Advertisements
Advertisements
प्रश्न
A voltage V = V0 sin ωt is applied to a series LCR circuit. Derive the expression for the average power dissipated over a cycle. Under what condition (i) no power is dissipated even though the current flows through the circuit, (ii) maximum power is dissipated in the circuit?
Advertisements
उत्तर
Voltage V=V0sinωt is applied to an series LCR circuit.
Current is π2" role="presentation" style="position: relative;">π2
`I_0=V_0/Z`
`phi=tan^(-1)((X_C-X_L)/R)`
Instantaneous power supplied by the source is
P=VI
=(V0sinωt)×(I0sin(ωt+ϕ)
`=(V_0I_0)/2[cosphi-cos(2omegat+phi)]`
The average power over a cycle is average of the two terms on the R.H.S of the above equation. The second term is time dependent; so, its average is zero.
`P=(V_0I_0)/2cosphi`
`=(V_0I_0)/(sqrt2sqr2)cosphi`
=VIcosϕ
P=I2Zcosϕ
cosϕ is called the power factor.
Case I
For pure inductive circuit or pure capacitive circuit, the phase difference between current and voltage is `pi/2`
`:.phi=pi/2,cosphi=0`
Therefore, no power is dissipated. This current is sometimes referred to as wattless current.
Case II
For power dissipated at resonance in an LCR circuit,
`X_C-X_L=0, phi=0`
∴ cos ϕ = 1
So, maximum power is dissipated.
APPEARS IN
संबंधित प्रश्न
The figure shows a series LCR circuit with L = 10.0 H, C = 40 μF, R = 60 Ω connected to a variable frequency 240 V source, calculate
(i) the angular frequency of the source which drives the circuit at resonance,
(ii) the current at the resonating frequency,
(iii) the rms potential drop across the inductor at resonance.
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.
A solenoid having inductance 4.0 H and resistance 10 Ω is connected to a 4.0 V battery at t = 0. Find (a) the time constant, (b) the time elapsed before the current reaches 0.63 of its steady-state value, (c) the power delivered by the battery at this instant and (d) the power dissipated in Joule heating at this instant.
An LR circuit with emf ε is connected at t = 0. (a) Find the charge Q which flows through the battery during 0 to t. (b) Calculate the work done by the battery during this period. (c) Find the heat developed during this period. (d) Find the magnetic field energy stored in the circuit at time t. (e) Verify that the results in the three parts above are consistent with energy conservation.
Draw a labelled graph showing a variation of impedance of a series LCR circuit with frequency of the a.c. supply.
In an L.C.R. series a.c. circuit, the current ______.
To reduce the resonant frequency in an LCR series circuit with a generator ______.
Consider the LCR circuit shown in figure. Find the net current i and the phase of i. Show that i = v/Z`. Find the impedance Z for this circuit.
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 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·
