Engineering Mathematics 4 Episode 6 ppsx

Findthe amplitude, periodic time, frequency andphase angle in degrees and minutes i D30 sin 100tC0.27 A, hence amplitude = 30 A.. circuit at any time tseconds is given by: i D5 sin 100t

In general, y= sin pt − a/ lags y =sin pt by a= p,

hence 7 sin2A  /3 lags 7 sin 2A by /3 /2, i.e

Problem 10 Sketch y D 2 cosωt  3/10

over one cycle

Amplitude D 2 and period D 2/ω rad

2 cosωt 3/10 lags 2 cos ωt by 3/10ω seconds

A sketch of y D 2 cosωt  3/10 is shown in

Fig 22.25

Now try the following exercise

Exercise 85 Further problems on sine and

cosine curves

In Problems 1 to 7 state the amplitude and

period of the waveform and sketch the curve

0

−2

p/2w p/w 3p/2w 2p/w t y

22.5 Sinusoidal form A sin ! t ± a/

In Fig 22.26, let OR represent a vector that isfree to rotate anticlockwise about O at a veloc-

ity of ω rad/s A rotating vector is called a sor After a time t seconds OR will have turned

pha-through an angle ωt radians (shown as angle TOR

in Fig 22.26) If ST is constructed perpendicular to

OR, then sin ωt D ST/OT, i.e ST D OT sin ωt

y 1.0

Figure 22.26

If all such vertical components are projected on

to a graph of y against ωt, a sine wave results ofamplitude OR (as shown in Section 22.3)

If phasor OR makes one revolution (i.e 2

radians) in T seconds, then the angular velocity,

ω D2/T rad/s,

from which, T = 2p=!seconds

Tis known as the periodic time.

The number of complete cycles occurring per

second is called the frequency, f

Frequency D number of cycles

second

D 1

T D

ω2 Hz

2p Hz

Hence angular velocity, ! =2p f rad/s

Amplitude is the name given to the maximum

or peak value of a sine wave, as explained in

Section 22.4 The amplitude of the sine wave shown

in Fig 22.26 has an amplitude of 1

A sine or cosine wave may not always start at

0° To show this a periodic function is represented

by y D sinωt š ˛ or y D cosωt š ˛ , where ˛

is a phase displacement compared with y D sin A

or y D cos A A graph of y D sinωt  ˛ lags

y Dsin ωt by angle ˛, and a graph of y D sinωt C

˛ leads y Dsin ωt by angle ˛

The angle ωt is measured in radians

hence angle ˛ should also be in radians

The relationship between degrees and radians is:

y D Asin ωt)

Problem 11 An alternating current is given

by i D 30 sin100t C 0.27 ... 36. 87° or0. 64 4 rad

Hence  D180° 36. 87°D 143 .13°

or  D 0. 64 4 D 2 .49 8 rad

which corresponds to length AB in Fig 23.9

Thus (1. 948 , −4. 057) in Cartesian co-ordinates

corresponds to (4. 5, 5.16