Circuits and Resonance

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1. An inductor and capacitor are used to make a resonant circuit. The inductor has a winding resistance that is relatively small compared with its reactance. Explain why a series connection and a parallel connection of the inductor and capacitor produce slightly different resonant frequencies in some detail.
Answer :
Capacitance and inductance have, in a sense, opposite effects, so they tend to cancel each other. The phenomenon of resonance occurs when the capacitive and inductive reactances are equal in a series or parallel combination of C and L, which happens at a frequency fo= 1/2π√(LC), called the resonant frequency. Resonance can be defined as zero phase angle, or as minimum or maximum impedance. For L and C in series, the zero phase angle resonant frequency is given by the formula above. When L and C are in parallel, the frequency of zero phase is slightly different, but very close to the value given by the formula if Q is large.
A resistance R must always be included in series with the inductance L of a coil to represent the losses in the core material as well as the DC resistance. For an air-core inductor, this resistance is small, but it is considerable for a ferrite-core inductor. This R is larger than the DC resistance of the winding, and is roughly proportional to frequency (since core losses are proportional to frequency, roughly). The ratio ωL/R is, therefore, roughly constant with frequency and is a useful parameter, denoted by Q. In a resonant, or tuned, circuit, this turns out to be the Q you are familiar with.
When L and C are connected in series, the impedance at resonance is due to R alone. Since the impedance is purely resistive, the phase angle between the voltage applied to the branch and the current through it is zero. At lower ...