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
Why does current in a steady state not flow in a capacitor connected across a battery? However momentary current does flow during charging or discharging of the capacitor. Explain.
The figure shows a series LCR circuit with L = 10.0 H, C = 40 μF, R = 60 Ω connected to a variable frequency 240 V source, calculate
(i) the angular frequency of the source which drives the circuit at resonance,
(ii) the current at the resonating frequency,
(iii) the rms potential drop across the inductor at resonance.
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.
Show that in an a.c. circuit containing a pure inductor, the voltage is ahead of current by π/2 in phase ?
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.
A coil of resistance 40 Ω is connected across a 4.0 V battery. 0.10 s after the battery is connected, the current in the coil is 63 mA. Find the inductance of the coil.
An LR circuit with emf ε is connected at t = 0. (a) Find the charge Q which flows through the battery during 0 to t. (b) Calculate the work done by the battery during this period. (c) Find the heat developed during this period. (d) Find the magnetic field energy stored in the circuit at time t. (e) Verify that the results in the three parts above are consistent with energy conservation.
An inductor of inductance 2.00 H is joined in series with a resistor of resistance 200 Ω and a battery of emf 2.00 V. At t = 10 ms, find (a) the current in the circuit, (b) the power delivered by the battery, (c) the power dissipated in heating the resistor and (d) the rate at which energy is being stored in magnetic field.
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.
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.
In series combination of R, L and C with an A.C. source at resonance, if R = 20 ohm, then impedence Z of the combination is ______.
In an LCR series a.c. circuit, the voltage across each of the components, L, C and R is 50V. The voltage across the LC combination will be ______.
In a series LCR circuit the voltage across an inductor, capacitor and resistor are 20 V, 20 V and 40 V respectively. The phase difference between the applied voltage and the current in the circuit is ______.
In series LCR AC-circuit, the phase angle between current and voltage is
The phase diffn b/w the current and voltage at resonance is
Consider the LCR circuit shown in figure. Find the net current i and the phase of i. Show that i = v/Z`. Find the impedance Z for this circuit.
When an alternating voltage of 220V is applied across device X, a current of 0.25A flows which lags behind the applied voltage in phase by π/2 radian. If the same voltage is applied across another device Y, the same current flows but now it is in phase with the applied voltage.
- Name the devices X and Y.
- 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.
