ENERGY-STORING COUPLING BETWEEN DOMAINS
M
ULTI
-P
ORT
E
NERGY
S
TORAGE
E
LEMENTS
Context: examine limitations of some basic model elements.
E
XAMPLE
:
open fluid container with deformable walls
P =
ρ
g h
h = A V
V = C
f
P
where C
f
=
A
ρ
g
—fluid capacitor
But when squeezed, h (and hence P) may vary with time even though V does not.
Seems to imply C
f
= C
f
(t)
i.e., C
f
=
A(t)
ρ
g
—apparently a “modulated capacitor”

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P
ROBLEM
!
E
p
=
V
2
2 C
f
V is constant, therefore no (pressure) work done
dV = 0
∴
PdV = 0
—yet (stored) energy may change
This would violate the first law (energy conservation)
P
V
where did
this energy
come from?
initial stored energy
—a BIG problem!
M
ODULATED ENERGY STORAGE IS PROSCRIBED
!

S
OLUTION
Identify another power port to keep track of the work done to change the stored energy
C
P
Q = dV/dt
v = dx/dt
F
introduces a new network element: a
multiport capacitor
Mathematical foundations:
Power variables:
Each power port must have properly defined conjugate power and energy variables.
Net input power flow is the sum of the products of effort and flow over all ports.
P =
∑
i
e
i
f
i
In vector notation:
e
∆
_
_
⎣
⎢
⎢
⎡
⎦
⎥
⎥
⎤
e
1
e
2
·
·
·
e
n
f
∆
_
_
⎣
⎢
⎢
⎡
⎦
⎥
⎥
⎤
f
1
f
2
·
·
·
f
n
P =
e
t
f
=
f
t
e

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