Trigonometric waveforms 145
(c) Whent=10ms
thenv=340sin
(
50 π
10
103
− 0. 541
)
=340sin( 1. 0298 )=340sin59◦
=291.4V
(d) Whenv=200volts
then 200=340sin( 50 πt− 0. 541 )
200
340
=sin( 50 πt− 0. 541 )
Hence( 50 πt− 0. 541 )=arcsin
200
340
= 36. 03 ◦or 0.6288rad
50 πt= 0. 6288 + 0. 541
= 1. 1698
Hence whenv=200V,
time,t=
1. 1698
50 π
=7.447ms
(e) When the voltage is a maximum,v=340V.
Hence 340=340sin( 50 πt− 0. 541 )
1 =sin( 50 πt− 0. 541 )
50 πt− 0. 541 =arcsin1
= 90 ◦or 1.5708rad
50 πt= 1. 5708 + 0. 541 = 2. 1118
Hence time,t=
2. 1118
50 π
=13.44ms
Asketchofv=340 sin( 50 πt− 0. 541 )volts is shown in
Fig. 14.30.
v 5 340 sin(50t 2 0.541)
v 5 340 sin 50t
(^0103040) t (ms)
7.447 13.44
2340
2 175.1
200
291.4
340
Voltage V
20
Figure 14.30
Now try the following exercise
Exercise 63 Further problemson the
sinusoidal formAsin(ωt±α)
In Problems 1 to 3 find the amplitude, periodic
time, frequency and phase angle (stating whether
it is leading or laggingAsinωt) of the alternating
quantities given.
- i=40sin( 50 πt+ 0. 29 )mA
[
40 , 0 .04s,25Hz, 0 .29rad
(or 16◦ 37 ′)leading 40sin50πt
]
- y=75sin( 40 t− 0. 54 )cm
[
75cm, 0 .157s, 6 .37Hz, 0 .54rad
(or30◦ 56 ′)lagging75sin 40t
]
- v=300sin( 200 πt− 0. 412 )V
[
300V, 0 .01s,100Hz, 0 .412rad
(or 23◦ 36 ′)lagging 300sin200πt
]
- A sinusoidal voltage has a maximum value of
120V and a frequency of 50Hz. At timet=0,
the voltage is (a) zero, and (b) 50V.
Express the instantaneous voltagevin the
formv=Asin(ωt±α).
[
(a)v=120sin100πtvolts
(b)v=120sin( 100 πt+ 0. 43 )volts
]
- An alternating current has a periodic time of
25ms and a maximum value of 20A. When
timet=0, currenti=−10 amperes. Express
the currentiin the formi=Asin(ωt±α).
[
i=20sin
(
80 πt−
π
6
)
amperes
]
- Anoscillatingmechanismhasamaximumdis-
placement of 3.2m and a frequency of 50Hz.
At timet=0 the displacement is 150cm.
Express the displacement in the general form
Asin(ωt±α).
[3.2 sin( 100 πt+ 0. 488 )m] - The current in an a.c. circuit at any time
tseconds is given by:
i=5sin( 100 πt− 0. 432 )amperes
Determine (a) the amplitude, periodic time,
frequency and phase angle (in degrees) (b) the
value of current att=0 (c) the value of current
att=8ms (d) the time when the current is first
a maximum (e) the time when the current first