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2 Modeling of DC machine • 2.1 Theorem of operation • Maximum torque is produced when two fluxes are in quadrature. • 2.2 Induced EMF • From Faraday’s law, the induced emf is dφf φf e=Z =Z dt t where t is the time taken by the conductors to cut φf 1flux lines.1 Therefore, t= 2 × frequency = p n 2( )( r ) 2 60 • Thus,(P: poles; Z: armature conductors; φf: a flux per pole; nr: rotation speed) e= Zφf Pn r 60 • If the armature conductors are divided into ‘a’ parallel paths, then e= Zφf Pn r 60 wave winding: a = 2; lap winding: a = P. • The usual expression e = Kφfωm where ωm=2πnr/60 rad/sec and K=(P/a)Z(1/2π) • If the field flux is constant, then emf is e = Kbωm 2.3 Equivalent circuit and electromagnetic torque • The terminal relationship is v = e + R a ia + La dia dt • In steady state, the armature current is constant and hence v = e + Raia • The power balance via = eia + Raia2 • • The power of mechanical form, Pa called the air gap power, is expressed in term of the electromagnetic torque and speed as Pa = ωmTe = eia • Hence, Te = eia K bωmia = = K bi a ωm ωm 2.4 Electromechanical modeling • The acceleration torque, Ta, drives the load and is given by J dωm + B1ωm = Te − T1 = Ta dt where J: a moment of inertia (kg-m2/sec2) B1: a viscous friction coefficient N·m/(rad/sec) T1: the load torque 2.5 State-space modeling • The dynamic equations in state-space form ⎡ Ra − ⎢ ⎡ pia ⎤ La = ⎢ pω ⎥ ⎢ K ⎣ m⎦ ⎢ b ⎢⎣ J Kb ⎤ ⎡ 1 − La ⎥ ⎡ i a ⎤ ⎢ La ⎥⎢ ⎥ + ⎢ B1 ⎥ ⎣ωm ⎦ ⎢ 0 − ⎢⎣ j ⎥⎦ ⎤ 0 ⎥ V ⎡ ⎤ ⎥⎢ ⎥ 1 ⎥ ⎣T1 ⎦ − J ⎥⎦ • The roots of the system are R a B1 R a B1 2 R a B1 K 2b ) −( + )± ( + ) − 4( + La J La J JLa JLa λ1, λ 2 = 2 2.6 Block diagrams and transfer functions • From (2.13) and (2.19), we get I a (s ) = V(s) − K bωm (s) R a + sLa K b Ia (s) − T1(s) ωm = ( B1 + sJ ) • The transfer functions Kb ω (s) G ωV (s) = m = 2 V(s) s ( JLa ) + a ( B1La + JR a ) + ( B1R a + K 2b ) ω (s ) − ( R a + sLa ) G ωl (s) = m = 2 Ts (s) s ( JLa ) + s( B1La + JR a ) + ( B1R a + K 2b ) • The speed response due to the simultaneous voltage input and load torque disturbance is ωm(s) = GωV(s)V(s) + Gωl(s)T1(s) 2.7 Field excitation • Separately excited dc machine • Shunt-excited dc machine • Series-excited dc machine • DC compound machine • Permanent-magnet dc machine 2.8 Measurement of motor constants • Armature resistance: It is measured between the armature terminals by applying a dc voltage. (need to subtract the brush and contact resistance) • Armature inductance: By the test schematic shown in Figure 2.11, the inductance is Va2 − R a2 fs: the frequency Ia La = 2 πf s Ra: the armature ac resistance • EMF constant Specified field voltage is applied and kept constant, and the shaft is rotated by another dc motor, and then the armature is connected a voltmeter. •