Week 3b EECS 42, Spring 2005 New topics – energy storage elements Capacitors Inductors
Week 3bEECS 42, Spring 2005
New topics – energy storage elementsCapacitorsInductors
Week 3bEECS 42, Spring 2005
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
Week 3bEECS 42, Spring 2005
The EECS 42 Supplementary Reader is now availableat Copy Central, 2483 Hearst Avenue (price: $12.99)
It contains selections from two textbooks thatwe 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.)
Reader
Week 3bEECS 42, Spring 2005
The CapacitorTwo conductors (a,b) separated by an insulator:
difference in potential = Vab=> equal & opposite charge Q on conductors
Q = CVab
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 = ε(F/m)
=> capacitance dAC ε
=
(stored charge in terms of voltage)
F(F)
Week 3bEECS 42, Spring 2005
Symbol:
Units: Farads (Coulombs/Volt)
Current-Voltage relationship:
or
Note: Q (vc) must be a continuous function of time
+vc–
ic
dtdCv
dtdvC
dtdQi c
cc +==
C C
(typical range of values: 1 pF to 1 µF; for “supercapa-citors” up to a few F!)
+
Electrolytic (polarized)capacitor
C
If C (geometry) is unchanging, iC = dvC/dt
Week 3bEECS 42, Spring 2005
Voltage in Terms of Current; Capacitor Uses
)0()(1)0()(1)(
)0()()(
00
0
c
t
c
t
cc
t
c
vdttiCC
QdttiC
tv
QdttitQ
+=+=
+=
∫∫
∫
Uses: Capacitors are used to store energy for camera flashbulbs,in filters that separate various frequency signals, andthey appear as undesired “parasitic” elements in circuits wherethey usually degrade circuit performance
Week 3bEECS 42, Spring 2005
Week 3bEECS 42, Spring 2005
Schematic Symbol and Water Model for a Capacitor
Week 3bEECS 42, Spring 2005
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 .221
21 CVQV =
Example: A 1 pF capacitance charged to 5 Volts has ½(5V)2 (1pF) = 12.5 pJ(A 5F supercapacitor charged to 5volts stores 63 J; if it discharged at aconstant rate in 1 ms energy isdischarged at a 63 kW rate!)
Stored EnergyCAPACITORS STORE ELECTRIC ENERGY
Week 3bEECS 42, Spring 2005
∫=
==∫
=
=∫
=
==⋅=
Final
Initial
c
Final
Initial
Final
Initial
ccc
Vv
VvdQ vdt
tt
tt
dtdQVv
Vvvdt ivw
2CV212CV
21Vv
Vvdv Cvw InitialFinal
Final
Initial
cc −∫=
===
+vc–
ic
A more rigorous derivation
Week 3bEECS 42, Spring 2005
Example: Current, Power & Energy for a Capacitor
dtdvCi =
–+
v(t) 10 µF
i(t)
t (µs)
v (V)
0 2 3 4 51
t (µs)0 2 3 4 51
1
i (µA) vc and q must be continuousfunctions of time; however,ic can be discontinuous.
)0()(1)(0
vdiC
tvt
+= ∫ ττ
Note: In “steady state”(dc operation), timederivatives are zero
C is an open circuit
Week 3bEECS 42, Spring 2005
vip =
0 2 3 4 51
w (J)–+
v(t) 10 µF
i(t)
t (µs)0 2 3 4 51
p (W)
t (µs)
2
0 21 Cvpdw
t
∫ == τ
Week 3bEECS 42, Spring 2005
Capacitors in Parallel
21 CCCeq +=
i(t)
+
v(t)
–
C1 C2
i1(t) i2(t)
i(t)
+
v(t)
–
Ceq
Equivalent capacitance of capacitors in parallel is the sumdtdvCi eq=
Week 3bEECS 42, Spring 2005
Capacitors in Series
i(t)C1
+ v1(t) –
i(t)
+
v(t)=v1(t)+v2(t)
–Ceq
C2
+ v2(t) –
21
111CCCeq
+=
Week 3bEECS 42, Spring 2005
Capacitive Voltage DividerQ: Suppose the voltage applied across a series combination
of capacitors is changed by ∆v. How will this affect the voltage across each individual capacitor?
21 vvv ∆+∆=∆
v+∆vC1
C2
+v2(t)+∆v2–
+v1+∆v1–+
–
Note that no net charge cancan be introduced to this node.Therefore, −∆Q1+∆Q2=0
Q1+∆Q1
-Q1−∆Q1
Q2+∆Q2
−Q2−∆Q2
∆Q1=C1∆v1
∆Q2=C2∆v2
2211 vCvC ∆=∆⇒
vCC
Cv ∆+
=∆21
12
Note: Capacitors in series have the same incremental charge.
Week 3bEECS 42, Spring 2005
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
FIXED OUTER PLATES
g1
g2
Week 3bEECS 42, Spring 2005
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
constgg
gggg
gA
gA
gA
gA
VV
VCCCCV
CCCVV
s
o
ssso
12
12
12
21
21
21
21
21
1 )2(
−=
+−
=+
−=
+−
=+
+−=
εε
εεC1
C2
Vs
–Vs
Vo
Circuit model
Week 3bEECS 42, Spring 2005
• 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
Practical Capacitors
Week 3bEECS 42, Spring 2005
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:
iLvL(t)
dtdiLtv L
L =)(
+
_
Note: In “steady state” (dc operation), timederivatives are zero L is a short circuit
Week 3bEECS 42, Spring 2005
Symbol:
Units: Henrys (Volts • second / Ampere)
Current in terms of voltage:
Note: iL must be a continuous function of time
+vL–
iL
∫ +=
=
t
tLL
LL
tidvL
ti
dttvL
di
0
)()(1)(
)(1
0ττ
L
(typical range of values: µH to 10 H)
Week 3bEECS 42, Spring 2005
Schematic Symbol and Water Model of an Inductor
Week 3bEECS 42, Spring 2005
Stored Energy
Consider an inductor having an initial current i(t0) = i0
20
2
21
21)(
)()(
)()()(
0
LiLitw
dptw
titvtp
t
t
−=
==
==
∫ ττ
INDUCTORS STORE MAGNETIC ENERGY
Week 3bEECS 42, Spring 2005
Inductors in Series
21 LLLeq +=
( )dtdiL
dtdiLL
dtdiL
dtdiLv eq=+=+= 2121
v(t)L1
+ v1(t) –
v(t)
+
v(t)=v1(t)+v2(t)
–
Leq
L2
+ v2(t) –
+–
+–
i(t) i(t)
Equivalent inductance of inductors in series is the sum
dtdiLv eq=
Week 3bEECS 42, Spring 2005
L1i(t)
i2i1
Inductors in Parallel
[ ]
)()()( with 111
)()(11
)(1)(1
0201021
020121
022
011
21
0
00
tititiLLL
titidvLL
i
tidvL
tidvL
iii
eq
t
t
t
t
t
t
+=+=⇒
++⎥⎦
⎤⎢⎣
⎡+=
+++=+=
∫
∫∫
τ
ττ
L2
+
v(t)
–
Leqi(t)
+
v(t)
–
)(10
0
tidvL
it
teq
+= ∫ τ
Week 3bEECS 42, Spring 2005
Capacitor
v cannot change instantaneouslyi can change instantaneouslyDo not short-circuit a chargedcapacitor (-> infinite current!)
n cap.’s in series:
n cap.’s in parallel:
Inductor
i cannot change instantaneouslyv can change instantaneouslyDo not open-circuit an inductor with current (-> infinite voltage!)
n ind.’s in series:
n ind.’s in parallel:
Summary
∑
∑
=
=
=
=
n
iieq
n
i ieq
CC
CC
1
1
11
2
21 Cvw
dtdvCi
=
=
2
21 Liw
dtdiLv
=
=
∑
∑
=
=
=
=
n
i ieq
n
iieq
LL
LL
1
1
11