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What is meant by half-power point frequencies? Obtain an expression for bandwidth in a LCR series circuit. Show half power point frequencies in a curve between alternating current and frequency.

Answer
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Hint: Bandwidth is the difference in the half power frequencies. To obtain an expression for bandwidth, find the values of the half power frequencies. Find the current when the power is half of the maximum in terms of the maximum current. Use V=imaxR and Z2=R2+(2πfL12πfC)2.

Formula used:
V=imaxR
Z2=R2+(2πfL12πfC)2

Complete answer:
Half power points frequencies for a given LCR are the frequencies for which the power in the circuit is half of the maximum power in the circuit.
The current in the circuit at maximum power is also maximum. Let it be imax.
Maximum current is obtained at resonance. At resonance, the frequency of the circuit is called resonance frequency (f0).
Every circuit has two half power frequencies f1 and f2.
The impedance of the circuit is Z=R.
Therefore, at resonanceV=imaxR.
imax=VR …. (i).
And the power is Pmax=imax2R
At the half frequencies, power of the circuit is P=Pmax2=imax2R2=(imax2)2R.
This means that the current in the circuit at half power frequencies is imax2.
Then, V=imax2Z.
Substitute the value of imax from (i).
V=(VR)2Z
2R=Z
2R2=Z2
But, Z2=R2+(2πfL12πfC)2
Therefore,
2R2=R2+(2πfL12πfC)2
R2=(2πfL12πfC)2
R=±(2πfL12πfC)
This proves that there are two values of half power frequencies.
 Therefore,
R=2πf1L12πf1C or R=12πf2C2πf2L.
R=4π2f12LC12πf1C
2πf1CR=4π2f12LC1 ….. (iii).
And
R=14π2f22LC2πf2C
2πf2CR=14π2f22LC ….. (iv).
Add (iii) and (iv).
2πf1CR+2πf2CR=4π2f12LC1+14π2f22LC
(f1+f2)R=2π(f12f22)L
(f1+f2)R=2π(f1+f2)(f1f2)L
(f1f2)=R2πL.
The difference in the half power frequencies i.e. (f1f2), is called bandwidth.
Therefore, we found an expression for the bandwidth.
The curve of alternating current with respect to the frequency of the source is shown below.

 
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Note:
The value of bandwidth can also be found if we know the expression for the quality factor (Q) of a given circuit. The quality factor is defined as Q=f0BW, f0 is the resonance frequency of the LCR circuit and BW is the bandwidth of the circuit.
The quality factor is also given as Q=2πf0LR.
Q=f0BW=2πf0LR
1BW=2πLR
BW=R2πL.