engineering electromagnetics-...

Department of Semiconductor Systems Engineering SoYoung Kim

Engineering Electromagnetics- 1 Lecture 15: Capacitance

SoYoung Kim

[email protected]

Department of Semiconductor Systems Engineering

College of Information and Communication Engineering

Sungkyunkwan University

mailto:[email protected]


Outline

Midterm results

Capacitance - Applications

Calculating Capacitance

Capacitance and Resistance

Method of Images


Capacitance Applications


Why Capacitance?

Delay

RC delay determines the timing and clock speed of a layout

Noise

Coupling cap causes noise to adjacent signal lines


Cap Extraction in VLSI Design Flow

Cap Extraction step generates RC network netlist file

STA tool calculates from the given RC network:

Interconnect delay

Path delay

Signal integrity

Cap extraction is a critical step for timing closure.


Basic Theory

The famous equation:

Q = CV

The potential of a conductor(V) is proportional to the total charge(Q), i.e., the capacitance(C) is constant.

C depends on the geometry of conductors and on the permittivity of the medium between them.


Analytic Cap Calculation


Field Solving Methods


Field-Solving Method

Volume-Based Method Finite element method (FEM)…

Mesh is created in space surrounding conductors

Problem is represented by a large sparse matrix

Raphael 2D, Raphael 3D

Surface-Based Method Boundary element method (BEM)…

Mesh points are placed on surface of conductors, and boundary between different dielectric materials

Representing matrices are smaller than FEM’s, but dense.

Raphael BEM 2D, Raphael BEM 3D

Random-Walk Method Randomly pick points, starting from a conductor and ending

on other conductors (random walk)

No mesh needed – much more efficient than FEM or BEM

QuickCap, Rapid3d(ranxt)


Capacitance Extraction Methodology

Field-Solver (FEM, BEM based)

Field-Solver (Randow walk based)


Capacitance Extraction Methodology

Asim Husain ISQED 2001

Configuration showing difference components of capacitances characterized for model library generation

Crossover structure showing 3D capacitance


Capacitance

Capacitance

For an isolated conductor:

The ratio of charge carried by the conductor and potential of the conductor

For two conductors with charges of same magnitude but opposite sign ( called ‘capacitor’):

The ratio of charge on one conductor and potential between the two conductors

1

1 22

V V V d E l


Procedures of Obtaining Capacitance

Methods of obtaining capacitance

Assuming Q and determining V in terms of Q (use Gauss’s law)

Assuming V and determining Q in terms of V (use Laplace’s equation)

Procedures for obtaining capacitance

Method #1

1. Choose a suitable coordinate system

2. Let the two conductors carry charges +Q and -Q

3. Determine E by Coulomb or Gauss’s law and find V.

4. Obtain C as Q/V

Method #2

1. Choose a suitable coordinate system

2. Let the two conductors have potential difference V0

3. Determine V by Laplace’s law, and find Q

4. Obtain C as Q/V


Parallel-Plate Capacitor - Method #1

Potential between two plates

Capacitance

Energy stored in the capacitor

In general,

)

1

2 0

(

S

Sx x

d

x x

Q

S

Q

S

Q QdV d dx

S S

E a a

E l a a

Q SC

V d

2 2 2 2

2 2 2 2

1 1

2 2 2 2 2E

v

Q Q Sd Q d QW dv QV

S S S C

221 1

2 2 2E

QW CV QV

C

1

2E

vW dv D E


– Laplace’s equation

– Boundary conditions

– Capacitance

Parallel-Plate Capacitor - Method #2

22

20

d VV

dx

V Ax B

0, 0 0 0

, 0o o

x V B

x d V V V Ad


Coaxial Capacitor


Spherical Capacitor

For isolated sphere, b

4C a


– Laplace’s equation

– Boundary conditions

– Capacitance

Spherical Capacitor - Method #2

2 2

2

10

d dVV r

r dr dr

AV B

r

, 0 0 or

1 1, o o

A Ar b V B B

b b

r a V V V Ab a

22

22

0 02

1 1

1 1 1 1

4 4sin

1 11 1 1 1

oo r r r

o r o o r o

o

VdV Ar bV V V

dr rr

a b a b

V V QQ d r d d C

Vr

a ba b a b

E a a a

E S


Connection of Capacitors

Series connection • Parallel connection

1 2

1 1 1

C C C 1 2C C C


Relation between R and C (I)

For two conductors separated by homogeneous medium,

Resistance between the two conductors

Capacitance between the two conductors

: Identical to the relaxation time Tr

RC


Relation between R and C (II)

Examples

Parallel capacitor

Cylindrical capacitor

Spherical capacitor

Isolated spherical capacitor

, S d

C Rd S

1n2 L

, 2

1n

b

aC Rb L

a

1 1

4,

1 1 4

a bC R

a b

14 ,

4C a R

a


Method of Image

“A given charge configuration above an infinite grounded

perfect conducting plane may be replaced by the charge configuration itself, its image, and an equipotential surface in place of the conducting plane.”


Method of Image

Two conditions need to be satisfied when method of image is used:

The image charge must be located in the conducting region

The potential on the conducting surface must be zero or constant


Example 1

A point charge above a grounded conducting plane

Electric field

1 2

3 3

1 24 4o o

Q Q

r r

r rE E E

1

2

( , , ) (0,0, ) ( , , )

( , , ) (0,0, ) ( , , )

x y z h x y z h

x y z h x y z h

r

r

0z


Example 1 (Cont’d)

Potential

Surface induced charge

1 2

1/2 1/22 2 2 2 2 2

4 4

1 1

4 ( ) ( )

o o

o

Q QV V V

r r

Q

x y z h x y z h


Example 2

A line charge above a grounded conducting plane

Electric field

1 2

1 2

1

2

2 2 2 2

2 2

( , , ) (0, , ) ( ,0, )

( , , ) (0, , ) ( ,0, )

( ) ( )

2 ( ) ( )

E E E a a

ρ

ρ

a a a aE

L L

o o

x z x zL

o

x y z y h x z h

x y z y h x z h

x z h x z h

x z h x z h


Example 2 (Cont’d)

Potential

Surface induced charge

11 2

2

1/22 2

2 2

1n 1n 1n2 2 2

( ) or 1n

2 ( )

L L L

o o o

L

o

V V V

x z hV

x z h

0 2 2

2 2

LS n o z z

Li S

hD E

x h

h dxdx

x h

/2

/2

Li

L

h d

h

By letting tanx h


Example 3

A point charge between two orthogonal semi-infinite conducting planes

Potential

29

1 2 3 4

1 1 1 1

4 o

QV

r r r r

1/22 2 2

1

1/22 2 2

2

1/22 2 2

3

1/22 2 2

4

( ) ( )

( ) ( )

( ) ( )

( ) ( )

r x a y z b

r x a y z b

r x a y z b

r x a y z b

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