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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
Define 'quality factor' of resonance in a series LCR circuit. What is its SI unit?
(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.
In a series LCR circuit, obtain the condition under which the impedance of the circuit is minimum ?
A series LCR circuit with R = 20 Ω, L = 1.5 H and C = 35 µF is connected to a variable-frequency 200 V ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?
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?
The resonant frequency of a RF oscillator is 1 MHz and its bandwidth is 10 kHz. The quality factor will be :
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.
Draw the phasor diagram for a series LRC circuit connected to an AC source.
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 ______.
What constitutes a series LCR circuit?
