modeling of system electromechanical, Armature Controlled D.C Motor -Reduced Order Field Controlled D.C Motor
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The document summarizes mathematical models of DC motor systems. It describes: 1) An armature-controlled DC motor model including electrical and mechanical subsystems and derived transfer functions relating input voltage to output speed and position. 2) A reduced-order model of the armature-controlled motor assuming small inductance. 3) A field-controlled DC motor model describing the electrical and mechanical subsystems and derived transfer functions relating input field voltage to output speed and position.
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Which yields following transfer function
Armature Controlled D.C Motor
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modeling of system electromechanical, Armature Controlled D.C Motor -Reduced Order Field Controlled D.C Motor
- 1. Modeling of Electromechanical System •Electrical and Mechanical System Basics •Armature Controlled D.C Motor -Reduced Order •Field Controlled D.C Motor 1
- 2. Electrical System 2 Component Laplace V-I Relation I-V Relation Resistor R Capacitor Inductor dt t di L t v L L ) ( ) ( dt t i C t v c c ) ( ) ( 1 R t i t v R R ) ( ) ( R t v t i R R ) ( ) ( dt t dv C t i c c ) ( ) ( dt t v L t i L L ) ( ) ( 1 Ls Cs 1
- 3. Basic motion concepts v adt ; x vdt ; dt f d mv ma N dt T d J J Nm J , Moment of inertia For rotational systems dt ; dt For translational systems 3 ..
- 6. oltage back-emf v where e , e dt di L i R u b b a a a a Mechanical Subsystem Bω ω J Tmotor Input: voltage u Output: Angular velocity Electrical Subsystem (loop method): Armature Controlled D.C Motor u ia T Ra La J B eb 6
- 7. Torque-Current: Voltage-Speed: a t motor i K T • Combing previous equations results in the following mathematical model: Power Transformation: ω K e b b 0 a t b a a a a i -K B ω J u ω K i R dt di L Armature Controlled D.C Motor u ia T Ra La J B eb 7
- 8. Taking Laplace transform of the system’s differential equations with zero initial conditions gives: Eliminating Ia yields the input-output transfer function b t a a a a t K K BR s BL JR Js L K U(s) Ω(s) 2 0 (s) I Ω(s)-K B Js U(s) Ω(s) K (s) I R s L a t b a a a Armature Controlled D.C Motor 8 t a K Ω(s) B Js (s) I /
- 9. Reduced Order Model Assuming small inductance, La 0 Armature Controlled D.C Motor 9 ) ( b t a a t K K BR s JR K U(s) Ω(s)
- 10. If output of the D.C motor is angular position θ then we know Which yields following transfer function Armature Controlled D.C Motor ) ( ) ( s s s or dt d u ia T Ra La J θ B eb 10 )] ( [ b t a a t K K BR s JR s K U(s) (s)
- 11. Applying KVL at field circuit Field Controlled D.C Motor if Tm Rf Lf J ω B Ra La ea ef dt di L R i e f f f f f Mechanical Subsystem Bω ω J Tm 11
- 12. Torque-Current: f f m i K T Combing previous equations and taking Laplace transform (considering initial conditions to zero) results in the following mathematical model: Power Transformation: ) ( ) ( ) ( ) ( ) ( ) ( s I K s B s Js s I sL s I R s E f f f f f f f where Kf: torque constant Field Controlled D.C Motor 12
- 13. If angular position θ is output of the motor Eliminating If(S) yields Field Controlled D.C Motor ) ( f f f f R s L B Js K (s) E Ω(s) if Tm Rf Lf J θ B Ra La ea ef ) ( f f f f R s L B Js s K (s) E (s) 13