EECS 40

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Announcements
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 (also a copy in
140 Cory lab bookcase) 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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New topics energy storage elements
Capacitors Inductors
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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 so-called capacitance of the
  structure,
• positive () charge is on the conductor at higher
  potential

Q

Vab
Q Magnitude of charge stored on each conductor
-
-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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Electrolytic (polarized) capacitor has sign on
one plate
Symbol Units Farads (Coulombs/Volt) Current-V
oltage relationship
or
C
C
(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(t) must be a continuous function of time
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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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Schematic Symbol and Water Model for a Capacitor
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Capacitor Uses
Capacitors are used to store energy for
camera flashbulbs in filters that separate
signals having different fre- quencies in
resonant circuits to tune a radio and
oscillators that generate a
time-varying voltage at a desired
frequency Capacitors also appear as undesired
parasitic elements in circuits where they
usually degrade circuit perfor- mance
(example, conductors on printed circuit boards
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Capacitors used in MEMS Airbag Deployment
Accelerometer (MEMS
MicroElectroMechanical Systems)
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)

g1
g2
FIXED OUTER PLATES
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Application Condenser Microphone
Vout dx Econst
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Capacitor Voltage in Terms of Current
Charge is integral of current through capacitor
and also equals capacitance C time capacitor
voltage
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Stored Energy
CAPACITORS STORE ELECTRIC ENERGY

• You might think the energy stored on a capacitor
  charged to voltage V 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

Example The energy stored in a 1 pF capacitance
charged to 5 Volts equals ½ (1pF) (5V)2 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 (s)
0
2
3
4
5
1
vc and q must be continuous functions of time
however, ic can be discontinuous.
i (mA)
t (s)
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 (s)
0
2
3
4
5
1
w (J)
t (s)
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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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
  flowing (-gt infinite voltage!)
• n ind.s in series
• n ind.s in parallel

q CvC
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Transformer Two Coupled Inductors


v1
v2
-
-
N1 turns
N2 turns
v2/v1 N2/N1
See Hambley pp. 712-4 on Ideal Transformer
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AC Power System

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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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High-voltage DC power systems
Highest DC voltage system /- 600 kV, in
Brazil brings 50 Hz power from 12,600 MW Itaipu
hydropower plant to 60 Hz network in Sao Paulo
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Summary of Electrical Quantities
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Summary of Electrical Quantities (concluded)