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R,L, and C Elements andthe Impedance Concept

1Basic Electrical Engineering

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To analyze ac circuits in the time domain is not

very practical. It is more practical to represent voltages and

currents as phasors, circuit elements as

impedances, and use complex algebra to analyze.

With this approach, ac circuits can be handledmuch like dc circuits - the relationships and laws

still apply.


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Acomplex number is in the f

orm a

+jb,wherej =

a is the real part and b is the imaginary part ofthe complex number.

This called the rectangular form.

A complex number may be representedgraphically with abeing the horizontalcomponent and bbeing the vertical

co

mpo

nent. 3

1

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IfC =a +jb in rectangular form, then C =

CU, where

4

a

btan

2

b

2

aC

sinCb

cosCa

!

!

!

!

BasicElectrical Engineering

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j 2 = -1

j 3 = -jj 4 = 1

1/j = -j

To add complex numbers, add the real partsand imaginary parts separately.

Subtraction is done similarly.


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To multiply or divide complex numbers, it

is best to convert to polar form first.

(AU)(BJ) = (AB)(U +J)

(AU)/(BJ) = (A/B)(U - J)

(1/CU) = (1/C)-U

The complex conjugate ofa +jb is a -jb.


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A voltages and currents can be

represented as phasors.

Since phasors have magnitude and angle,

they can be viewed as complex numbers.

A voltage given as 100 sin(314t+30) canbe written as 10030.


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We can represent a source by its phasor

equivalent from the start.

The phasor representation contains allthe information we need except for the

angular velocity.

By doing this, we have transformed fromthe time domain to the phasor domain.

KVL and KCL apply in both time

domain and phasor domain. 8

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To add or subtract waveforms in time

domain is very tedious.

This can be done easier by converting to

phasors and adding as complex numbers.

Once the waveforms are added, thecorresponding time equation and

companion waveform can be determined.


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Until now, we have used peak values when

writing voltages and current in phas

or fo

rm. It ismore common that they be written as RMS values.

To add or subtract sinusoidal voltages or currents,

convert to phasor form, add or subtract, then

convert back to sinusoidal form.Quantities expressed as phasors are said to be in

phasor domain orfrequency domain.

13

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R, L, and Ccircuit elements each have quitedifferent electrical properties.

These differences result in different voltage-current relationships.

When a circuit is connected to a sinusoidal

source, all currents and voltages in the circuitwill be sinusoidal.

These sine waves will have the samefrequency as the source and will differ from itonly in terms of their magnitudes and angles.


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In a purely resistive circuit, Ohms Law

applies; the current is proportional to thevoltage.

Current variations follow voltage variations,each reaching their peak values at the sametime.

The voltage and current of a resistor are inphase.


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For a resistor, the voltage and current are in

phase.

If the voltage has a phase angle, the current

has the same angle.

The impedance of a resistor is equal toR0.


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The voltage of an inductor is proportional to

the rate of change of the current.Because the voltage is greatest when the

rate of change (or the slope)of the current

is greatest, the voltage and current are not in

phase.

The voltage phasor leads the current by 90

for an inductor.17

Basic Electrical Engineering

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Inductive reactance, XL, represents the

opposition that inductance presents tocurrent for the sinusoidal ac case.

XL

is frequency-dependent.

XL = V/Iand has units ofohms.X

L= [L = 2TfL


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For an inductor, voltage leads current by

90.

If the voltage has an angle of 0, the current

has an angle of -90.

The impedance of an inductor is XL90.


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In a capacitive circuit, the current is

proportional to the rate of change of thevoltage.

The current will be greatest when the rateof change of the voltage is greatest, so the

voltage and current are out of phase.

For a capacitor, the current leads thevoltage by 90.


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Capacitive reactance, XC, represents the

opposition that capacitance presents tocurrent for the sinusoidal case.

XC

is frequency-dependent. As the

frequency increases,X

C decreases.X

C= V/Iand has units ofohms.

23

fX

T[ 2

11!!

Basic Electrical Engineering

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For a capacitor, the current leads the

voltage by 90.If the voltage has an angle of 0, the current

has an angle of 90.

The impedance of a capacitor is given asX

C-90.


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The opposition that a circuit elementpresents to current is the impedance, Z.

Z = V/I, is in units ofohms

Z in phasor form is ZU where U is thephase difference between the voltageand current.


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