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प्रश्न
Derive an expression for the average power dissipated in a series LCR circuit.
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उत्तर
We have seen that a voltage v = vm sin ωt applied to a series RLC circuit drives a current in the circuit given by `"i" = "i"_"m" sin(omega t + ∅)` Where
`"I"_"m" = "v"_"m"/"z" and ∅ = tan^-1 (("X"_"c"-"X"_"L")/("R"))`
Therefore, the instantaneous power p supplied by the source is
`"p" = "vi" = ("v"_"m"sin omega t) xx ["i"_"m" sin(omega t + ∅)]`
= `("v"_"m""i"_"m")/(2)[cos ∅ - cos(2omega t + ∅)]`
The average power over a cycle is given by the average of the two terms in R.H.S. of the above equation. It is only the second term which is time-dependent. Its average is zero (the positive half of the cosine cancels the negative half). Therefore,
`"P" = ("v"_"m""i"_"m")/(2) cos ∅ = ("v"_"m")/sqrt2 ("i"_"m")/sqrt2 cos ∅`
= `"V" "I" cos ∅`
This can also be written as
`"P" = "I"^2 "Z"cos ∅`
So, the average power dissipated depends not only on the voltage and current but also on the cosine of the phase angle ∅ between them. The quantity cos ∅ is called the power factor.
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संबंधित प्रश्न
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.
Keeping the source frequency equal to the resonating frequency of the series LCR circuit, if the three elements, L, C and R are arranged in parallel, show that the total current in the parallel LCR circuit is minimum at this frequency. Obtain the current rms value in each branch of the circuit for the elements and source specified for this frequency.
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 a series LCR circuit supplied with AC, ______.
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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.
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.
Draw the impedance triangle for a series LCR AC circuit and write the expressions for the impedance and the phase difference between the emf and the current.
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