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
In a series LCR circuit, obtain the condition under which watt-less current flows in the circuit ?
The time constant of an LR circuit is 40 ms. The circuit is connected at t = 0 and the steady-state current is found to be 2.0 A. Find the current at (a) t = 10 ms (b) t = 20 ms, (c) t = 100 ms and (d) t = 1 s.
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.
The potential difference across the resistor is 160V and that across the inductor is 120V. Find the effective value of the applied voltage. If the effective current in the circuit be 1.0 A, calculate the total impedance of the circuit.
Derive an expression for the average power dissipated in a series LCR circuit.
In series combination of R, L and C with an A.C. source at resonance, if R = 20 ohm, then impedence Z of the combination is ______.
A coil of 40 henry inductance is connected in series with a resistance of 8 ohm and the combination is joined to the terminals of a 2 volt battery. The time constant of the circuit is ______.
In series LCR circuit, the phase angle between supply voltage and current is ______.
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 ______.
Three students, X, Y and Z performed an experiment for studying the variation of ac with frequency in a series LCR circuit and obtained the graphs as shown below. They all used
- an AC source of the same emf and
- inductance of the same value.
- Who used minimum resistance?
- In which case will the quality Q factor be maximum?
- What did the students conclude about the nature of impedance at resonant frequency (f0)?
- An ideal capacitor is connected across 220 V, 50 Hz, and 220 V, 100 Hz supplies. Find the ratio of current flowing through it in the two cases.
