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Chapter 3Inductance and Capacitance
Goal1. Current (voltage) for a Capacitance or Inductance
given the voltage (current) as a function of time.
2. Capacitance of a Parallel-plate Capacitor.
3. Stored Energy in a Capacitance or Inductance.Passive Elements : Resistor, Capacitor, Inductor
Not Generate Energy, But Store Energy
4. Typical Physical Construction of Capacitors and Inductors
5. Voltages across mutually coupled inductances in terms of the currents.
CAPACITANCE
Positive Charge is Balanced by Negative Charge of Equal Amount
Water Pressure : PotentialWater Amount : Charge
dtdvCi =
)(Cvdtd
dtdqi ==
Cvq =q : stored charge
C : capacitance
Stored Charge
at fixed Voltage
Unit : Farad = Coulombs / Volt
∫ +=t
ttvdtti
Ctv
0
)()(1)( 0
Ctqtv )()( 0
0 =∫ +=t
t Ctqdtti
Ctv
0
)()(1)( 0
∫ +=t
ttqdttitq
0
)()()( 0
Voltage in Terms of Current
Cvq =
Initial voltage
Example of Current in terms of Voltage
)(10)( 6 tvCvtq −==
dttdv
dttdvCti )(10)()( 6−== sV
dttdv /105)( 6×=
AdttdvCti 5)()( ==
sVdttdv /10)( 7−=
AdttdvCti 10)()( −=
Example of Voltage in terms of Current
)10sin(5.0)( 4 tti =
∫ +=t
ttqdttitq
0
)()()( 0
∫ −×==t
ttdtttq
0
))10cos(1(105.0)10sin(5.0)( 444
))10cos(1(500)()( 4 tCtqtv −×==
Ctq
2)(2
=
)()(21 tqtv=
)(21 2 tCv=
∫=)(
0
tvCvdv∫=
t
tdt
dtdvCv
0∫=t
tdttptw
0
)()(
dtdvCvtp =)()()()( titvtp =
Stored Energy
powerdtdvCi =
energy
dtdvCi 11 =
CAPACITANCES IN SERIES AND PARALLEL
dtdvCi 22 = dt
dvCi 33 =
321 iiii ++=
dtdvC
dtdvC
dtdvC 321 ++=
dtdvCCC )( 321 ++=
321 CCCCeq ++=dtdvCeq=
Capacitances in Parallel
321 1111
CCCCeq ++
=
Capacitances in Series
∫=t
tdtti
Ctv
0
)(1)(
KCL
KVL 321 vvvv ++=
∫++=t
tdtti
CCC 0
)(]111[321
∫=t
teq
dttiC 0
)(1
PHYSICAL CHARACTERISTICS OF CAPACITORS
0εεε r=
dAC ε
=
rε0ε = 8.85 ⅹ 10-12 F/m
: relative dielectric constant 78.54.33.47.05.51.0
WaterQuartzPolysterMicaDiamondAir
Materials rε
Real Capacitor
Parasitic Element Rs, Ls, Rp
INDUCTANCE
dtdiLtv =)( dttv
Ldi )(1=
∫∫ =t
t
ti
tidttv
Ldi
00
)(1)(
)(
)()(1)( 00
tidttvL
tit
t+= ∫
L : Inductance Unit : HH : Volt sec/Ampere
)(21)( 2 tLitw =∫=
)(
0
tiLidi∫=
t
tdt
dtdiLi
0∫=t
tdttptw
0
)()(
dtditLitp )()( =)()()( tvtitp =
Stored Energy
dtdiLtv =)(power
energy
Inductor Current with Constant Voltage
t=0 : i=0 because Open Switch
)()(1)( 00
tidttvL
tit
t+= ∫
i(to)=0
051021)(
0
>== ∫ tAdttit
t
dtdiLtv =)(
If Open Switch at t=1s, di/dt=-Infinite Infinite Voltage : Impulse Occur (Surge)
INDUCTANCES IN SERIES AND PARALLEL
Practical Inductor
Examples of Series & Parallel
Practical Application : Electronic Flash
Power Transfer From Battery to Flash is not Possible1) Battery Voltage : a few ten volts2) Maximum Power Transfer : 1W
Electronic Switch : 10,000 times / sec ON/OFFduring ON : Battery cause to Build up Current in Inductorduring OFF : Inductor force Currents to flow through diode to Charge Capacitor
Diode Prevent Charge from CapacitorMultiple On/OFF Build-up several Hundred Voltage at CapacitorFlash Switch On Flash Discharge
Electronic Switch
Diode
Capacitor
Flash Tube
Battery
Voltage : Water PressureCurrent : Water Amount
MUTUAL INDUCTANCE
LVDT (Linear Variable Differential Transformer)
)cos()( tKxtvo ω=
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