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Energy-Storage Elements
Capacitance and Inductance
ELEC 308
Elements of Electrical Engineering
Dr. Ron Hayne
Images Courtesy of Allan Hambley and Prentice-Hall
Energy-Storage Elements
 Remember

Resistors convert electrical energy into heat

Cannot store energy!
 Inductors and Capacitors can store energy and
later return it to the circuit


Do NOT generate energy!
Passive elements, like resistors
 Capacitance is a circuit property that accounts
for energy STORED in ELECTRIC fields
 Inductance is a circuit property that accounts for
energy STORED in MAGNETIC fields
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Inductance and Capacitance Uses
 Microphones

Capacitance changes with sound pressure
 Linear variable differential transformer

Position of moving iron core converted into voltage
 Conversion from DC-AC, AC-DC, AC-AC
 Electrical signal filters

Combinations of inductances and capacitances in
special circuits
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Capacitors
 Constructed by separating two sheets of
CONDUCTOR (usually metallic) by a thin layer of
INSULATING material

Insulating material called a DIELECTRIC

Can be air, Mylar®, polyester, polypropylene, mica, etc.
 Parallel-plate
Capacitor:
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Fluid-Flow Analogy
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Stored Charge in Terms of Voltage
 In an IDEAL capacitor

Stored charge, q, is proportional to the voltage
between the plates:
q  Cv

Constant of proportionality is the capacitance, C
Units are farads (F)
 Units equivalent to Coulombs per volt
 Farad is a VERY LARGE amount of capacitance

 Usually deal with capacitances from 1 pF to 0.01 F
 Occasionally, use femtofarads (in computer chips)
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Current in Terms of Voltage
 Remember that current is the
time rate of flow of charge
 In an IDEAL capacitor

The relationship between
current and voltage is
dq d
dv
i
 Cv  C
dt dt
dt
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dv(t )
i (t )  C
dt
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Example 3.1
 Plot the current vs. time
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Stored Energy in a Capacitor
 Remember: pt   v t it 
dv
 For an ideal capacitor: pt   Cv
dt

 For an ideal, uncharged capacitor (v(t0) = 0):
wt   1 Cv 2 t 
2
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Example 3.3
 Plot current, power delivered and energy stored
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Capacitances in Parallel
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Capacitances in Series
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Parallel-Plate Capacitors
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Parallel-Plate Capacitors
 If d<<W and d<<L, the capacitance is approx.
A WL
C
d

d
where ε is the dielectric constant of the material
BETWEEN the plates
 For vacuum,
the dielectric constant is

  0  8.85 1012 F/m
 For other materials,   r0
where εr is the relative dielectric constant
 See Table 3.1 on page 135 of textbook

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Practical Capacitors
 Dimensions of 1μF parallel-plate capacitors are TOO LARGE
for portable electronic devices
 Plates are rolled into smaller area
 Small-volume capacitors require very thin dielectrics (with
HIGH dielectric constant)


Dielectric materials break down when electric field intensity is
TOO HIGH (become conductors)
Real capacitors have MAXIMUM VOLTAGE RATING
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Electrolytic Capacitors





One plate is metallic aluminum or tantalum
Dielectric is OXIDE layer on surface of the metal
Other “plate” is ELECTROLYTIC SOLUTION
Metal plate is immersed in the electrolytic solution
Gives high capacitance per unit volume

Requires that ONLY ONE polarity of voltage can be
applied
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Inductors
 Constructed by coiling a wire around some
type of form
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Voltage in Terms of Current
 In an IDEAL inductor


Voltage across the coil is
proportional to the time rate of
change of the current
Constant of proportionality is
the inductance, L
Units are henries (H)
 Units equivalent to volt-seconds
per amperes
 Usually deal with inductances
from 0.001μH to 100 H

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Stored Energy in an Inductor
 Remember: pt   v t it 
di
 For an ideal inductor: pt   Lit 
dt

 For an ideal inductor with i(t0) = 0:
1 2
w
t   Li t 
2
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Example 3.6
 Plot voltage, power, and energy
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Equivalent Inductance
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Practical Inductors
 Cores (metallic iron forms) are made of thin
sheets called laminations
 Voltages are induced in the core by the
changing magnetic fields

Cause eddy currents to flow in the core



Dissipate energy
Results in UNDESIRABLE core loss
Can reduce eddy-current core loss



Laminations
Ferrite (iron oxide) cores
Powdered iron with insulating binder
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Electronic Photo Flash
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Mutual Inductance
 Several coils wound on the same form


Magnetic flux produced by one coil links the others
Time-varying current flowing through one coil
induces voltages on the other coils
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Mutual Inductance
 Flux of one coil aids the flux produced by the other
coil
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Ideal Transformers
v 2 (t ) 
V2 rms
N2
v1 (t )
N1
N2

V1rms
N1
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Ideal Transformers
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Power Transmission Losses
 Power Line Losses

2
Ploss  RlineI rms
 Large Voltages and Small Currents
 Smaller Line Loss
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Power Transmission
 Step-Up and Step-Down Transformers

99% Efficiency (vs. 50% with no transformers)
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U.S. Power Grid
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Summary
 Capacitance






 Inductance
Voltage
Current
Power
Energy
Series
Parallel



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


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Voltage
Current
Power
Energy
Series
Parallel
Mutual Inductance
Transformers
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