EECS 40

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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 Newtons 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
• gt 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

Q

Vab
(stored charge in terms of voltage)
-
-Q

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

F
(F)
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Symbol Units Farads (Coulombs/Volt) Current-V
oltage relationship
C
or
C
Electrolytic (polarized) capacitor
(typical range of values 1 pF to 1 mF for
supercapa- citors up to a few F!)
ic
vc
If C (geometry) is unchanging, iC dvC/dt
Note Q (vc) must be a continuous function of
time
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Voltage in Terms of Current Capacitor Uses
Uses Capacitors are used to store energy for
camera flashbulbs, in filters that separate
various frequency signals, and they appear as
undesired parasitic elements in circuits
where they usually degrade circuit performance
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Schematic Symbol and Water Model for a Capacitor
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Stored Energy
CAPACITORS STORE ELECTRIC ENERGY

• 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.

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!)
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A more rigorous derivation
ic
vc
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Example Current, Power Energy for a Capacitor
i(t)
v (V)

v(t)
10 mF
1
t (ms)
0
2
3
4
5
1
vc and q must be continuous functions of time
however, ic can be discontinuous.
i (mA)
t (ms)
0
2
3
4
5
1
Note In steady state (dc operation),
time derivatives are zero ? C is an open circuit
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p (W)
i(t)

v(t)
10 mF
t (ms)
0
2
3
4
5
1
w (J)
t (ms)
0
2
3
4
5
1
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Capacitors in Parallel
i1(t)
i2(t)
v(t)
i(t)
C1
C2
v(t)
i(t)
Ceq
Equivalent capacitance of capacitors in parallel
is the sum
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Capacitors in Series
v1(t)
v2(t)
v(t)v1(t)v2(t)
C1
C2
i(t)
i(t)
Ceq

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Capacitive Voltage Divider

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

DQ1C1Dv1
Note that no net charge can can be introduced to
this node. Therefore, -DQ1DQ20
Q1DQ1
v1Dv1
C1
-Q1-DQ1
vDv

v2(t)Dv2
Q2DQ2
C2
-Q2-DQ2
DQ2C2Dv2
Note Capacitors in series have the same
incremental charge.
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MEMS Airbag Deployment Accelerometer
Chip about 1 cm2 holding in the middle an
electromechanical accelerometer around which
are electronic test and calibration circuits
(Analog Devices, Inc.) Hundreds of millions
have been sold.
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

g1
g2
FIXED OUTER PLATES
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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

Circuit model
Vs
C1
Vo
C2
Vs
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Application Condenser Microphone
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Practical Capacitors

• 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

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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

vL(t)
iL
_
Note In steady state (dc operation),
time derivatives are zero ? L is a short circuit
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Symbol Units Henrys (Volts second /
Ampere) Current in terms of voltage
L
(typical range of values mH to 10 H)
iL
vL
Note iL must be a continuous function of time
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Schematic Symbol and Water Model of an Inductor
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Stored Energy
INDUCTORS STORE MAGNETIC ENERGY

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

)
(
)
(
)
(
t
i
t
v
t
p
t
ò


t
t
)
(
)
(
d
p
t
w
t
0
1
1
2
-

2
)
(
Li
Li
t
w
0
2
2
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Inductors in Series
v1(t)
v2(t)
v(t)v1(t)v2(t)
L1
L2
i(t)
i(t)


v(t)
v(t)
Leq
Equivalent inductance of inductors in series is
the sum
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Inductors in Parallel
v(t)
v(t)
i2
i1
i(t)
i(t)
Leq
L1
L2
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Summary

• Capacitor
• v cannot change instantaneously
• i can change instantaneously
• Do not short-circuit a charged
• capacitor (-gt 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 (-gt
  infinite voltage!)
• n ind.s in series
• n ind.s in parallel

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Transformer Two Coupled Inductors


v1
v2
-
-
N1 turns
N2 turns
v2/v1 N2/N1
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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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Summary of Electrical Quantities
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 1mA to 1kA 1A 1C/s
Voltage V volt V 1mV to 500kV 1V 1N-m/C
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Summary of Electrical Quantities (concluded)
Power P watt W 1mW to 100MW 1W 1J/s P dU/dt PIV
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 W 1W to 10MW 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 1mH to 1H v L(di/dt) U (1/2)LI2 L