The FM radio band covers the frequency range 88-108 MHz. If the variable capacit
ID: 2213861 • Letter: T
Question
The FM radio band covers the frequency range 88-108 MHz. If the variable capacitor in an FM receiver ranges from 13.9pF to 20.9 pF , what inductor should be used to make an LC circuit whose resonant frequency spans the FM band? Thanks!Explanation / Answer
Q You have an inductor that you are planning to use in series with a variable capacitor C in the tuning section of a radio. (a) If you have a fixed inductance L = 2.5 mH, find the maximum and minimum capacitances the variable capacitor must be able to reach in order that the resonant frequencies of the circuit cover the entire AM band: 550 - 1600 kHz. Neglect the internal resistance of the inductor. Cmax = ? F Cmin = ? F (b) For the parameters of part (a), if a current is present in the circuit with peak value 3 µA, calculate the maximum voltage that appears across the inductor and capacitor, respectively, at the upper end of the AM frequency band. VL max = ? V VC max = ? V (c) Suppose now you wish to build a variable LC circuit whose resonant frequencies cover the full FM band: 88 - 108 MHz. If you choose a fixed inductance L = 29 µH, what are the maximum and minimum capacitances the variable capacitor must reach in this case? C'max = ? F C'min = ? F ans A. First we put the frequencies into rad/s. w1 = 5.5E5*2pi = 3455751.91894877 rad/s w2 = 1.6E6*2pi = 10053096.4914873 rad/s (BTW, I'm not showing off with all these sig. figs, it's just easier to copy/paste straight from my calculator.) Then we solve for C: w = 1/sqrt(LC) ==> C = 1/(w^2L) C1 = 3.34946061627564D-11 F C2 = 3.95785873602883D-12 F B. We know that at resonance, XL = -XC, so the peak V and I are the same in the capacitor and the inductor. Arbitrarily choosing XL to work with, V = I*XL (Note, these are not Roman numerals; nor is C!) XL = wL VL max = VC max = I*XL = I*w2*L = 0.075398223686155 V C. Just repeat step A with the two FM frequencies and the new value for L. I hope you're starting to get the hang of these problems. Given the questions you've already asked and gotten answers to, this one should have been relatively easy. A quick review: XL = wL XC = -1/(wC) X = XL+XC At resonance (as in a tuner), XL = -XC which leads to (1) Resonant frequency w = 1/sqrt(LC) rad/s (2) X = 0 At any frequency, the reactive voltage Vx = IX, just like the resistive voltage Vr = IR. The difference is that X changes with frequency and R doesn't. Also, in an RLC circuit not at resonance, Vx and Vr differ in phase by 90 deg or pi/2 rad. The total voltage V = sqrt(Vx^2+Vr^2). In a series-resonant LC circuit Vx is the algebraic sum of V(L) and V(C). V(L) and V(C) may be, and at resonance are, much larger than Vx, but they are phase-opposed and thus sum to a smaller value. In a parallel-resonant LC circuit, the same statements can be made about the currents Ix, I(L) and I(C). I hope this will help you have more confidence in, and place more reliance on, your own work. Email me if you want to discuss this further.
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