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Circuits and Resonance

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.

2. A 50nF capacitor and a 100mH inductor with 25Ω winding resistance are used to form a resonant circuit.

a. What is the approx resonant frequency.
b. Compared with true resonance, how do the operating conditions in a parallel-connected circuit vary when the reactances of the inductor and capacitor are equal.
c. How do the series and parallel Q values change as the inductors winding resistance increases.

3. State the operating condition that occurs when an LC circuit is at resonance

Solution Preview

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 ...

Solution Summary

The solution is an detailed discussion of the questions. Each question has an answer of approximately 250 words each.

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