Transcript Appendix A: Tensors - Lamar University Electrical Engineering

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Appendix A: Tensors
Instructor:
Dr. Gleb V. Tcheslavski
Contact: [email protected]
Office Hours: TR
Class web site:
http://www.ee.lamar.edu/gleb/
em/Index.htm
“tensor” by Kevin McCormick
ELEN 3371 Electromagnetics
Fall 2008
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The Euler’s formulas
e  cos( )  j sin( )
j
e
 j
 cos( )  j sin( )
j
 j
e

e
cos( ) Re{e j } 
2
j
 j
e

e
j
sin( ) Im{e } 
2j
ELEN 3371 Electromagnetics
Fall 2008
(A.2.1)
(A.2.2)
(A.2.3)
(A.2.4)
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Phasors (re-visited)
Example: Express the loop eqn for a circuit in phasors if v(t) = V0 cos(t)
di
1
v(t )  L  Ri   idt '
dt
C
- The loop eqn.
(A.3.1)
(A.3.2)
Since cos is a reference, we can express the current
similarly to (A.3.1):
i(t )  I 0 cos(t   )
(A.3.3)
Combining (A.3.1), (A.3.2), and (A.3.3), we arrive at:
d ( I 0 cos(t   ))
1
V0 cos(t )  L
 RI 0 cos(t   )   I 0 cos(t   )dt
dt
C
1


 I 0   L sin(t   )  R cos(t   ) 
sin(t   ) 
C


ELEN 3371 Electromagnetics
Fall 2008
(A.3.4)
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Phasors (re-visited 2)
Perhaps, we can solve the last equation for I0 and … Let’s
use phasors instead!
Recall:
then
v(t )  V0 cos(t )  Re V0e jt   Re V (  )e jt 
i (t )  Re  I 0e j e jt   Re  I (  )e jt 
(A.4.1)
(A.4.2)
Here V (  ) V0 and I (  )  I 0e are phasors that correspond to the voltage
and current respectively. The beauty of this notation is that phasors are free of time
dependence! To express the loop equation in phasors, we replace the derivative and
the integral in the loop eqn (A.3.4) by j and 1/j respectively.
j

V ( )   R 

ELEN 3371 Electromagnetics
1 

j L 
 I (  )
C 

Fall 2008
(A.4.3)
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Tensors
So far, we have learned that quantities either have a
direction (vectors) or they don’t (scalars)… the truth,
however, is that a quantity may have MORE THAN ONE
direction!
A dyadic (also referred to as a dyadic tensor), like a vector,
is a quantity that has magnitude and direction but unlike the
vector, the dyadic has a dual directionality.
ELEN 3371 Electromagnetics
Fall 2008
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Tensors (cont)
A dyadic in multilinear algebra is formed by juxtaposing pairs of vectors, i.e.
placing pairs of vectors side by side. Each component of a dyadic is a dyad.
A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient.
As an example:
Aaux  buy and X xux  yuy aretwovectors
A juxtaposition of A and X is
AX axuxux  ayuxuy  bxuyux  byuyuy
The dyadic tensor
uyux  uxuy is a 90
o
rotation operator in 2D.
In 3D, the juxtaposition would have 9 components.
ELEN 3371 Electromagnetics
Fall 2008
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Tensors (cont 2)
As we discussed, while a vector has a single direction, a
dyad is a dual-directional quantity. A tensor – a multidirectional quantity – is a further generalization. A
“dimensionality” of a tensor is called a rank.
In fact, a scalar can be viewed as a tensor of rank 0; vector
is a tensor of rank 1, a dyadic is a tensor of rank 2, etc.
Finally, similarly to scalar fields and vector fields, there are
also tensor fields!
ELEN 3371 Electromagnetics
Fall 2008
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Why bother?
• General relativity is formulated completely in the language of tensors.
• For anisotropic dielectrics (those having different physical properties
in different directions), such as in crystalline materials, dielectric
properties are expressed in terms of tensors. Piezoelectric and
magnetostrictive materials used for acoustic transducers are examples
of anisotropic materials.
• Wave propagation in the ionosphere and other plasma media
constitute further examples of the use of tensors.
• Perhaps the most important engineering examples are the stress
tensor and strain tensor, which are both 2nd rank tensors, and are
related in a general linear material by a fourth rank elasticity tensor.
ELEN 3371 Electromagnetics
Fall 2008