New topics – energy storage elements
Capacitors
Inductors
EE 42 and 100, Fall 2005
Week 3b
1
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.
EE 42 and 100, Fall 2005
Week 3b
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
(stored charge in terms of voltage)
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
A
C
d
EE 42 and 100, Fall 2005
(F)
F
Week 3b
3
+
or
Symbol:
C
C
C
Electrolytic (polarized)
capacitor
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 (vc) must be a continuous function of time
EE 42 and 100, Fall 2005
Week 3b
4
Voltage in Terms of Current; Capacitor
Uses
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
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
EE 42 and 100, Fall 2005
Week 3b
5
EE 42 and 100, Fall 2005
Week 3b
6
Schematic Symbol and Water Model for a Capacitor
EE 42 and 100, Fall 2005
Week 3b
7
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.
Thus, energy is 1 QV 
2
1
CV 2
2
.
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!)
EE 42 and 100, Fall 2005
Week 3b
8
A more rigorous derivation
ic
+
vc
–
t  t Final
v  VFinal dQ
v  VFinal
w
v c  ic dt 
dt 

 vc
 v c dQ
t  t Initial
v  VInitial dt
v  VInitial
v  VFinal
1
1
2
2
w
Cv
dv

CV

CV
 c c
Final
Initial
2
2
v  VInitial
EE 42 and 100, Fall 2005
Week 3b
9
Example: Current, Power & Energy for a Capacitor
t
v (V)
1
0
1
2
4
5
1
2
EE 42 and 100, Fall 2005
3
4
v(t)
10 mF
t (ms)
vc and q must be continuous
functions of time; however,
ic can be discontinuous.
dv
iC
dt
i (mA)
0
3
i(t)
–
+
1
v(t )   i( )d  v(0)
C0
5
Week 3b
t (ms)
Note: In “steady state”
(dc operation), time
derivatives are zero
 C is an open circuit
10
p (W)
i(t)
0
1
2
3
4
5
–
+
v(t)
10 mF
t (ms)
p  vi
w (J)
0
t
1
2
EE 42 and 100, Fall 2005
3
4
5
Week 3b
t (ms)
1 2
w   pd  Cv
2
0
11
Capacitors in Parallel
i(t)
i1(t)
i2(t)
+
C1
C2
v(t)
–
+
Ceq
i(t)
v(t)
Ceq  C1  C2
–
dv
i  Ceq
dt
Equivalent capacitance of capacitors in parallel is the sum
EE 42 and 100, Fall 2005
Week 3b
12
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, Fall 2005
Week 3b
13
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?
Dv  Dv1  Dv2
DQ1=C1Dv1
Q1+DQ1
C1
v+Dv
+
–
-Q1DQ1
Q2+DQ2
C2
Q2DQ2
+
v1+Dv1
–
Note that no net charge can
can be introduced to this node.
Therefore, DQ1+DQ2=0
+
v2(t)+Dv2
–
 C1Dv1  C2Dv2
C1
Dv2 
Dv
C1  C2
DQ2=C2Dv2 Note: Capacitors in series have the same incremental
charge.
EE 42 and 100, Fall 2005
Week 3b
14
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.
EE 42 and 100, Fall 2005
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
15
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) MEMS = micro•
electro-mechanical systems
g1
g2
FIXED OUTER PLATES
EE 42 and 100, Fall 2005
Week 3b
16
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
C1  C2
Vo  Vs 
(2Vs ) 
Vs
C1  C2
C1  C2
C1
Vo
C2
–Vs
EE 42 and 100, Fall 2005
A A

Vo g1 g 2 g 2  g1 g 2  g1



Vs A  A g 2  g1
const
g1 g 2
Week 3b
17
Application: Condenser Microphone
Condenser microphone
G
Sound waves
Econst
x
Electret microphone
E c onst
X1
G
Vout
x
Cylindrical air-filled cavity
Flexible conducting diaphragm
Conducting rigid cup
Vout ~ x E c onst
EE 42 and 100, Fall 2005
Vout
Week 3b
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
18
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, Fall 2005
Week 3b
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, Fall 2005
Week 3b
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, Fall 2005
Week 3b
21
Schematic Symbol and Water Model of an Inductor
EE 42 and 100, Fall 2005
Week 3b
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, Fall 2005
Week 3b
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, Fall 2005
Week 3b
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, Fall 2005
Week 3b
25
Summary
Capacitor
Inductor
dv
iC
dt
1 2
w  Cv
2
di
vL
dt
1 2
w  Li
2
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, Fall 2005
i cannot change instantaneously
v can change instantaneously
Do not open-circuit an inductor with
current (-> 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
26
Transformer – Two Coupled Inductors
+
+
v1
v2
-
-
N1 turns
N2 turns
|v2|/|v1| = N2/N1
EE 42 and 100, Fall 2005
Week 3b
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, Fall 2005
Week 3b
28
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
EE 42 and 100, Fall 2005
Week 3b
29
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, Fall 2005
Week 3b
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, Fall 2005
Week 3b
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, Fall 2005
F
H
1fF to 5F
1mH to 1H
Week 3b
Q = CV; i =
C(dv/dt);U =
(1/2)CV2
v = L(di/dt);
U = (1/2)LI2

C


L


32