Advertisements
Advertisements
प्रश्न
Derive an expression for the average power dissipated in a series LCR circuit.
Advertisements
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.
APPEARS IN
संबंधित प्रश्न
An inductor-coil of resistance 10 Ω and inductance 120 mH is connected across a battery of emf 6 V and internal resistance 2 Ω. Find the charge which flows through the inductor in (a) 10 ms, (b) 20 ms and (c) 100 ms after the connections are made.
(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.
Use the expression for Lorentz force acting on the charge carriers of a conductor to obtain the expression for the induced emf across the conductor of length l moving with velocity v through a magnetic field B acting perpendicular to its length.
In a series LCR circuit supplied with AC, ______.
Which of the following components of an LCR circuit, with a.c. supply, dissipates energy?
To reduce the resonant frequency in an LCR series circuit with a generator ______.
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 ______.
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.
Draw a labelled graph showing variation of impedance (Z) of a series LCR circuit Vs frequency (f) of the ac supply. Mark the resonant frequency as f0·
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 ______.
