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
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 ac circuit as shown in the figure has an inductor of inductance L and a resistor or resistance R connected in series. Using the phasor diagram, explain why the voltage in the circuit will lead the current in phase.
If an LCR series circuit is connected to an ac source, then at resonance the voltage across ______.
The resonant frequency of a RF oscillator is 1 MHz and its bandwidth is 10 kHz. The quality factor will be :
To reduce the resonant frequency in an LCR series circuit with a generator
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.
An alternating voltage of 220 V is applied across a device X. A current of 0.22 A flows in the circuit and it lags behind the applied voltage in phase by π/2 radian. When the same voltage is applied across another device Y, the current in the circuit remains the same and it is in phase with the applied voltage.
- Name the devices X and Y and,
- Calculate the current flowing in the circuit when the same voltage is applied across the series combination of X and Y.
The net impedance of circuit (as shown in figure) will be ______.
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 ______.
Which of the following combinations should be selected for better tuning of an L-C-R circuit used for communication?
