department of electronic engineering basic electronic engineering inductance and capacitance

Department of Electronic EngineeringBASIC ELECTRONIC ENGINEERING

Inductance and Capacitance


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


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


Capacitors and Capacitance• Capacitor Construction

– Plates

– Dielectric


Capacitor Charge• Electrostatic Charge Develops• Electrostatic Field Stores energy

Insert Figure 12.2


Capacitor Discharge

Insert Figure 12.3


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


Capacitance

Insert Figure 12.4


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


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


Capacitance of the Parallel-Plate Capacitor

WLAd

AεC

mF 1085.8 120

ε

0 r


Capacitance

CvQ

dt

dvCi

0)(0)(

ti

t

tvFor DC

It acts as a voltage source

t

vC

t

Cv

t

Q


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


Stored Energy

)()()( titvtp t

tvCti

)(

)(

t

vCvtp

)(

t

t

tv

tv

t

t o oo

Cvdvtwdtdt

dvCvtwdttptw

)(

)()(,)(,)()(

)(2

1)(

2

1)( 22

otCvtCvtw


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


Parallel Capacitors

• Connecting Capacitors in Parallel

where Cn = the highest-numbered capacitor in the parallel

circuit

nT CCCC .....21


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


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


Inductance

dt

diLtv

0)(0)(

tvdt

tdiFor DC

It acts as a short circuit


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


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


Inductance

Insert Figure 10.8


Connecting Inductors in Series

• Series-Connected Coils

where Ln = the highest-numbered inductor in the circuit

nT LLLLL ...321


Characteristic of Capacitor and Inductor Under AC Excitation


Connecting Inductors in Parallel• Parallel-Connected Coils

where Ln = the highest-numbered inductor in the circuit

n

T

LLL

L1

....11

1

21


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


AC Coupling and DC Isolation• AC Coupling – DC offset is blocked

Insert Figure 12.7


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



• Sine-Wave Values of

– reaches its maximum value when v = 0

Insert Figure 12.8

dt

dv

dt

dvCiC

dt

dv


The Phase Relationship Between Capacitor Current and Voltage

• Current leads voltage by 90°

• Voltage lags current by 90°

)90sin(

cos/

sin

otCV

tCVdtCdvi

tVv


Capacitive Reactance (XC)• Series and Parallel Values of XC

Insert Figure 12.18


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


Capacitive Reactance (XC)

• Calculating the Value of XC

CfX

I

VX C

rms

rmsC 2

1or


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


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


The Phase Relationship Between Inductor Current and Voltage

• Voltage leads current by 90°

• Current lags voltage by 90°

)90sin(

cos/

sin

otLI

tLIdtiLdv

tIi


Inductive Reactance (XL)• Inductor Opposes Current

Insert Figure 10.15

kmA

V

I

VOpposition

rms

rms 101

10


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


Inductive Reactance (XL)

• XL and Ohm’s Law– Example: Calculate the total current below

mAK

V

X

VI

L

s 121

12


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)



Insert Figure 12.10


Figure 4.23

Euler’s identity

tfAtA

f

t

2coscos

2

In Euler expression,

A cos t = Real (Ae j t )


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)


Figure 4.29

The impedance element


Figure 4.33

Impedances of R, L, and C in the complex plane


Figure 4.37


Figure 4.41

An AC circuit


Figure 4.44

AC equivalent circuits


Figure 4.45

Rules for impedance and admittance reduction

department of electronic engineering basic electronic engineering inductance and capacitance

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