week 3bee 42 and 100, fall 20051 new topics – energy storage elements capacitors inductors
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Week 3bEE 42 and 100, Fall 2005
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New topics – energy storage elements Capacitors Inductors
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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.
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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
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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
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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
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Schematic Symbol and Water Model for a Capacitor
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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
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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
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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
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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
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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
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Capacitors in Series
i(t)C1
+ v1(t) –
i(t)
+
v(t)=v1(t)+v2(t)
–
Ceq
C2
+ v2(t) –
21
111
CCCeq
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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.
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Week 3bEE 42 and 100, Fall 2005
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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.
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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
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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
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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
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Week 3bEE 42 and 100, Fall 2005
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• 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
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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
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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)
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Schematic Symbol and Water Model of an Inductor
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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
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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
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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
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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
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Transformer – Two Coupled Inductors
N1 turns N2 turns
+
-
v1
+
v2
-
|v2|/|v1| = N2/N1
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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
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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
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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.)
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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
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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)