Advertisements
Advertisements
प्रश्न
A series LCR circuit with L = 0.12 H, C = 480 nF, R = 23 Ω is connected to a 230 V variable frequency supply.
(a) What is the source frequency for which current amplitude is maximum. Obtain this maximum value.
(b) What is the source frequency for which average power absorbed by the circuit is maximum. Obtain the value of this maximum power.
(c) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies?
(d) What is the Q-factor of the given circuit?
Advertisements
उत्तर
Inductance, L = 0.12 H
Capacitance, C = 480 nF = 480 × 10−9 F
Resistance, R = 23 Ω
Supply voltage, V = 230 V
Peak voltage is given as:
V0 = `sqrt2 xx 230` = 325.22 V
(a) Current flowing in the circuit is given by the relation,
I0 = `"V"_0/(sqrt("R"^2 + (ω"L" - 1/(ω"C"))^2`
Where,
I0 = maximum at resonance
At resonance, we have
`ω_"R""L" - 1/(ω_"R""C")` = 0
Where,
ωR = Resonance angular frequency
∴ ωR = `1/sqrt("LC")`
= `1/sqrt(0.12 xx 480 xx 10^-9)`
= 4166.67 rad/s
∴ Resonant frequency, vR = `ω_"R"/(2π) = 4166.67/(2 xx 3.14)` = 663.48 Hz
And, maximum current `("I"_0)_"Max" = "V"_0/"R" = 325.22/23` = 14.14 A
(b) Maximum average power absorbed by the circuit is given as:
`("P"_"av")_"Max" = 1/2 ("I"_0)_"Max"^2 "R"`
= `1/2 xx (14.14)^2 xx 23`
= 2299.3 W
Hence, resonant frequency (vR) is 663.48 Hz.
(c) The power transferred to the circuit is half the power at resonant frequency.
Frequencies at which power transferred is half, = ωR ± Δω
= `2π ("v"_"R" ± Δ"v")`
Where,
Δω = `"R"/(2"L")`
= `23/(2 xx 0.12)`
= 95.83 rad/s
Hence, change in frequency, Δv = `1/(2π)Δω = 95.83/(2π)` = 15.26 Hz
∴ vR + Δv = 663.48 + 15.26 = 678.74 Hz
And, vR − Δv = 663.48 − 15.26 = 648.22 Hz
Hence, at 648.22 Hz and 678.74 Hz frequencies, the power transferred is half.
At these frequencies, current amplitude can be given as:
I' = `1/sqrt2 xx ("I"_0)_"Max"`
= `14.14/sqrt2`
= 10 A
(d) Q-factor of the given circuit can be obtained using the relation, Q = `(ω_"R""L")/"R"`
= `(4166.67 xx 0.12)/23`
= 21.74
Hence, the Q-factor of the given circuit is 21.74.
संबंधित प्रश्न
Define 'quality factor' of resonance in a series LCR circuit. What is its SI unit?
A series LCR circuit is connected to an ac source. Using the phasor diagram, derive the expression for the impedance of the circuit. Plot a graph to show the variation of current with frequency of the source, explaining the nature of its variation.
Derive an expression for the average power consumed in a series LCR circuit connected to a.c. source in which the phase difference between the voltage and the current in the circuit is Φ.
Find the value of t/τ for which the current in an LR circuit builds up to (a) 90%, (b) 99% and (c) 99.9% of the steady-state value.
An inductor-coil of inductance 17 mH is constructed from a copper wire of length 100 m and cross-sectional area 1 mm2. Calculate the time constant of the circuit if this inductor is joined across an ideal battery. The resistivity of copper = 1.7 × 10−8 Ω-m.
A solenoid having inductance 4.0 H and resistance 10 Ω is connected to a 4.0 V battery at t = 0. Find (a) the time constant, (b) the time elapsed before the current reaches 0.63 of its steady-state value, (c) the power delivered by the battery at this instant and (d) the power dissipated in Joule heating at this instant.
In a series, LCR circuit, obtain an expression for the resonant frequency.
Draw a labelled graph showing a variation of impedance of a series LCR circuit with frequency of the a.c. supply.
Answer the following question.
In a series LCR circuit connected across an ac source of variable frequency, obtain the expression for its impedance and draw a plot showing its variation with frequency of the ac source.
Answer the following question.
Draw the diagram of a device that is used to decrease high ac voltage into a low ac voltage and state its working principle. Write four sources of energy loss in this device.
A series LCR circuit with R = 20 Ω, L = 1.5 H and C = 35 µF is connected to a variable-frequency 200 V ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?
Figure shows a series LCR circuit connected to a variable frequency 230 V source. L = 5.0 H, C = 80 µF, R = 40 Ω.
- Determine the source frequency which drives the circuit in resonance.
- Obtain the impedance of the circuit and the amplitude of current at the resonating frequency.
- Determine the rms potential drops across the three elements of the circuit. Show that the potential drop across the LC combination is zero at the resonating frequency.
A series LCR circuit containing 5.0 H inductor, 80 µF capacitor and 40 Ω resistor is connected to 230 V variable frequency ac source. The angular frequencies of the source at which power transferred to the circuit is half the power at the resonant angular frequency are likely to be ______.
Which of the following combinations should be selected for better tuning of an LCR circuit used for communication?
In series LCR circuit, the plot of Imax vs ω is shown in figure. Find the bandwidth and mark in the figure.
An alternating voltage of 220 V is applied across a device X. A current of 0.22 A flows in the circuit and it lags behind the applied voltage in phase by π/2 radian. When the same voltage is applied across another device Y, the current in the circuit remains the same and it is in phase with the applied voltage.
- Name the devices X and Y and,
- Calculate the current flowing in the circuit when the same voltage is applied across the series combination of X and Y.
A series LCR circuit is connected to an ac source. Using the phasor diagram, derive the expression for the impedance of the circuit.
Draw the impedance triangle for a series LCR AC circuit and write the expressions for the impedance and the phase difference between the emf and the current.
A resistance of 200Ω and an inductor of \[\frac {1}{2π}\]Н are connected in series to a.c. voltage of 40 V and 100 Hz frequency. The phase angle between the voltage and current is ______.
