Advertisements
Advertisements
प्रश्न
In a series, LCR circuit, obtain an expression for the resonant frequency.
Derive an expression for resonant frequency of series resonant circuit.
Advertisements
उत्तर
In a series LCR circuit,
I = `E/sqrt(R^2+(omega L - 1/(omega c))^2`
From this equation, we may observe that when ω = 0, I becomes 0, and again when ω = ∞, I = 0.
∴ This implies there must be a value of ω at which I is maximum.
So when I is maximum, `sqrt(R^2 + (omega L - 1/(omega c))^2` is minimum.
For this either R = 0, or `omega L - 1/(omega c)` = 0
So when `omega L -1/(omega c)` = 0, this is called the resonance condition.
∴ At resonance,
ωL = `1/(omega c)`
⇒ ω2 = `1/(L c)`
⇒ ωr = `1/sqrt(L c)`
⇒ 2 π fr = `1/sqrt(L c)`
∴ Frequency of resonance (fr) = `1/(2 pi) * 1/sqrt(L c)`
i.e., fr = `1/(2 pi) * 1/sqrt(L c)`
This is the required expression for resonant frequency.
संबंधित प्रश्न
In a series LCR circuit connected to an a.c. source of voltage v = vmsinωt, use phasor diagram to derive an expression for the current in the circuit. Hence, obtain the expression for the power dissipated in the circuit. Show that power dissipated at resonance is maximum
In a series LCR circuit, obtain the condition under which the impedance of the circuit is minimum ?
In a series LCR circuit, obtain the condition under which watt-less current flows in the circuit ?
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.
A series LCR circuit is connected to a source having voltage v = vm sin ωt. Derive the expression for the instantaneous current I and its phase relationship to the applied voltage.
Obtain the condition for resonance to occur. Define ‘power factor’. State the conditions under which it is (i) maximum and (ii) minimum.
An LR circuit having a time constant of 50 ms is connected with an ideal battery of emf ε. find the time elapsed before (a) the current reaches half its maximum value, (b) the power dissipated in heat reaches half its maximum value and (c) the magnetic field energy stored in the circuit reaches half its maximum value.
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.
The potential difference across the resistor is 160V and that across the inductor is 120V. Find the effective value of the applied voltage. If the effective current in the circuit be 1.0 A, calculate the total impedance of the circuit.
The selectivity of a series LCR a.c. circuit is large, when ______.
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.
If an LCR series circuit is connected to an ac source, then at resonance the voltage across ______.
Assertion: When the frequency of the AC source in an LCR circuit equals the resonant frequency, the reactance of the circuit is zero, and so there is no current through the inductor or the capacitor.
Reason: The net current in the inductor and capacitor is zero.
In series LCR circuit, the phase angle between supply voltage and current is ______.
At resonant frequency the current amplitude in series LCR circuit is ______.
In an LCR circuit having L = 8 henery. C = 0.5 µF and R = 100 ohm in series, the resonance frequency in radian/sec is
As the frequency of an ac circuit increases, the current first increases and then decreases. What combination of circuit elements is most likely to comprise the circuit?
- Inductor and capacitor.
- Resistor and inductor.
- Resistor and capacitor.
- Resistor, inductor and capacitor.
In series LCR circuit, the plot of Imax vs ω is shown in figure. Find the bandwidth and mark in the figure.
For an LCR circuit driven at frequency ω, the equation reads
`L (di)/(dt) + Ri + q/C = v_i = v_m` sin ωt
- Multiply the equation by i and simplify where possible.
- Interpret each term physically.
- Cast the equation in the form of a conservation of energy statement.
- Integrate the equation over one cycle to find that the phase difference between v and i must be acute.
A series LCR circuit driven by 300 V at a frequency of 50 Hz contains a resistance R = 3 kΩ, an inductor of inductive reactance XL = 250 πΩ, and an unknown capacitor. The value of capacitance to maximize the average power should be ______.
A series LCR circuit containing a resistance of 120 Ω has angular resonance frequency 4 × 105 rad s-1. At resonance the voltage across resistance and inductance are 60 V and 40 V respectively. At what frequency the current in the circuit lags the voltage by 45°. Give answer in ______ × 105 rad s-1.
A series LCR circuit is connected to an ac source. Using the phasor diagram, derive the expression for the impedance of the circuit.
Three students, X, Y and Z performed an experiment for studying the variation of ac with frequency in a series LCR circuit and obtained the graphs as shown below. They all used
- an AC source of the same emf and
- inductance of the same value.
- Who used minimum resistance?
- In which case will the quality Q factor be maximum?
- What did the students conclude about the nature of impedance at resonant frequency (f0)?
- An ideal capacitor is connected across 220 V, 50 Hz, and 220 V, 100 Hz supplies. Find the ratio of current flowing through it in the two cases.
To reduce the resonant frequency in an L-C-R series circuit with a generator ______.
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 ______.
What constitutes a series LCR circuit?
Which law is applied around the loop to obtain the circuit equation?
