Problem #0265 Electricity & Magnetism Sub-menu Problem #0267 Chapters Chapters

536.

Problem 39.21 (RHK)

We have to show that the fractional width of the resonance curves in LCR circuits is given, to a close approximation, by

in which is the resonant frequency and is the width of the resonance peak at . Note that this expression may be written as , which shows that clearly that a “high-Q” circuit has a sharp resonance peak, that is, a small value of .

Solution:             Click For PDF Version

Let an LCR circuit be driven by a sinusoidal emf

,

and let the current in the circuit be given by the function

.

In an LCR circuit the current amplitude and the emf amplitude are related to each other by the impedance

,

as

The peak value of is at the resonance frequency

.

Let at frequencies and the peak current be half of , that is .

We, therefore, have the equation from which the frequencies can be calculated:

We assume

We thus have the equation

Therefore,

.

Therefore, the width of the resonance peak at half maximum is

where “quality” of an LCR is defined as