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
”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.
EE 42 and 100, Spring 2006
Week 3b. Prof. White
1
New topics – energy storage elements
Capacitors
Inductors
EE 42 and 100, Spring 2006
Week 3b. Prof. White
2
The Capacitor
Two conductors (a,b) separated by an insulator:
difference in potential = Vab
=> equal & opposite charges Q on conductors
+Q
-Q
+
-
Vab
Q = CVab
Q = Magnitude of charge
stored on each conductor
where C is the so-called 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
A
C
d
EE 42 and 100, Spring 2006
F (F)
Week 3b. Prof. White
3
EE 42 and 100, Spring 2006
Week 3b. Prof. White
4
or
Symbol:
C
C
Electrolytic (polarized)
capacitor has + sign on
one plate
Units: Farads (Coulombs/Volt)
(typical range of values: 1 pF to 1 mF; for “supercapacitors” up to a few F!)
Current-Voltage relationship:
dQ
dvc
dC
ic 
C
 vc
dt
dt
dt
If C (geometry) is unchanging, iC
= dvC/dt
ic
+
vc
–
Note: Q(t) must be a continuous function of time
EE 42 and 100, Spring 2006
Week 3b. Prof. White
5
EE 42 and 100, Spring 2006
Week 3b. Prof. White
6
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
EE 42 and 100, Spring 2006
Week 3b. Prof. White
7
Schematic Symbol and Water Model for a Capacitor
EE 42 and 100, Spring 2006
Week 3b. Prof. White
8
Capacitor Uses
Capacitors are used to:
store energy for camera flashbulbs;
in filters that separate signals having different frequencies
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 performance (example, conductors on printed circuit boards
EE 42 and 100, Spring 2006
Week 3b. Prof. White
9
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.
EE 42 and 100, Spring 2006
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.
Week 3b. Prof. White
10
Application Example: MEMS Accelerometer
to 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
EE 42 and 100, Spring 2006
Week 3b. Prof. White
11
Application: Condenser Microphone
Condenser microphone
G
Sound waves
Econst
Electret microphone
E c onst
X1
Vout
x
Cylindrical air-filled cavity
Flexible conducting diaphragm
Conducting rigid cup
Vout ~ x E c onst
Vout = x Econst
EE 42 and 100, Spring 2006
Week 3b. Prof. White
x
Vout
G
X1
Vout ~ x Econst
Electret: insulator
(e.g., teflon) that was
bombarded with electrons
that remain imbedded
in it to “bias” the
condenser.
Widely used in telephone handsets;
available at RadioShack
12
Capacitor Voltage in Terms of Current
Charge is integral of current through capacitor and also
equals capacitance C time capacitor voltage:
t
Q (t )   ic (t )dt  Q (0)
0
t
t
1
Q ( 0) 1
vc (t )   ic (t )dt 
  ic (t )dt  vc (0)
C0
C
C0
EE 42 and 100, Spring 2006
Week 3b. Prof. White
13
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
Thus, energy is 1 QV 
2
1
CV 2
2
.
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!)
EE 42 and 100, Spring 2006
Week 3b. Prof. White
14
A more rigorous derivation
ic
t  t Final
v  VFinal dQ
v  VFinal
w
v c  ic dt 
dt 

 vc
 v c dQ
dt
t  t Initial
v  VInitial
v  VInitial
+
vc
–
v  VFinal
1
1
2
2
w
Cv
dv

CV

CV
 c c
Final
Initial
2
2
v  VInitial
EE 42 and 100, Spring 2006
Week 3b. Prof. White
15
Example: Current, Power & Energy for a Capacitor
t
1
v(t )   i( )d  v(0)
C0
0
1
2
4
5
1
2
EE 42 and 100, Spring 2006
3
4
10 mF
t (s)
vc and q must be continuous
functions of time; however,
ic can be discontinuous.
dv
iC
dt
i (mA)
0
3
v(t)
–
+
v (V)
1
i(t)
5
t (s)
Week 3b. Prof. White
Note: In “steady state”
(dc operation), time
derivatives are zero
 C is an open circuit
16
p (W)
i(t)
0
1
2
3
4
5
–
+
v(t)
10 mF
t (s)
p  vi
w (J)
0
t
1
2
EE 42 and 100, Spring 2006
3
4
5
t (s)
Week 3b. Prof. White
1 2
w   pd  Cv
2
0
17
Capacitors in Parallel
i(t)
i1(t)
i2(t)
+
C1
C2
v(t)
–
+
i(t)
Ceq
v(t)
Ceq  C1  C2
–
dv
i  Ceq
dt
Equivalent capacitance of capacitors in parallel is the sum
EE 42 and 100, Spring 2006
Week 3b. Prof. White
18
Capacitors in Series
+ v1(t) – + v2(t) –
i(t)
C1
C2
+
i(t)
Ceq
v(t)=v1(t)+v2(t)
–
1
1
1
 
Ceq C1 C2
EE 42 and 100, Spring 2006
Week 3b. Prof. White
19
The Inductor
• An inductor is constructed by coiling a wire around some
type of form.
+
vL(t)
iL
_
• 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:
Note: In “steady state” (dc operation), time
derivatives are zero  L is a short circuit
EE 42 and 100, Spring 2006
Week 3b. Prof. White
diL
vL (t )  L
dt
20
Symbol:
L
Units: Henrys (Volts • second / Ampere)
(typical range of values: mH to 10 H)
Current in terms of voltage:
1
diL  vL (t )dt
L
t
1
iL (t )   vL ( )d  i (t0 )
L t0
iL
+
vL
–
Note: iL must be a continuous function of time
EE 42 and 100, Spring 2006
Week 3b. Prof. White
21
Schematic Symbol and Water Model of an Inductor
EE 42 and 100, Spring 2006
Week 3b. Prof. White
22
Stored Energy
INDUCTORS STORE MAGNETIC ENERGY
Consider an inductor having an initial current i(t0) = i0
p(t )  v(t )i(t ) 
t
w(t )   p( )d 
t0
1 2 1 2
w(t )  Li  Li0
2
2
EE 42 and 100, Spring 2006
Week 3b. Prof. White
23
Inductors in Series
di
v  Leq
dt
+ v1(t) – + v2(t) –
+
v(t) –
L1
i(t)
L2
i(t)
v(t) +
–
Leq
+
v(t)=v1(t)+v2(t)
–
di
di
di
di
v  L1  L2  L1  L2   Leq
dt
dt
dt
dt
Leq  L1  L2
Equivalent inductance of inductors in series is the sum
EE 42 and 100, Spring 2006
Week 3b. Prof. White
24
Inductors in Parallel
+
i1
i(t)
L1
+
i2
i(t)
v(t) L2
Leq
v(t)
–
–
t
t
t
1
1
i  i1  i2   vd  i1 (t0 )   vd  i2 (t0 )
L1 t0
L2 t0
1
i
vd  i (t0 )

Leq t0
 1 1 t
i      vd  i1 (t0 )  i2 (t0 )
 L1 L2  t0
1
1 1

 
with i (t0 )  i1 (t0 )  i2 (t0 )
Leq L1 L2
EE 42 and 100, Spring 2006
Week 3b. Prof. White
25
Summary
Capacitor
Inductor
di
vL
dt
1 2
w  Li
2
q = CvC
dv
dt
1
w  Cv 2
2
iC
v cannot change instantaneously
i can change instantaneously
Do not short-circuit a charged
capacitor (-> infinite current!)
n cap.’s in series:
n
1
1
n ind.’s in series:

Ceq i 1 Ci
n cap.’s in parallel: Ceq 
EE 42 and 100, Spring 2006
i cannot change instantaneously
v can change instantaneously
Do not open-circuit an inductor with
current flowing (-> infinite voltage!)
n
n
C
i 1
i
Leq   Li
i 1
n
1
1

n ind.’s in parallel:
Leq i 1 Li
Week 3b. Prof. White
26
Transformer – Two Coupled Inductors
+
+
v1
v2
-
-
N1 turns
N2 turns
|v2|/|v1| = N2/N1
See Hambley pp. 712-4 on Ideal Transformer
EE 42 and 100, Spring 2006
Week 3b. Prof. White
27
AC Power System
Flowing water
or
Steam produced
from
Fuel oil,
Natural gas,
Coal,
or
Nuclear energy
35,000 volts
Turbine
130,000 volts
Generator
Generating Plant
Step-up Transformer
(for efficient transmission at higher voltage, lower current)
21,000 volts
Step-down Transformer (for local
distribution)
Transmission-line support towers
120-240
volts
Step-down transformer mounted
on power pole
Customer
Simplified representation of the transmission and distribution systems that bring electric
power from a generating station to customers. In the generating station, a turbine driven by any of the
means indicated at the upper left is coupled to a generator, turning it to produce three-phase alternating
voltages.
EE 42 and 100, Spring 2006
Week 3b. Prof. White
28
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.)
EE 42 and 100, Spring 2006
Week 3b. Prof. White
29
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
EE 42 and 100, Spring 2006
Week 3b. Prof. White
30
Summary of Electrical Quantities
Quantity
Variable
Unit
Unit
Symbol
Typical
Values
Defining
Relations
Charge
Q
coulomb
C
1aC to 1C
magnitude of
6.242 × 1018
electron
charges qe = 1.602x10-19 C
Current
I
ampere
A
1mA to 1kA
1A = 1C/s
Important
Equations
i = dq/dt
N node
I
n 1
Voltage
V
volt
V
1mV to 500kV
1V = 1N-m/C
0
n
N loop
V
n 1
EE 42 and 100, Spring 2006
Week 3b. Prof. White
Symbol
n
0
31
Summary of Electrical Quantities (concluded)
Power
P
watt
W
1mW 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
Time
t
second
s
Resistance
R
ohm

1N = 1kgm/s2
1 to 10M
V = IR; P =
V2/R = I2R
R

Capacitance
Inductance
C
L
farad
henry
EE 42 and 100, Spring 2006
F
1fF to 5F
H
1mH to 1H
Week 3b. Prof. White
Q = CV; i =
C(dv/dt);U =
(1/2)CV2
v = L(di/dt);
U = (1/2)LI2

C


L


32