Capacitance
Capacitor
FQ
CV
F C/V1 1
C [F]
Note: The “A” conductor has
the positive charge (connected to anode).
Q = QA
V = VAB
Notation:
(both positive)
Note: The value of C is always positive!
+Q
-Q
+
-V r
A
B
++++++++++++++++++
- - - - - - - - - - - - - - - - -
2
Leyden JarThe Leyden Jar was one of the earliest capacitors. It was invented in 1745 by Pieter van Musschenbroek at the University of Leiden in the Netherlands (1746).
3
Typical Capacitors
Ceramic capacitors
The ceramic capacitor is often manufactured in the shape of a disk. After leads are attached to each side of the capacitor, the capacitor is completely covered with an insulating moisture-proof coating. Ceramic capacitors usually range in value from 1 picofarad to 0.01 microfarad and may be used with voltages as high as 30,000 volts.
4
Typical Capacitors (cont.)
Paper capacitors
A paper capacitor is made of flat thin strips of metal foil conductors that are separated by waxed paper (the dielectric material). Paper capacitors usually range in value from about 300 picofarads to about 4 microfarads. The working voltage of a paper capacitor rarely exceeds 600 volts.
5
This type of capacitor uses an electrolyte (sometimes wet but often dry) and requires a biasing voltage (there is an anode and a cathode). A thin oxide layer forms on the anode due to an electrochemical process, creating the dielectric. Dry electrolytic capacitors vary in size from about 4 microfarads to several thousand microfarads and have a working voltage of approximately 500 volts.
Note: One lead (anode) is often longer than the other to indicate polarization.
Electrolytic capacitors
Typical Capacitors (cont.)
+
-
6
Supercapacitors (Ultracapacitors)
Typical Capacitors (cont.)
Maxwell Technologies "MC" and "BC" series supercapacitors (up to 3000 farad capacitance)
Compared to conventional electrolytic capacitors, the energy density is typically on the order of thousands of times greater.
7
Current - Voltage Equation
dQi
dtd
Cvdt
dvC
dt
Note: Q is the charge that flows from
left to right, into plate A.
9
-
-
-
+Q -Q +
+
+
+i i
+ -
v (t)
A B
-
-
-
-
The reference directions for voltage and current correspond to “passive sign convention” in circuit theory.
Hence we have dvi t C
dt
“Passive sign convention”
Current - Voltage Equation (cont.)
10
i i
+ -
v (t)
Example
Method #1 (start with E)
Ideal parallel plate capacitor
Assume:0ˆ xE x E
0
0 0
0
B h h
AB x x
A
x
V V E dr E dx E dx
E h
Find C
A
B
xh [m]
A [m2]
r Ex0
11
QC
V
Example (cont.)
Hence
0
0
0
0 0
ˆ
ˆ
ˆ
As A
r A
r
r x
r x
Q A A D n
A n E
A x E
A E
A E
0 0
0
r x
x
AEQC
V E h
0 Fr
AC
h
A
B
xh [m]
A [m2]
r Ex0
Note: C is only a function of the geometry and the permittivity!
12
Example (cont.)
Method #2 (start with s)
A
B
x
+s0
-s0
0sQ A
0
0
0
0 0
ˆ
ˆ ˆ
AA s s
x s
x s
r x s
D n
xD x
D
E
0
As s
13
QC
V
00
0
sx x
r
E E
Hence
Example (cont.)
Hence
0
0 0
0
B h h
AB x x
A
x
V V E dr E dx E dx
h E
Therefore,
00
0
sx
r
V h E h
0
0
0
s
s
r
AC
h
0 Fr
AC
h
14
Example
h [m]
A [m2]
rr = 6.0 (mica)A = 1 [cm2]h = 0.01 [mm]
0
412
3F
1 108.854 10 6.0
0.01 10
r
AC
h
pF nF530.12 0.53012C
15
Example
2
2 0
ˆ
ˆs
As
Ax x s
As
D n
D x
D D
Q A
1 2 0x x xD D D
r1 h1
xh2r2 2
1
Ex2
Ex1
A
B
Dx0
B. C. on plate A:
Find C
Goal: Put Q and V in terms of Dx0.
Starting assumption:
16
QC
V
0xQ D A
Example (cont.)
1 1 2 2
1 21 2
1 2
0 01 2
1 2
1 20
1 2
x x
x x
x x
x
V E h E h
D Dh h
D Dh h
h hD
17
1 20
1 2x
h hV D
Example (cont.)
Hence:
0
1 20
1 2
1 2 1 2
1 2 1 2
1 2
1 2
1
11 1
x
x
D AQC
V h hD
Ah h h h
A A
A A
h h
1 2
11 1
C
C C
11
1
AC
h
2
22
AC
h
1 2
1 1 1
C C C
or
where
Hence,
18
Example
1 2 0x x xE E E
0xV E h
1 1 1 1 1 0
2 2 2 2 2 0
As x x x
As x x x
D E E
D E E
A1 A2
r1
xr2
Dx1 Dx2
x = h
A
B
Ex0
+ + + + + + + + + + + + + + + +
Find C
Goal: Put Q and V in terms of Ex0.
Starting assumption:
19
QC
V
Example (cont.)
1 1 2 2
1 0 1 2 0 2
s s
x x
Q A A
E A E A
1 1 0 2 2 0
0
x x
x
A E A EQC
V E h
1 1 2 2A AC
h h
1 2C C C
20
1 11
AC
h
2 22
AC
h
where
Example
-l0
l0
a
b
r0 0
0
ˆ ˆ2 2
l l
r
B
A
b
a
E
V E dr
E d
Find Cl
(capacitance / length)
Coaxial cabler
21
0llC
V
0 0: / 2as l a Note
Example (cont.)
0
0
0
0
1
2
ln2
b
r a
r
V d
b
a
Hence we have
0 0
0
0
ln2
l
r
CV b
a
Therefore
We then have
0 F/m2
[ ]ln
rlC
ba
22
Example (cont.)
0l
l
LZ
C
Formula from ECE 3317:
(characteristic impedance)
23
The characteristic impedance is one of the most important numbers that characterizes a transmission line.
Coax for cable TV: Z0 = 75 []Twin lead for TV: Z0 = 300 []
Coaxial cable Twin lead
Example (cont.)
Formula from ECE 3317:
1 1p
l l
vL C
phase velocity
Hence
0l
l
LC
0 H/mln [ ]2l
bL
a
0 Assume
Using our expression for Cl for the coax gives:
24
0 r
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