ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Lecture 11Inductance and Capacitance
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Goals1. Find the current (voltage) for a capacitance or inductance given the voltage (current) as a function of time.
2. Compute the capacitances of parallel-plate capacitors.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
3. Compute the stored energies in capacitances or inductances.
4. Describe typical physical construction of capacitors and inductors and identify parasitic effects.
5. Find the voltages across mutually coupled inductances in terms of the currents.
Goals
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Capacitor
Energy is stored in the electric field that exists between the plates when the capacitor is charged.
vqC =
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Capacitance
( ) ( ) ( )0
0
tqdttitqt
t
+= ∫
( ) ( ) ( )0
0
1 tvdttiC
tvt
t
+= ∫
Cqv
vqC ==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Does DC current flow through a capacitor? No! Only AC current flows through a capacitor!
Capacitance
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Determining Current for a Capacitance Given Voltage
Find i(t) given v(t):
dttdv
dttdvCti
tvtCvtq)(10)()(
)(10)()(
6
6
−
−
==
==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Determining Current for a Capacitance Given Voltage
( ) Adt
tdvCti
sV
sxV
dttdv
tidtdv
Axxdt
tdvCti
sVx
sxV
dttdv
101010)()(
1010110)(
0)(0
510510)()(
10510210)(
76
76
66
66
−=−==
−=−
=
=→=
===
==
−
−
−
−From 0 to 2 μs:
from 2 to 4 μs:
from 4 to 5 μs:
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Determining Voltage for a Capacitance Given Current
0)0()10sin(5.0)( 4 === tqtti
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Determining Voltage for a Capacitance Given Current
[ ]1)10cos(105.0
)10cos(105.0
)10sin(5.0
)0()()(
440
44
0
4
0
−−=
−=
=
=+=
−
−
∫
∫
tx
tx
dtt
tqdttitq
t
t
t
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Determining Voltage for a Capacitance Given Current
[ ])10cos(150010
)()()( 47 ttq
Ctqtv −===
−
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Stored Energy
CtqtqtvtCvvdvCdt
dtdvvCdttptw
dtdvvCtitvtp
tvt
t
t
t2
)()()(21)(
21)()(
)()()(
22
)(
000
======
==
∫∫∫
Ctqtv
tvtqC )()()()(
==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Current, Power and Energy for a Capacitance
53105
310101010)(
)()(
53)5(500311000101000)(
3
3
<<−=
<<=<<=
=
<<−=<<=<<=
−
−
tx
ttforxti
dttdvCti
ttttforttv
C = 10μF
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
53)5(5.23101010
)()()(
<<−=<<=<<=
=
ttttfort
titvtp
Current, Power and Energy for a Capacitance
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Current, Power and Energy for a Capacitance
53)5(25.1
315105
)(21)(
2
2
2
<<−=
<<=<<=
=
tt
ttfort
tCvtw
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 3.2The current through a 0.1 μF capacitor is shown below. Find the charge, voltage, power and stored energy as functions of time and plot them to scale versus time.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
∫ ∫
∫
∫
−
−
−
<<+−=+−=
<<==
=+=
−−−−
−−
t
x
x
x
t
mstmsxtxdtxdtx
msttxdtx
tqdttitq
3
3
3
102
63102
0
33
102
0
33
0
42104101101101
20101101
)0()()(
Charge, Voltage, Power and Stored Energy
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Charge, Voltage, Power and Stored Energy
mstmst
mstt
Ftq
Ctqtv
421040
2010
1.0)()()(
4
4
<<−=
<<=
==μ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Charge, Voltage, Power and Stored Energy
mstmstx
mstttvtitp
42101040
2010)()()(
3 <<+−=
<<==
−
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Charge, Voltage, Power and Stored Energy
mstmstx
mstt
dttptwt
42)1040(105.0
205
)()(
247
20
<<−=
<<=
=
−
∫
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Capacitances in Parallel
( )
321
321321321
33
22
11
CCCCdtdVC
dtdVCCC
dtdVC
dtdVC
dtdVCiiii
dtdVCi
dtdVCi
dtdVCi
eq
eq
++=
=++=++=++=
=
=
=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Capacitances in Series
321
03210302010
030201221
1111
)(111)(1)(1)(1)(1
)(1)(1)(1
CCCC
dttiCCC
dttiC
dttiC
dttiC
dttiCC
Qv
dttiC
dttiC
dttiC
vvvv
eq
ttttt
eqeq
ttt
++=
⎥⎦
⎤⎢⎣
⎡++=++===
++=++=
∫∫∫∫∫
∫∫∫
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Capacitance of the Parallel-Plate Capacitor
WLAdAC == ε
mF 1085.8 120
−×≅ε
0εεε r=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Capacitance of the Parallel-Plate Capacitor
( ) nFFxmx
mm
FxdAC
mxmmd
mmxmxcmcmWLA
77.110177010102.010854.8
1011.0
02.0)1020)(1010()20)(10(
124
212
0
4
222
==⎥⎥⎦
⎤
⎢⎢⎣
⎡==
==
====
−−
−
−
−−
ε
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Design a 1 μF Capacitor
mmmFx
mxFxW
CdL
polyestermxmd
mcmWd
WLdAFxFC
r
r
rr
93.24)02.0)(/10854.8)(4.3(
)1015)(101(
)(4.3101515
02.02
1011
12
66
0
6
006
===
===
==
====
−
−−
−
−
εε
εμ
εεεεμ
Impractical!
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Parasitic Elements
Rs = Resistivity of the capacitor plates
LS = inductance due to current flow into and out of the capacitor
RP = Leakage resistance of the dielectric
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Parasitic Elements
At t = 0 C1 is charged up to 100V, C2 is discharged
The total energy wtotal:
mJwwww
mJVFvCw
total 50
5)100)(10(21
21
21
2
262111
=+==
=== −
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Parasitic Elements
At t=0 the switch is closed, and the charge q on the capacitor plates redistributes:
Cqqqq
CVFxvCq
total μ
μ
1000
100)100)(101(
21
2
6111
=+==
=== −
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Parasitic Elements
mJwww
mJVFvCw
mJVFvCw
VFC
Cq
v
FCCC
total
eq
eq
eq
teq
eq
5.2
25.1)50)(10(21
21
25.1)50)(10(21
21
502
100
2
21
26222
26211
21
=+=
===
===
===
=+=
−
−
μμ
μ
Where did the missing energy go?
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Inductance
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Joseph Henry (1797 – 1878)
Writes Henry: "I may however mention one fact which I have not seen noticed in any work ...when a small battery is moderately excited... if a wire thirty or forty feet long be used instead of the short wire, though no spark will be perceptible when the connection is made, yet when it is broken by drawing one end of the wire from its cup of mercury, a vivid spark is produced."
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
The polarity of the voltage is such as to oppose the change in current (Lenz’s law).
Inductance
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
( )dtdiLtv =
( ) ( ) ( )0
0
1 tidttvL
tit
t
+= ∫
( ) ( )tLiLididtdtdiLidttptw
tit t2
)(
00 021)( ∫∫ ∫ ====
Inductance
dtditLititvtp )()()()( ==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Inductor Current with Constant Applied Voltage
tAdtVH
tidttvL
titt
t
)5()10(21)()(1)(
00
0
==+= ∫∫
What happens if we open the switch at t = 1 sec?
dtdiLtv =)(
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Water Hammer
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Find the Current given the Voltage
AtitidtL
tistfor
AtHxs
Vxtdtx
Ltistfor
tidttvL
tit
t
1.0)0()2(01)(42
1.0)10150(2
)105.7()105.7(1)(20
)()(1)(
4
2
2
0
22
06
66
0
0
====+=<<
===<<
+=
∫
∫
∫
−
μ
μ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Series Inductances
( )
321
321321221)(
LLLL
dtdiLLL
dtdiL
dtdiL
dtdiLvvv
dtdiLtv
eq
eq
++=
++=++=++==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Parallel Inductances
321
0321030201
320
1
11
)(1)(1)(1)(1
)0()(1)(
LLLL
dttvLLL
dttvL
dttvL
dttvL
iiitidttvL
ti
eq
tttt
t
eq
++=
⎟⎟⎠
⎞⎜⎜⎝
⎛++
=++=
++==+=
∫∫∫∫
∫
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Equivalent Inductance
HHHLeq 532 =+=HHH
HHHHLeq 5.2
1025
55)5)(5( 2
==+
=
HHHLeq 5.35.21 =+=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
CP = Capacitance between the coil windings
RP = Resistance of the coil windings
RS = Resistance of connecting wires