new topics – energy storage elements capacitors inductorsee42/sp05/pdf/week 3b.pdf · you might...
TRANSCRIPT
![Page 1: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/1.jpg)
Week 3bEECS 42, Spring 2005
New topics – energy storage elementsCapacitorsInductors
![Page 2: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/2.jpg)
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
![Page 3: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/3.jpg)
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
![Page 4: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/4.jpg)
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)
![Page 5: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/5.jpg)
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
![Page 6: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/6.jpg)
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
![Page 7: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/7.jpg)
Week 3bEECS 42, Spring 2005
![Page 8: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/8.jpg)
Week 3bEECS 42, Spring 2005
Schematic Symbol and Water Model for a Capacitor
![Page 9: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/9.jpg)
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
![Page 10: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/10.jpg)
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
![Page 11: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/11.jpg)
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
![Page 12: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/12.jpg)
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
![Page 13: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/13.jpg)
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=
![Page 14: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/14.jpg)
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
+=
![Page 15: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/15.jpg)
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.
![Page 16: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/16.jpg)
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
![Page 17: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/17.jpg)
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
![Page 18: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/18.jpg)
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
![Page 19: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/19.jpg)
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
![Page 20: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/20.jpg)
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)
![Page 21: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/21.jpg)
Week 3bEECS 42, Spring 2005
Schematic Symbol and Water Model of an Inductor
![Page 22: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/22.jpg)
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
![Page 23: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/23.jpg)
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=
![Page 24: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/24.jpg)
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
![Page 25: New topics – energy storage elements Capacitors Inductorsee42/sp05/pdf/Week 3b.pdf · You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules](https://reader030.vdocument.in/reader030/viewer/2022040214/5eabca49d414377a713a25d6/html5/thumbnails/25.jpg)
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