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Unformatted text preview: Note 06. DC Machines Spring . BOSE2410 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 inputoutput 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 inputoutput 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 _)? Ernestrant
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 moanlull: flold. the
magnum: m pm a torqu
Mﬁch turns the Theturrung mime _ ﬁeldde
ollhemmoris ] ﬂomlha Nam
proportionale mammaoum
nugnellcﬂeld. pole. http :ﬂhy‘perphysics.phyastr.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: parpandwla:
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‘nyperphysicsphyastr. 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 splitring 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 springloaded 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.phyastr.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—ladleCmcoil 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.
 Spring '07
 WOZNY

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