barcode reader code in c# net Introduction to Electric Machines in Software

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17
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Introduction to Electric Machines
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is fairly common practice to refer to the speed of rotation of an electric machine in units of revolutions per minute (rev/min)1 In this book, we shall uniformly use the symbol n to denote angular speed in rev/min; the following relationship should be committed to memory: 60 (1711) m (rad/s) 2 If the speed is expressed in rev/min, the armature constant changes as follows: n (rev/min) = Eb = ka n where pN (1713) 60M Having introduced the basic equations relating torque, speed, voltages, and currents in electric machines, we may now consider the interaction of these quantities in a DC machine at steady state, that is, operating at constant speed and eld excitation Figure 1715 depicts the electrical circuit model of a separately excited DC machine, illustrating both motor and generator action It is very important to note the reference direction of armature current ow, and of the developed torque, in order to make a distinction between the two modes of operation The eld excitation is shown as a voltage, Vf , generating the eld current, If , that ows through a variable resistor, Rf , and through the eld coil, Lf The variable resistor permits adjustment of the eld excitation The armature circuit, on the other hand, consists of a voltage source representing the back emf, Eb , the armature resistance, Ra , and the armature voltage, Va This model is appropriate both for motor and for generator action When Va < Eb , the machine acts as a generator (Ia ows out of the machine) When Va > Eb , the machine acts as a motor (Ia ows into the machine) Thus, according to the circuit model of Figure 1715, the operation of a DC machine at steady state (ie, with the inductors in the circuit replaced by short circuits) is described by the following equations: ka = If + Vf =0 Rf and and V a R a Ia E b = 0 V a + R a Ia E b = 0 (motor action) (1714) (generator action) (1712)
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If Lf Field circuit + Vf _ Ia
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Ra Armature La circuit + + Eb _ T m TL Va
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Mechanical coupling (a) Motor reference direction If Lf Field circuit + Vf _ _ T m Mechanical coupling TL Ia Ra Armature La circuit + + Eb Va
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Vf If + =0 Rf
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(b) Generator reference direction
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Figure 1715 Electrical circuit model of a separately excited DC machine
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Equation pair 1714 together with equations 176 and 178 may be used to determine the steady-state operating condition of a DC machine The circuit model of Figure 1715 permits the derivation of a simple set of differential equations that describe the dynamic analysis of a DC machine The dynamic equations describing the behavior of a separately excited DC machine are as follows: dIa (t) Va (t) Ia (t)Ra La Eb (t) = 0 (armature circuit) (1715a) dt Vf (t) If (t)Rf Lf
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1 Note
dIf (t) =0 dt
( eld circuit)
(1715b)
that the abbreviation RPM, although certainly familiar to the reader, is not a standard unit, and its use should be discouraged
Part III
Electromechanics
These equations can be related to the operation of the machine in the presence of a load If we assume that the motor is rigidly connected to an inertial load with moment of inertia J and that the friction losses in the load are represented by a viscous friction coef cient, b, then the torque developed by the machine (in the motor mode of operation) can be written as follows: T (t) = TL + b m (t) + J d m (t) dt (1716)
where TL is the load torque TL is typically either constant or some function of speed, m , in a motor In the case of a generator, the load torque is replaced by the torque supplied by a prime mover, and the machine torque, T (t), opposes the motion of the prime mover, as shown in Figure 1715 Since the machine torque is related to the armature and eld currents by equation 176, equations 1716 and 1717 are coupled to each other; this coupling may be expressed as follows: T (t) = ka Ia (t) or ka Ia (t) = TL + b m (t) + J d m (t) dt (1718) (1717)
The dynamic equations described in this section apply to any DC machine In the case of a separately excited machine, a further simpli cation is possible, since the ux is established by virtue of a separate eld excitation, and therefore = Nf If = kf If R (1719)
where Nf is the number of turns in the eld coil, R is the reluctance of the structure, and If is the eld current
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