New topics – energy storage elements
Capacitors
Inductors
EECS 42, Spring 2005
Week 3b
Books on Reserve for EECS 42 in
Engineering Library
“The Art of Electronics” by Horowitz and Hill (1st and 2nd
editions) -- A terrific source book on 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
“Electrical Engineering: Principles and Applications” by
Hambley (3rd edition) – Backup copy of text for
EECS 42
EECS 42, Spring 2005
Week 3b
Reader
The EECS 42 Supplementary Reader is now available
at Copy Central, 2483 Hearst Avenue (price: $12.99)
It contains selections from two textbooks that
we will use when studying semiconductor devices:
Microelectronics: An Integrated Approach
(by Roger Howe and Charles Sodini)
Digital Integrated Circuits: A Design Perspective
(by Jan Rabaey et al.)
EECS 42, Spring 2005
Week 3b
The Capacitor
Two conductors (a,b) separated by an insulator:
difference in potential = Vab
=> equal & opposite charge Q on conductors
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
EECS 42, Spring 2005
A
C
d
(F)
F
Week 3b
+
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
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 2005
Week 3b
EECS 42, Spring 2005
Week 3b
Schematic Symbol and Water Model for a Capacitor
EECS 42, Spring 2005
Week 3b
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!)
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 2005
Week 3b
Example: Current, Power & Energy for a Capacitor
t
v (V)
1
0
1
2
4
5
1
EECS 42, Spring 2005
2
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
p (W)
i(t)
0
1
2
3
4
5
–
+
v(t)
10 mF
t (ms)
p  vi
w (J)
0
t
1
EECS 42, Spring 2005
2
3
4
5
Week 3b
t (ms)
1 2
w   pd  Cv
2
0
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
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 2005
Week 3b
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.
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 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
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
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 2005
Week 3b
diL
vL (t )  L
dt
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
EECS 42, Spring 2005
Week 3b
Schematic Symbol and Water Model of an Inductor
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 2005
Week 3b
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
EECS 42, Spring 2005
Week 3b
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 
EECS 42, Spring 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