my EE notes 4 non-ee

8/8/2019 my EE notes 4 non-ee

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AC Alternating Current

it is time varying signal the movement (or flow) of electric charge periodically reverses

direction.

An electric charge would for instance move forward,then backward, then forward, then backward, over and over again

Different types of signals

Used generically, AC refers to the form in which electricity is delivered tobusinesses and residences. The usual waveform of an AC power circuit is asine wave , however in certain applications, different waveforms are used,such as triangular or square waves .

Types of Alternating signals
http://en.wikipedia.org/wiki/Electricityhttp://en.wikipedia.org/wiki/Waveformhttp://en.wikipedia.org/wiki/AC_powerhttp://en.wikipedia.org/wiki/Sine_wavehttp://en.wikipedia.org/wiki/Triangle_wavehttp://en.wikipedia.org/wiki/Square_wavehttp://en.wikipedia.org/wiki/Electricityhttp://en.wikipedia.org/wiki/Waveformhttp://en.wikipedia.org/wiki/AC_powerhttp://en.wikipedia.org/wiki/Sine_wavehttp://en.wikipedia.org/wiki/Triangle_wavehttp://en.wikipedia.org/wiki/Square_wave


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The sine wave is the fundamental type of alternating current (ac) and alternatingvoltage. It is also referred to as a sinusoidal wave, or, simply, sinusoid.

Sine waves, or sinusoidals, are produced by two types of sources: rotating electricalmachines (ac generators) or electronic oscillator circuits. The electrical service provided by the power companies is in the form of sinusoidal voltage and current.

Why sinusoidal?1. is easy to generate, transmit and distribute2. is easy to handle mathematically3. Simplifies the design of electric system

A sinusoid is a signal that has the form of the sine or cosine function. Sinusoidal varying function can be expressed with either the sine function or

the cosine function Sinusoidal function produces both (natural) transient response and a (forced)

steady-state responseThe natural response dies out with time so that only thesteady-state response remains after a long time.

Sinusoidal varying function; t Y Y m sin= the waveform is representing threebasic waveforms: the waveform of a n AC voltage, the waveform of an ACcurrent, & also the waveform of induced flux.
http://en.wikipedia.org/wiki/File:Waveforms.svg


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t f

1= Where;

t = coefficient of time,period

f t

22

== f = frequency, Hz (cycle/sec)

f 2 = = angular freq. of the sine wave, radian/s t

radianof .180

degreeof .0

nono

=

One way to express the intensity or magnitude (amplitude) of an AC waveformquantity is to measure its peak height (maximum height).

Peak value or Maximum value

Another way is to measure the total height between opposite peaks. This is knownas the peak-to-peak value of an AC waveform


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Peak-to-Peak value

A sine wave varies with time (t) in a definable manner. The time required for agiven sine wave to complete one full cycle is called the period (T).

FREQUENCY


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Frequency is the number of cycles that a sine wave completes in onesecond. Frequency (f) is measured in units of hertz . One hertz (Hz) is equivalent to

one cycle per second .

The relationship between frequency and period is important.

The formulas for this relationship are as follows:

t

1 f ;

1 == f

t

There is a reciprocal relationship between frequency and period.

The more cycles completed in one second, the higher the frequency.

Polarity of a Sine Wave

A sine wave changes polarity at its zero value; that is, it alternatesbetween positive and negative values.


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Average value of a Sinusoid

The average value of a sine wave when taken over one complete cycle is alwayszero because the positive values (above the zero crossing) offset the negativevalues (below the zero crossing).

The average value of a sine wave is defined over a half-cycle rather than over afull cycle

dt t YmT

Yave T sin2/

1 2/0

=

2/

0cos

2 T t

Tw

Ym =

Ym

Yave2=

Root-mean-square (RMS) value of a Sinusoid

The term RMS stands for root mean square . It refers to the mathematical

process used to derive this value. The RMS value is also referred to as theeffective value . Most AC voltmeters display the rms value of a voltage.

The RMS value of a sinusoidal voltage is actually a measure of the heatingeffect of the sine wave.

The RMS value of a sinusoidal voltage is equal to the dc voltage thatproduces the same amount of heat in a resistance as does thesinusoidal voltage

t Y mT Y r m s

T

22

0s in

1 =


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T t

T t

T

Y m

0

2

) ]2) ( s i n2([

2

=

2

YmYrms =

Form factor

In the form factor of an alternating current waveform (signal) is theratio of the RMS (root-mean-square) value to the average value(mathematical mean of absolute values of all points on the waveform).

In case of a sinusoidal wave form, the form factor is approximately1.11.

11.1Factor Form ==YaveYrms

Phase angle difference

General Expression of Sinusoidal: )sin( = t Y Y m


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Two waveform with different phases. Comparing two different waveforms Y 1 and Y 2 .

Waveform Y 2 is lagging Y 1 by electrical degrees. Or simply , Y 2 lags Y 1 by

Waveform Y 1 is leading Y 2 by electrical degrees. Or simply , Y 1 leads Y 2 by

Respond of the current to the alteration of the source voltage

When a sinusoidal voltage source (Vs) is applied to a resistive circuit, an alternatingsinusoidal current results.

During the positive alternation of the source voltage Vs, the current is in the CW direction.


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When the voltage changes polarity, the current correspondingly changes direction asindicated.

During a negative alternation of the source voltage, the current is in the opposite(CCW) direction.

Elements of AC Circuit

for pure element

IRV =

RV

I =

dt di

LV L =

= L

dt V i L L


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= C dt i

V LC

dt dV C iC C =

Voltage-Current relationship for pure elements

RESISTOR Terminal voltage is t V V mt sin=

Solving for the current, Ri

Rt V

im

R

sin=

t RV

i m R sin=

Where, R

Vm=Im

t I i m R sin=

*** The voltage and the current are in phase

INDUCTOR

Terminal voltage across the inductor is t V V mt sin=

Solving for the current Li

= Ldt V

i L L Where,

=

L

dt t V i m L

sin

L

V i m

m =

Lt V

i m L

cos=

Lt V

i m L

)90sin( 0=


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)90sin( 0= t L

V i m L

)90sin( 0= t ii m L

*** The current is lagging the voltage by 900

CAPACITOR

Terminal voltage across the capacitor is t V V mt sin=

solving for the current, C i

dt dV

C i C C =

dt t dV

C i mC sin=

)cos( t V C i mC =

C

t V i mC 1

cos=

)90sin()

1(

0+= t

C

V i mC

Where,)

1(

C

V i mm

=

)90sin( 0+= t ii mC

*** The current is leading the voltage by 90 0

Ohms Law in AC Circuit

Resistor load

t V V mt sin=


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t RV

i m R sin=

Inductor Load

t V V mt

sin= )90sin( 0= t

LV

i m L

Inductive Reactance, X L ; Ohms

L X L = where; f 2 =

Lf 2 = L X

Capacitor Load

t V V mt sin=

)90sin()

1(

0+= t

C

V i mC

Capacitive Reactance, X C ; Ohms

C X C

1= where; f 2 =

Cf 21

=C X

Characteristics of Passive Element

What is a resistor?

A resistor is a two- terminal electric component that produces a voltage across its terminals that is proportional to the electric current through it inaccordance with Ohm's law :

V = IR
http://en.wikipedia.org/wiki/Terminal_(electronics)http://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Proportionality_(mathematics)#Direct_proportionhttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Ohm's_lawhttp://en.wikipedia.org/wiki/Terminal_(electronics)http://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Proportionality_(mathematics)#Direct_proportionhttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Ohm's_law


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Resistors are elements of electrical networks and electronic circuits. Practicalresistors can be made of various compounds and films, as well as resistance wire (wire made of a high-resistivity alloy, such as nickel/chrome).

The primary characteristics of a resistor are the resistance , the tolerance ,

maximum working voltage and the power rating. Other characteristicsinclude temperature coefficient .

Resistance

The electrical resistance of an object is a measure of its opposition to thepassage of a steady electric current . An object of uniform cross section willhave a resistance proportional to its length and inversely proportional to itscross-sectional area, and proportional to the resistivity of the material.

Resistivity is a measure of the material's ability to oppose electric current.Resistivity is a measure of the material's ability to oppose electric current.

A L

R =

For a wide variety of materials and conditions, the electrical resistance doesnot depend on the amount of current through or the amount of voltageacross the object, meaning that the resistance R is constant for the giventemperature and material.

AC resistance

If a wire conducts high-frequency alternating current then the effective crosssectional area of the wire is reduced because of the skin effect . If severalconductors are together, then due to proximity effect , the effectiveresistance of each is higher than if that conductor were alone.

The skin effect is the tendency of an alternating electric current (AC) to distribute itself within a conductor so that the current density near the surface of the conductor isgreater than that at its core. That is, the electric current tends to flow at the "skin" of the conductor. The skineffect causes the effective resistance of the conductor toincrease with the frequency of the current. Skin effect isdue to eddy currents set up by the AC current
http://en.wikipedia.org/wiki/Electrical_networkshttp://en.wikipedia.org/wiki/Resistance_wirehttp://en.wikipedia.org/wiki/Resistance_wirehttp://en.wikipedia.org/wiki/Electrical_resistancehttp://en.wikipedia.org/wiki/Engineering_tolerance#Electrical_component_tolerancehttp://en.wikipedia.org/wiki/Electric_power#In_circuitshttp://en.wikipedia.org/wiki/Temperature_coefficienthttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Constanthttp://en.wikipedia.org/wiki/Skin_effecthttp://en.wikipedia.org/wiki/Proximity_effect_(electromagnetism)http://en.wikipedia.org/wiki/Alternating_currenthttp://en.wikipedia.org/wiki/Alternating_currenthttp://en.wikipedia.org/wiki/Conductor_(material)http://en.wikipedia.org/wiki/Electrical_resistancehttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Eddy_currentshttp://en.wikipedia.org/wiki/Electrical_networkshttp://en.wikipedia.org/wiki/Resistance_wirehttp://en.wikipedia.org/wiki/Resistance_wirehttp://en.wikipedia.org/wiki/Electrical_resistancehttp://en.wikipedia.org/wiki/Engineering_tolerance#Electrical_component_tolerancehttp://en.wikipedia.org/wiki/Electric_power#In_circuitshttp://en.wikipedia.org/wiki/Temperature_coefficienthttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Constanthttp://en.wikipedia.org/wiki/Skin_effecthttp://en.wikipedia.org/wiki/Proximity_effect_(electromagnetism)http://en.wikipedia.org/wiki/Alternating_currenthttp://en.wikipedia.org/wiki/Alternating_currenthttp://en.wikipedia.org/wiki/Conductor_(material)http://en.wikipedia.org/wiki/Electrical_resistancehttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Eddy_currents


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What is an Inductor?

An inductor is a passive electrical component that can store energy in a

magnetic field created by the electric current passing through it. Aninductor's ability to store magnetic energy is measured by its inductance , inunits of henries . Typically an inductor is a conducting wire shaped as a coil,the loops helping to create a strong magnetic field inside the coil due toFaraday's law of induction . Inductors are one of the basic electroniccomponents used in electronics where current and voltage change with time,due to the ability of inductors to delay and reshape alternating currents.

Inductance (L) (measured in henries ) is an effect resulting from the magnetic field that forms around a current-carrying conductor that tends to resistchanges in the current. Electric current through the conductor creates a

magnetic flux proportional to the current.

A change in this current creates a change in magnetic flux that, in turn, byFaraday's law generates an electromotive force (EMF) that acts to opposethis change in current. Inductance is a measure of the amount of EMFgenerated per unit change in current.

For example, an inductor with an inductance of 1 henry produces an EMF of 1 volt when the current through theinductor changes at the rate of 1 ampere per second. Thenumber of loops, the size of each loop, and the material it iswrapped around all affect the inductance.

Ideal and real inductors

An "ideal inductor" has inductance, but no resistance , and does not dissipateor radiate energy. A real inductor may be partially modeled by a combinationof inductance, resistance (due to the resistivity of the wire and losses in corematerial).

Inductance is the property in an electrical circuit where a change in theelectric current through that circuit induces an electromotive force (EMF) that opposes the change in current.

In honor of Joseph Henry , the unit of inductance has been given the nameHenry (H) :
http://en.wikipedia.org/wiki/Passive_componenthttp://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Inductancehttp://en.wikipedia.org/wiki/Henry_(unit)http://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Inductancehttp://en.wikipedia.org/wiki/Henry_(unit)http://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Electrical_conductorhttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Magnetic_fluxhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Electromotive_forcehttp://en.wikipedia.org/wiki/Electrical_resistancehttp://en.wikipedia.org/wiki/Electrical_circuithttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Electromotive_forcehttp://en.wikipedia.org/wiki/Joseph_Henryhttp://en.wikipedia.org/wiki/Passive_componenthttp://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Inductancehttp://en.wikipedia.org/wiki/Henry_(unit)http://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Inductancehttp://en.wikipedia.org/wiki/Henry_(unit)http://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Electrical_conductorhttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Magnetic_fluxhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Electromotive_forcehttp://en.wikipedia.org/wiki/Electrical_resistancehttp://en.wikipedia.org/wiki/Electrical_circuithttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Electromotive_forcehttp://en.wikipedia.org/wiki/Joseph_Henry


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What is a capacitor?

A capacitor or condenser is a passive electric component consisting of apair of conductors separated by a dielectric . When a voltage potential difference exists between the conductors, an electric field is present in the

dielectric. This field stores energy and produces a mechanical force betweenthe plates. The effect is greatest between wide, flat, parallel, narrowlyseparated conductors.

An ideal capacitor is characterized by a single constant value, capacitance ,which is measured in farads . This is the ratio of the electric charge on eachconductor to the potential difference between them. In practice, thedielectric between the plates passes a small amount of leakage current . Theconductors and leads introduce an equivalent series resistance and thedielectric has an electric field strength limit resulting in a breakdown voltage .

Capacitors are widely used in electronic circuits to block the flow of direct current while allowing alternating current to pass, to filter out interference,to smooth the output of power supplies , and for many other purposes. Theyare used in resonant circuits in radio frequency equipment to selectparticular frequencies from a signal with many frequencies.

In electromagnetism and electronics , capacitance is the ability of a body tohold an electrical charge. Capacitance is also a measure of the amount of electric charge stored (or separated) for a given electric potential . A commonform of charge storage device is a parallel-plate capacitor . In a parallel platecapacitor capacitance is directly proportional to the surface area of the

conductor plates and inversely proportional to the separation distancebetween the plates. If the charges on the plates are +Q and Q, and V givesthe voltage between the plates, then the capacitance is given by

The SI unit of capacitance is the farad ; 1 farad = 1 coulomb per volt .

The energy (measured in joules ) stored in a capacitor is equal to the work done to charge it. Consider a capacitance C, holding a charge +q on oneplate and -q on the other. Moving a small element of charge d q from oneplate to the other against the potential difference V = q/C requires the workd W :
http://en.wikipedia.org/wiki/Passivity_(engineering)http://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Electrical_conductorhttp://en.wikipedia.org/wiki/Dielectrichttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Potential_differencehttp://en.wikipedia.org/wiki/Potential_differencehttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Capacitancehttp://en.wikipedia.org/wiki/Faradhttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Leakage_(electronics)http://en.wikipedia.org/wiki/Equivalent_series_resistancehttp://en.wikipedia.org/wiki/Breakdown_voltagehttp://en.wikipedia.org/wiki/Direct_currenthttp://en.wikipedia.org/wiki/Direct_currenthttp://en.wikipedia.org/wiki/Alternating_currenthttp://en.wikipedia.org/wiki/Power_supplyhttp://en.wikipedia.org/wiki/LC_circuithttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Electric_potentialhttp://en.wikipedia.org/wiki/Capacitorhttp://en.wikipedia.org/wiki/SIhttp://en.wikipedia.org/wiki/Faradhttp://en.wikipedia.org/wiki/Coulombhttp://en.wikipedia.org/wiki/Volthttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Passivity_(engineering)http://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Electrical_conductorhttp://en.wikipedia.org/wiki/Dielectrichttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Potential_differencehttp://en.wikipedia.org/wiki/Potential_differencehttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Capacitancehttp://en.wikipedia.org/wiki/Faradhttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Leakage_(electronics)http://en.wikipedia.org/wiki/Equivalent_series_resistancehttp://en.wikipedia.org/wiki/Breakdown_voltagehttp://en.wikipedia.org/wiki/Direct_currenthttp://en.wikipedia.org/wiki/Direct_currenthttp://en.wikipedia.org/wiki/Alternating_currenthttp://en.wikipedia.org/wiki/Power_supplyhttp://en.wikipedia.org/wiki/LC_circuithttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Electric_potentialhttp://en.wikipedia.org/wiki/Capacitorhttp://en.wikipedia.org/wiki/SIhttp://en.wikipedia.org/wiki/Faradhttp://en.wikipedia.org/wiki/Coulombhttp://en.wikipedia.org/wiki/Volthttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Joule


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where W is the work measured in joules, q is the charge measured incoulombs and C is the capacitance, measured in farads.

We can find the energy stored in a capacitance by integrating this equation.Starting with an uncharged capacitance ( q =0) and moving charge from one

plate to the other until the plates have charge +Q and -Q requires the workW :

What is an Impedance?

Impedance is a comprehensive expression of any and all forms of opposition to electron flow, including both resistance and reactance. It ispresent in all circuits, and in all components. When alternating current goesthrough an impedance, a voltage drop is produced that is somewherebetween 0 o and 90 o out of phase with the current. Impedance ismathematically symbolized by the letter Z and is measured in the unit of ohms (), in complex form.

Perfect resistors (Figure below ) possess resistance, but not reactance.Perfect inductors and perfect capacitors (Figure below ) possess reactancebut no resistance. All components possess impedance, and because of thisuniversal quality, it makes sense to translate all component values(resistance, inductance, capacitance) into common terms of impedance asthe first step in analyzing an AC circuit.

Perfect resistor, inductor, and capacitor.

The impedance phase angle for any component is the phase shift betweenvoltage across that component and current through that component. For aperfect resistor, the voltage drop and current are always in phase with eachother, and so the impedance angle of a resistor is said to be 0 o . For anperfect inductor, voltage drop always leads current by 90 o , and so aninductor's impedance phase angle is said to be +90 o . For a perfect capacitor,
http://en.wikipedia.org/wiki/Integralhttp://www.ibiblio.org/kuphaldt/electricCircuits/AC/AC_5.html#02347.pnghttp://www.ibiblio.org/kuphaldt/electricCircuits/AC/AC_5.html#02347.pnghttp://en.wikipedia.org/wiki/Integralhttp://www.ibiblio.org/kuphaldt/electricCircuits/AC/AC_5.html#02347.pnghttp://www.ibiblio.org/kuphaldt/electricCircuits/AC/AC_5.html#02347.png


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voltage drop always lags current by 90 o, and so a capacitor's impedancephase angle is said to be -90 o .

Power in AC circuit

Power dissipation in a resistor

Instantaneous AC power in a pure resistive circuit is always positive.

Note that the power is never a negative value. When the current is positive(above the line), the voltage is also positive, resulting in a power (p=ie) of apositive value. Conversely, when the current is negative (below the line), thevoltage is also negative, which results in a positive value for power (anegative number multiplied by a negative number equals a positivenumber).

This consistent polarity of power tells us that the resistor is alwaysdissipating power, taking it from the source and releasing it in the form of heat energy. Whether the current is positive or negative, a resistor stilldissipates energy.

Power in an inductor


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Note: Current lags voltage by 90 o in a pure inductive circuit.

In a pure inductive circuit, instantaneous power may be positive or negative

Because instantaneous power is the product of the instantaneous voltageand the instantaneous current (p=ie), the power equals zero whenever theinstantaneous current or voltage is zero. Whenever the instantaneouscurrent and voltage are both positive (above the line), the power is positive.Because the current and voltage waves are 90 o out of phase, there are timeswhen one is positive while the other is negative, resulting in equally frequentoccurrences of negative instantaneous power .

But what does negative power mean? It means that the inductor is releasingpower back to the circuit, while a positive power means that it is absorbingpower from the circuit. Since the positive and negative power cycles are

equal in magnitude and duration over time, the inductor releases just asmuch power back to the circuit as it absorbs over the span of a completecycle. What this means in a practical sense is that the reactance of aninductor dissipates a net energy of zero, quite unlike the resistance of aresistor, which dissipates energy in the form of heat. Mind you, this is forperfect inductors only, which have no wire resistance.


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Power in a capacitor

Note: Voltage lags current by 90 o in a pure capacitive circuit.

In a pure capacitive circuit, the instantaneous power may be positive or negative.

As with the simple inductor circuit, the 90 degree phase shift betweenvoltage and current results in a power wave that alternates equally betweenpositive and negative. This means that a capacitor does not dissipate poweras it reacts against changes in voltage; it merely absorbs and releasespower, alternately.


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