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Inductance and Capacitance
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Objectives
1. 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
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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
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Capacitors and Capacitance
• Capacitor Construction
– Plates
– Dielectric
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Capacitor Charge
• Electrostatic Charge Develops
• Electrostatic Field Stores energy
Insert Figure 12.2
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Capacitor Discharge
Insert Figure 12.3
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Capacitors and Capacitance
• Capacity – amount of charge that a
capacitor can store per unit volt applied
Q
C
V
or
Q  CV
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
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Capacitance
Insert Figure 12.4
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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
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Capacitors and Capacitance
• Physical Characteristics of Capacitors
C  (8,85 x10
12
A
) r
d
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)
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Capacitance of the ParallelPlate Capacitor
εA
C
d
ε0  8.85 10
   r 0
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A  WL
12
Fm
Capacitance
Q  Cv
Q Cv
v

C
t
t
t
dv
iC
dt
For DC
v (t )
 0 i (t )  0
t
It acts as a voltage source
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Voltage in terms of Current
t
qt   
q
q  Cv, v 
C
i t dt  qt0  , q(to) is the initial charge
t0
1
v t  
C
t

t0
qt0 
i t dt 
C
1
v t  
C
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qt0 
v t0  
C
t
 it dt  vt 
0
t0
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Stored Energy
p(t )  v(t )i(t )
w(t ) 

t
to
v(t )
i (t )  C
t
v
p(t )  Cv
t
t
dv
p(t )dt, w(t )  Cv dt, w(t ) 
to
dt

1 2
1 2
w(t )  Cv (t )  Cv (to )
2
2
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
v(t )
Cvdv
v ( to )
Series Capacitors
• Series Capacitors
CT 
1
1
1
1

 .....
C1 C2
Cn
Where
CT = the total series capacitance
Cn = the highest-numbered capacitor in the string
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Parallel Capacitors
• Connecting Capacitors
in Parallel
CT  C1  C2  .....  Cn
where
Cn = the highest-numbered capacitor in the parallel
circuit
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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
di
vL  L
dt
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Insert Figure 10.5
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Inductance
• Induced Voltage
di
vL  L
dt
where
vL = the instantaneous value of induced voltage
L = the inductance of the coil, measured in henries (H)
di
= the instantaneous rate of change in inductor current
dt
(in amperes per second)
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Inductance
di
v t   L
dt
For DC
di(t )
 0 v (t )  0
dt
It acts as a short circuit
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Current in terms of Voltage
di
1
v  L , di  vdt
dt
L

i (t )
1
di 
i ( to )
L

t
to
1
i(t )  i(to ) 
L
1
i (t ) 
L
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
t
to
v(t )dt

t
v(t )dt
to
v(t )dt  i(to )
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Stored Energy
di(t )
v (t )  L
dt
p(t )  v(t )i(t )
di(t )
di(t )
p ( t )  i (t ) v (t )  i ( t ) L
 Li (t )
dt
dt
t
t
i (t )
di
w(t )  p(t )dt, w(t )  Li dt, w(t ) 
Lidi
to
to
i ( to )
dt


1 2
1 2
w(t )  Li (t )  Li (to )
2
2
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
Inductance
Insert Figure 10.8
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Connecting Inductors in Series
• Series-Connected Coils
LT  L1  L2  L3  ...  Ln
where
Ln = the highest-numbered inductor in the circuit
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Characteristic of Capacitor and Inductor
Under AC Excitation
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Connecting Inductors in Parallel
• Parallel-Connected Coils
LT 
1
1 1
1
  .... 
L1 L2
Ln
where
Ln = the highest-numbered inductor in the circuit
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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
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AC Coupling and DC Isolation
• AC Coupling – DC offset is blocked
Insert Figure 12.7
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Capacitor Current
dv
iC  C
dt
where
iC = the instantaneous value of capacitor current
C = the capacity of the component(s), in farads
dv
= the instantaneous rate of change in capacitor voltage
dt
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Alternating Voltage and Current
Characteristics
• Sine-Wave Values of
dv
dt
dv
iC  C
dt
dv
–
reaches its maximum value when v = 0
dt
Insert Figure 12.8
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The Phase Relationship Between
Capacitor Current and Voltage
• Current leads voltage by 90°
• Voltage lags current by 90°
v  V sin t
i  Cdv / dt  CV cost
 CV sin(t  90 o )
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Capacitive Reactance (XC)
• Series and Parallel Values of XC
Insert Figure 12.18
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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
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Capacitive Reactance (XC)
• Calculating the Value of XC
Vrms
XC 
I rms
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or
1
XC 
2 f C
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Capacitive Reactance (XC)
• XC and Ohm’s Law
– Example: Calculate the total current below
Insert Figure 12.17
Xc 
1
2 f

1
V
10V
 121, I c  s 
 8.26mA
2 (60 Hz )( 22 F )
X c 121
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The Phase Relationship Between
Inductor Current and Voltage
di
• Sine-Wave Values of v L  L
dt
–
di reaches its maximum value when i = 0
dt
Insert Figure 10.9
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The Phase Relationship Between Inductor
Current and Voltage
• Voltage leads current by 90°
• Current lags voltage by 90°
i  I sin t
v  Ld i / dt  LI  cos t
 LI  sin(t  90 o )
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Inductive Reactance (XL)
• Inductor Opposes Current
Vrms 10V
Opposition 

 10k
I rms 1mA
Insert Figure 10.15
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Inductive Reactance (XL)
• Inductive Reactance (XL) – the opposition
(in ohms) that an inductor presents to a changing
current
• Calculating the Value of XL
Vrms
XL 
I rms
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or
X L  2 f L
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Inductive Reactance (XL)
• XL and Ohm’s Law
– Example: Calculate the total current below
Vs
12V
I

 12mA
X L 1K
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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)
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Alternating Voltage and Current
Characteristics
Insert Figure 12.10
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Euler’s identity
  t
  2 f
A cos  t  A cos 2 f t
Figure 4.23
In Euler expression,
A cos t = Real (Ae j t )
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y  Ae jt
dy
 jAe jt
dt
dv
For Capacitor, i  C
dt
if v  Ae j t , then i  CjAe j t , that is , i  Cj v
v
1

 Z C ( it is called the impedance of a capacitor)
i j C
di
For Inductor , v L
dt
if i Ae j t , then v  Lj Ae j t .
v
v  jL i,  jL  Z L ( it is called the impedance of an inductor)
i
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The impedance element
Figure
4.29
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Impedances of R, L, and C in the complex plane
Figure
4.33
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Figure 4.37
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An AC circuit
Figure
4.41
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AC equivalent circuits
Figure 4.44
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Rules for impedance and admittance reduction
Figure
4.45
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