**HIGHLIGHTS**

1. Modern alternators produce an e.m.f. which is for all practical purposes sinusoidal {i.e. a sine curve), the equation between the e.m.f. and time being

e = E _{max} sin rot

where e = instantaneous voltage ; E max = maximum voltage;

wt = angle through which the armature has turned from neutral.

2. The r.m.s. value of an alternating current is given by that steady (D.C.) current which when flowing through a given circuit produces the same heat as is produced by the alternating current when flowing through the same circuit for the same time.

I _{rms} = 0.707 _{max}.

3. The average or mean value of an alternating current is expressed by that steady current which transfers across any circuit the same charge as is transferred by that alternating current during the same time.

au = 0.637 I _{max}

Form factor is the ratio of r.m.s. value to average value of the wave form.

Peak factor is the ratio of maximum value to the r.m.s. value of the wave form.

4. The frequency (series circuit) of the voltage which gives the maximum value of the current in the circuit is called resonant frequency({,) and the circuit is said to be resonant.

F r = 1 / 2π √ LC

5. Q-factor of a series circuit is defined as equal to the voltage magnification in the circuit at resonance.

Q-factor = 1 / R √ L / C

6. Q-factor of a parallel circuit is defined as the ratio of the current circulating between its two branches to the line current drawn from the supply or simply, as the current magnification.

Q-factor = 1 / R √ L / C

**OBJECTIVE TYPE QUESTIONS**

Choose the Correct Answer :

1. A sine wave has a frequency of 50 Hz. Its angular frequency is …… radian/second.

(a) 100 π (b) 50 π

(c) 25 π (d) 5 π.

2. The reactance offered by a capacitor to alternating current of frequency 50 Hz is 20 n. If frequency is increased to 100 Hz, reactance becomes …… ohms.

(a) 2.5 (b) 5

(c) 10 (c) 15

3. The period of a wave is

(a) the same as frequency (b) time required to complete one cycle

(c) expressed in amperes (d) none of the above.

4. The form factor is the ratio of

(a) peak value to r.m.s. value (b) r.m.s. value to average value

(c) average value to r.m.s. value (d) none of the above.

5. The period of a sine wave is 1 / 50 seconds. Its frequency is

(a) 20Hz (b) 30Hz

(c) 40Hz (d) 50 Hz

6. An A. C. current is even by i = 200 sin 100 πt. It will achieve a value of 100 A after …… second.

(a) 1 / 900 (b) 1 / 800

(c) 1 / 700 (d) 1 / 600

7. A heater is rated as 230 V, 10 kW, A.C. The value 230 V refers to

(a) average voltage (b) r.m.s. voltage

(c) peak voltage (d) none of the above.

8. If two sinusoids of the same frequency but of different amplitudes and phase angles are subtracted, the resultant is

(a) a sinusoid of the same frequency (b) a sinusoid of half the original frequency

(c) a sinusoid of double the frequency (d) not a sinusoid.

9. The peak value of a sine wave is 200 V. Its average value is

(a) 127.4 V (b) 141.4 V

(c) 282.8 V (d) 200 V.

10. If two sine waves of the same frequency have a phase difference of1t radians, then

(a) both will reach their minimum values at the same instant

(b) both will reach their maximum values at the same instant

(c) when one wave reaches its maximum value, the other will reach its minimum value

(d) none of the above.

11. The r.m.s. value of a sine wave is 100 A. Its peak value is

(a) 70.7 A (b) 141.4 A

(c) 150 A (d) 282.8 A.

12. If two waves are expressed as e1 = Em1 sin (rot+ a!) and e2 = Em2 sin (rot+ az), then

(a) e1 is leading e2 by < (a_{2} – a_{1}) (b) e2 is leading et by < (a_{2} – a_{1})

(c) e2 is leading et by < (a_{1} – a_{2}) (d) e1 is in phase with e_{2}.

13. The voltage of domestic supply is 220 V. This figure represents

(a) mean value (b) r.m.s. value

(c) peak value (d) average value.

14. Two waves of the same frequency have opposite phase when the phase angle between them is

(a) 360° (b) 180°

(c) 90° (d) 0°

15. The power consumed in a circuit element will be least when the phase difference between the current and voltage is

(a) 180° (b) 90°

(c) 60° (d) 0°

16. The r.m.s. value and mean value is the same in the case of

(a) triangular wave (b) sine wave

(c) square wave (d) half wave rectified sine wave.

17. For the same peak value which of the following wave will have the highest r.m.s. value?

(a) square wave (b) half wave rectified sine wave

(c) triangular wave (d) sine wave.

18. For the same peak value, which of the following wave has the least mean value ?

(a) half wave rectified sine wave (b) triangular wave

(c) sine wave (d) square wave.

19. For a sine wave with peak value I _{max} = the r.m.s. value is

(a) 0.5 I _{max} (b) 0.707 I _{max}

(c) 0.9 I _{max } (d) 1.414 I _{max.}

20. Form Factor is the ratio of

(a) average value/r.m.s. value (b) average value/peak value

(c) r.m.s. value/average value (d) r.m.s. value/peak value.

21. Form factor for a sine wave is

(a) 1.414 (b) 0.701

(c) 1.11 (d) 0.637.

22. For a sine wave with peak value E max = the average value is

(a) 0.636 E _{max} (b) 0.707 E _{max}

(c) 0.43f E _{max} (d) 1.414 E _{max}

23. The current in a circuit is given by : i = 100 sin 314 t amperes The maximum value and frequency of current are

(a) 50 √2 A, 100Hz (b) 100 √2 A, 100Hz

(c) 100 A, 50 Hz (d) 70.7 A, 50 Hz.

24. For a frequency of 200 Hz, the time period will be

(a) 0.05 s (b) 0.005 s

(c) 0.0005 s (d) 0.5 s.

25. The phase difference between voltage and current wave through a circuit element is given as 30 The essential condition is that

(a) both waves must have same frequency

(b) both waves must have identical peak values

(c) both waves must have zero value at the same time

(d) none of the above.

26. An A. C. voltage of 50 Hz has a maximum value of 50 V. Its value after ll600 second after the instant the current is zero, will be

(a) 5 V (b) 12.5 V

(c) 25 V (d) 43.3 v.

27. When two waves are in phase they have peak values at an interval of

(a) 180″ (b) 120″

(c) 90″ (d) none of the above.

28. For 200 V r.m.s. value triangular wave, the peak voltage will be

(a) 200 V (b) 222 V

(c) 282 V (d) 346 v.

29. A sine wave of voltage varies from zero to maximum of 200 V. How much is the voltage at the instant of 30″ of the cycle ?

(a) 50 V (b) 82.8 V

(c) 100 V (d) 173.2 v.

30. How much r.m.s. current does a 300 W 200 V bulb take from the 200 V, 50 Hz power line ?

(a) 0.5 A (b) 1.5 A

(c) 2 A (d) 3 A.

31. Two sinusoidal currents are given by i 1 = 100 sin (wt + π/3), and i2 = 150 sin (wt- π/4)

The phase difference between them is …… degrees.

(a) 15 (b) 50

(c) 60 (d) 105.

32. The r.m.s. value of a half-wave rectified current is 100 A. Its value for full-wave rectification would be …… amperes.

(a) 141.4 (b) 200

(c) 200 π (d) 40π.

33. From the two voltages equations e1 = E max sin 100Ttt, and e2 = E max sin (100 T + 7rl6), it is obvious that

(a) 1leads 2 by 30° (b) 2lags behind 1

(c) 2 achieves its maximum value second before 1 does

(d) 1 achieves its zero value second before 2.

34. The r.m.s. value of a sinusoidal A. C. current is equal to its value at an angle of …… degrees.

(a) 90 (b) 60

(c)45 (d) 30.

35. Capacitive reactance is more when

(a) capacitance is less and frequency of supply is less

(b) capacitance is less and frequency of supply is more

(c) capacitance is more and frequency of supply is less

(d) capacitance is more and frequency of supply is more

36. Time constant of a capacitive circuit Increases with the

(a) increase of capacitance and decrease of resistance

(b) increase of capacitance and increase of resistance

(c) decrease of capacitance and decrease of resistance

(d) decrease of capacitance and increase of resistance.

**THEORETICAL QUESTIONS**

1. Define the following terms :

Circuit, Electrical network, Active network, Node and Branch.

2. What are the limitations of ohm’s law ?

3. State and explain Kirchhofl’s laws.

4. Discuss briefly application of Kirchhofl’s laws.

5. Explain the nodal voltage method for solving networks. How are the nodal equations written ?

6. Explain Cramer’s rule used for solving equations by determinants.

7. State and explain Superposition theorem.

8. State Norton’s theorem. List the steps for finding the current in a branch of a network with the help of this theorem.

9. State Thevenin’s theorem.

10. State the maximum power transfer theorem and explain its importance.

11. Define the following terms as applied to an alternating current :

Cycle, frequency, time period, amplitude.

12. What do you mean by the term “Phase difference” ?

13. Explain the following terms relating alternating current :

(i) R.M.S. value (ii) Average value

(iii) Form factor (iv) Peak factor.

14. Explain briefly the following as applied to A. C. series and parallel circuits :

(i) Resonance frequency (ii) Q-factor.

15. What do you mean by transient disturbances ?

16. Define single energy and double energy transients.

** **

**UNSOLVED EXAMPLES**

1. An alternating current of frequency 60Hz has a maximum value of 120 A. Write down the equation for its instantaneous value. Reckoning time from the instant the current is zero and is becoming positive, find:

(i) The instantaneous value after 1 / 360 second

(ii) The time taken to reach 96 A for the first time.

[Ans. 103.9 A, 0.00245 second]

2. An alternating current of frequency 50 Hz has a maximum value of 100 A. Calculate:

(i) Its value 1/600 second after the instant the current is zero and its value decreasing thereafter

wards.

(ii) How many seconds after the instant the current is zero (increasing thereafter wards) will the current attain the value of 86.6 A ?

Ans. – 50 A, 1/300 s)

3. Calculate the r.m.s. value, the form factor of a periodic voltage having the following values for equal time intervals changing suddenly from one value to the next : 0, 5, 10, 20, 50, 60, 50, 20, 10, 5, 0, 5, 10 V etc. What would be the r.m.s. value of sine wave having the same peak value ?

Ans. 31 V ; 23 V ; 1.35 ; (app.) ; 42.2 VI

4. A sinusoidally varying alternating current has an average value of 127.4 A. When its value is zero, then its rate of change is 62,800 A/s. Find the analytical expression for the sine wave.

Ans. i = 200 sin 100 ]

5. A coil of resistance 10 Q and inductance 0.1 H is connected in series with a 150 11F capacitor across a 200 V, 50 Hz supply. Calculate (i) the inductive reactance, (ii) the capacitive reactance, (iii) the impedance (iv) the current, (v) the power factor, (vi) the voltage across the coil and the capacitor respectively.

[Ans. (i) 31.4 n, (ii) 21.2 n, (iii) 14.3 n, (iv) 14 A, (v) 0.7 lag (vi) 460 V, 297 VI

6. A circuit is made up of 10 Q resistance, 12 m H inductance and 281.5 11F capacitance in series. The supply voltage is 100 V (constant). Calculate the value of the current when the supply frequency is (i) 50 Hz and (ii) 150 Hz.

[Ans. 8 A leading ; 8 A lagging)

7. A coil having a resistance of 10 Q and an inductance of 0.2 His connected in series with a capacitor of 50.7 11F. The circuit is connected across a 100 V, 50 Hz A.C. supply. Calculate (i) the current flowing

(ii) the voltage across the capacitor (iii) the voltage across the coil. Draw a vector diagram to scale.

[Ans. (i) 10 A (ii) 628 V (iii) 635 VI

8. A coil is in series with a 20 ~ capacitor across a 230 V, 50 Hz supply. The current taken by the circuit is 8 A and the power consumed is 200 W. Calculate the inductance of the coil if the power factor of the circuit is (i) leading and (ii) lagging. Sketch a vector diagram for each condition and calculate the coil power factor in each case.

[Ans. 0.415 H; 0.597 H; 0.0238 ; 0.0166)

9. An A. C. series circuit has a resistance of 10 n, an inductance of 0.2 H and a capacitance of 60 F Calculate:

(i) the resonant frequency (ii) the current and

(iii) the power at resonance.

Given that the applied voltage is 200 V.

[Ans. 46 Hz ; 20 A ; 4 kW]

10. A circuit consists of an inductor which has a resistance of 10 n and an inductance of 0.3 H, in series with a capacitor of 30 F capacitance. Calculate :

(i) The impedance of the circuit to currents of 40 Hz ;

(ii) The resonant frequency ;

(iii) The peak value of stored energy in joules when the applied voltage is 200 Vat the resonant frequency.

[Ans. 58.31 n; 53 Hz; 120 J]

11. A resistor and a capacitor are connected in series with a variable inductor. When the circuit is connected to a 240 V, 50 Hz supply, the maximum current given by varying the inductance is 0.5 A. At this current, the voltage across the capacitor is 250 V. Calculate the values of the following

(i) The resistance ; (ii) The capacitance ;

(iii) The inductance.

Neglect the resistance of the inductor. [Ans. 480 n, 6.36 ; 1.59 HI

12. A resistance, a capacitor and a variable inductance are connected in series across a 200 V, 50 Hz supply. The maximum current which can be obtained by varying the inductance is 314 rnA and the voltage across the capacitor is then 300 V. Calculate the capacitance of the capacitor and the values of the inductance and resistance. [Ans. 3.33 3.04 H, 637 Q]

13. A circuit consisting of a coil of resistance 12 n and inductance 0.15 H in series with a capacitor of 12 is connected to a variable frequency supply which has a constant voltage of 24 V.

Calculate : (i) The resonant frequency, (ii) The current in the circuit at resonance, (iii) The voltage across the capacitor and the coil at resonance. [Ans. (i) 153 Hz, (ii) 2 A, (iii) 224 V)

14. A resistance of24 n, a capacitance of 150 and an inductance of0.16 Hare connected in series with each other. A supply at 240 V, 50 Hz is applied to the ends of the combination. Calculate (i) the current in the circuit (ii) the potential differences across each element of the circuit (iii) the frequency to which the supply would need to be changed so that the current would be at unity power-factor and find the current at this frequency.

[Ans. (i) 6.37 A (ii) VR = 152.8 V, Vc = 320 V, VL = 123.3 V, (iii) 32Hz; 10 A]

15. A coil-A of inductance 80 mH and resistance 120 n is connected to a 230 V, 50 Hz single-phase supply. In parallel with it is a 16 JlF capacitor is series with a 40 n non-inductive resistor B. Determine

(i) The power factor of the combined circuit,

(ii) The total power taken from the supply.

[Ans. (i ) 0.945 lead (ii) 473 W)

16. A choking coil of inductance 0.08 Hand resistance 12 ohm, is connected in parallel with a capacitor of 120 . The combination is connected to a supply at 240 V, 50 Hz. Determine the total current from the supply and its power factor. Illustrate your answers with a phasor diagram.

[Ans. 3.94 A, 0.943 lag)

17. A choking coil having a resistance of20 n and an inductance of 0.07 henry is connected with a capacitor of 60 capacitance which is in series with a resistor of 50 n. Calculate the total current and the phase angle when this arrangement is connected to 200 V, 50 Hz mains. [Ans. 7.15 A, 24 39′ lag)

18. A coil of resistance of 15 n and inductance 0.05 His connected in parallel with a non-inductive resistance of 20 n. Find (i) the current in each branch ; (ii) the total current (iii) the phase angle of whole arrangement for an applied voltage of 200 Vat 50 Hz.

[Ans. 9.22 A ; 10 A ; 22.1 •]

19. A sinusoidal 50 Hz voltage of200 V (r.m.s.) supplies the following three circuits which are in parallel: (i) a coil of inductance 0.03 Hand resistance 3 n; (ii) a capacitor of 400 JlF in series with a resistance of 100 n ; (iii) a coil of inductance 0.02 H and resistance 7 n in series with a 300 JlF capacitor. Find the total current supplied and draw a complete vector diagram.

20. In a series-parallel circuit, the two parallel branches A and B are in series with C. The impedances are ZA = (10-j8) n, Z8 = (9-j6) n and Zc = (100 + j0). Find the currents IA and 18 and the phase difference between them. Draw the phasor diagram.

[Ans. !A = 12.71 L- 3Ci• 58′ 18 = 15 L- 35• 56′; 4581]