Week 3bEE 42 and 100, Fall 20051 New topics – energy storage elements Capacitors Inductors

Week 3bEE 42 and 100, Fall 2005

1

New topics – energy storage elements Capacitors Inductors


2

Books on Reserve for EE 42 and 100 in the Bechtel Engineering Library

“The Art of Electronics” by Horowitz and Hill (2nd edition) -- A terrific source book on practical

electronics“Electrical Engineering Uncovered” by White and Doering (2nd edition) – Freshman intro to aspects of engineering and EE in particular”Newton’s Telecom Dictionary: The authoritative resource for Telecommunications” by Newton (“18th edition – he updates it annually) – A place to find definitions of all terms and acronyms connected with telecommunications. TK5102.N486 Shelved with

dictionaries to right of entry gate.


3

The Capacitor

Two conductors (a,b) separated by an insulator:difference in potential = Vab

=> equal & opposite charges Q on conductors

Q = CVab

where C is the capacitance of the structure, positive (+) charge is on the conductor at higher potential

Parallel-plate capacitor:• area of the plates = A (m2)• separation between plates = d (m) • dielectric permittivity of insulator = (F/m)

=> capacitance d

AC

(stored charge in terms of voltage)

F(F)

Vab

+

-

+Q

-Q


4

Symbol:

Units: Farads (Coulombs/Volt)

Current-Voltage relationship:

or

Note: Q (vc) must be a continuous function of time

+vc

–

ic

dt

dCv

dt

dvC

dt

dQi c

cc

C C

(typical range of values: 1 pF to 1 F; for “supercapa-citors” up to a few F!)

+

Electrolytic (polarized) capacitor

C

If C (geometry) is unchanging, iC = dvC/dt


5

Voltage in Terms of Current; Capacitor Uses

)0()(1)0(

)(1

)(

)0()()(

00

0

c

t

c

t

cc

t

c

vdttiCC

Qdtti

Ctv

QdttitQ

Uses: Capacitors are used to store energy for camera flashbulbs,in filters that separate various frequency signals, andthey appear as undesired “parasitic” elements in circuits wherethey usually degrade circuit performance


6


7

Schematic Symbol and Water Model for a Capacitor


8

You might think the energy stored on a capacitor is QV = CV2, which has the dimension of Joules. But during charging, the average voltage across the capacitor was only half the final value of V for a linear capacitor.

Thus, energy is .22

1

2

1CVQV

Example: A 1 pF capacitance charged to 5 Volts has ½(5V)2 (1pF) = 12.5 pJ

(A 5F supercapacitor charged to 5 volts stores 63 J; if it discharged at a constant rate in 1 ms energy is discharged at a 63 kW rate!)

Stored EnergyCAPACITORS STORE ELECTRIC ENERGY


9

Final

Initial

c

Final

Initial

Final

Initial

ccc

Vv

VvdQ vdt

tt

tt

dt

dQVv

Vvvdt ivw

2CV2

12CV2

1Vv

Vvdv Cvw InitialFinal

Final

Initial

cc

+vc

–

ic

A more rigorous derivation


10

Example: Current, Power & Energy for a Capacitor

dt

dvCi

–+

v(t) 10 F

i(t)

t (s)

v (V)

0 2 3 4 51

t (s)0 2 3 4 51

1

i (A) vc and q must be continuousfunctions of time; however,ic can be discontinuous.

)0()(1

)(0

vdiC

tvt

Note: In “steady state”(dc operation), timederivatives are zero C is an open circuit


11

vip

0 2 3 4 51

w (J)–+

v(t) 10 F

i(t)

t (s)0 2 3 4 51

p (W)

t (s)

2

0 2

1Cvpdw

t


12

Capacitors in Parallel

21 CCCeq

i(t)

+

v(t)

–

C1 C2

i1(t) i2(t)

i(t)

+

v(t)

–

Ceq

Equivalent capacitance of capacitors in parallel is the sumdt

dvCi eq


13

Capacitors in Series

i(t)C1

+ v1(t) –

i(t)

+

v(t)=v1(t)+v2(t)

–

Ceq

C2

+ v2(t) –

21

111

CCCeq


14

Capacitive Voltage Divider

Q: Suppose the voltage applied across a series combination of capacitors is changed by v. How will this affect the voltage across each individual capacitor?

21 vvv

v+v

C1

C2

+ v2(t)+v2

–

+ v1+v1

–+–

Note that no net charge cancan be introduced to this node.Therefore, Q1+Q2=0

Q1+Q1

-Q1Q1

Q2+Q2

Q2Q2

Q1=C1v1

Q2=C2v2

2211 vCvC

vCC

Cv

21

12

Note: Capacitors in series have the same incremental charge.


15

MEMS Airbag Deployment Accelerometer

Chip about 1 cm2 holding in themiddle an electromechanicalaccelerometer around which areelectronic test and calibrationcircuits (Analog Devices, Inc.) Hundreds of millions have beensold.

Airbag of car that crashed into the back of a stopped Mercedes. Within 0.3 seconds after deceleration the bag is supposed to be empty. Driver was not hurt in any way; chassis distortion meant that this car was written off.


16

Application Example: MEMS Accelerometerto deploy the airbag in a vehicle collision

• Capacitive MEMS position sensor used to measure acceleration (by measuring force on a proof mass) MEMS = micro-

• electro-mechanical systems

FIXED OUTER PLATES

g1

g2


17

Sensing the Differential Capacitance

– Begin with capacitances electrically discharged– Fixed electrodes are then charged to +Vs and –Vs

– Movable electrode (proof mass) is then charged to Vo

const

gg

gg

gg

gA

gA

gA

gA

V

V

VCC

CCV

CC

CVV

s

o

ssso

12

12

12

21

21

21

21

21

1 )2(

C1

C2

Vs

–Vs

Vo

Circuit model


18

Flexible conducting diaphragm

Sound waves

Cylindrical air-filled cavity

Conducting rigid cup

Condenser microphone Electret microphone

Electret: insulator(e.g., teflon) that wasbombarded with electronsthat remain imbeddedin it to “bias” thecondenser.Widely used in tele-phone handsets;available at RadioShack

G

X1Econst

xVout

Vout ~

x

x Econst

Econst

X1

GG

GVout

Vout ~ x Econst

Application: Condenser Microphone


19

• A capacitor can be constructed by interleaving the plates with two dielectric layers and rolling them up, to achieve a compact size.

• To achieve a small volume, a very thin dielectric with a high dielectric constant is desirable. However, dielectric materials break down and become conductors when the electric field (units: V/cm) is too high.– Real capacitors have maximum voltage ratings– An engineering trade-off exists between compact size and

high voltage rating

Practical Capacitors


20

The Inductor

• An inductor is constructed by coiling a wire around some type of form.

• Current flowing through the coil creates a magnetic field and a magnetic flux that links the coil: LiL

• When the current changes, the magnetic flux changes a voltage across the coil is induced:

iLvL(t)

dt

diLtv L

L )(

+

_

Note: In “steady state” (dc operation), timederivatives are zero L is a short circuit


21

Symbol:

Units: Henrys (Volts • second / Ampere)

Current in terms of voltage:

Note: iL must be a continuous function of time

+vL

–

iL

t

t

LL

LL

tidvL

ti

dttvL

di

0

)()(1

)(

)(1

0

L

(typical range of values: H to 10 H)


22

Schematic Symbol and Water Model of an Inductor


23

Stored Energy

Consider an inductor having an initial current i(t0) = i0

20

2

2

1

2

1)(

)()(

)()()(

0

LiLitw

dptw

titvtp

t

t

INDUCTORS STORE MAGNETIC ENERGY


24

Inductors in Series

21 LLLeq

dt

diL

dt

diLL

dt

diL

dt

diLv eq 2121

v(t)

L1

+ v1(t) –

v(t)

+

v(t)=v1(t)+v2(t)

–

Leq

L2

+ v2(t) –

+–

+–

i(t) i(t)

Equivalent inductance of inductors in series is the sum

dt

diLv eq


25

L1i(t)

i2i1

Inductors in Parallel

)()()( with 111

)()(11

)(1

)(1

0201021

020121

022

011

21

0

00

tititiLLL

titidvLL

i

tidvL

tidvL

iii

eq

t

t

t

t

t

t

L2

+

v(t)

–

Leqi(t)

+

v(t)

–

)(1

0

0

tidvL

it

teq


26

Capacitor

v cannot change instantaneously

i can change instantaneously

Do not short-circuit a chargedcapacitor (-> infinite current!)

n cap.’s in series:

n cap.’s in parallel:

Inductor

i cannot change instantaneously

v can change instantaneously

Do not open-circuit an inductor with current (-> infinite voltage!)

n ind.’s in series:

n ind.’s in parallel:

Summary

n

iieq

n

i ieq

CC

CC

1

1

11

2

2

1Cvw

dt

dvCi

2

2

1Liw

dt

diLv

n

i ieq

n

iieq

LL

LL

1

1

11


27

Transformer – Two Coupled Inductors

N1 turns N2 turns

+

-

v1

+

v2

-

|v2|/|v1| = N2/N1


28

Flowing wateror

Steam producedfrom

Fuel oil, Natural gas,

Coal,

Nuclear energy

Turbine Generator

Generating Plant Step-up Transformer(for efficient trans-mission at higher volt-age, lower current)

35,000 volts 130,000 volts

Transmission-line support towersStep-down Trans-former (for local distribution)

21,000 volts

Step-down trans-former mountedon power pole

120-240 volts

Customer

Simplified representation of the transmission and distribution systems that bring electricpower from a generating station to customers. In the generating station, a turbine driven by any of themeans indicated at the upper left is coupled to a generator, turning it to produce three-phase alternatingvoltages.

or

AC Power System


29

High-Voltage Direct-Current Power Transmission

http://www.worldbank.org/html/fpd/em/transmission/technology_abb.pdf

Highest voltage +/- 600 kV, in Brazil – brings 50 Hz power from12,600 MW Itaipu hydropower plant to 60 Hz network in Sao Paulo


30

Relative advantages of HVDC over HVAC power transmission

• Asynchronous interconnections (e.g., 50 Hz to 60 Hz system)

• Environmental – smaller footprint, can put in underground cables more economically, ...

• Economical -- cheapest solution for long distances, smaller loss on same size of conductor (skin effect), terminal equipment cheaper

• Power flow control (bi-directional on same set of lines)• Added benefits to the transmission (system stability,

power quality, etc.)


31

Quantity Variable Unit UnitSymbol

Typical Values

DefiningRelations

ImportantEquations

Symbol

Charge Q coulomb C 1aC to 1C magnitude of 6.242 × 1018 electron charges qe = -1.602x10-19 C

i = dq/dt

Current I ampere A 1A to 1kA 1A = 1C/s

Voltage V volt V 1V to 500kV 1V = 1N-m/C

Summary of Electrical Quantities

01

nodeN

nnI

01

loopN

nnV


32

Power P watt W 1W to 100MW

1W = 1J/s P = dU/dt; P=IV

Energy U joule J 1fJ to 1TJ 1J = 1N-m U = QV

Force F newton N 1N = 1kg-m/s2

Time t second s

Resistance R ohm 1 to 10M V = IR; P = V2/R = I2R

R

Capacitance C farad F 1fF to 5F Q = CV; i = C(dv/dt);U =

(1/2)CV2

C

Inductance L henry H 1H to 1H v = L(di/dt); U = (1/2)LI2

L

Summary of Electrical Quantities (concluded)

Documents

Week 3bEE 42 and 100, Fall 20051 New topics – energy storage elements Capacitors Inductors