Advertisements
Advertisements
Question
Derive an expression for the average power dissipated in a series LCR circuit.
Solution
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
RELATED QUESTIONS
In a series LCR circuit connected to an a.c. source of voltage v = vmsinωt, use phasor diagram to derive an expression for the current in the circuit. Hence, obtain the expression for the power dissipated in the circuit. Show that power dissipated at resonance is maximum
In a series LCR circuit, obtain the condition under which watt-less current flows in the circuit ?
An L-R circuit has L = 1.0 H and R = 20 Ω. It is connected across an emf of 2.0 V at t = 0. Find di/dt at (a) t = 100 ms, (b) t = 200 ms and (c) t = 1.0 s.
An inductor-coil of inductance 17 mH is constructed from a copper wire of length 100 m and cross-sectional area 1 mm2. Calculate the time constant of the circuit if this inductor is joined across an ideal battery. The resistivity of copper = 1.7 × 10−8 Ω-m.
An LR circuit having a time constant of 50 ms is connected with an ideal battery of emf ε. find the time elapsed before (a) the current reaches half its maximum value, (b) the power dissipated in heat reaches half its maximum value and (c) the magnetic field energy stored in the circuit reaches half its maximum value.
An inductor of inductance 2.00 H is joined in series with a resistor of resistance 200 Ω and a battery of emf 2.00 V. At t = 10 ms, find (a) the current in the circuit, (b) the power delivered by the battery, (c) the power dissipated in heating the resistor and (d) the rate at which energy is being stored in magnetic field.
Choose the correct answer from given options
The selectivity of a series LCR a.c. circuit is large, when
To reduce the resonant frequency in an LCR series circuit with a generator
A series LCR circuit containing 5.0 H inductor, 80 µF capacitor and 40 Ω resistor is connected to 230 V variable frequency ac source. The angular frequencies of the source at which power transferred to the circuit is half the power at the resonant angular frequency are likely to be ______.
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 ______.
