Energy-Storage ElementsCapacitance and Inductance

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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-plateCapacitor:

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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:

Constant of proportionality is the capacitance, CUnits are farads (F)Units equivalent to Coulombs per voltFarad 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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q Cv

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

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i dq

dt

d

dtCv C

dv

dt

dt

tdvCti

)()(

Example 3.1

Plot the current vs. time

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Stored Energy in a Capacitor

Remember:

For an ideal capacitor:

For an ideal, uncharged capacitor (v(t0) = 0):

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p t v t i t

p t Cvdv

dt

tCvtw 2

2

1

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.

where ε is the dielectric constant of the material BETWEEN the plates

For vacuum, the dielectric constant is

For other materials, where εr is the relative dielectric constant

See Table 3.1 on page 135 of textbook

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0 8.85 10 12 F/m

r0

C A

d

WL

d

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, LUnits are henries (H)Units equivalent to volt-seconds per

amperesUsually deal with inductances from

0.001μH to 100 H

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Stored Energy in an Inductor

Remember:

For an ideal inductor:

For an ideal inductor with i(t0) = 0:

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p t v t i t

p t Li t di

dt

w t 1

2Li2 t

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

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rmsrms VN

NV

tvN

Ntv

11

22

11

22 )()(

Ideal Transformers

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Power Transmission Losses

Power Line Losses

Large Voltages and Small CurrentsSmaller Line Loss

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2rmslineloss IRP

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 Voltage Current Power Energy Series Parallel

Inductance Voltage Current Power Energy Series Parallel Mutual Inductance Transformers

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