Tài liệu Fundamentals of electromechanical energy conversion docx

ENGNG 2024 Electrical Engineering  E Levi, 2002 1 FUNDAMENTALS OF ELECTROMECHANICAL ENERGY CONVERSION 1. PRELIMINARY CONSIDERATIONS Electromechanical energy conversion is achievable in a number of ways. These possibilities rely on different fundamental laws of electrical engineering. As the only method that has importance on the large scale is electromechanical conversion achieved by means of electromagnetic converters, this section is fully devoted to the analysis of basic principles involved in electromagnetic electromechanical conversion. Electromechanical energy conversion is achieved by devices that are usually called electric machines. In principle, laws of electromagnetics can be used to design converters with the linear and with the rotary motion. Converters with linear motion are called linear electric machines, while those that rely on rotating motion are called rotating electric machines. Vast majority of existing machinery belong to the category of rotating electric machines. These include all the machines used to generate the electricity, as well as the most of the machines used in industry to perform some useful work while converting electric into mechanical energy. Linear machines are used relatively rare for somewhat specialised applications. It is for this reason that only rotating electric machines will be dealt with here. Prefix ‘rotating’ will be omitted and the converters will be called simply electric machines, implying that devices under consideration are characterised with rotational movement. Operating principles of electric machines involve two basic laws of electromagnetism, namely the law of the electromagnetic induction (Faraday’s law) and the law of force creation in an electromagnetic field (Bio-Savart’s law). Consider the situation shown in Fig. 1. A conductor is connected to an electric source and it carries current I. It is placed in the magnetic field of certain flux density B (which is of course a vector; hence the arrow above the symbol in Fig. 1). Interaction of the flux density and the conductor current leads to the creation of an electromagnetic force BlIF e ×= (1) where l is the conductor length. This electromagnetic force will cause the movement of the conductor, which will start travelling at certain linear speed v to the left. The electromagnetic force will be balanced by another, mechanical force that acts in the opposite direction (to the right). The equilibrium will be established when the two forces are mutually equal and the conductor will then travel at a constant speed. Note that the magnitude of force in (1) will simply be F e =IlB, since the angle between the conductor and the flux density is 90 degrees. Once when the conductor moves, according to the law of electromagnetic induction an electromotive force will be induced in the electric circuit, ( ) lBve •×= (2) The magnitude of this emf is simply e=vBl, since the angle between the speed vector and flux density vector is 90 degrees, while the angle between the conductor length vector and the vector product is zero degrees. A process of electromechanical conversion is established in this way. The energy will be converted from electrical to mechanical and the process is called motoring. In order for the motoring to happen it is necessary to: i) establish flux density, using permanent magnets for example; ii) create an electric circuit, that is connected to a voltage source and is placed in the flux density. This will lead to establishment of the electromagnetic force, which causes linear movement of the conductor. This movement is counterbalanced by the applied mechanical ENGNG 2024 Electrical Engineering  E Levi, 2002 2 force (not shown in Fig. 1) and the equilibrium is established when the conductor travels at a constant speed. Under this condition the electromagnetic force and the mechanical force are mutually equal, but act in the opposite directions. Consider next Fig. 2, where the same conductor is placed in the same flux density. However, the conductor is now not connected to the electric source; instead, the electric circuit is closed by using, say, an external resistance. The conductor is now dragged through the flux density using mechanical force at certain speed and this is the origin of the movement in this case. The sequence of events now reverses. An electromotive force, given with (2), is at first induced in the conductor. Since the circuit is closed, a current starts flowing. Interaction of the current and the flux density causes creation of the electromagnetic force. This force again acts in the opposite direction to the mechanical force and the equilibrium is established when the two forces are equal but act in the opposite direction. Note that in this case the source of motion is the supplied mechanical energy. The mechanical energy is now converted into electrical energy and the process is called generation. B I F e v Fig. 1 – Illustration of motoring. B I F m v F e Fig. 2 – Illustration of generation. It is important to note here that the process of electromechanical energy conversion is reversible. This means that either electric energy can be converted into mechanical energy, or mechanical energy can be converted into electric energy, by means of the same physical assembly. Note as well that both the expression for electromagnetic force acting on a conductor and the expression for induced electromotive force due to relative movement of conductor with respect to flux density, which are vectorial, reduce to very simple expressions due to the relative position of flux density vector, conductor and speed of motion. This is exactly the situation that is encountered in electric machines. Therefore equations (1) and (2), which contain scalar and vectorial multiplications, reduce to a very simple form of F e =IlB and e=vBl. Nothing changes in principle when rotational movement is under consideration instead of the linear movement. Table I gives the analogy between the linear and the rotational movement. Creation of torque in the case of rotational movement is illustrated in Fig. 3. Table I – Analogy between linear and rotational movement. Speed Source of motion Road travelled Power Linear motion Linear, v [m/s] Force, [N] Linear, s [m] Fv Rotational motion Angular, ω [rad/s] Torque, [Nm] Angle, θ [rad] T ω Suppose once more that there is a certain flux density, in which a structure is placed. This structure can rotate and is of radius r. Assume that there are two conductors placed on the structure, 180 degrees apart, as shown in Fig. 3, and let these two conductors carry current in designated (mutually opposite) directions. An electromagnetic force, F e =IlB,is created on each of the two conductors. However, one of these two forces acts to the left, while the other ENGNG 2024 Electrical Engineering  E Levi, 2002 3 one acts to the right (due to opposite directions of the current flow in the two conductors). Now, a torque is created on each of the two conductors, that equals the product of the force and the radius. However, since forces act in opposite directions at opposite sides of the structure, the torques will both act in anticlockwise direction, initiating the rotation of the structure in anticlockwise direction. The total electromagnetic torque will in general be the sum of all the individual torques acting on individual conductors. Current in Current out B F e F e Fig. 3 – Torque creation in the rotating structure. Every electric machine consists of ferromagnetic iron cores and windings mounted on the iron cores, these elements being of essential importance for electromechanical conversion. An electric machine consists of a stationary element, called stator, and a rotating element (such as the one in Fig. 3) called rotor. The winding is placed in slots of the stationary stator and/or in slots of rotational rotor. The winding consists of an appropriate number of turns. A turn is composed of two conductors which are placed in such a way that the induced electromotive forces in them sum up. The current therefore flows in the opposite direction, as illustrated in Fig. 3. As already noted and explained, the operation of electric machines relies on Faraday's law of electromagnetic induction and on Bio-Savar's law of electromagnetic force (torque). One important point to note is that the induced emf will be described with (2) only if the current in the system is pure constant DC current. A more general expression for the induced emf says that, if the total flux through the electric circuit is changed, an electromotive force is induced, () θ ω θθ ψ ψ d dL i dt di Le dtdddLidtdiLdtdLidtdiLe Li dtde +=− −−=−−= = −= (3) The first term in this expression will exist only in circuits with AC currents and it is called transformer emf. The second term is what corresponds to (2) and it is the induced emf due to the movement of a conductor in certain flux density. It is called rotational emf. Note that, according to (3), a rotational emf will be induced only if the inductance of an electromagnetic structure is a function of the rotor position θ . This may sound awkward but will be clarified later on. In deriving (3) the use was made of the correlation between the angle travelled by the rotor and its speed of rotation, ENGNG 2024 Electrical Engineering  E Levi, 2002 4 = dt ωθ (4) that reduces for a constant speed of rotation to θ = ω t. Chain differentiation rule was applied as well. The total flux of the winding is called flux linkage and is denoted with ψ in (3). It depends on the flux seen by each conductor Φ and on the number of turns N. Flux linkage is ψ =N Φ . Electromotive force in an electric machine is induced either due to rotation of a winding in the flux density, or due to rotation of the flux density with respect to a stationary winding. Change of flux linkage can be caused either by mechanical motion or by change of current in time. This is reflected in (3) and will be elaborated in detail later on. Let us further clarify the two operating regimes of electric machines, generating and motoring. Generating is discussed first. Due to the action of the prime mover (which delivers mechanical energy to the machine’s shaft) rotational part of the machine is forced to rotate (Fig. 4). Consequently, the speed of rotation is constant (n =const.)andT e = T PM .Voltageat machine terminals and induced emf differ because of the voltage drop in the winding; for generating induced emf is greater than terminal voltage (in the sense of rms values in AC machines, i.e. in the sense of average values in DC machines). Note that in generation direction of the speed of rotation coincides with the direction of the mechanical (prime mover’s) torque, while the electromagnetic torque of the machine opposes motion. During motoring (Fig. 4) created electromagnetic torque, which is a consequence of electric energy delivered to the machine, acts as the source of motion, i.e. it causes the rotor rotation. In this case the direction of speed and the electromagnetic torque coincide, while the mechanical torque (that is now load torque) acts against the direction of rotation. Once more the speed of rotation is constant (n =const.)andT e = T L . During motoring induced emf has the opposite polarity since it balances the applied voltage. It is therefore usually called counter-electromotive force. The terminal voltage is greater than the counter-emf in motoring. n n T PM T L T e T e Fig. 4 – Torque and speed directions in generation (left) and motoring (right). In what follows a generalised electromechanical converter is discussed at first. The analysis is valid for any type of electric machine; the only constraint is that there is only one degree of freedom for mechanical motion (i.e. rotor can rotate along one axis only). 2. GENERAL MODEL OF AN ELECTRIC MACHINE 2.1 Losses and efficiency Efficiency of an electric machine is defined in the same way as for any other device, as ratio of the output to input power ENGNG 2024 Electrical Engineering  E Levi, 2002 5 11 < + −= + == lossout loss lossout out inout PP P PP P PP η (5) where the difference between the input and the output power is the loss in the machine, that consists of three components: winding loss (or copper loss) that is caused by the current flow in windings of the machine (in general, it appears at both stator and rotor), iron (or core) loss that appears in the ferromagnetic structure of the machine that is exposed to AC flux, and mechanical loss that takes place due to friction in bearings and rotation of the rotor in the air: mechlossFeCuoutinloss PPPPPP − ++=−= (6) Copper losses are of standard RI 2 form and total winding loss is given with the summation of losses for all individual windings. Iron or core loss depends on the flux density and frequency and comprises hysteresis and eddy-current losses, () 22 mFeFe BffmP ξς += (7) It is, according to (7), proportional to the mass of the ferromagnetic material. Mechanical loss can be taken as proportional to the speed of rotation squared, 2 ω kP mechloss = − (8) Since mechanical power is a product of torque and speed, this means that the mechanical loss torque is taken as proportional to the speed of rotation. One important point to note is that the nature of the input and output power depends on the role of the machine. In motoring the input power is electrical, while the output power is mechanical. In generation it is the other way round, the input power is mechanical while the output power is electrical (in generation, there may be some windings that take electrical power as well, while some other windings generate electrical power). It has to be remembered that the rated power of the machine (power for which the machine has been designed), which is always given on the nameplate of the machine, is the output power. Hence, in generation the known rated power (always identified further on with an index n) is the output electrical power, while in motoring it is the output mechanical power. 2.2 Power flow in an electric machine Since the role of the input and the output power is dependent on the function that the machine performs, the two cases are treated separately. In what follows lower case symbols are used for all the quantities, meaning that instantaneous time-domain variables are under consideration. The idea behind the subsequent development is to develop a general mathematical model that is valid for any rotating electric machine. It is for this reason that the number of windings is not specified. Instead, it is taken as being equal to n, where this is an arbitrary number. The electric machine is for the time being a black box. There are two doors that enable access to the machine, electrical door and mechanical door. The power can be either delivered to the machine, or taken away from the machine, through these two doors. Electromechanical conversion takes place inside the box and the converted power is p c .Fig.5 illustrates power flows inside an electric machine for motoring and generating. Apart from these two doors there are two windows that are unwanted outputs only. These windows are outputs for winding losses and mechanical losses, which are inevitably created within a machine, and which represent lost power. Note that the iron (or core) loss is omitted from this representation. The reason is that it is of electromagnetic nature and it does not take place in the windings. The existence of the iron loss can be accounted for at a later stage, in an approximate manner, as it is done in transformer theory. Both doors can be either inputs or outputs (depending on whether the machine operates as a motor or as a generator), while windows are outputs only. Normally, one door will be the input while the other door will be the output, although in generation some windings make consume electrical energy, while ENGNG 2024 Electrical Engineering  E Levi, 2002 6 other windings are generating it (as shown in Fig. 5). The role of doors is thus reversible, as the machine can operate both as a generator and as a motor. Electrical input power Mechanical output power Copper losses in windings M echanical loss Converted power Electromagnetic energy storage M echanical energy storage Electrical output power Mechanical input power Copper losses in windings Mechanical loss Converted power Electromagnetic energy storage Mechanical energy storage Small electrical input power Fig. 5 – Power flow in an electric machine for motoring and generation, respectively. As can be seen from Fig. 5, apart from input and output power and losses, there are two internal storages of energy inside the machine. The first one is the stored electromagnetic energy, while the second one is the stored mechanical energy. Stored mechanical energy is the energy stored in rotating masses (kinetic energy) and it is in every aspect analogous to the energy stored under linear movement (which is 2 2 1 mvW mech = ,wherem is the mass of the body). Rotating bodies are characterised with so-called inertia (that is function of the mass and dimensions) J [kgm 2 ], while instead of the linear velocity one uses angular velocity. Hence 2 2 1 ω JW m = (9) Stored energy in the electromagnetic system is function of the inductances and the currents of the windings (or flux linkages and currents). For example, in the case of a single winding ENGNG 2024 Electrical Engineering  E Levi, 2002 7 iLiW e ψ 2 1 2 1 2 == (10a) If the machine has two windings, then the stored energy is () 112222 212111 22112112 2 22 2 11 2 1 2 1 2 1 iLiL iLiL iiiiLiLiLW e += += +=++= ψ ψ ψψ (10b) Taking index e for electrical power and index m for mechanical power in Fig. 5, one can write the following power balance equations: Motoring: m m mechlossc c e Cue p dt dW pp p dt dW pp ++= ++= − (11) Generation: 21 ee e Cuc c m mechlossm pp dt dW pp p dt dW pp −++= ++= − (12) Note that storages are energies, as defined in (9)-(10). Powers are time derivatives of energies and this is taken into account in formulation of (11)-(12). In generation some windings make take the power ( p e2 ), while other winding actually generate the power (p e1 ). Equations (11)-(12) enable formulation of the converted power that is defined as mec tp ω = (13) in terms of other known powers and derivation of the equation for motion of rotating masses in terms of known parameters and inputs of the machine. This is a tedious procedure for the generalised n-winding converter and most of the derivations will be therefore omitted. Only the starting equations and the final equations are presented in the next sub-section. It is to be noted that all the powers, as well as all the other variables (currents, flux linkages) were denoted with lower-case letters in this section. These are instantaneous time domain quantities, and the same approach is used in the following sub-section. This enables creation of a general mathematical model, in terms of time-domain instantaneous quantities, that is valid for all possible existing types of electric machines with rotational movement. 2.3 Mathematical model Each of the n windings of the machine is a piece of wire. Hence each winding can be characterised with its resistance and inductance. In addition, there are mutual inductances between any two windings. An induced emf appears in general in each winding. Hence the voltage equilibrium equation for one particular winding can be written as niniiii iiiiii iLiLiLiL dtdiReiRv ++++= +=−= 21211 ψ ψ (14) There is one flux linkage equation and one voltage equation for each of the n windings. It is convenient to use further on matrix notation to express these and other equations, since matrix notation will enable substitution of n equations with a single matrix equation. Hence for all the n windings one has (matrices are underlined): ENGNG 2024 Electrical Engineering  E Levi, 2002 8 iL dt d iRv = += ψ ψ (15) where = − n n R R R R R 1 2 1 = nnnnn n n n LLLL LLLL LLLL LLLL L 321 3333231 223221 113121 (16a) = n v v v v v 3 2 1 = n i i i i i 3 2 1 = n ψ ψ ψ ψ ψ 3 2 1 (16b) Note that in any electrical machine L ij = L ji . Input electrical power in motoring is viivivivp T nne =+++= 2211 (17) Note that current matrix in (17) has to be transposed to satisfy the rules of matrix multiplication. In generation the output power is viivivivivivp T nnkkkke =−−−+++= ++ 112211 (18) where winding 1… k generate electricity, while windings k+1…n consume electric energy. Current vector in (18) has positive currents for the windings that generate and negative currents for the windings that consume electric power. Winding losses can be expressed as iRiiRiRiRp T nnCu =+++= 22 22 2 11 (19) Stored electromagnetic energy is iLiW iiLiiLiiLiiLiiLiLiLiLW T e nnnnnnnne 2 1 2 1 2 1 2 1 1)1(32231131132112 22 22 2 11 = ++++++++++= −− (20) Current sign in voltage equation (15) is such that the current is positive when it flows into the winding. Hence in generation all the windings that generate will have negative currents since the current flow will be in the opposite direction from assumed positive current flow. Mechanical power and mechanical loss are governed with ω ω ω kt tp tp mechloss mechlossmechloss mm = = = − −− (21) and the correlation between speed of rotation and the angle travelled is ENGNG 2024 Electrical Engineering  E Levi, 2002 9 dtddt θωωθ == (22) Angular speed of rotation is related to the speed n in [rpm] through n 60 2 π ω = .Angle θ is the mechanical angle measured with respect to certain stationary defined axis in the machine’s cross-section. Mechanical torque in (21) can be the load torque (in motoring) or the prime mover torque (in generation). Stored mechanical energy remains to be given with 2 2 1 ω JW m = (23) Whatremainstobedoneistosubstituteallthepowersandderivativesofstored energies into the power balance equations (11)-(12). This enables, first of all, calculation of the converted power and the electromagnetic torque. Regardless of which of the two regimes is considered, the converted power is found to be i dt Ld ip T c 2 1 = (24) Since according to (13) converted power is ω ec tp = and since one can write using chain differentiation rule that ()()() ωθθθ dLddtddLddtLd ==/ , one finds the electromagnetic torque in the form i d Ld i p t T c e θω 2 1 == (25) Electromagnetic torque is positive for the motoring, while it has negative value in generation (as it opposes motion). From the second of (11) or the first of (12) one finds the equation of motion of the rotor in the form generation motoring ω ω ω ω k dt d JtT k dt d JTt ePM Le +=− +=− (26a) On the left-hand one has the difference between the driving torque (electromagnetic torque in motoring, prime mover torque in generation) and the opposing torque (load torque in motoring and electromagnetic torque in generation). On the right-hand side the first term is the acceleration/deceleration torque (that exists only during transients and is zero in steady- state) and the second term is the torque that describes mechanical losses. This particular torque can be always taken as part of the load (or prime mover) torque since it is mechanical in nature. One then arrives at the equation of mechanical motion in the frequently used form generation motoring dt d JtT dt d JTt ePM Le ω ω =− =− (26b) which shows that in any steady state (at constant speed) generation0 motoring0 =− =− ePM Le tT Tt (27) The meaning of (27) is simple. It is the basic law of action and reaction. In any steady state thetwotorquesareofthesameabsolutevaluebutactintheoppositedirection. The equations presented in this sub-section fully describe any rotational electrical machine, in terms of the instantaneous time-domain variables. The full mathematical model is summarised in the following sub-section. ENGNG 2024 Electrical Engineering  E Levi, 2002 10 2.4 Summary of the mathematical model Any rotating electric machine, regardless of the actual structure of the stator and rotor and regardless of the number of windings, is completely described with the following set of equations: iL dt d iRv = += ψ ψ < >− =+=+ generation0 motoring0, ePM eL mme tT tT tk dt d Jtt ω ω (28) i d Ld it T e θ 2 1 = dtd θω = where J and k are parameters of the machine and = − n n R R R R R 1 2 1 = nnnnn n n n LLLL LLLL LLLL LLLL L 321 3333231 223221 113121 = n v v v v v 3 2 1 = n i i i i i 3 2 1 = n ψ ψ ψ ψ ψ 3 2 1 (29) Equations (28)-(29) constitute the mathematical model of a generalised n-winding electromechanical energy converter. Note once more that all the variables (voltages, currents, flux linkages, electromagnetic torque, speed of rotation) are instantaneous time-domain quantities. Note as well that voltage equation is valid for current flowing into the winding. Hence in generation some of the currents will be negative since they will be flowing out of the machine. 2.5 Existence of converted power and electromagnetic torque and average torque Equation (25) shows that power will be converted if and only if the machine rotates. This means that at zero speed converted power is always zero. Further, one can see that electromagnetic torque can exist at zero speed (a machine will always start from standstill and the torque at zero speed is called starting torque). In order for an electromagnetic torque to exist it is necessary that at least some windings carry current and that at least some inductances of the machine are functions of the rotor angular position. Note that unless this [...]... torque will be zero The issue of dependence of machine’s inductances on angular position of the rotor will be discussed later Although an electromagnetic torque will exist if appropriate currents flow in the machine and there are inductances that depend on the rotor position, this is still not sufficient to realise useful electromechanical energy conversion Assume that the torque of a hypothetical electric... complete time-domain mathematical model of a generalised electromechanical converter with n windings and define all the matrices of the model b) State the general condition of average torque existence in a two-winding structure (define all the symbols used) c) A two-winding system has stator inductance of 0.1 [H], rotor inductance of 0.04 [H] and mutual inductance of 0.05 cos ϑ [H] If the rotor rotates... maximum value of the mutual inductance between the two winding be M Flux linkage of the stator and the rotor winding can be expressed as ψ s = Ls is + Lsr ir (35) ψ r = Lr ir + Lsr is For the sake of explanation, let us assume that rotor current is constant DC and let us investigate the contribution of this rotor current to the flux linkage in stator winding The maximum value of the contribution of the rotor... with two-phase system of currents, such that i1 = 10 cos ωt i 2 = 10 sin ωt Sketch cross-sectional view of the machine and identify the type of the machine Calculate the average torque for load angle equal to 45 degrees and explain its nature Sketch the dependence of average torque on load angle δ, identify motoring and generating part and define the region of stable operation Q4 An electromechanical converter... DC, of 15 A Stator windings are fed with two-phase system of currents, such that i1 = 10 cos ωt i 2 = 10 sin ωt Sketch cross-sectional view of the machine and identify the type of the machine Calculate the average torque for load angle equal to 60 degrees and explain its nature Sketch the dependence of average torque on load angle δ, identify motoring and generating part and define the region of stable... define the region of stable operation Q6 a) State the complete time-domain mathematical model of a generalised electromechanical converter with n windings and define all the matrices and vectors of the model b) A two-winding system has stator inductance of 0.15 [H], rotor inductance of 0.05 [H] and mutual inductance of 0.1 cos ϑ [H] If the rotor rotates at 250 rad/s and stator current is known to be 15 sin... Rotor winding current is constant DC, of 10 A Stator windings are fed with two-phase system of currents, such that i s1 = 10 cos ω s t i s 2 = 10 sin ω s t a) Sketch a cross-sectional view of the machine and identify the type of the machine b) Develop the expression for the instantaneous and average torque produced by the machine under the assumption that the condition of average torque existence is satisfied... explain its nature c) Sketch the dependence of average torque on load angle δ, identify motoring and generating part and define the region of stable operation Solution: In this example a so-called two-phase machine is considered The example will show that with the specific two-phase winding structure on stator it is possible to realize electromechanical energy conversion with the time independent constant... and they act along the defined magnetic axis of the winding From (41) one notices that each of the three m.m.f.’s is varying in time The values of the  E Levi, 2002 17 ENGNG 2024 Electrical Engineering three phase m.m.f.’s in the given instant of time correspond to those met in any three phase system a c b Fa Fb Fc b c a Fig 8 - Individual phase m.m.f.’s of a three-phase winding The resultant magneto-motive... magneto-motive force that stems from the three phase system of currents flowing through spatially displaced windings is the sum of the individual contributions of the three phases The summation is done in the cross-section of the machine, and it is necessary to observe the spatial displacement between the three m.m.f.’s One may regard the cross-section of the machine as a Cartesian co-ordinate system in which . 2002 1 FUNDAMENTALS OF ELECTROMECHANICAL ENERGY CONVERSION 1. PRELIMINARY CONSIDERATIONS Electromechanical energy conversion is achievable in a number of ways devoted to the analysis of basic principles involved in electromagnetic electromechanical conversion. Electromechanical energy conversion is achieved by