Understanding Engineering Mathematics

In terms ofXwe can writex(t)as:

x(t)=Im(Xejωt)

wherein lies the usefulness of phasors – the frequency behaviour of x(t) is sepa-
rated out.
Phasors representing sinusoids of the same frequency can be added and subtracted
by complex algebra. Thus, if the phasorsX 1 =A 1 ejφ^1 ,X 2 =A 2 ejφ^2 represent signals
x 1 (t)=A 1 sin(ωt+φ 1 ),x 2 (t)=A 2 sin(ωt+φ 2 )respectively then the phasorX 1 +X 2
represents the signalx 1 (t)+x 2 (t)and the phasorX 1 −X 2 represents the signalx 1 (t)−
x 2 (t).

1. Sketch and write down the phasors corresponding to the sinusoids 2 sin(ωt+π/ 4 ),
   sin(ωt+ 4 π/ 3 ),3cos(ωt+π/ 3 ),4sin(ωt−π/ 6 )– write these in Cartesian forma+
   jb.


2. Sketch and write down the phasors corresponding to the sinusoids:

(i) 3 sin(ωt+π/ 6 ) (ii) 2 sin(ωt+π/ 2 )

(iii) sin(ωt−π/ 4 ) (iv) 3 cos(ωt−π/3)

Express the phasors ina+jbform.

Answers:

(i) 3ejπ/^6 ,

3

√

3

2

+j

3

2

(ii) 2ejπ/^2 ,2j

(iii) e−jπ/^4 ,

1

√

2

−j

1

√

2

(iv) 3ejπ/^6 ,

3

√

3

2

+j

3

2

3. Find the sinusoidal functions (assume frequencyω) corresponding to the following
   phasors ina+jbform

(i) 3+ 3 j (ii) − 4 + 2

√

3 j (iii) − 4 − 3 j (iv) 3− 5. 2 j

Answers:

(i) 3

√

2sin(ωt+π/ 4 ) (ii) 2

√

7sin(ωt+ 2. 43 )

(iii) 5 sin(ωt+ 3. 79 ) (iv) 6 sin(ωt− 1. 05 )

NB. All angles in radians.

4. Express 2 cosωtand

√

2cos(ωt+π/ 4 )in phasor form and hence determine the ampli-

tude and phase of

2cosωt+

√

2cos(ωt+π/ 4 )