Higher Engineering Mathematics, Sixth Edition

224 Higher Engineering Mathematics

2. Two impedances, Z 1 =( 3 +j 6 ) and
   Z 2 =( 4 −j 3 )are connected in series to
   a supply voltage of 120V. Determine the
   magnitude of the current and its phase angle
   relative to the voltage.
   [15.76A, 23. 20 ◦lagging]


3. If the two impedances in Problem 2 are con-
   nected in parallel determine the current flow-
   ing and its phase relative to the 120V supply
   voltage. [27.25A, 3. 37 ◦lagging]


4. A series circuit consists of a 12resistor, a
   coil of inductance 0.10H and a capacitance of
   160 μF. Calculate the current flowing and its
   phase relative to the supply voltage of 240V,
   50Hz. Determine also the power factor of the
   circuit. [14.42A, 43. 85 ◦lagging, 0.721]


5. For the circuit shown in Fig. 20.11, determine
   the currentIflowing and its phase relative to
   the applied voltage. [14.6A, 2. 51 ◦leading]


6. Determine, using complex numbers, the mag-
   nitude and direction of the resultant of the
   coplanar forces given below, which are act-
   ingat a point.ForceA, 5N acting horizontally,
   ForceB, 9N acting at an angle of135◦toforce
   A,ForceC, 12N acting at an angle of 240◦to
   forceA. [8.394N, 208.68◦from forceA]

I

R 1530 V

R 3525 V

V 5200 V

R 2540 V XL^550 V

XC 520 V

Figure 20.11

7. A delta-connected impedance ZA is given
   by:

ZA=

Z 1 Z 2 +Z 2 Z 3 +Z 3 Z 1

Z 2

Determine ZA in both Cartesian and polar

form givenZ 1 =( 10 +j 0 ),

Z 2 =( 0 −j 10 )andZ 3 =( 10 +j 10 ).

[( 10 +j 20 ), 22. 36 ∠ 63. 43 ◦]

8. In the hydrogen atom, the angular momen-
   tum, p, of the de Broglie wave is given

by: pψ=−

(

jh

2 π

)

(±jmψ). Determine an

expression forp.

[

±

mh

2 π

]

9. An aircraftPflying at a constant height has
   a velocity of( 400 +j 300 )km/h. Another air-
   craftQat the same height has a velocity of
   ( 200 −j 600 )km/h. Determine (a) the veloc-
   ity ofPrelative toQ, and (b) the velocity of
   Qrelative toP. Express the answers in polar
   form, correct to the nearest km/h.[
   (a) 922km/h at 77. 47 ◦
   (b) 922km/h at− 102. 53 ◦

]

10. Three vectors are represented byP,2∠ 30 ◦,
    Q,3∠ 90 ◦and R,4∠− 60 ◦. Determine in
    polar form the vectors represented by (a)
    P+Q+R,(b)P−Q−[R.
    (a) 3. 770 ∠ 8. 17 ◦
    (b) 1. 488 ∠ 100. 37 ◦

]

11. In a Schering bridge circuit,
    ZX=(RX−jXCX),Z 2 =−jXC 2 ,

Z 3 =

(R 3 )(−jXC 3 )

(R 3 −jXC 3 )

andZ 4 =R 4

whereXC=

1

2 πfC

At balance:(ZX)(Z 3 )=(Z 2 )(Z 4 ).

Show that at balance RX=

C 3 R 4

C 2

and

CX=

C 2 R 3

R 4