HSC PYQ 13 AC Circuits – Maharashtra Board Solutions

13 AC Circuits

Multiple Choice Questions

1., In the purely resistive A.C. circuit

(A) current leads e.m.f. by a phase angle of

radians.

(B) current leads e.m.f. by a phase angle of radians.

(D) current lags behind e.m.f. by a phase angle of radians.

Ans. (C) current and e.m.f. are in phase

2. In LCR series circuit, at resonance, the power factor is

(A) Zero
(B) 0.5
(C) 1
(D)

Ans. (C) 1

3. In series LCR circuit, at resonance, applied e.m.f. and the current are

(A) out of phase

(B) in phase

(D) differ in phase by radian

Ans. (B) in phase

4. A current in A.C. circuit lags behind the applied emf by . The circuit contains

(A) only resistance

(B) only capacitance

(D) only inductance

Ans. (D) only inductance

5. Reactance of a coil is . On connecting the coil across a source of frequency , the current lags behind e.m.f. by . The inductance of the coil is
(A)
(B)
(C)
(D)

Ans. (A)

6. If A.C. voltage is applied to a pure capacitor, then voltage across the capacitor

(A) leads the current by phase angle .

(B) leads the current by phase angle .

(D) lags behind the current by phase angle ( ) rad.

Ans. (C) lags behind the current by phase angle .

7. In series LCR circuit at resonance, phase difference between current and e.m.f. of source is

(A)
(B)
(C)
(D) zero rad

Ans. (D) zero rad

8. In series LCR circuit , , the impedance of the circuit is:

(A)
(B)
(C)
(D)

Ans. (C)

9. The average value of alternating current over a full cycle is always

( Peak valve of current)
(A) zero
(B)
(C)
(D)

Ans. (A) zero

Theory Questions

13.2 AC Generator

What is the average value of alternating current over a complete cycle?

Ans: Average value of alternating current over a complete cycle is zero.

13.5 Different types of AC circuits

Define:

i. Inductive reactance

ii. Capacitive reactance

iii. Impedance

Ans:

i. Inductive reactance: The opposing nature of an inductor to the flow of alternating current is called inductive reactance.

ii. Capacitive reactance: The capacitive reactance of a capacitor is defined as the ratio of r.m.s voltage (e.m.f) across the capacitor to the corresponding r.m.s current.

iii. Impedance: The ratio of rms voltage to the rms value of current is called impedance. The SI unit of impedance is ohm .

13.6 Power in AC circuit

Obtain an expression for average power dissipated in a purely resistive A.C. circuit. Define power factor of the circuit and state its

Ans: value for purely resistive A.C. circuit. [Oct 15]

i. Expression for average power in purely resistive circuit:

a. Let, be applied e.m.f across a resistor of resistance ‘ ‘ as shown in figure. At certain instant, current is flowing through the resistor. In this case both ‘ ‘ and ‘ ‘ are in phase.

b. Instantaneous power in circuit is given by,

c. Average power for a complete cycle can be obtained by integrating equation (1).

is also called as apparent power.

ii. Power factor for a circuit is given by,

In purely resistive circuit,

Obtain an expression for average power dissipated in series LCR A.C. circuit. Hence obtain an expression for power factor of the circuit.

Ans:

i. Expression for average power dissipated in LCR circuit:

a. Suppose that an alternating e.m.f is applied across an circuit containing L, C and as shown in figure.

b. In such circuit, there is a phase difference between applied emf and current. Therefore, the instantaneous current is given by,

c. Instantaneous power in the circuit is given by,

This expression shows that varies with .

d. Average power dissipated in the circuit can be obtained by integrating equation (1),

e. Also,

f. Substitute equations (3) and (4) in equation (2), we get,

Equation (5) represents required average power or true power dissipated in the circuit.

ii. Power factor:

a. Average power dissipated in LCR series circuit is given by,

b. This implies that, the average power dissipated in LCR series circuit depends on the rms values of current and emf as well as the phase difference between them. The term is called power factor.

c. It is given by,

d. The average power is also termed as true power and product is called apparent power.

Power factor,

e. Consider impedance triangle for series LCR circuit as shown in figure,

13.8 Electric Resonance

What is series LCR resonant circuit? Obtain the expression for impedance. Hence state the conditions for series resonance and derive the expression for resonant frequency.

Ans:

i. A circuit in which inductance L, capacitance C and resistance are connected in series and the circuit admits maximum current corresponding to a given frequency of , is called a series resonance circuit.

ii. The impedance (Z) of an LCR circuit is given by,

iii. At very low frequencies, inductive reactance is negligible but capacitive reactance is very high.

iv. As we increase the applied frequency then increases and decreases.

v. At some angular frequency

i.e.,

Where is called the resonant frequency.

vi. At this particular frequency , since we get . This is the least value of .

vii. Thus, when the impedance of an LCR circuit is minimum, circuit is said to be purely resistive, current and voltage are in phase and hence the current is maximum. This condition of the LCR circuit is called resonance condition and this frequency is called series resonant frequency.

Assuming expression for impedance in a parallel resonant circuit, state the conditions for parallel resonance. Define resonant frequency and obtain an expression for it.

Ans:

i. A parallel resonance circuit is an AC circuit in which a parallel combination of an inductor and capacitor is connected to a source of an alternating e.m.f.

ii. Let the alternating emf supplied by the source be,

iii. In case of an inductor, the current lags behind the applied emf by a phase angle of , then the instantaneous current through is given by

iv. Similarly in a capacitor, as current leads the emf by a phase angle of , we can write

v. Therefore, the total current in the circuit at this instant is,

vi. Therefore, when minimum,

Where is called the resonant frequency. vii. Conditions for parallel resonance to occur,

i.e., .

Numericals

13.5 Different types of circuits

An alternating e.m.f. of peak value and frequency is connected across LCR series circuit with and . Calculate inductive reactance, capacitive reactance and impedance of the circuit.

Solution

Given:

To find: i. Inductive reactance

ii. Capacitive reactance

iii. Impedance

Formulae: i.

ii.

iii.

Calculation: From formula (i),

From formula (ii),

From formula (iii),

Ans: i. The value of inductive reactance is .

ii. The value of capacitive reactance is 127.31 .

iii. The value of impedance is .

An A. C. supply of frequency is supplied to a series combination of condenser,

0.1 henry inductor and resistor.

Calculate inductive and capacitive reactance. Also find impedance of the circuit.

Solution

Given:

To find: i. Inductive reactance

ii. Capacitive reactance

iii. Impedance

Formulae:
i.
ii.

Calculation: From formula (i),

From formula (ii),

From formula (iii),

Ans: i. Inductive reactance is

ii. Capacitive reactance is

iii. Impedance of circuit is

A capacitor of capacitance is connected to a source of alternating e.m.f. of frequency . What is the capacitive reactance?

Solution:

Given:

To find: Capacitive reactance

Formula:

Calculation: From formula,

Ans: The capacitive reactance of a capacitor is .

An alternating voltage given by sin (314.2 ) is connected across a pure resistor of .

Calculate :

i. the frequency of the source

ii. the r.m.s current through the resistor

Solution:

Given: i. On comparing it with standard equation,

We get

ii. Given:

Ans: i. The frequency of the source is .

ii. The rms current through the resistor is

13.6 Power in AC circuit

A resistor is connected to a , supply.

Calculate:

i. r.m.s. value of current and

ii. net power consumed over the full cycle

Solution:

Given:

To find: i. rms current )

ii. Net power consumed

Formulae: i.

ii.

Calculation: From formula (i),

From formula (ii),

Ans: i. The rms current in the circuit is .

ii. Net power consumed over a full cycle is .

13.8 Electric Resonance

An series combination has , and . Determine:

(i) The resonant frequency

(ii) the current in the circuit, and

(iii) Voltages across and , when an alternating voltage of , operating at the resonant frequency, is applied to the series combination.