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प्रश्न
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.
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उत्तर
Consider the LCR circuit
An AC source E with voltage `ν = ν_m` sin ωt is applied across LCR circuit
As the inductor, capacitor, and the resistor are connected in series so the current through all of them is the same (same amplitude and same phase)
Let the current be 1= Im sin wt
The voltage across each component has a different phase relation with the current.
- Let the maximum voltage across the resistor be VR = ImR that is in the same phase of the current hence it is represented by OA in the phasor diagram.
- Let the maximum voltage across the inductor be VL = ImXL and that leads the current by `pi/2` it is represented by OD in the phasor diagram.
- Let the maximum voltage across the capacitor be Vc = ImXc and that lags behind the current by `pi/2`, it is represented by OC in the phasor diagram.
Resultant voltage can be found by using the vector sum of the phasors. The resultant voltage is represented by OF.
It can be written as:
`V_m = sqrt(V_R^2 + (V_L - V_c)^2)`
`V_m = sqrt((I_mR)^2 + (I_mX_L - I_mX_c)^2)`
`V_m = I_m sqrt(R^2 + (X_L - X_c)^2)`
`Z = V_m/I_m = sqrt(R^2 + (X_L - X_c)^2)`
or, `Z = V_m/I_m = sqrt(R^2 + (ωL - 1/(ωC))^2)`
Variation of impedance Z with frequency f:
संबंधित प्रश्न
In a series LCR circuit, obtain the condition under which the impedance of the circuit is minimum ?
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 constant current exists in an inductor-coil connected to a battery. The coil is short-circuited and the battery is removed. Show that the charge flown through the coil after the short-circuiting is the same as that which flows in one time constant before the short-circuiting.
Consider the circuit shown in figure. (a) Find the current through the battery a long time after the switch S is closed. (b) Suppose the switch is again opened at t = 0. What is the time constant of the discharging circuit? (c) Find the current through the inductor after one time constant.
(i) An a.c. source of emf ε = 200 sin omegat is connected to a resistor of 50 Ω . calculate :
(1) Average current (`"I"_("avg")`)
(2) Root mean square (rms) value of emf
(ii) State any two characteristics of resonance in an LCR series circuit.
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?
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.
Define Impedance.
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.
