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Calculating resonant frequency
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The formula for calculating resonant frequency fo, in terms of the inductance L in henrys and the capacitance C in farads, is fo 0.159/(LC)1/2
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The 1/2 power is the square root. If you know L and C in henrys and farads, and you want to find fo, do these calculations in this order: First, find the product LC, then take the square root, then divide 0.159 by this value. The result is fo in hertz. The formula will also work to find fo in megahertz (MHz), when L is given in microhenrys ( H) and C is in microfarads ( F). These values are far more common than hertz, henrys, and farads in electronic circuits. Just remember that millions of hertz go with millionths of henrys and millionths of farads. This formula works for both series-resonant and parallel-resonant RLC circuits.
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320 Power and resonance in ac circuits
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Find the resonant frequency of a series circuit with an inductance of 100 H and a capacitance of 100 pF. First, convert the capacitance to microfarads: 100 pF 0.000100 F. Then find the product LC 100 0.000100 0.0100. Take the square root of this, getting 0.100. Finally, divide 0.159 by 0.100, getting fo 1.59 MHz.
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Find the resonant frequency of a parallel circuit consisting of a 33- H coil and a 47-pF capacitor. Again, convert the capacitance to microfarads: 47 pF 0.000047 F. Then find the product LC 33 0.000047 0.00155. Take the square root of this, getting 0.0394. Finally, divide 0.159 by 0.0394, getting fo 4.04 MHz. There are times when you might know the resonant frequency fo that you want, and you need to find a particular inductance or capacitance instead. The next two problems illustrate this type of situation.
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A circuit must be designed to have fo 455 kHz. A coil of 100 H is available. What size capacitor is needed Convert kHz to MHz: 455 kHz 0.455 MHz. Then the calculation proceeds in the same way as with the preceding problem: fo 0.159/(LC)1/2 0.455 0.159/(100 C)1/2 0.4552 0.1592/(100 C) 0.207 0.0253/(100 C) 0.207 100 C 0.0253 20.7 C 0.0253 C 0.0253/20.7 0.00122 F 1220 pF
A circuit must be designed to have fo 9.00 MHz. You have a 33-pF fixed capacitor available. What size coil will be needed to get the desired resonant frequency Use the formula fo 0.159/(LC)1/2, and plug in the values. Convert the capacitance to microfarads: 33 pF 0.000033 F. Then just manipulate the numbers, using familiar rules of arithmetic, until the value of L is ferreted out : 9.00 0.159/(L 0.000033)1/2 9.002 0.1592/(0.000033 L) 81.0 0.0253/(0.000033 L) 81.0 0.000033 L 0.0253 0.00267 L 0.0253 L 0.0253/0.00267 9.48 H
AM FL Y
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Resonant devices 321 In practical circuits, variable inductors and/or variable capacitors are often placed in tuned circuits, so that small errors in the frequency can be compensated for. The most common approach is to design the circuit for a frequency slightly higher than fo, and to use a padder capacitor in parallel with the main capacitor (Fig. 17-13).
17-13 Padding capacitors (Cp) allow adjustment of resonant frequency in a series LC circuit (A) or a parallel LC circuit (B).
Resonant devices
While resonant circuits often consist of coils and capacitors in series or parallel, there are other kinds of hardware that exhibit resonance. Some of these are as follows.
Crystals
Pieces of quartz, when cut into thin wafers and subjected to voltages, will vibrate at high frequencies. Because of the physical dimensions of such a crystal, these vibrations occur at a precise frequency fo, and also at whole-number multiples of fo. These multiples, 2fo, 3fo, 4fo, and so on, are called harmonics. The frequency fo is called the fundamental frequency of the crystal. Quartz crystals can be made to act like LC circuits in electronic devices. A crystal exhibits an impedance that varies with frequency. The reactance is zero at fo and the harmonic frequencies.
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