Lesson 14: Transfer Functions of Dc Motors

10/28/2015

lesson14et438a.pptx

LESSON 14: TRANSFER

FUNCTIONS OF DC MOTORS

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ET 438a Automatic Control Systems Technology

lesson14et438a.pptx

LEARNING OBJECTIVES

After this presentation you will be able to: Write the transfer function for an armature

controlled dc motor. Write a transfer function for a dc motor that

relates input voltage to shaft position. Represent a mechanical load using a mathematical

model. Explain how negative feedback affects dc motor

performance.

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lesson14et438a.pptx

STEADY-STATE OPERATION OF SEPARATELY EXCITED DC MOTORS

Consider steady-state model

ia = armature current

eb= back emf

ea= armature terminal voltage

wm

wm = motor speed (rad/sec)

T = motor torque

Tf = static friction torque Ra = armature resistance La = armature inductance Jm = rotational inertia Bm = viscous friction

Review the steady-state relationships

Of machine

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lesson14et438a.pptx

STEADY-STATE OPERATION OF SEPARATELY EXCITED DC MOTORS

Relationships of Separately Excited Dc Motor

Torque-Current Curve

Back EMF Curve

T KT=DT/Dia

eb KT=DT/Dia

DT

Dia ia

-Tf

wm

wm=wnl ? (Dwm/DT)T

Speed-Torque Curve wnl

Dwm

DT

T

DT Dia

wm

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lesson14et438a.pptx

STEADY-STATE MOTOR EQUATIONS

Developed Torque

T = KT ia Tf N - m

T = motor torque KT = torque constant Tf = motor friction torque ia = armature current

Back EMF

eb = Ke wm V

wm= shaft speed (rad/s) eb = back emf Ke = back emf constant

KVL in Armature Circuit

ea = ia Ra eb V

ea= armature voltage eb = back emf Ra = armature resistance

Developed Power

P = wm T W

P = shaft power

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lesson14et438a.pptx

STEADY-STATE MOTOR EQUATIONS

Combining the previous equations gives:

wm

=

KT

ea

(T Tf KT Ke

)Ra

(1)

wm

=

ea

ia Ra Ke

(2)

If the load torque is zero (T=0) then the above equation (1) gives the no-load speed

wnl

=

KT

ea (Tf ) KT Ke

Ra

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STEADY-STATE MOTOR OPERATION

Example 14-1: An armature-controlled dc motor has the following ratings: Tf=0.012 N-m, Ra=1.2 ohms, KT=0.06 N-m/A, Ke=0.06 V-s/rad. It has a maximum speed of 500 rad/s with a maximum current of 2 A. Find: a) maximum output torque, b) maximum mechanical output power, c) maximum armature voltage, d) no-load speed at maximum armature voltage.

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lesson14et438a.pptx

lesson14et438a.pptx

EXAMPLE 14-1 SOLUTION (1)

Define given variables

a) Tmax occurs at Imax so....

b) Find Pmax

Answer

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Answer

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EXAMPLE 14-1 SOLUTION (2)

c) Find maximum back emf

Answer d) Find no-load motor speed

At no-load, T=0. Load torque is zero.

T=0

lesson14et438a.pptx

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lesson14et438a.pptx

TRANSFER FUNCTION OF ARMATURECONTROLLED DC MOTOR

Write all variables as time functions

Write electrical equations and mechanical equations. Use the electromechanical relationships to couple the two equations.

Ra + ia(t) ea(t)

La

+ T(t)

eb(t)

Jm Bm

Consider ea(t) and eb(t) as inputs and ia(t) as output. Write KVL around armature

ea (t)

=

Ra

ia

(t

)

L

dia (t dt

)

eb (t)

Mechanical Dynamics

T(t)

=J

m

dwm (t) dt

Bm

wm

(

t

)

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