Note 9 DC Machines-1 - Note 06. DC Machines Spring ....

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Unformatted text preview: Note 06. DC Machines Spring . BOSE-2410 Signals & Systems (Wozny) II. Introduction. Direct current motors and generators are common actuating devices in control systems, used for example, to position radar antennas, telescopes, or robots. This note presents the basic equations of these machines. Basic Principles Magnetic Characteristics. A current i applied to a coil wound around an iron core sets up a magnetic field it about the coil according to the magnetization curve showfiiin Figure 1. We will assume that o is proportional toi i.e., (1). Generator Action. When a voltage e! is applied to the field circuit, a current i, sets up a magnetic field it! in the stator windings. Applying an external torque Ti. to the shaft causes the rotor winding torotate at an angular velocity to through the stator magnetic field (9,, inducing a voltage it‘ across the rotor windings (armature circuit). This induced (generated) voltage is given by asham- Motor Action. The currents if and i_ cause magnetic fields (live, to interact and create a torque T on the rotor shaft. This output torque is T = Km, (3). Note that Equations (2) and (3) are nonlinear (products of variables). In most control system applications, one of the variables in each equation is held constant, resulting in a linear relationship. Now let’s consider the various permutations of the variables if ,iva'l , keeping two constant each time. and derive the input-output characteristics (transfer functions) of four practical machine configurations. III. Transfer Functions of Various Machine Configurations _ _ 4r - «w: Generator (m=constant). The basic DC machine equaticn for this configuration is e: =IKItfwlwm, (4). Applying an input voltage e , results in an output voltage 9‘. Generators are used to create large output power for driving heavy devices. The Laplace transform of the equations describing the generator input—output behavior are E,(s) = I,(s)(R, +sL,) E'(s)' = K,If(s). It is convenient to draw the block diagram (Figure 5), from which we obtain the generator overall transfer . E'(s) __ Kt E,(s)_RI+st' function Field ghnmllgd Motor ( t" =constant). The mechanical elements are J, the inertia of the motor shaft and B the coefficient of viscous friction. The basic machine equation is Emit,me (5). The system equations in this case are: - . E,(s)=1,(s)[a, +st] 6+3: WW‘ TU) = K ,1 ,e (s) 1;“) 415%— T(s) = only + as] -7 “' HE}: J OH 1' a ) Note that the first equation is an electrical circuit equti’ti‘on, the Second is the basic motor equation (which represents the conversion of electrical energy into mechanical energy), and the third is the mechanical equation. For such problems we can use block diagrams to facilitate the writing of the overall transfer function: First define the input-output variables and then write the equations in a form so that the intermediate variables drop out (as shown by the arrows above). The block diagram follows directly, K and the overall uansfer function is 9(3) = ~———--='——. . E f (s) 501'). + sL)(B+sJ) . The overall transfer function is " Annatw'e Controlled Motor (if =constant). In this case we must extend the previous concepts to include the “reaction force". When the current i, is applied to the armature circuit, the interaction of 1p, and or creates a torque causing the rotor to turn. However, the rotor is naming in the stator magnetic fieid of causing a voltage to be induced in the rotor circuit. According to Lenz’s law this voltage will be of such a polarity as to oppose the “force” that originally created it, namely, the armature current i. . r = Kzififlkmm = Kg, (6) ' t- E‘=K1(ot‘l =K,tu. (7) f _)? Ernest-rant introduce ‘Teedback” in the block diagram. an 9'3 Note that the machine equations, now"?- 6:) 313 61" The system equations are EhI (s) =input E. —- E I“ m = ,(s) ,(s) T(s)= K320)“ 9(5) = fl—T“) 503 + B) E‘ (s) = K ,3 9(3) output: 9(5) 9(s) = K. Emir) s[JR,s+(R,B+K,K_)]' The motor transfer function in this case is Emfit A Torque Controlled Generator (i)r =constant). The behavior of this configuration is analogous to the previous case in that there. exists a reaction to the input “force”. When a torque fl}, is applied to the generator shaft, the armature winding rotates in the magnetic field q), of the field circuit. Thus a I " voltage e, is induced in the armature. If the T. Tb ammonecitcuithasaloaMRchenacurrent i, “it” flows, creating a magnetic field on . This magnetic field at, interacts with the field circuit magnetic field do! to create a “back torque" T, which opposes the applied torque Th. The machine equatiOns are the same as (6) and ('7). The system equations are r, (s) = J529(s)+ ms) Tb(s) = [(1145) 15,“) = la(s)(R. + 11;): lama 53(3) = K ,to E (s) K,R rho) = JRs+K,K, ' IV. DC Tachometer. Tachometers -- which measure the speed of'a rotating shaft -- normally have a permanent magnet stator (p, =constant) and are constructed so that the back torque and inertia] are negligible. Connecting the tachometer to a high impedance -- load also reduces the back torque. Thus its . . E(s) tranf fucu ‘ z . ser n onxs (0(5) M Summary The above conditions can be condensed into the fofloudng three general blocks. magnetic General behavior (Lenz’s law): field The “input variable” creates the “generated variable” that in turn not only produces the output, but also creates the ‘Teaction variable”, that opposes the . original “input variable”. Motor : The interaction of magnetic fields created by inputs £6.13, produce output T,n). The output to causes the rotor winding to cut the magnetic field ' 1', created by if, and produce a’ reaction “back voltage” (also called back emf) that opposes the original input current 1". Generator: The inputs T,co cause the rotor to rotate a) in the magnetic field produced by if, creating output 13,. The application of E ‘ across a load ' ' creates a current i_. The magnetic field created by i, interacts with the magnetic field created by if to produce a “back torque” that opposes the original input torque. 5 Current in DC Motor When Mic current passes through a can In a magnum: field. we magnate farm produces a torque which turns the DC tumor IMcurrarItaam half Wufim to mums Mammogtm amame minimum mmallymw d _ . Magnetic Field in DC Motor When ma: current passes mrough a call In a moan-lull: flold. the magnum: m pm a torqu- Mfich turns the Theturrung mime _ fieldde ollhemmoris ] flomlha Nam proportionale mammaoum nugnellcfleld. pole. http :flhy‘perphysics.phy-astr.gsu.edumbase/magneticlmotdc.html#c 1 Force in DC Motor When mic current passes through a call In a magnum fluid. the magnetic {owe promwes a tumult which turns the DC motor F=ILB ants: parp-andwla: lo bath wire and WIGM ‘ F I ' When Hectic- curmn! - passes throng: a call in a mag mile field, the magnetic force \ produces a want ‘ which turns "19 Torque = force . x lever arm = Huang] sin e x 2 sides - a ILBW sin a = m sin 9 http :ffl‘nyperphysicsphy-astr. gsu.edufhbasefmagneticfmotdc .html#c l Commutator and Brushes on DC Motor To keep the torgue on a DC motor from reversing every time the coil moves through the plane perpendicular to the magnetic field, a split-ring device called a commutator is used to reverse the current at that point. The electrical contacts to the rotating ring are called "brushes" since copper brush contacts were used in early motors. Modern motors normally use spring-loaded carbon contacts, but the historical name for the contacts has persisted. http:ffliyperphysics.phy—astr.gsu.edu/hbase/magneticr’motdc .html#c 1 f ............... .. l change in the magnetic environment of a coil of wire will cause a voltage EKernf) to be "induced" in the coil. No matter how the change is produced, the ivoltage will be generated. The change could be produced by changing the imagnetic field strength, moving a magnet toward or away from the coil, gmoving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc. £332 2 . . m x “in is Faraday S Law summarizes the ways VOIIHQB can be 5 generated. (3th “a in magnetic fietd , 5—A = 02 mesh l at _ _. . r _ _ V = -3 x 0.2T x 02 mass ——— = 0.2 mate i Rotating l coil in ‘ magmatic vgma- cox 0.2T x 0.2 mats field it w ~5 I: once max 0.4 T rs ” ’0'” “"5 9°“ = 43.004 volts i a E .Further comments on these examples | ; WW". I EFaraday's law is a fundamental relationship which comes from Maxwell‘s i . ............................................. ...... ...... Index EFaradajg's 1 concepts ieg uations. It serves as a succinct summary of the ways a voltage (or emf) may lbe generated by a changing magnetic environment. The induced emf in a coil Ilis equal to the negative of the rate of change of maggetic flux times the inumber of turns in the coil. It involves the interaction of charge with magnetic ilfield. http:/fhyperphysics.phy-astr.gsu.edu/hbasefmagnetic/motdc.html#cl Go" 0, am A (Magnetic Faraday‘s Law field away will N tume from _ r} = I N A? i t 5‘ l Lam‘s l i Induced c - L“ where N = number of turns - A call at wire moving into a Eb a BA .—. magnetic flux "1391896 field is one example 3 = ma} magan [595d 1: %_ at an em! generated according A = area a; mi; i g i to Faraday's Laws The current 5 Induced will cream a menace The minus Sign denotes Lenz's Law. sale which opposes the buildup Emf is the term for generated or ; liengs—ladleCmc-oil cilani "" Te éFaraday's Law and Auto I i H erPh sics*****Electricit and ma etism R Nave 100 Back {2 ' ___________________________ .. Lenz s Law ; I When an emf is generated by a change in magnetic flux according to Faraday's j Iggy, the polarity of the induced emf is such that it produces a current whose magnetic field Opposes the change which produces it. The induced magnetic ifield inside any loop of wire always acts to keep the magnetic flux in the loop iconstant. In the examples below, if the B field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction Iof the applied field to try to keep it constant. Index %' ..................... _ .............. ......................... ...
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This note was uploaded on 04/10/2008 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.

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Note 9 DC Machines-1 - Note 06. DC Machines Spring ....

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