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there can be many such frequencies The lowest frequency at which resonance occurs is called the resonant frequency, symbolized fo Refer to the schematic diagram of Fig 17-11 You should recognize this as a series RLC circuit At some particular frequency, XL = XC This is inevitable if L and C are finite and nonzero This frequency is fo for the circuit At fo, the effects of capacitive reactance and inductive reactance cancel out The result is that the circuit appears as a pure resistance, with a value that is theoretically equal to R If R = 0, that is, if the resistor is a short circuit, then the circuit is called a series LC circuit, and the impedance at resonance will be theoretically 0 + j 0 The circuit will offer no opposition to the flow of alternating current at the frequency fo This condition is series resonance In a practical series LC circuit, there is always a little bit of loss in the coil and capacitor, so the real part of the complex impedance is not exactly equal to 0 (although it can be extremely small)

The last two properties state the the joint distribution is nondecreasing in both arguments The joint probability density is de ned as pvw (V, W ) = 2 Fvw (V, W ) V W (418)

Parallel Resonance Refer to the circuit diagram of Fig 17-12 This is a parallel RLC circuit Remember that, in this sort of situation, the resistance R should be thought of as a conductance G, with G = 1/R Then the circuit can be called a parallel GLC circuit At some particular frequency fo, the inductive susceptance BL will exactly cancel the capacitive susceptance BC ; that is, BL = BC This is inevitable for some frequency fo, as long as the circuit contains finite, nonzero inductance and finite, nonzero capacitance At the frequency fo, the susceptances cancel each other out, leaving theoretically zero susceptance The admittance through the circuit is then very nearly equal to the conductance, G, of the resistor If the circuit contains no resistor, but only a coil and capacitor, it is called a parallel LC circuit, and the admittance at resonance will be theoretically 0 + j0 That means the circuit will offer great opposition to alternating current at fo, and the complex impedance will theoretically be infinite! This condition is parallel resonance In a practical parallel LC circuit, there is always a little bit of loss in the coil and capacitor, so the real part of the complex impedance is not infinite (although it can be extremely large) Calculating Resonant Frequency The formula for calculating resonant frequency fo, in terms of the inductance L in henrys and the capacitance C in farads, is as follows:

The joint density has the following properties: pvw (V, W ) 0

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