School of Mechanical Engineering Purdue University. ME375 ElectroMechanical - 1

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1 Electro-Mechanical Systems DC Motors Principles of Operation Modeling (Derivation of fg Governing Equations (EOM)) Block Diagram Representations Using Block Diagrams to Represent Equations in s - Domain Block Diagram Representation of DC Motors Example ME375 ElectroMechanical - 1

2 DC Motors Motors are actuation devices (actuators) that generate torque as actuation. Terminology Rotor : the rotating part of the motor. Stator : the stationary part of the motor. Field System : the part of the motor that provides the magnetic flux. Armature : the part of the motor which carries current that interacts with the magnetic flux to produce torque. Brushes : the part of the electrical circuit through which the current is supplied to the armature. Commutator : the part of the rotor that is in contact with the brushes. ME375 ElectroMechanical - 2

3 DC Motors - Principles of Operation Torque Generation B B df dl i Force will act on a conductor in a magnetic field with current flowing through the conductor. d f i dl B a Integrate over the entire length: f Total torque generated: Coil B i ME375 ElectroMechanical - 3

4 DC Motors - Principles of Operation Let N be the number of coils in the motor. The total torque generated from the N coils is: m N ( 2 i B L R) a For a given motor, (N, B,, L, R) ) are fixed. We can define K T 2 N B L R Nm /A as the Torque Constant of the motor. The torque generated by a DC motor is proportional to the armature current i a : a m K T i a For a DC motor, it is desirable to have a large K T. However, size and other physical limitations often limits the achievable K T. Large K T : Large (N, L, R). (N, L, R) is limited by the size and weight of the motor. Large B: Need to understand the methods of generating flux... ME375 ElectroMechanical - 4

5 DC Motors - Principles of Operation Back-EMF Generation Electromotive force (EMF) is generated in a conductor moving in a magnetic field: deemf ( v B ) dl v B Integrate over the entire length L: v e emf Since the N armature coils are in series, the total EMF is: Define the Back Eemf 2 N ( R ) L v Back-EMF Constant K b : K N R L The Back-EMF generated due to the rotation of the motor armature is opposing the applied voltage and is proportional to the angular speed of the motor: E emf K b B b 2 B V / (rad / sec) Note: K T = K b is true only if consistent SI units are used! ME375 ElectroMechanical - 5

6 DC Motors - Principles of Operation Generating Magnetic Flux Permanent Magnet Permanent-Magnet DC Motors (PMDC) Field Coil Induced Magnetic Field (a) Series Wound DC Motor High starting torque and no-load speed Unidirectional (b) Shunt Wound DC Motor Low starting torque and no-load speed Good speed regulation Unidirectional (c) Compound DC Motor High starting torque & good speed regulation (d) Separately Excited DC Motor ME375 ElectroMechanical - 6

7 DC Motors - Modeling Schematic + e i (t) _ + e Ra + e La Element Laws: Electrical Subsystem R A i A L A + E emf _ m J A L FBD: B Mechanical Subsystem Interconnection Laws: ME375 ElectroMechanical - 7

8 DC Motors - Modeling Derive I/O Model: I/O Model from e i (t) and L to angular speed : LAJ K T A F HG L B RJ RB A A A A Kb ei() t K K K I/O Model from e i (t) and L to angular position : LAJ K T A T T I F KJ HG T I KJ b L K R A L A L F L B R J I F RB I L R A A A A b A L A L g Kb ei t HG KT KT KJ HG KT KJ KT () T g ME375 ElectroMechanical - 8

9 DC Motors - Modeling Transfer Function: () s E() s () s i L () s E () s () s i L Q: Let the load torque be zero (No Load), what is the steady state t speed (No-Load ds Speed) of the motor for a constant input voltage V? Q: Let the load torque L = T, what is the steady state speed of the motor for a constant input voltage V? ME375 ElectroMechanical - 9

10 Block Diagram Representation Differential Equation Transfer Signal Addition/Subtraction Function (System & Signals) Y ( s ) U 1( s) U 2( s ) Y() s G() s U() s U 1 (s) + Y(s) U(s) Y(s) G(s) ( ) Input Signal Output Signal U 2 (s) Ex: Draw the block diagram for the following DE: J Ex: Draw the block diagram for the following DE: J B ME375 ElectroMechanical - 10

11 Block Diagram Representation Transfer Function in Series Multiple Inputs Ys ( ) G 2 ( s ) Y 1 ( s ), Y 1 ( s ) G 1 ( s ) U ( s ) Y 1 ( s ) G 1 ( s ) U 1 ( s ), Y 2 ( s ) G 2 ( s ) U 2 ( s ) Ys () bg2() sg1() sgus () Ys () Ys 1() Y2() s G1() s U1() s G2() s U2() s U(s) Y 1 (s) Y(s) G 1 (s) G 2 (s) U Input Output 1 (s) Y 1 (s) G Signal 1 (s) Signal Y(s) Transfer Function in Parallel X1() s G1() s U() s, X2() s G2() s U() s Ys () X() s X() s b 1 2 G() s G () s U() s 1 2 g U 2 (s) Input Signals G 2 (s) Y 2 (s) Output Signal U(s) G 1 (s) X 1 (s) Y(s) Input Signal G 2 (s) X 2 (s) Output Signal ME375 ElectroMechanical - 11

12 Block Diagram Representation Feedback Loop U(s) Input Signal X (s) G (s) Y(s) Output Signal H(s) ME375 ElectroMechanical - 12

13 Block Diagram Representation of DC Motors Schematic + e Ra + e La Electro-Mechanical Coupling: Take Laplace Transform of the Eqs. + e i (t) _ R A i A L A + E emf _ m J A L B Governing Equations: d LA ia RAiA Eemf ei( t) ( ) dt J B ( ) A m L m KT ia E emf Kb ME375 ElectroMechanical - 13

14 Block Diagram Representation of DC Motors Electrical Subsystem: Take Laplace Transform of the Eqs. Mechanical Subsystem: Take Laplace Transform of the Eqs. Q: Now that we generated a block diagram of a voltage driven DC Motor, can we derive the transfer function of this motor from its block diagram? ( This is the same as asking you to reduce the multiblock diagram to a simpler form just relating inputs e i (t) and L to the output, either or.) ME375 ElectroMechanical - 14

15 Block Diagram Reduction From Block Diagram to Transfer Function Label each signal and block 1 L A s +R A K T 1 J A s +B K b Write down the relationships between signals ME375 ElectroMechanical - 15

16 Block Diagram Reduction Solve for the output signal in terms of the input signals Substitute tute the transfer functions label with the actual formula and simplify () s KT Ei () s LJs ( BL RJ) s( RBKK) LAs RA L() s LJs ( BL RJ) s( RBKK) 2 2 A A A A A A b T A A A A A A b T ( s) KT LAs RA Ei () s sljs ( ( BL RJ) s( RBKK)) sljs ( ( BL RJ) s( RBKK)) 2 2 L A A A A A A b T A A A A A A b T () s ME375 ElectroMechanical - 16

17 Example (A) Given the following specification of a DC motor and assume there is no load, find its transfer function from input voltage to motor angular speed L A = 10 mh R A = 10 K T = 0.06 Nm/A J A = Kg m 2 B = Nm/(rad/sec) (B) Find the poles of the transfer function. (C) Plot the Bode diagram of the transfer function ME375 ElectroMechanical - 17

18 Example 20 Phase (deg) Magnitude (db) Frequency (rad/sec) Q:If we are only interested t in the system response up to 1000 rad/sec, can we simplify our model? How would you simplify the model? ME375 ElectroMechanical - 18

19 Model Reduction Neglect Electrical Dynamics Derive the model for the DC motor, if the armature inductance L A is neglected: 1 L A s+r A K T 1 J A s+b K b () s KT Ei () s LJs ( BL RJ) s( RBKK) Ls A RA L() s LJs ( BL RJ) s( RBKK) 2 2 A A A A A A b T A A A A A A b T KT () s E() s () s By neglecting the effect of armature inductance, we reduced the order of our model from two to one. i L ME375 ElectroMechanical - 19

20 Model Reduction Q: Physically, what do we mean by neglecting armature inductance? By neglecting the armature inductance, we are assuming that it takes no time for the current to reach its steady state value when there is astep change in input voltage, i.e., a sudden change in input voltage will result in a sudden change in the armature current, which in turn will result in a sudden change in the motor torque output. This is equivalent to having direct control over the motor current. Mathematically: L d dt i R i E e () t J E A A A A emf i B K i U K V A m L m T A emf b W From Block Diagram: 1 L A s + R A K T ME375 ElectroMechanical - 20

21 Example Media Advance System in InkJet Printers The figure on the right shows the media advance system of a typical inkjet printer. The objective of the system is to precisely and quickly position the media such that ink droplets can be precisely dropped on to the media to form nice looking images. The system is driven by a DC motor through two sets of gear trains. You, as the new kid on the development team, are given the task of specifying a motor and designing the control system that will achieve the desirable performance. Some time your manager will also walk by your desk and ask you if a certain level of performance is achievable. How would you start your first engineering g project? ME375 ElectroMechanical - 21

22 Example Schematic + e i (t) _ + e Ra R A i A DC Motor + e La L A + E emf _ m J A N 2 N 1 L L J L B L Assumptions: Gears and shafts are rigid and massless. B Block diagram of the load inertia: Block diagram of the gear train: ME375 ElectroMechanical - 22

23 Example Block diagram of the DC motor subsystem: 1 L A s + R A K T 1 J A s + B K b Reduce the mechanical portion of the block diagram: ME375 ElectroMechanical - 23

24 Example Simplified block diagram: 1 L A s + R A K T K b Transfer Function from input voltage E i (s) to the angular position of the load (s): G E i () s () s E() s i Q: Is this system stable? Q: What command (voltage voltage) ) would you use to move the roller s angular position by, say 60 o? ME375 ElectroMechanical - 24

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