4.Electrical Variational System.pdf

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Electrical Variational System

❖ Electrical variational system

  ,

Premultiply by allowable variation of

0 or fixed conditions const. along conductors E



 

=

= − 



  D = (   )  =  

 V D dV  V dV

( )

1 1

1 0

1 1

d ,

( ) d 0

E E

i i V

E E

V

E E

U W

U U

D E V q

D E E V W

   



= =

=

− =

 

 

( )

S

V D D V D dV V dV

D

dV d S

  





• − •  =

  



D

E

2

U

E

1

U

E

1-42

V D •  EdV = V   dV

 

𝟎

(discrete)

i i

q 



= 

 

divergence theorem

divFdv F ndA

(2)

Electrical Variational System

❖ Non-complementary principle

- Allow variation of free charge distribution - Constant with E equal

D •  D = 

- Pre-multiply by 𝜑

( )

2 2

2 0

0

linear relationship

V V

i i

V i

E E

E D V

D EdV dV

D E

dV

q

U W

U E D d DdV D E

  







• =

•

=

− =

= •

=

 

 

 

( )

V V

V V V

D dV dV

dV dV dV

D

D D

   

     



 • =

 • − •  =

 

  

1

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❖ Non-complimentary principle (continued)

PMTPE PMTCPE

Complementary

Non-Complementary

( )

1 1

2 2

1 1

2 2

0 0 0 0

M M

E E

U W U W U W U W



− =

1 2

E E

U = D E • − U

Mechanical , , , , u S T f E D

 

 

 

1-44

(4)

❖ Combined Electrical-Mechanical

From thermodynamics

1 2

1 2 1 2

1 1

2

TOT M E

TOT M E M E

M V E

V

U U U

U U U W W W

U T SdV

U E DdV

     

 

= +

= + = + =

=



^T

( )

^{S D}^,

1

^st

law of thermodynamics,

TOT TOT

dU dQ dW

dU  ds T d S E d D

= +

= + +

, , ,

, ,

S D s D S s

U U U

dU ds d S d D

s S D

U U U

T E

s S D



  

= + +

  

     

=      =      =      (

,

)

E S D

entropy

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Electrical Variational System

❖ Combined Electrical-Mechanical (continued)

: Helmholtz Free Energy

Displacement, Electrical Potential

, ,

,

, ,

S D D

S

A U s dA sd T d S E d D

A A A

s T E

s D



 



= − = − + +

  

     

=      =      =     

temperature

1 1

1 2 2 2 2

2 2

,

0 ,

M E M E

G U s T S dG sd S dT E d D

G W U U W W

G U s D E dG sd T d S D d E

 

     

 

= − − = − − +

= → − + + − =

= − − = − + −

: Gibbs Free Energy

1 M 1 E 1 M 1 E 0

U U W W

 −  −  +  =

1-46

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❖ Consider

0

q CV A V x

AE E q

A



=

+q +q

V

x dx

2 0

2

0 0 2 2

0 2

0

1 2 1 2

1 2

V x

q const

U E dV

q Adx A

U q

x A

F U Eq

x



 



=

 =



  

= −      = −



(7)

❖ Consider constant voltage

2 0

2 0 0

1 2 1 2 1 2

Vol

x

U E dVol

V dVol x

V dx A x



=

 

=    

 

=     



2

2 0

2

0 2

2

0 2

1 1

2 2

1 2 ( ) 1

2

V const

U A V CV

x

U V

x A x

U V

F A

x x



=

= =

 = −



= −  =



What charge is required for V to be constant?

0

0 0

0

V Ex q x A

q x

dV dx dq

A A

dq q dx x



 

= =

= = +

= −

1-48

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Electrical Variational System

❖ Total Energy

The force

0 2 2

2 2

0 0

2 2

( 1 )

2

1 1

2 , 2



 

= −

= 



= −

− = =

= = − =

dU dU Vdq dU U dx

x

dU A V dx

x

q A

Vdq Udx V dx

x x

A dU A

dU V F V

x dx x

: electrical enthalpy



= −

= − 



U U q

U E D

 

  

= − = = −

  

U U U

q

F x x x

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❖ Total Energy (continued) More formally

: total energy

,



   

 

= −

= − −

 

= +

 

   

=      =     

D S

U U q

U U q q

q F x

U U

dU dS dq

S q

U U

T E

S D

*

( )



 

 

=      + 

= +

U U

q

dU dq dx

x x

dq Fdx

: energy per volume U EdD TdS

 = −

1-50

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Gibbs free energy Elastic free energy Electric free energy Helmholtz free energy

enthalpy Elastic enthalpy Electric enthalpy

1 2

: : : : : : :

ij ij m m

ij ij

m m

G U T S E D G U T S

G U E D A U

H U T S E D H U T S

H U E D



= − − −

= − −

= −

= −  −

= − 

= −

temperature

❖ Total Energy (continued)

1 2

1

ij ij m m

dG d S dT D dE dG d S dT E dD dG d T dS D dE dA d T dS E dD dH d S dT D dE dH d S dT E dD dH d T dS D dE

 

 

= − − −

= − − +

= − + −

= − + +

= − −

= − +

= + −

enthalpy

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,

E,

,

1 2

( , , )

( , ) ( , )

ijkl

mn

ij m ij m

ij m

T E

ij

ij T

m

m E

ij kl

T

m m

G G

dG T E d dT dE

T E

G

S G

T D G

E

S s T f T E

D E f T E



  



 



  

= + +

  

  

= −     

  

= −       

  

= −     

= +

1-52

Ref. 1.”Dynamics and Mechanics of Electrical Systems” by Crandall - Chap. 6

2. “Principle and Applications of Ferroelectrics and Related Material” by M.E.

Lines & A.M. Glass - Chap. 3

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2 ,

Expand



 

   

=      +     

 

   

=      = −       

ij

E T

ij ij

ij kl m

kl m

E E E ij

ijkl

kl ij kl

S

S S

dS dT dE

T E

S G

s T T T

1

1 1 1

2 T ,

2

Piezo free

Coefficient of T l :

h m

: er a

T T ij

ijk

k ij k

T E T E ij T E

ij

S G

d

n Strain

Expa o

E T E

S G nsi

T



  

 

   

=      = −       

 

   

=      = −       

Linearize →

→

m m

dD D

,

ij ij m m

ij m

E T

ij m

dG S dT D dE

G G

dG dT dE

T E

G G

S D

T E

= − −

 

= +

 

     

= −        = −     