Alter Voltage and Current Lec17 20

7/30/2019 Alter Voltage and Current Lec17 20

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EE1101 BASIC ELECTRICAL

TECHNOLOGY

ec ure s - :Alternating Voltage and Current

EEE1101 Basic Electrical Technology Rev by Shalyn Lim 03/11


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Learning Outcome

Distinguish various types of AC voltages andcurrents and how to interpret its value.

EEE1101 Basic Electrical Technology1


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Alternating Waveforms The term alternating indicates only that the waveform alternates

between two prescribed levels in a set time sequence.

The sinusoidal waveform (sine wave) is the fundamental

alternating current (ac) and alternating e.m.f (voltage) waveform



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Generation of an Alternating e.m.f.

The elementary AC generator consists of a conductor,or wire loop in a magnetic field and it can be rotated in

a stationary magnetic field to produce induced e.m.f inthe loop.

Two ends of the loop are connected to slip rings, and


t ey are n contact w t two rus es. Sliding contacts (brushes) connect the loop to an

external circuit load in order to pick up the inducede.m.f.

When the loop rotates it cuts magnetic lines of force,first in one direction and then the other.


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Sinusoidal voltages are produced by ac generators.

When a conductor rotates in a constant magneticfield, a sinusoidal wave is generated.



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1st Rotation


2nd Rotation


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3rd Rotation


4th Rotation


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At the instant the loop is in the vertical position, the

coil sides are moving parallel to the field and do not

cut magnetic lines of force.

In this instant, there is no voltage induced in the

-


.

direction, the coil sides will cut the magnetic lines of

force in opposite directions.

The direction of the induced voltages depends on the

direction of movement of the coil.


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Alternating e.m.f. Instantaneous value of e.m.f generated in a coil is

v = Vm sin

Where Vp or Vm Maximum value of e.m.f. generated in a coil

- angle of loop from position of zero e.m.f.


sinVvp

=

pV

pV


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Alternating e.m.f. Most electrical energy is provided by rotating a.c.

generators.

The e.m.f and the resulting voltages and currents are

usually sinusoidal but there are also circuits operating


waveform.


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Alternating e.m.f.


Alternating current waveforms


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Sinusoidal Alternating WaveformsUseful terms and definition of alternating systems:

Cycle repetition of a variable quantity at equalintervals

Period (T) duration of one cycle

Fre uenc number of c cles that occur in 1


second

Tf

1= Hertz


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Sinusoidal Alternating Waveforms Peak value (Ep or Vp ,Ip) or (Em or Vm ,Im) maximum

instantaneous value measured from its zero value. Known

as peak amplitude maximum instantaneous value

measured from the mean value.

Peak-to-peak value (Epp or Vpp ,Ipp) maximum variation


between the maximum positive and maximum negativeinstantaneous value.


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Average values The average or mean value of a symmetrical

alternating quantity, (such as a sine wave), is theaverage value measured over a half cycle, (since

over a complete cycle the average value is zero)


m

av

I2I =

Iav = 0.637Im A (sinusoidal waveform)

Note:

1. Generally, Ip = Im


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Average values Similarly, the average value of voltage is found as

m

av

V2V =


Vav = 0.637 Vm V (sinusoidal waveform)


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R.M.S. values The effective value or the root mean square (r.m.s.)

value of an alternating current is that current which

will produce the same heating effect as an equivalent

direct current.



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R.M.S. values The r.m.s value of current,

m

m I7071.02

II == (sinusoidal waveform)


Similarly, the r.m.s value of voltage is found as

mm V7071.02

VV ==


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R.M.S. values The r.m.s. value is always greater than the average

value (except for a rectangular wave, r.m.s value =

average value).


A sine wave, over one cycle. The dashed line

represents the r.m.s, average and peak value.


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ExampleFor the waveform shown, the same power would be deliveredto a load with a dc voltage of ?

45 V

60 V


0 V

30 V

-30 V

-45 V

-60 V

t( s)0 25 37.5 50.0


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Example1. If the effective voltage of an ac receptacle is 120V,

what is the peak-to-peak voltage?

2. What is the effective voltage if v = 10 sin( - 50)?



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Representation of an Alternating

Quantity by a Phasor

A phasor is a rotating line whose projection on a

vertical axis can be used to represent sinusoidally

varying quantities.



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Representation of an Alternating

Quantity by a PhasorThe instantaneous value of the alternating waveform is given byx = A

sin

hypotenuse

rightangle

opposite side

A

o

x

B


OA = Ip (maximum value of current)

Assume OA to rotate anti-clockwise about 0 at a uniform angular

velocity ().

AB = OA sin

=Im sin

= i (instantaneous current)


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Angular Velocity The rate at which the generator coil rotates is called its

angular velocity,.

If the coil rotates through an angle of 30 in one

second, for example, its angular velocity is 30 per


.

Normally angular velocity is expressed in radians per

second (rad/s) instead of degrees per second.

In general,

t =


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Angular Measurement In practice, is usually expressed in rad/s, where

radians and degrees are related by the identity

degrees360

rad2rad

=


radrad2

deg =


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V and I as Functions of TimeRelationship between , T, and f



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V and I as Functions of Time Recap the equation of a sinusoidal waveform,

Since, = t

v = Vm sin


v = Vm sin t

Similarly,

v =m

s n tor

i = Im sin t i = Im sin2ftor


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Sine wave equation

A plot of sinusoidal waveform (peak at 25 V) is shown. The instantaneous

voltage at 50o is 19.2.

V and I as Functions of Time

v= = 19.2 VVpsinVp

500

= 50

Vp

Vp

= 25 V

26 EEE1101 Basic Electrical Technology


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Example1. Find the amplitude and frequency of 42.1sin(377t+30o).

2. A current sine wave has a peak of 58mA and a radian

frequency of 90 rad/s. Find the instantaneous current at

t=23ms.



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V and I with Phase Shifts If a sine wave does not pass through zero at t=0 s, it

has a phase shift. Waveforms may be shifted to the

left or to the right.

For a waveform shifted left as


For a waveform shifted right as

v =m

s n t +

v = Vm sin (t - )


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Phase Difference Phase difference refers to the angular displacement

between different waveforms of the same frequency.

If the angular displacement is 0 as in (a), the

waveforms are said to be in phase; otherwise, they


.

When describing a phase difference, select one

waveform as reference. Other waveforms then lead,

lag, or are in phase with this reference


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Phase Difference For example, in (b), for reasons to be discussed in the

next paragraph, the current waveform is said to lead

the voltage waveform, while in (c) the current

waveform is said to lag.


Illustrating phase difference. In these examples, voltage is taken as reference.


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Phase Difference The terms lead and lag can be understood in terms of

phasors. If the observing phasors rotating, the one that

passing first is leading and the other is lagging.

Phasor Im leads phasor Vm; thus current i(t) leads


.


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Phase Difference



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ExampleWrite the general voltage equation that describes this waveform



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Complex Numbers and Polar Notation A complex number is a number of the form C = a + jb,

where a and b are real numbers and j = . The number

a is called the real part of C and b is called its

imaginary part.

1


geometrically, either in rectangular form or inpolar form as points on a two-dimensional plane

called the complex plane .


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Complex Numbers and Polar Notation E.g. The complex number C

= 6 + j8,represents a point

whose coordinate on the realaxis is 6 and whose

coordinate on the imaginary


ax s s . s orm o

representation is called the

rectangular form.


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Complex Numbers and Polar Notation Complex numbers may also

be represented in polar form

by magnitude and angle.Thus, C = 1053.13 is a

complex number with


magn tu e an ang e

53.13.

This magnitude and angle

representation is just an

alternate way of specifyingthe location of the point

represented by C = a + jb.


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Complex Numbers and Polar Notation Conversion between Rectangular and Polar Forms



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Complex Numbers and Polar Notation Reciprocals


The conjugate of a complex number (denoted by an

asterisk *) is a complex number with the same real part

but the opposite imaginary part.


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Complex Numbers and Polar Notation Powers of j are frequently required in calculations.

Here are some useful powers



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Addition and Subtraction of PhasorsArithmetic of complex numbers:

If phasorA1 = a1 + jb1 and phasorA2 = a2 + jb2:

Addition


1 2 1 1 2 2

= (a1 + a2) +j( b1 + b2)

A1 - A2 = (a1 + jb1)-( a2 + jb2)= (a1 - a2) +/-j( b1 - b2)

Subtraction


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Multiplication and Division of Phasors

A1 xA2 = (a1 + jb1)( a2 + jb2)

= a1a2 + j2b1b2 + ja1b2 + ja2b1

= (a1a2 - b1b2) + j(a1b2 + a2b1) since j2 = -1

Multiplication


22

11

2

1

jba

jba

A

A

+

+=

( )( )

( )( )2222

2211

jbajba

jbajba

+

+=

( ) ( )2

2

2

2

21122121

bababajbbaa

+

++=

Division


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Multiplication and Division of Phasors These operations are usually performed in polar form.

For multiplication, multiply magnitudes and add

angles algebraically.

For division, divide the magnitude of the denominator


,

algebraically the angle of the denominator from that ofthe numerator.


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Multiplication and Division of Phasors

Multiplication of phasors:

A xB= AB (+)


Division of phasors:

( )

=

B

A

B

A


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Example1. Convert the following numbers to polar form:

a. 6+j9

b. -21+j33.3


2. Find the product of )604)(253( oo


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Example1. Convert the following numbers to complex form.

a.

b.

o202.10

o3041.6


2. Find the product of (0.3+j0.4)(-5+j6)

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Alter Voltage and Current Lec17 20