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Electromechanical Dynamics
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Woodson, Herbert H., and James R. Melcher. Electromechanical Dynamics.
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Appendix
E
SUMMARY OF PARTS I AND II
AND USEFUL THEOREMS
IDENTITIES
Ax B.C= AB x C,
A x (B x C) = B(A. C) - C(A
V(o + y) = V
.
B)
+ Vy,
V. (A + B) = V A + V B,
V x (A + B)= V x A + V x B,
V(ov) =
V P + v vo,
V-(VA)= A -VyB +
V
.
A,
V.(A x B)= B.V x A - A V x B,
v. v-
= Vý,
V-V x A= 0,
V x V
=0,
V x (V x A) = V(V
.
A) - V 2 A,
(V x A) x A = (A . V)A -
lV(A . A),
V(A . B)= (A . V)B + (B - V)A + A x (V x B) + B x (V x A)
V x (OA)= Vo x A + OV x A,
V x (A x B) = A(V . B) - B(V
.
A) + (B . V)A El
(A . V)B.
THEOREMS
b
Výk - Al
Divergence theorem
Stokes's theorem
=
r
sA .nda=J V.AdV
A
A dl =(V
ý,C
x A) . n da
S(Vx
~daI
bi
Table 1.2
Summary of Quasi-Static Electromagnetic Equations
Differential Equations
Magnetic field system
V x H = J,
V. J=
0
VXE=
-
(1.1.1)
aB
Integral Equations
H - dl =
B.nda = 0
B
(1.1.21)
SJ, - n da = 0
(1.1.22)
f
E'.dl =-
Tt
(1.1.20)
J" - n da
B . n da
(1.1.23)
where E' = E + v X B
Electric field system
VX E =0
(1.1.11)
(1.1.12)
V.D= Pf
V.J=
(1.1.14)
at
VXH=J 1 +
aD
(1.1.15)
(1.1.24)
EgE.dl= 0
D
D'nda=fvpp dV
(1.1.25)
Jf'-nda
(1.1.26)
H'.dlT =
d vpdV
J .n
da +
where J' = J= - pfv
H'=H-
v XD
D - n da
(1.1.27)
Table 2.1
Summary of Terminal Variables and Terminal Relations
Magnetic field system
Electric field system
V
C
Definition of Terminal Variables
Charge
Flux
qk= f
B.nda
A =
pdV
Voltage
Current
f -n'da
ik ý
Vk
=
fE
di
Terminal Conditions
dqk
dAk
Sdt
iA, = 4.(i ...
i, = ik(. 1
dt
iN; geometry)
AN; geometry)
qk = qk(v 1 • • • vv; geometry)
vk = vk(ql1 " "qN; geometry)
Table 3.1
Energy Relations for an Electromechanical Coupling Network with N Electrical
and M Mechanical Terminal Pairs*
Electric Field Systems
Magnetic Field Systems
Conservation of Energy
N
J
dWm =
N
31
f
ijdAj -
j=1
j=1
N
M
e
(a)
dxj
dWe
31
v dq, -
-=
j=1
dWi = I
(b)
j=1
M
NV
(c)
dW2 = I A di, + I fe dx3=1
i=1
f e d-j
'
qj dv
+ I Ly dx3
j=1
(d)
t=1
Forces of Electric Origin,j = 1, ... M
e=
-a
ANl
Ax
.
f
M
axj
at(i .x.
...
)
1 t)
(e-)
hf =
(g)
he
aWe(ql.... q; x 1 .
je
x(e
_
.....VN
((a
=.
.
(f)
x)
x,1 )
l ...
; X1,
(h)
Relation of Energy to Coenergy
N
N
Wm + w" =
(i)
ij
(j)
W e + We'= =vq
3=1
3=1
Energy and Coenergy from Electrical Terminal Relations
Wm=.
j=
N
(
,
2'0
d
0;4)
(k)
(
We =
1
W
N
i
i,...
Wm
1, 0i,...,0; xI,.... x)
ij-1,
di
(mi)
W
=
v(ql.
f-
.
q 0
,q,0,...
,
; x1 .
x) dq
0;xj,...m)dq
()
(1)
0; x...x) dv
(n)
P't
q(vl,...
I,
vj1',
0 ...
0
j=1 o=1
* The mechanical variables f and x, can be regarded as thejth force and displacement or thejth torque T, and angular displacement
Q0.
Table 6.1
Differential Equations, Transformations, and Boundary Conditions for Quasi-static Electromagnetic Systems with
Moving Media
Differential Equations
V x H = J
Magnetic
(1.1.1)
(6.1.35)
n X(Ha -Hb)
=
K
(6.2.14)
a
(1.1.2)
B' = B
(6.1.37)
n.(B --Bb) = 0
V.
(1.1.3)
J =
(6.1.36)
n. (J, -_Jfb) + V,
(1.1.5)
E' = E + vr x B
(6.1.38)
n X (Ea - Eb) =
B = p0 (H + M)
(1.1.4)
M' = M
(6.1.39)
V x E= 0
(1.1.11)
E' = E
(6.1.54)
n X (E a - Eb) = 0
(6.2.31)
V.D = p,
(1.1.12)
D' = D
(6.1.55)
n.(D a - Db)=rr
(6.2.33)
(1.1.14)
P = Pf
r
J = Jf - pfv
(6.1.56)
.J =
(6.1.58)
aD
V X H = Jf+ (1.1.15)
H'= H - v x D
(6.1.57) n X (Ha - Hb) = K, + van X [n x (Da - Db)]
D = eoE + P
P' = P
(6.1.59)
= 0
aB
Vx E =
field
systems
H' = H
Boundary Conditions
V. B =O0
systems
Electric
Transformations
-
-a
(1.1.13)
(6.2.7)
v,
K7 = 0
(6.2.9)
(Ba - Bb)
(6.2.22)
.
n* (Jf a -_ Jb) + V.,- Kf = V((pfa
fb) _ "
at
(6.2.36)
(6.2.38)