CapacitorsCapacitors store charge

+ ++ ++

– – – ––

E

+Q

–Q

++++++ +++++

–––––––– ––––––––

d

dA

C

Q=CV

dV

E

11120 mAsV10854.8 dt

dQI

dtdV

CI

Parallel plate capacitor

plate area

Dielectric permittivity ε=εrε0

Units: farad [F]=[AsV-1] more usually μF, nF, pF

1

Capacitor typesRange Maximum voltage

Ceramic 1pF–1μF 30kV

Mica 1pF–10nF 500V (very stable)

Plastic 100pF–10μF 1kV

Electrolytic 0.1μF–0.1F 500V

Parasitic ~pF Polarised

2

CapacitorsCapacitors in parallel: C1 C2 C3

321n

nTotal CCCCC

Capacitors in series: C1 C2 C3

321n nT C

1C1

C1

C1

C1

21 V

2

2

V

1

1

T CQ

CQ

CQ

V V1 V2

+Q1 –Q1 +Q2 –Q2

Q1=Q2

3

Capacitors – energy stored

2

21

V

0

t

0

t

0

t

0

CV'dV'VC

'dt'tV'dt'tdV

C'dt'tV'tI'dt'tPW

C2Q

QVCVW2

212

21

Energy stored as electric field

4

RC circuits

+V0

R

C

1

2

Initially VR=0 VC=01

Switch in Position “1” for a long time.

Then instantly flips at time t = 0.

Capacitor has a derivative!How do we analyze this?

There are some tricks…

1.) What does the Circuit do up until the switch flips?(Switch has been at pos. 1 for a VERY long time. easy)

2). What does the circuit do a VERY long time AFTER the switch flips? (easy)

3). What can we say about the INSTANT after switch flips? (easy if you know trick)

I

+

+

ALWAYS start by asking these three questions:

5

The Trick!!• Remember • Suppose the voltage on a 1 farad Capacitor changes

by 1 volt in 1 second.– What is the current?– What if the same change in V happens in 1 microsecond?– So…What if the same change in V happens instantly?

• Rule: It is impossible to change the voltage on a capacitor instantly!– Another way to say it:

The voltage at t = 0-e is the same as at t = 0+ .e

Note for Todd : Put problem on board and work through these points! 6

RC circuits

+V0

R

C

1

2

Initially at t = 0- VR=0 VC=0 I=0

1

2

CI

dtdI

R0

C

QIR

VVV

VVV

CR

CR

0

0

0

differentiate wrt t

Then at t = 0+ Must be that VC=0

If VC=0 then KVL says VR=V0 and I=V0/R.Capacitor is acting like a short-circuit.

Finally at t = +∞VR=0 VC=V0 I=0

I

7

CI

dtdI

R0

dtdI

RCI

dII1

dtRC

1

bIln

blnIln

aIlnRC

t

bIe RCt

RCt

0

RCt

eI

eb1

I

8

t0 eRV

tI+V0

R

RC

9

Integrator Capacitor as integrator

R

CVi Vint

t

int C 0

ti C

C 0

t

i0

1V V Idt '

C1 V V

V dt 'C R

1Vdt '

RC

If RC>>t

VC<<Vi

10

Differentiation

RC

Vi Vdiff

dtdV

dtdV

RCRIVV RiRdiff

Small RC dt

dVdtdV Ri

dtdV

RCV idiff

11

Inductors

I

Solenoid N turns

NI

B 0

Electromagnetic induction

B

Vind

BdA flux Magnetic

dtd

Vind

dtdI

NN

AV

cetanInduc

21

0ind

12

Self Inductance

Mutual Inductance

I

dtdI

LV

2

0

NAL

I

dtdI

MV M

21

0

NNAM Units: henrys [H] = [VsA-1]

for solenoid

13

InductorsWire wound coils - air core

- ferrite core

Wire loops

Straight wire

14

Series / parallel circuitsL1 L2 L3

Inductors in series: LTotal=L1+L2+L3…

n

nT LL

L1 L2

Inductors in parallel

21

n nT

L1

L1

L1

L1

dtdI

LdtdI

LdtdI

LV 321

I

I1 I2

2

2

1

1

21

LVdtdI

LVdtdI

LVdtdI

III

15

Inductors – energy stored

t

0'dt'tPW

221

I

0

t

0

t

0

LI

'IdIL

'dtdt

'tdIL'tI

'dt'tV'tI

Energy stored as magnetic field

16

The Next Trick!!• Remember • Suppose the current on a 1 henry Inductor changes

by 1 amp in 1 second.– What is the voltage?– What if the same change in I happens in 1 microsecond?– So…What if the same change in I happens instantly?

• Rule: It is impossible to change the current on an inductor instantly!– Another way to say it:

The current at t = 0-e is the same as at t = 0+ .e– Does all this seem familiar? It is the concept of

“duality”.17

RL circuits

+V0

R

L

1

2

20R

Initially t=0-

VR=V0 VL=0 I=V0/R

And at t=∞ (Pos. “1”)VR=0 VL=0 I=0

At t=0+ Current in L MUST be the sameI=V0/R So…

VR=V0 VR20=20V0

and VL = -21V0 !!!

I

+

++

dt

dILIR

dt

dILIRIR

VVV LRR

210

200

0 20

For times t > 0

texp0ItI

ttI

I

dt

I

dI

IL

R

dt

dI

00

021

t0ItI

ln

R

L

21

00tI R

VI 00

t

R

VtI exp0 0

R

VtI 00

19

t

R

VtI exp0

+V0

R

L

Notice Difference in Scale! 20

+V0

R2

R1L

I2

t<0 switch closed I2=?t=0 switch openedt>0 I2(t)=?

voltage across R1=?

Write this problem down and DO attempt it at home 21

Physics 3 June 2001 Notes for Todd: Try to do it cold and consult the backupslides if you run into trouble.

22

LCR circuit

+V0

R

C

1

2

L

V(t)

0t

0tfor

V

0tV

0

I(t)=0 for t<0

0ILC1

dtdI

LR

dtId

VCQ

dtdI

LIR

VVVV

0t

2

2

0

0CLR

23

R, L, C circuits

• What if we have more than just one of each?

• What if we don’t have switches but have some other input instead?– Power from the wall socket comes as a

sinusoid, wouldn’t it be good to solve such problems?

• Yes! There is a much better way!– Stay Tuned!

24

BACKUP SLIDES

25

26

27

28

29

RL circuits

+V0

R

L

1

2Initially VR=0 VL=01

2

dtdi

LiR

VVV LR0

texp0ItI

t

0

tI

0I

dtidi

0iLR

dtdi

t0ItI

ln

RL

00tI

RVt

exp0ItI 0

RV

0I 0

texp1

RV

tI 0

LV

iLR

dtdi 0

30

texp1

RV

tI 0+V0

R

L

RL

31

This way up32

V0

R1

R1

R0

L

R1

R0

L

R1

0

1

V

R

R0

L

0

1

V

R1R

2R0

L

0V

2

1R

2

Norton

0 10

V R dII R L

2 2 dt 33

This way up

0 10

V R dII R L

2 2 dt

34

This way up35

VolH2

NIBH HA

IN

A2

LIW

20

0

202

1

20

20

221

Energy stored per unit volumeInductor Capacitor

VolE2

VdA

CVW

20

2021

221

dA

C 0

dV

E

36