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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
(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.
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.
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.
The selectivity of a series LCR a.c. circuit is large, when ______.
In series LCR circuit, the phase angle between supply voltage and current is ______.
To reduce the resonant frequency in an LCR series circuit with a generator
A series LCR circuit containing a 5.0 H inductor, 80 µF capacitors, and 40 Ω resistor is connected to a 230 V variable frequency ac source. The angular frequencies of the source at which power is transferred to the circuit are half the power at the resonant angular frequency are likely to be ______.
Which of the following statements about a series LCR circuit connected to an ac source is correct?
A series LCR circuit (L = 10 H, C = 10 µF, R = 50 Ω) is connected to V = 200 sin (100t). If ν0 is the resonant frequency and ν is the source frequency, then ______.
To reduce the resonant frequency in an L-C-R series circuit with a generator ______.
