Download κ1 κ2 d2 d1 width = w V L1 L2 x - Electrical and Computer Engineering

University of Rochester
Department of Electrical & Computer Engineering
EE234/434
Homework #4
due 13 Feb, 2008
-----------------------------------------------------------------1)
Consider the geometry below of a pair of electrodes
containing a liquid dielectric of dielectric constant k2 in a
surrounding ambient medium k1 (also a liquid). The piston at
the right prevents the liquid from moving. Note that the
position of the left surface of the liquid is at defined to
be x.
L1
d1
x
V
k1
L2
k2
d2
Fext
width = w
a)
Find an expression for the system capacitance as a
function of the position of the left surface of the
liquid x.
b)
What is the net force Fext that must be exerted upon the
piston on the right side to keep the liquid from moving?
2)
Work problem #3.3 from Woodson and Melcher for the non-linear
dielectric. NOTE: because the dielectric in this device is
non-linear, the usual capacitance cannot be defined.
Instead, you must use the integral form of Gauss’s Law to
find a terminal relation between the net charge and the
voltage and mechanical displacement. This relation will be
non-linear but you can integrate to obtain the coenergy.
3)
Work problem #3.16 from Woodson & Melcher. NOTE: all
electrical and mechanical constraints external to the
lossless electromechanical coupling can be imposed only after
an expression for the force of electrical origin fe has been
obtained in terms of the chosen state variables.
2 of 2 pages
4)
The coaxial, rotating capacitive transducer shown below in
cross-section, has length L.
concentric dielectric arc
d = thickness
k = dielectric const.
k
q
R
cross-sectional
view
gap
g << L,R
(a)
Find an expression for the device capacitance C(q), assuming
that 0 < q < p. Ignore all fringing fields.
(b)
Use coenergy to determine the torque of electrical origin Te.