Prof. David R. JacksonECE Dept.

Fall 2014

Notes 25

ECE 2317 Applied Electricity and Magnetism

1

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

MEMS capacitor

Typical Capacitors (cont.)

Variable capacitor Capacitors for substations

8

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

Example (cont.)

Hence, the characteristic impedance of thecoaxial cable transmission line is:

00 ln [ ]

2 r

bZ

a

where0

00

376.7603 [ ]

120 F/m8.854187 10 [ ]

70 H/m4 10 [ ]

25

0l

l

LZ

C

0 F/m2

[ ]ln

rlC

ba

0 H/mln [ ]2l

bL

a

(intrinsic impedance of free space)