4. ELECTROSTATICS, Part 27e Applied EM by Ulaby and Ravaioli

Electrostatics 2

4.7 Dielectrics (유전체)

4.8 Boundary conditions (경계조건)

4.9 Capacitance (정전용량)

4.10 Electrostatic energy (정전기장 에너지)

4.11 Image method (영상법)

4.7 Dielectric Materials (유전체)

4.7.1 Polarization (분극)

P = electric flux density induced by E

ε = ε0εr : permittivity (유전율)ε0 = 8.854·10-12 F/m (진공의 유전율)εr : relative permittivity (비유전율) = dielectric constant (유전상수)

Electric or Dielectric Breakdown (절연파괴)Dielectric Strength (유전강도, 절연내력, 절연강도)

Insulation, Electrical insulation (전기절연)

절연강도: 공기 = 30 kV/cm, 팔리스타이린 = 2 MV/cm

Discharge and Arc

Dielectric Breakdown Curve

Dielectric Breakdown Terminology

Paschen's law: breakdown voltage vs gas pressure

Avalanche breakdown

Townsend avalanche

Static spark gap

Discharge:

Dark discharge: no light

Glow discharge: a plasma formed by the current through a low-pressure gas

Arc discharge: a lightning volt, a single or localized set of discharges

Corona discharge: distributed arc discharge

Partial discharge: localized discharge in an insulation system

Ionization energy: 1-3 eV

4.8 Boundary Conditions (경계조건)

- 전기장의 경계조건: 매질 경계면에서 전기장의 접선성분과 법선성분이

만족해야 할 조건

- 접선성분: 폐경로를 따라서 전기장 적분 → 전위차 0

- 법선송분: 폐곡면에서 전속밀도 적분 → 가우스법칙 적용

Summary of Boundary Conditions

Remember E = 0 in a good conductor

Field Lines at Conductor Boundary

At conductor boundary, E field direction is always perpendicular to conductor surface

4.9 Capacitance (정전용량)

Capacitance

For any two-conductor configuration:

For any resistor:

아래 구조의 저항으로부터 커패시턴스를 구하라.

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Fringing field (누설장) and fringing capacitance (누설 커패시턴스)

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커패시터의 직렬/병렬 연결시 합성 정전용량

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1 2

1 2 1 2

1 1 1 d dC C C S S

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병렬로 중첩된 콘덴서

콘덴서의 직병렬연결에 의한 정전용량의 계산은,

병렬, 직렬연결 저항계산법과 동일하다.

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Application of Gauss’s law gives:

Q is total charge on inside of outer cylinder, and –Q is on outside surface of inner cylinder

구형축전기의 정전용량

- 단일 구의 정전용량: 11 1, 4 4b C a

a b

4 1 1 1 1 1,4

abCb a C a b

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직렬로 적층된 구형 커패시터: 음극면은 무한구면

1 2 1 1 0 1

1 1 1 1 1 1 1 14 4C C C a r r

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Tech Brief 8: Supercapacitors

For a traditional parallel-plate capacitor, what is the maximum attainable energy density?

= permittivity of insulation materialV = applied voltage = density of insulation materiald = separation between plates

Mica has one of the highest dielectric strengths ~2 x 10**8 V/m. If we select a voltage rating of 1 V and a breakdown voltage of 2 V (50% safety), this will require that d be no smaller than 10 nm. For mica, = 60 and = 3 x 10**3 kg/m3 .

Hence:

W = 90 J/kg = 2.5 x10**‒2 Wh/kg.

By comparison, a lithium-ion battery has W = 1.5 x 10**2 Wh/kg, almost 4 orders of magnitude greater

Energy density is given by:

A supercapacitor is a “hybrid” battery/capacitor

Users of Supercapacitors

Energy Comparison

4.10 Electrostatic Potential Energy (전기장 에너지)

전하를 이동시키는데 소요된 에너지

전하를 이동시키는데 소요된 에너지

정전기장의 에너지

+

두 식을 합한다. 여기서 V1 = V1,2 + V1,3 + ··· + V1,N

정전기장의 에너지

커패시터에 저장된 에너지

0

1 1

2 2 0

: volume outside the electrodes, : volume and surface of the negative electrode with potential = 0, : volume and surface of the postive electrode with potential =

1 12 2

12

v sV S

VV SV S

W dV dS

0 1 2

1

2 20 0 0

20 0

1 0 ( 0)2

1 0 ( 0)21 1 1 ( )2 2 2

1 1 2 2

v v vV V V V

sS

s sS S

dV dV

dS

dS dS Q

W Q C

Energy stored in a capacitor

전계강도를 이용한 정전기장 에너지 계산

1 1 ( ) ( , potential)2 2v v

V VW dV dV D D

( ) ( ) ( ) ( ) ( ) ( ) ( ) D D D D D D D D E

1 1 1 1( ) ( ) ( )2 2 2 2V V S V

W dV dV d dV D D E D S D E

2

3

1 (surface density of electric energy, J/m )2

1 (volume density of electric energy, J/m )2

s

v

w

w

D n

D E

전기장에 저장된 에너지

0

0 1 2 0

2

1 0

0

0

2

0

1 0 since 0 in 2

1 1 1 12 2 2 2

0 since 0 on

1 1 12 2 2

12

v vV

vV S S V

S

S V

V

dV V

dV d d dV

d S

d Q dV

W dV

D S D S D E

D S

D S D E

D E

Electrostatic potential energy density (Joules/volume)

Total electrostatic energy stored in a volume

유전체 손실

1) 커패시터를 이용한 유도

2

*

2

1 1 (capacitor stored energy)2 2

1, , ( )2

1 ( ) | |2

C

C c c c C C

C

qw vqC

dvp vi i C P V I I C j Vdt

P j C V

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2 2

*

2

1 1( ) ( ) | | ( ) | |2 2

1, ,2

1 1 1| | Re( ) ( )2 ( )

C

R R R c R R

R C

S S SC j P V j Vd d d

v Vp vi i P V I IR R

S d dP V P RR R d S S

,( )

j

S dC Rd S

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2) 전기장 전력밀도를 이용한 유도

3

* * 2

2 2

2

1 1 (J/m , electric field energy density)2 21 1 1= ( ) ( ) | | (phasor for energy density)4 4 4

1 1( 2 ) ( ) | | ( ) | | (phasor for power density)2 2

1 1( ) | | | |2 2r

w DE

W DE E j E j E

P j W E j E

P E E

D E

2

2

1 (loss power density)2

1 ( ) | | (stored power density)2i

JE

P E

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4.11 Image Method

Image method simplifies calculation for E and V due to charges near conducting planes.

1. For each charge Q, add an image charge –Q2. Remove conducting plane3. Calculate field due to all charges

Image Method Applied to Capacitance Calculation

Image Method: Two-cylinder problem

Image method: two-sphere problem, one sphere grounded1. First charge: at the center of the left sphere

Left sphere: equipotential2. Second charge: inside the right sphere

Let V at the right sphere be zero.3. Do 2 for the left sphere. Repeat the above.

Image method: two-sphere problem, two spheres floated

Tech Brief 9: Capacitive Sensors

Wheatstone Bridge Formula

Humidity Sensor (습도센서)습도(수분, 물)에 따른 커패시턴스 변화 측정

Pressure Sensor (압력센서)압력에 따라 커패시터 극판간격 변화에 따른커패시턴스 측정

Planar capacitors (평판커패시터)터치(물체접촉) 센서: 물체유무에 따른커패시턴스 변화측정. 스마트폰 등 HMI

Fingerprint Imager (지문센서)지문의 요철을 커패시터 센서로 인식Fusitsu MBF200: 256x300 pixels, 500 dpi