Understanding Engineering Mathematics

3.In the relationship

(R+jpL)

(

S−j

1

pC

)

=

P

Q

all the quantities are real exceptj. Show that

p=

√

R

LSC

and findRin terms ofC,L,P,Q,S.

4.In electrical circuit theory impedance is represented by a complex variable in which
the resistanceRis the real part and the reactanceXis the imaginary part:

Z=R+jX

The combined impedance,Z=R+jXof two impedance’sZ 1 =R 1 +jX 1 ,Z 2 =

R 2 +jX 2 in parallel is given by the usual result:

1

Z

=

1

Z 1

+

1

Z 2

Obtain explicit expressions forRandXin terms ofR 1 ,R 2 ,X 1 ,X 2.

5.A typical sinusoidal function

x(t)=Asin(ωt+φ)

describing an oscillating signal with period 2π/ω, amplitudeAand phase angleφmay

be represented, with the frequencyωunderstood, by a line of lengthAmaking angle

φwith a horizontal axis in an Argand plane. This is called aphasor diagram, the line

being thephasorof the signalx(t).

b = A sin f

a = A cos f

A

f

The phasorXof x(t)is then represented mathematically by a complex number of
magnitudeAand argumentφ:

X=a+jb=Acosφ+jAsinφ=Aeiφ