department of electronic engineering basic electronic engineering inductance and capacitance
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Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Inductance and Capacitance
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Objectives1. Find the current (voltage) for a capacitance
or inductance given the voltage (current) as a function of time.
2. Compute the capacitance of a parallel-plate capacitor.
3. Compute the stored energy in a capacitance or inductance.
4. Describe typical physical construction of capacitors and inductors
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitors and Capacitance
• Capacitance – the ability of a component to store energy in the form of an electrostatic charge
• Capacitor – is a component designed to provide a specific measure of capacitance
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitors and Capacitance• Capacitor Construction
– Plates
– Dielectric
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitor Charge• Electrostatic Charge Develops• Electrostatic Field Stores energy
Insert Figure 12.2
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitor Discharge
Insert Figure 12.3
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitors and Capacitance
• Capacity – amount of charge that a capacitor can store per unit volt applied
where C = the capacity (or capacitance) of the component, in
coulombs per volt Q = the total charge stored by the component V = the voltage across the capacitor corresponding to the
value of Q
CVQV
QC or
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitance
Insert Figure 12.4
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitance
• Unit of Measure – farad (F) = 1 coulomb per volt (C/V)
• Capacitor Ratings– Most capacitors rated in the picofarad (pF) to
microfarad (F) range– Capacitors in the millifarad range are commonly rated
in thousands of microfarads: 68 mF = 68,000 F– Tolerance
• Usually fairly poor• Variable capacitors used where exact values required
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitors and Capacitance• Physical Characteristics of Capacitors
where C = the capacity of the component, in farads
(8.85 X 10-12) = the permittivity of a vacuum, in farads per meter (F/m) or expressed as o
r = the relative permittivity of the dielectric A = the area of either plate d = the distance between the plates (i.e., the thickness
of the dielectric)
d
AxC r)1085,8( 12
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitance of the Parallel-Plate Capacitor
WLAd
AεC
mF 1085.8 120
ε
0 r
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitance
CvQ
dt
dvCi
0)(0)(
ti
t
tvFor DC
It acts as a voltage source
t
vC
t
Cv
t
Q
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Voltage in terms of Current
C
tqdtti
Ctv
t
t
0
0
1
0
0
tqdttitqt
t
, q(to) is the initial charge
C
qvCvq ,
0
0
1tvdtti
Ctv
t
t
C
tqtv 00
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Stored Energy
)()()( titvtp t
tvCti
)(
)(
t
vCvtp
)(
t
t
tv
tv
t
t o oo
Cvdvtwdtdt
dvCvtwdttptw
)(
)()(,)(,)()(
)(2
1)(
2
1)( 22
otCvtCvtw
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Series Capacitors
• Series Capacitors
Where CT = the total series capacitance Cn = the highest-numbered capacitor in the string
n
T
CCC
C1
.....11
1
21
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Parallel Capacitors
• Connecting Capacitors in Parallel
where Cn = the highest-numbered capacitor in the parallel
circuit
nT CCCC .....21
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Inductance
• Unit of Measure – Henry (H)– Inductance is measured in volts per rate of change in
current– When a change of 1A/s induces 1V across an inductor,
the amount of inductance is said to be 1 H
Insert Figure 10.5dt
diLvL
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Inductance• Induced Voltage
where vL = the instantaneous value of induced voltage L = the inductance of the coil, measured in henries (H)
= the instantaneous rate of change in inductor current (in amperes per second)
dt
diLvL
dt
di
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Inductance
dt
diLtv
0)(0)(
tvdt
tdiFor DC
It acts as a short circuit
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Current in terms of Voltage
dttvL
dit
t
ti
ti oo )(
1)(
)(
vdtL
didt
diLv
1,
t
to
o
dttvL
titi )(1
)()(
)()(1
)( o
t
ttidttv
Lti
o
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Stored Energy
)()()( titvtp dt
tdiLtv
)()(
dt
tditLi
dt
tdiLtitvtitp
)()(
)()()()()(
t
t
ti
ti
t
t o oo
Liditwdtdt
diLitwdttptw
)(
)()(,)(,)()(
)(2
1)(
2
1)( 22
otLitLitw
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Inductance
Insert Figure 10.8
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Connecting Inductors in Series
• Series-Connected Coils
where Ln = the highest-numbered inductor in the circuit
nT LLLLL ...321
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Characteristic of Capacitor and Inductor Under AC Excitation
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Connecting Inductors in Parallel• Parallel-Connected Coils
where Ln = the highest-numbered inductor in the circuit
n
T
LLL
L1
....11
1
21
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Alternating Voltage and Current Characteristics
• AC Coupling and DC Isolation: An Overview– DC Isolation – a capacitor prevents flow of charge once
it reaches its capacity
Insert Figure 12.6
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
AC Coupling and DC Isolation• AC Coupling – DC offset is blocked
Insert Figure 12.7
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitor Current
where iC = the instantaneous value of capacitor current C = the capacity of the component(s), in farads
= the instantaneous rate of change in capacitor voltage
dt
dvCiC
dt
dv
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Alternating Voltage and Current Characteristics
• Sine-Wave Values of
– reaches its maximum value when v = 0
Insert Figure 12.8
dt
dv
dt
dvCiC
dt
dv
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
The Phase Relationship Between Capacitor Current and Voltage
• Current leads voltage by 90°
• Voltage lags current by 90°
)90sin(
cos/
sin
otCV
tCVdtCdvi
tVv
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitive Reactance (XC)• Series and Parallel Values of XC
Insert Figure 12.18
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitive Reactance (XC)• Capacitor Resistance
– Dielectric Resistance – generally assumed to be infinite
– Effective Resistance – opposition to current, also called capacitive reactance (XC)
Insert Figure 12.15
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitive Reactance (XC)
• Calculating the Value of XC
CfX
I
VX C
rms
rmsC 2
1or
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitive Reactance (XC)• XC and Ohm’s Law
– Example: Calculate the total current below
Insert Figure 12.17
mAV
X
VI
FHzfX
c
scc 26.8
121
10,121
)22)(60(2
1
2
1
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
The Phase Relationship Between Inductor Current and Voltage
• Sine-Wave Values of
– reaches its maximum value when i = 0
Insert Figure 10.9
dt
di
dt
diLvL
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
The Phase Relationship Between Inductor Current and Voltage
• Voltage leads current by 90°
• Current lags voltage by 90°
)90sin(
cos/
sin
otLI
tLIdtiLdv
tIi
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Inductive Reactance (XL)• Inductor Opposes Current
Insert Figure 10.15
kmA
V
I
VOpposition
rms
rms 101
10
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Inductive Reactance (XL)
• Inductive Reactance (XL) – the opposition (in ohms) that an inductor presents to a changing current
• Calculating the Value of XL
LfXI
VX L
rms
rmsL 2or
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Inductive Reactance (XL)
• XL and Ohm’s Law– Example: Calculate the total current below
mAK
V
X
VI
L
s 121
12
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Capacitive Versus Inductive Phase Relationships
• Voltage (E) in inductive (L) circuits leads current (I) by 90° (ELI)
• Current (I) in capacitive (C ) circuits leads voltage (E) by 90° (ICE)
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Alternating Voltage and Current Characteristics
Insert Figure 12.10
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Figure 4.23
Euler’s identity
tfAtA
f
t
2coscos
2
In Euler expression,
A cos t = Real (Ae j t )
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
C
tjtj
tj
tj
ZCji
v
vCjiisthatAeCjithenAevif
dt
dvCiCapacitorFor
Aejdt
dy
Aey
1
,,,
,
( it is called the impedance of a capacitor)
L
tjtj
ZLji
vijLv
AeLjvthenAeiif
dt
diLvInductorFor
,
.,
,
( it is called the impedance of an inductor)
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Figure 4.29
The impedance element
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Figure 4.33
Impedances of R, L, and C in the complex plane
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Figure 4.37
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Figure 4.41
An AC circuit
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Figure 4.44
AC equivalent circuits
Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING
Figure 4.45
Rules for impedance and admittance reduction
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