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प्रश्न
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.
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उत्तर
Let an alternating Emf E = E0 sinωt is applied to a series combination of inductor L, capacitor C and resistance R. Since all three of them are connected in series the current through them is same. But the voltage across each element has a different phase relation with current.
The potential difference VL, VC and VR across L, C and R at any instant is given by
VL = IXL, VC = IXC and VR = IR
Where I is the current at that instant.
XL is inductive reactance and
XC is capacitive reactance.
VR is in phase with I. VL leads I by 90° and VC lags behind I by 90°
In the phases diagram,
VL and VC are opposite to each other. If VL > VC then resultant (VL − VC) is represent by OD. OR represent the resultant of VR and (VL − VC). It is equal to the applied Emf E.
`E^2 = V_R^2 + (V_L -V_C)^2`
`E^2 =I^2 +[R^2+(X_L -X_c)^2]`
`or I =E/sqrt (R^2 + (X_2 -X_c)^2)`
The term `sqrt(R^2 +(X_2 - X_c))` is called impedance Z of the LCR circuit.
`Z = sqrt(R^2 +(X_2 -X_c)^2) =sqrt(R^2 +(L omega-1/(comega))^2)`
Emf leads current by a phase angle Φ
`tan phi = (V_L -V_C)/R = (X_L - X_c)/R =(Lomega -1/(comega))/R`
When resonance takes place
`omegaL= 1/(omegac)`
Impedance of circuit becomes equal to R. Current becomes maximum and is equal to `E/R`
`omega_0 = 1/sqrt(LC)`
`f_0 = omega_0/(2pi) = 1/(2pisqrt(LC))`
This is the condition for resonance.
When at resonance f = f0 the current in the circuit is maximum and hence impedance of the circuit is maximum for values of f less than or greater than f0 comparatively small current flames in the circuit.
संबंधित प्रश्न
A voltage V = V0 sin ωt is applied to a series LCR circuit. Derive the expression for the average power dissipated over a cycle. Under what condition (i) no power is dissipated even though the current flows through the circuit, (ii) maximum power is dissipated in the circuit?
(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 C1, to be connected in parallel with the capacitor C, in order to make the power factor of the circuit unity.
Two coils A and B have inductances 1.0 H and 2.0 H respectively. The resistance of each coil is 10 Ω. Each coil is connected to an ideal battery of emf 2.0 V at t = 0. Let iA and iBbe the currents in the two circuit at time t. Find the ratio iA / iB at (a) t = 100 ms, (b) t = 200 ms and (c) t = 1 s.
Answer the following question.
What is the phase difference between the voltages across the inductor and the capacitor at resonance in the LCR circuit?
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.
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 an LCR circuit having L = 8 henery. C = 0.5 µF and R = 100 ohm in series, the resonance frequency in radian/sec is
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 ______.
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.
