7. Modeling of Electromechanical Systems

Electromechanical systems consist of an electrical subsystem and a mechanical subsystem with mass and possibly elasticity and damping.

In some devices, such as motors and speakers, the mass is driven by a force generated by the electrical subsystem.

In other devices, such as microphones, the motion of the mass generates a voltage or current in the electrical subsystem.

DC MOTORSDC MOTORS

There are many types of electric motors, but the two main categories are direct current (dc) motors and alternating current (ac) motors.

Within the dc motor category there are the armature-controlled motor and the field-controlled motor.

We aim to control the speed or motion of dc motors.

There are many different types of servo-drivers. Most are designed to control the speed of dc motors, which improves the efficiency of operating servomotors.

Here we shall discuss only armature control of a dc motor and obtain its mathematical model.

The basic elements of a motor, as shown in the Figure are the stator, the rotor, the armature, and the commutator.

The stator is stationary and provides the magnetic field.

The rotor is an iron core that is supported by bearings and is free to rotate.

The coils are attached to the rotor, and the combined unit is called the armature.

Elements of DC Motor Elements of DC Motor

The direction of the force (F) due to a magnetic field (B) is perpendicular to the direction of motion.

Right Hand Rule for Magnetic Field Right Hand Rule for Magnetic Field

B

F

r

aq

The majority of electromechanical devices utilize a magnetic field.

The basic principle of Dc motor is based on a wire carrying a current within a magnetic field: a force is exerted on the conductor by the field.

rFTm Tm: Motor torque (moment)

We will use right hand rule to find the direction of the force of a magnetic field

Basic Principle of DC Motor Basic Principle of DC Motor

The product of the magnetic force (F) and the radius (r) will generate the motor moment.

Example 7.1 System with DC Motor

Motor +-

Vk

Jm , Bm

Ra , La

Ki , Kb

1 2By

K2z2

z1

JL

3 4

2

Ra : Motor’s resistance

Vk : Motor’s supply voltage

aq : Motor’s current

aim qKT

mbb KV

m1

23L22 )(K

2

1E in shaft 2:

K2: Rotational spring constant of shaft numbered 2

2

1

zz

N 3mN

JL : Load’s mass moment of inertiaBy : Rotational damping coefficient in bearings

L4

(Rigid shaft)m2

La : Motor’s inductance

Jm : Motor’s mass moment of inertia

Bm : Motor’s rotational damping coefficient

Ki : Motor’s torque constantKb : Motor’s back emf constant

The torque Tm developed by the motor is proportional to the product of Motor’s torque constant and the current . When the sign of the current is reversed, the sign of the torque will be reversed.

The torque Tm developed by the motor is proportional to the product of Motor’s torque constant and the current . When the sign of the current is reversed, the sign of the torque will be reversed.

When the armature is rotating, the voltage (back emf) Vb is directly proportional to the angular velocity of the motor.

When the armature is rotating, the voltage (back emf) Vb is directly proportional to the angular velocity of the motor.

DC Motor +-Vk

Jm , Bm

Ra , La

Ki , Kb

1 2By

K2z2

z1

JL

3 4

2

aim qKT mbb KV

21m (Rigid shaft)

23L22 )(K

21

E In shaft 2 :

2

1

z

zN

3mN

Energy equations for Lagrange equation:

2aa1 qL

21

E 2mmJ

21

2LLJ

21

2mL22 )N(K

21

E

ak qVW aaa qqR maiqK amb qK mmmB

mmyB mmyB )N()N(B mmy

LLyB

mmy2

myaimmambaak )BNB2qKB(q)KqRV(W

LLyB

Input : Vk Lagrange Equation→

Homework 07-Problem 1; Generalized variables : qa, θm, θL

b/2

k/2

k/2

b/2

fa(t)

x(t)

xd

dC)x(C

0

00

Example 7.2 Movable plate capacitor

Inputs: Vk(t) ve fa(t)

Generalized variables: q(t) ve x(t)

R

C

Fixed

Movable, m

Vk

+

-

q

The force fa is applied to movable plate. The displacement of movable plate is x(t). The value of the capacitor depends on the changing of the distance between the plates changes.

The plate of the capacitor at left hand side is fixed. The other plate is movable. The moving plate is fixed to the body with the elements of the spring k and damper c.

Vk is the power supply. Vk is connected to the lines with the resistor R and the capacitor C in serial.

,C0 0d are the constants.

b/2

k/2

k/2

b/2

fa(t)

x(t) Inputs: Vk(t) ve fa(t)

Generalized variables: q(t) ve x(t)

21 xm

2

1E

2

00

022 q

dC)xd(

21

x2k

21

2E

xx2

b2xfqqRqVW ak

R

C

Sabit

Movable, m

Vk

+

-

q xd

dC)x(C

0

00

For the electromechanical system, We can write the energy and virtual work equation as follows.

b/2

k/2

k/2

b/2

fa(t)

x(t) Inputs: Vk(t) ve fa(t)

Generalized variables: q(t) ve x(t)

21 xm

2

1E 2

00

022 q

dC

)xd(

2

1x

2

k

2

12E

xx2

b2xfqqRqVW ak

The equations of motion of the system are obtained by applying the Lagrange equation to the general variables.

xbfqdC2

1kxxm a

2

00

qRVqdC

)xd(k

00

0

Set of non-linear differential equations

Runge-Kutta methodLinearization

Homework 07- Problem 2: Movable core inductance

R

C

Sabit

Movable, m

Vk

+

-

q