Lecture Electromechanical energy conversion: Electromechanical – energy – conversion principles - Nguyễn Công Phương

This chapter presents the following content: Forces and torques in magnetic field systems, energy balance, energy in singly – excited magnetic field systems, determination of magnetic force and torque from energy and coenergy, multiply – excited magnetic field systems, forces and torques in systems with permanent magnets, dynamic equations, analytical techniques.

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Nguyễn Công Phương

ELECTROMECHANICAL ENERGY

CONVERSION

Electromechanical – Energy –

Conversion Principles

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I Magnetic Circuits and Magnetic Materials

II Electromechanical Energy Conversion

Principles

III Introduction to Rotating Machines

IV Synchronous Machines

V Polyphase Induction Machines

VI DC Machines

VII.Variable – Reluctance Machines and Stepping

Motors

VIII.Single and Two – Phase Motors

IX Speed and Torque Control

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4 Determination of Magnetic Force and Torque

from Energy and Coenergy

5 Multiply – Excited Magnetic Field Systems

6 Forces and Torques in Systems with Permanent

Magnets

7 Dynamic Equations

8 Analytical Techniques

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Forces and Torques in Magnetic

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Field Systems (2)

Ex.

A nonmagnetic rotor contains a single – turn coil, it

is in a uniform magnetic field The rotor is of radius

R and of length l Find the θ – directed torque as a

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Field Systems (3)

Lossless magnetic energy storage system

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Field Systems (4)

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Magnets

7 Dynamic Equations

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Energy Balance

Energy is neither created or destroyed, it is merely changed in form

dW  eidt  dW  dW

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Magnets

7 Dynamic Equations

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Energy in Singly – Excited Magnetic Field Systems (1)

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Energy in Singly – Excited Magnetic Field Systems (2)

2

1 ( ) 2

fld

W  L x i

2

2 0

i

h g

d l



x d

g

Magnetic flux

Ex.

A relay is made of infinitely – permeable magnetic

material, h >> g Compute the magnetic energy as

a function of plunger position?

0 2

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4 Determination of Magnetic Force and Torque

from Energy and Coenergy

Magnets

7 Dynamic Equations

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Determination of Magnetic Force and

Torque from Energy and Coenergy (1)

2 ( ) 2 ( )

fld

dL x f

fld

i dL x f

dx

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Determination of Magnetic Force and Torque from Energy and Coenergy (2)

Ex 1

Plot the force as a function of position for a current of 1A

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Ex 2

Plot the force as a function of position for a flux of 2mWb

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W T

2 ( ) 2 ( )

fld

dL T

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fld

i dL T

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Lossless –

N turns coil

+ –

i

h g

d l



Ex 3

A relay is made of infinitely – permeable magnetic

material, h >> g Compute the magnetic energy as

a function of plunger position, if i(x) = I0x/d (A)?

2

0 (1 / ) ( )

fld

i dL x f

dx



0( )

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Find the torque acting on the rotor as a function of

the dimensions and the magnetic field in the two

air gap, suppose that the reluctance of the steel is

negligible (μ → ∞) The axial length is h.

1Energy density :

2 BH

2

1 Energy density of the core : 0

steel steel steel

Energy density of the air-gap :

[2 ( 0.5 ) ] 2

ag ag

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0 2( )

Find the inductance as a function of θ, then extract

the expression for the torque acting on the rotor as

a function of i & θ The axial length is h.

g

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5 Multiply – Excited Magnetic Field Systems

Magnets

7 Dynamic Equations

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Multiply – Excited Magnetic

Field Systems (1)

Mechanicalterminal

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Field Systems (2)

Mechanicalterminal

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1 , are const

1 2 2

i

fld fld

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fld fld

i i

fld

W T

fld fld

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6 Forces and Torques in Systems with

Permanent Magnets

7 Dynamic Equations

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Forces and Torques in Systems

with Permanent Magnets (1)

f

I

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Forces and Torques in Systems with Permanent Magnets (2)

Fictitiouswinding

Find an expression for:

a) the coenergy of the system as a function of plunger

R m

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Ex 1

Find an expression for:

a) the coenergy of the system as a function of plunger

position x?

b) the force on the plunger as a function of x?

Fictitiouswinding

R m

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Externalmagneticcircuit

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Ex 2

a) Find the x – directed force on the plunger when the

current in the excitation winding is zero and x = 3 mm?

b) Find the current in the excitation winding required to

N L

2

equiv fld

m x

Ni W



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Ex 2

a) Find the x – directed force on the plunger when the

current in the excitation winding is zero?

b) Find the current in the excitation winding required to

2

equiv fld

m x

Ni W

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Magnets

7 Dynamic Equations

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K B

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K B

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K B

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Dynamic Equations (4)

Ex.

Extract the dynamic equations of motion of the

electromechanical system?

Length of flux path in the direction of field

(area of flux path perpendicular to field)

daN L

d g

Coil

Spring, K

Cylindrical steel plunger,

M

Applied force, f t

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( , )

fld fld

d g

Coil

Spring, K

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d g

Coil

Spring, K

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Magnets

7 Dynamic Equations

8 Analytical Techniques

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 

0If

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 

0If

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2 0

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