274 Higher Engineering Mathematics
Hence,v 2 =v−v 1 =30sin100πt
−20sin( 100 πt− 0. 59 )
= 30 ∠ 0 − 20 ∠− 0 .59rad
=( 30 +j 0 )−( 16. 619 −j 11. 127 )
= 13. 381 +j 11. 127
= 17. 40 ∠ 0 .694 rad
Hence, by using complex numbers, the resultant
in sinusoidal form is:v−v 1 =30sin100πt−20sin( 100 πt− 0. 59 )
= 17 .40sin(ωt+ 0. 694 )volts(b) Supply frequency,f=ω
2 π=100 π
2 π=50 Hz(c) Periodic time,T=1
f=1
50= 0 .02sor20 ms(d) R.m.s. value of supply voltage,= 0. 707 × 30
= 21 .21voltsNow try the following exerciseExercise 111 Further problems on
resultant phasors by complexnumbersInProblems1to4,expressthecombinationofperi-
odic functions in the formAsin(ωt±α)by using
complex numbers:- 8sinωt+5sin
(
ωt+π
4)[12.07sin(ωt+ 0. 297 )]- 6sinωt+9sin
(
ωt−π
6)[14.51sin(ωt− 0. 315 )]- v=12sinωt−5sin
(
ωt−π
4)[9.173sin(ωt+ 0. 396 )]- x=10sin
(
ωt+π
3)
−8sin(
ωt−3 π
8)[16.168sin(ωt+ 1. 451 )]- The voltage drops across two compo-
nents when connected in series across
an a.c. supply are:v 1 =240sin314. 2 t and
v 2 =150sin( 314. 2 t−π/ 5 )volts respectively.
Determine the:
(a) voltage of the supply (given byv 1 +v 2 )
in the formAsin(ωt±α).
(b) frequency of the supply.
[(a) 371.95sin( 314. 2 t− 0. 239 )V
(b) 50Hz] - If the supply to a circuit isv=25sin200πt
volts and the voltage drop across one of
the components isv 1 =18sin( 200 πt− 0. 43 )
volts, calculate the:
(a) voltage drop across the remainder of
the circuit, given byv−v 1 , in the form
Asin(ωt±α).
(b) supply frequency.
(c) periodic time of the supply.
[(a) 11.44sin( 200 πt+ 0. 715 )V
(b) 100 Hz (c) 10 ms] - The voltages across three components in a
series circuit when connected across an a.c.
supply are:
v 1 =20sin(
300 πt−π
6)
volts,v 2 =30sin(
300 πt+π
4)
volts,andv 3 =60sin(
300 πt−π
3)
volts.Calculate the:
(a) supply voltage, in sinusoidal form, in the
formAsin(ωt±α).
(b) frequency of the supply.
(c) periodic time.
(d) r.m.s. value of the supply voltage.
[(a) 79.73sin( 300 π− 0. 536 )V
(b) 150 Hz(c) 6.667 ms(d) 56.37 V]