OUTLINE

1
Lecture 9

• OUTLINE
• The capacitor
• The inductor
• Chapter 3

Reading
2
The Capacitor

• Two conductors (a,b) separated by an insulator
• difference in potential Vab
• gt 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

(stored charge in terms of voltage)

• Parallel-plate capacitor
• area of the plates A
• separation between plates d
• dielectric permittivity of insulator ?
• gt capacitance

3
Symbol Units Farads (Coulombs/Volt) Current-V
oltage relationship
C
C
(typical range of values 1 pF to 1 mF)
ic
vc
Note Q (vc) must be a continuous function of
time
4
Voltage in Terms of Current
5
Op-Amp Integrator
C
ic
vC
R
in
vin

vn
vo
vp
6
Stored Energy

• You might think the energy stored on a capacitor
  is QV, 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.

Example A 1 pF capacitance charged to 5 Volts
has ½(5V)2 (1pF) 12.5 pJ
7
A more rigorous derivation
ic
vc
8
Example Current, Power Energy for a Capacitor
i(t)
v (V)

v(t)
10 mF
1
t (ms)
0
2
3
4
5
1
i (mA)
vc must be a continuous function of time
however, ic can be discontinuous.
t (ms)
0
2
3
4
5
1
Note In steady state (dc operation),
time derivatives are zero ? C is an open circuit
9
p (W)
i(t)

v(t)
10 mF
t (ms)
0
2
3
4
5
1
w (J)
t (ms)
0
2
3
4
5
1
10
Capacitors in Parallel
i1(t)
i2(t)
v(t)
i(t)
C1
C2
v(t)
i(t)
Ceq
Equivalent capacitance of capacitors in parallel
is the sum
11
Capacitors in Series
v1(t)
v2(t)
v(t)v1(t)v2(t)
C1
C2
i(t)
i(t)
Ceq

12
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?

DQ1C1Dv1
Note that no net charge can can be introduced to
this node. Therefore, -DQ1DQ20
Q1DQ1
v1Dv1
C1
-Q1-DQ1
vDv

v2(t)Dv2
Q2DQ2
C2
-Q2-DQ2
DQ2C2Dv2
Note Capacitors in series have the same
incremental charge.
13
Application Example MEMS Accelerometer

• Capacitive position sensor used to measure
  acceleration (by measuring force on a proof mass)

g1
g2
FIXED OUTER PLATES
14
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
Vo
C2
Vs
15
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

16
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

vL(t)
iL
_
Note In steady state (dc operation),
time derivatives are zero ? L is a short circuit
17
Symbol Units Henrys (Volts second /
Ampere) Current in terms of voltage
L
(typical range of values mH to 10 H)
iL
vL
Note iL must be a continuous function of time
18
Stored Energy

• Consider an inductor having an initial current
  i(t0) i0

)
(
)
(
)
(
t
i
t
v
t
p
t
ò


t
t
)
(
)
(
d
p
t
w
t
0
1
1
2
-

2
)
(
Li
Li
t
w
0
2
2
19
Inductors in Series
v1(t)
v2(t)
v(t)v1(t)v2(t)
L1
L2
i(t)
i(t)


v(t)
v(t)
Leq
Equivalent inductance of inductors in series is
the sum
20
Inductors in Parallel
v(t)
v(t)
i2
i1
i(t)
i(t)
Leq
L1
L2
21
Summary

• Capacitor
• v cannot change instantaneously
• i can change instantaneously
• Do not short-circuit a charged
• capacitor (-gt infinite current!)
• n cap.s in series
• n cap.s in parallel

• Inductor
• i cannot change instantaneously
• v can change instantaneously
• Do not open-circuit an inductor with current (-gt
  infinite voltage!)
• n ind.s in series
• n ind.s in parallel