An alternating voltage and the corresponding current in a circuit are given by ε = 110 sin ⁡ ω t + π 6 and I = 5 sin ⁡ ω t - π 6 respectively. Find the impedance and the average power dissipation in it.

Given, Voltage in the circuit ε = 110 sin ⁡ ω t + π 6 Comparing with ε = ε 0 sin ⁡ ω t + δ 1 ε 0 = 110 V , δ 1 = π 6 Current in the circuit I = 5 sin ⁡ ω t - π 6 Comparing with I = I 0 sin ⁡ ω t + δ 2 I 0 = 5 A , δ 2 = - π 6 Phase difference between voltage and current ϕ = δ 1 - δ 2 = π 3 Impedance of the circuit Z = ε 0 I 0 = 110 5 = 22 Ω Average power dissipated P a v g = ε 0 I 0 2 cos ϕ P a v g = 110 × 5 2 × cos π 3 = 137.5 W

(i) Find the value of the phase difference between the current and the voltage in the series LCR circuit shown below. Which one leads in phase: current or voltage? (ii) Without making any other change, find the value of the additional capacitor C 1 , to be connected in parallel with the capacitor C, in order to make the power factor of the circuit unity.

Given, Inductance of the circuit L = 100 m H Capacitance of the circuit C = 2 μ F Resistance of the circuit R = 400 Ω Voltage of the source V = V 0 sin 1000 t + ϕ So, angular frequency, ω = 1000 r a d s - 1 . Inductive reactance X L = ω L = 1000 × 100 × 10 - 3 Ω = 100 Ω . Capacitive reactance X C = 1 ω C = 1 1000 × 2 × 10 - 6 Ω = 500 Ω . (i) Phase value ϕ between current and voltage is given by, tan ⁡ ϕ = X L - X C R = 100 - 500 400 = - 1 ⇒ ϕ = - π 4 ϕ is negative, implies X L > X C or the current leads the voltage by π / 4 rad. (ii) Power factor is unity when X C = X L or 1 ω C ' = 100 Ω ⇒ C ' = 1 100 ω = 1 100 × 1000 F = 10 μ F ⇒ C ' = C + C 1 or 10 = 2 + C 1 ∴ Required capacitance need to be added C 1 = 8 μ F .

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12th CBSE

IMPORTANT

Earn 100

Calculate the current and power factor of the circuit shown in figure.

Important Questions on Alternating Current and Electrical Machines

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12th CBSE

IMPORTANT

PHYSICS A Reference Book for Class 12 Volume 2>Chapter 1 - Alternating Current and Electrical Machines>Problems For Practice>Q 5

In the circuit shown in figure, the potential differences across resistance, capacitance and inductance are given. Find the emf of the source of alternating current and power factor of the circuit.

Question Image

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12th CBSE

IMPORTANT

PHYSICS A Reference Book for Class 12 Volume 2>Chapter 1 - Alternating Current and Electrical Machines>Problems For Practice>Q 6

A group of electric bulbs has a power rating of

300 W

. An a.c. voltage,

V = 141.4 s i n (314 t + π / 3)

is applied to the group. Calculate the effective current.

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12th CBSE

IMPORTANT

PHYSICS A Reference Book for Class 12 Volume 2>Chapter 1 - Alternating Current and Electrical Machines>Problems For Practice>Q 7

An alternating voltage and the corresponding current in a circuit are given by

ε = 110 \sin (ω t + \frac{π}{6})

and

I = 5 \sin (ω t - \frac{π}{6})

respectively. Find the impedance and the average power dissipation in it.

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12th CBSE

IMPORTANT

PHYSICS A Reference Book for Class 12 Volume 2>Chapter 1 - Alternating Current and Electrical Machines>Problems For Practice>Q 8

An inductor

200 m H

, capacitor

500 μ F

, resistor

10 Ω

are connected in series with a

100 V

, variable frequency a.c. source. Calculate the (i) frequency at which the power factor of the circuit is unity, (ii) current amplitude at this frequency, (iii) Q-factor.

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12th CBSE

IMPORTANT

PHYSICS A Reference Book for Class 12 Volume 2>Chapter 1 - Alternating Current and Electrical Machines>Problems For Practice>Q 9

An inductor of unknown value, a capacitor of

100 μ F

and a resistor of

10 Ω

are connected in series to a

200 V

50 H z

a.c. source. It is found that the power factor of the circuit is unity. Calculate the inductance of the inductor and the current amplitude.

HARD

12th CBSE

IMPORTANT

PHYSICS A Reference Book for Class 12 Volume 2>Chapter 1 - Alternating Current and Electrical Machines>Problems For Practice>Q 10

(i) Find the value of the phase difference between the current and the voltage in the series LCR circuit shown below. Which one leads in phase: current or voltage?

Question Image

(ii) Without making any other change, find the value of the additional capacitor $C_{1}$ , to be connected in parallel with the capacitor C, in order to make the power factor of the circuit unity.

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12th CBSE

IMPORTANT

PHYSICS A Reference Book for Class 12 Volume 2>Chapter 1 - Alternating Current and Electrical Machines>Problems For Practice>Q 11

A circuit containing a

80 m H

inductor and a

60 μ F

capacitor in series is connected to a

230 V

50 H z

supply. If the circuit has a resistance of

15 Ω

, obtain the average power transferred to each element of the circuit and the total power absorbed.

MEDIUM

12th CBSE

IMPORTANT

PHYSICS A Reference Book for Class 12 Volume 2>Chapter 1 - Alternating Current and Electrical Machines>Problems For Practice>Q 12

A resistor $R$ and an element $X$ are connected in series to an ac source of voltage. The voltage is found to lead the current in phase by $π / 4$ . If X is replaced by another element $Y$ , the voltage lags behind the current by $π / 4$ .
(i) Identify elements $X$ and $Y$ .

(ii) When both $X$ and $Y$ are connected in series with $R$ to the same source, will the power dissipated in the circuit be maximum or minimum? Justify your answer.

Calculate the current and power factor of the circuit shown in figure.

Important Questions on Alternating Current and Electrical Machines

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