated asWc ( i, x ) λ ( i, x )di λ ( ξ, x )dξi0 For a linear electromagnetic system, the λi plots arestraightline relationships. Thus, for the singly excitedmagnetically linear system, λ ( i, x ) L ( x ) i , whereL(x) is the inductance. Let’s evaluate Wf(i,x). dλ λ ( i, x ) di with dx 0 idλ L ( x ) di1Wf ( i,x ) ξL ( x )dξ L ( x ) i 202iActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig107
The field energy is a state function and the expressiondescribing the field energy in terms of the statevariables is valid regardless of the variations in thesystem variables. Wf expresses the field energy regardless of thevariations in L(x) and i. The fixing of the mechanicalsystem so as to obtain an expression for the fieldenergy is a mathematical convenience and not arestriction upon the result.1Wf ( i,x ) ξL ( x )dξ L ( x ) i 202iActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig108
In the case of a multiexcited electromagnetic system,an expression for the field energy may be obtained byevaluating the following relation with dx 0:JWf i jdλ jj 1 Since the coupling field is considered conservative,this expression may be evaluated independent of theorder in which the flux linkages or currents arebrought to their final values. Let’s consider a doubly excited electric system withone mechanical input.Wf ( i1 ,i 2 , x ) i1dλ1 ( i1 ,i 2 , x ) i 2 dλ 2 ( i1 ,i 2 , x ) Actuators & Sensors in MechatronicsElectromechanical Motion Fundamentalswith dx 0Kevin Craig109
The result is:Wf ( i1 ,i 2 , x ) i10 i20 λ1 ( ξ, 0, x )ξdξ ξ λ1 ( i1 , ξ, x ) λ 2 ( i1 , ξ, x ) ξ i1 dξ ξ ξ The first integral results from the first step of theevaluation with i1 as the variable of integration andwith i2 0 and di2 0. The second integral comesfrom the second step of the evaluation with i1 equal toits final value (di1 0) and i2 as the variable ofintegration. The order of allowing the currents toreach their final state is irrelevant.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig110
Let’s now evaluate the energy stored in amagnetically linear electromechanical system withtwo electrical inputs and one mechanical input.λ1 ( i1 ,i 2 , x ) L11 ( x ) i1 L12 ( x ) i 2λ 2 ( i1 ,i 2 , x ) L 21 ( x ) i1 L 22 ( x ) i 2 The self-inductances L11(x) and L22(x) include theleakage inductances. With the mechanical displacement held constant (dx 0):dλ1 ( i1 ,i 2 , x ) L11 ( x ) di1 L12 ( x ) di 2dλ 2 ( i1 ,i 2 , x ) L21 ( x ) di1 L22 ( x ) di 2Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig111
Substitution into:Wf ( i1 ,i 2 , x ) i10 i20 λ1 ( ξ, 0, x )ξdξ ξ λ1 ( i1 , ξ, x ) λ 2 ( i1 , ξ, x ) ξ i1 dξ ξ ξ Yields:Wf ( i1 ,i 2 , x ) ξL11 ( x ) d ξ i1L12 ( x ) ξL 22 ( x ) dξ00112 L11 ( x ) i1 L12 ( x ) i1i 2 L22 ( x ) i 2222i1Actuators & Sensors in MechatronicsElectromechanical Motion Fundamentalsi2Kevin Craig112
It follows that the total field energy of a linearelectromagnetic system with J electric inputs may beexpressed as:1 J JWf ( i1 ,K ,i j , x ) L pqi p iq2 p 1 q 1Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig113
Electromagnetic and Electrostatic Forces Energy Balance Equation:JWf efji jdt f e dxj 1JdWf efji jdt f edxj 1Jf e dx efji jdt dWfj 1 To obtain an expression for fe, it is first necessary toexpress Wf and then take its total derivative. The totaldifferential of the field energy is required here.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig114
The force or torque in any electromechanical systemmay be evaluated by employing: dWf dWe dWm We will derive the force equations for electromechanical systems with one mechanical input and Jelectrical inputs.J For an electromagnetic system: f e dx i jdλ j dWfj 1 Select ij and x as independent variables: W W ( ri , x )rr Wf i , xJ Wf i,xdWf di j dx i j x j 1 rr λ j i , xJ λj i,xdλ j di n dx i n x n 1 ( )( )Actuators & Sensors in MechatronicsElectromechanical Motion Fundamentals( )ffrλj λj i,x( )( )Kevin Craig115
The summation index n is used so as to avoidconfusion with the subscript j since each dλj must beevaluated for changes in all currents to account formutual coupling between electric systems. Substitution:rr Wf i , xJ Wf i,xdWf di j dx i j x j 1 rr λ j i , xJ λj i,xdλ j di n dx i n x n 1 ( )( )Actuators & Sensors in MechatronicsElectromechanical Motion Fundamentals( )( )intoJf e dx i jdλ j dWfj 1Kevin Craig116
rr J λj i , x λ j i , xJr f e i , x dx i j di n dx x j 1 n 1 i n rr Wf i , xJ Wf i,x di j dx i j x j 1 rr Wf i , x J λi,xrj f e i , x dx i j dx x x j 1 rr Wf i , xJ J λ j i , x i j di n di j i n i j j 1 n 1 Result:( )( )( )( )( )( )Actuators & Sensors in MechatronicsElectromechanical Motion Fundamentals( )( )( )( )Kevin Craig117
This equation is satisfied provided that:rr Wf i , xJ λrj i,x f e i , x i j x x j 1 rr Wf i , xJ J λ j i , x 0 i j di n di j i n i j j 1 n 1 ( )( )( )( )( ) The first equation can be used to evaluate the force onthe mechanical system with i and x selected asindependent variables.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig118
We can incorporate an expression for coenergy andJobtain a second force equation:Wc i jλ j Wfj 1 Since i and x are independent variables, the partialderivative with respect to x is:r Wc i , x( ) x Substitution:rr Wf i , xJ λj i,x i j x x j 1 ( )( )rrr Wf i , xJ λ Wc i , xrj i,x f e i , x i j x x xj 1 ( )Actuators & Sensors in MechatronicsElectromechanical Motion Fundamentals( )( )( )Kevin Craig119
Note:– Positive fe and positive dx are in the same direction– If the magnetic system is linear, Wc Wf. Summary:rr Wf i , xJ λrj i,x f e i , x i j x x j 1 r Wc i , xrrJ λfe i , x rj i, θ xTe i , θ i j θj 1 rfeTe Wc i , θrTe i , θ xθ θ( )( )( )( )( )( )( )Actuators & Sensors in MechatronicsElectromechanical Motion Fundamentals( )( )r Wf i, θ θ ( )Kevin Craig120
By a similar procedure, force equations may be derivedwith flux linkages λ1, , λj of the J windings and x asindependent variables. The relations, given withoutproof, are:rr( )( )J i j λ , x Wc λ, xr f e λ , x λ j x xj 1 r Wf λ, xrfe λ, x r xJ i j λ, θrTe λ , θ λ j θj 1 r Wf λ, θrTe λ , θ θ( )( )( )( )( )Actuators & Sensors in MechatronicsElectromechanical Motion Fundamentals( )( )r Wc λ, θ θ ( )Kevin Craig121
One may prefer to determine the electromagneticforce or torque by starting with the relationshipdWf dWe dWmrather than by selecting a formula. Example:– Given:λ 1 a ( x ) i 2– Find fe(i,x)Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig122
Elementary Electromagnet The system consists of:– stationary core with a winding of N turns– block of magnetic material is free to slide relative to thestationary memberx x(t)Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig123
dλv ri voltage equation that describes the electric systemdtλ Nφflux linkagesφ φl φm(the magnetizing flux is common toφl leakage fluxboth stationary and rotating members)φm magnetizing fluxNiφl ℜlNiφm ℜmIf the magnetic system is considered to belinear (saturation neglected), then, as in thecase of stationary coupled circuits, we canexpress the fluxes in terms of reluctances.Actuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig124
N2 N2 λ i ℜl ℜm ( Ll Lm ) iℜm ℜi 2ℜgℜiℜi liµ riµ 0 A ixℜg µ0 A gℜgflux linkagesLl leakage inductanceL m magnetizing inductancereluctance of the magnetizing pathtotal reluctance of the magnetic materialof the stationary and movable membersreluctance of one of the air gapsAssume that the cross-sectional areas ofthe stationary and movable members areequal and of the same materialActuators & Sensors in MechatronicsElectromechanical Motion FundamentalsKevin Craig125
Ag AiThis may be somewhat of an oversimplification,but it is sufficient for our purposes.ℜm ℜi 2ℜg 1 li 2x µ 0 Ai µ ri 2NLm 1 li 2x µ 0 Ai µ ri Assume that the leakage inductanceis constant.The magnetizin
Electromechanical System – Comprises Electric system Mechanical system Means whereby the electric and mechanical systems can interact – Interactions can take place through any and all electromagnetic and electrostatic fields which are common to both systems, and energy is transferred as a result of this interaction.
For practical devices magnetic medium is most suitable. When we speak of electromechanical energy conversion, however, we mean either the conversion of electric energy into mechanical energy or vice versa. Electromechanical energy conversion is
Electromechanical Feeders The high-capacity performers Syntron MF Heavy-Duty Electromechanical Feeders are the heavy-weights of bulk material handling and are used for higher capacity requirements. The ten heavy-duty models handle capacities from 600 to 4,000 tons per hour.* Syntron Heavy-Duty Electromechanical Feeders combine
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energy conversion scheme using both wind and photovoltaic energy sources. 1.1 Wind Energy Systems Wind energy conversion systems convert the kinetic energy associated with wind speed into electrical energy for feeding power to the grid. The energy is captured by the blades of wind turbines whose rotor is connected to the shaft of electric .
Electromechanical Motion Fundamentals K. Craig 1 Electromechanical Motion Fundamentals Electric Machine – device that can convert either . often used in the design of electric machines and transformers to simplify the complex design pro
Laboratory at the NASA George C. Marshall Space Flight Center (MSFC) is involved in a program to develop electromechanical actuators for the purpose of testing and TVC system implementation. Through this effort, an electromechanical thrust vector control actuator has been designed and assembled.
Electric Power and Machine department Ain Shams University. Cairo, Egypt Mahmoud M. Kashef Electric Power and Machine department Ain Shams University. Cairo, Egypt ABSTRACT Many type of electrical machine can be used as an electromechanical battery for low earth orbit satellite. Electromechanical battery is motor generator mode coupling
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