Download Problem 6.16 The parallel-plate capacitor shown in Fig. P6.16 is

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Transcript
Problem 6.16 The parallel-plate capacitor shown in Fig. P6.16 is filled with a lossy
dielectric material of relative permittivity εr and conductivity σ . The separation
between the plates is d and each plate is of area A. The capacitor is connected to
a time-varying voltage source V (t).
I
A
V(t)
ε, σ
d
Figure P6.16: Parallel-plate capacitor containing a lossy dielectric material (Problem 6.16).
(a) Obtain an expression for Ic , the conduction current flowing between the plates
inside the capacitor, in terms of the given quantities.
(b) Obtain an expression for Id , the displacement current flowing inside the
capacitor.
(c) Based on your expressions for parts (a) and (b), give an equivalent-circuit
representation for the capacitor.
(d) Evaluate the values of the circuit elements for A = 4 cm2 , d = 0.5 cm, εr = 4,
σ = 2.5 (S/m), and V (t) = 10 cos(3π × 103t) (V).
Solution:
(a)
R=
(b)
d
,
σA
Ic =
V
VσA
=
.
R
d
∂D
∂ E εA ∂V
V
,
Id =
· A = εA
=
.
d
∂t
∂t
d ∂t
(c) The conduction current is directly proportional to V , as characteristic of a
resistor, whereas the displacement current varies as ∂ V/∂ t, which is characteristic
E=
of a capacitor. Hence,
R=
d
σA
C=
and
εA
.
d
I
+
ε, σ
V(t)
R
Ic
Id
+
C
V(t)
-
-
Actual Circuit
Equivalent Circuit
Figure P6.16: (a) Equivalent circuit.
(d)
0.5 × 10−2
= 5 Ω,
2.5 × 4 × 10−4
4 × 8.85 × 10−12 × 4 × 10−4
= 2.84 × 10−12 F.
C=
0.5 × 10−2
R=