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Unformatted text preview: Resonance and Electric Resonance and Electric Filters Filters Chapter 9 Artemio Artemio P. P. Magabo Magabo Professor of Electrical Engineering Professor of Electrical Engineering Department of Electrical and Electronics Engineering University of the Philippines - Diliman Revised by Luis G. Sison, Mar 7, Revised by Luis G. Sison, Mar 7, 2004 2004 Revised Jhoanna Pedrasa Sept 2005 Revised Jhoanna Pedrasa Sept 2005 Department of Electrical and Electronics Engineering - 2 Network Sensitivity to Frequency Recall the reactance of an inductor or a capacitor. L X L ω = in Ω , for an inductor C 1 X C ω = in Ω , for a capacitor Plots of X L and X C versus ω are shown. ω X L X C Note: The inductive reactance increases with ω . The capacitive reactance decreases with ω . Reactance Department of Electrical and Electronics Engineering - 3 Consider next the series RLC network. Z eq R L C The equivalent impedance is C j 1 L j R Z eq ω + ω + = A plot of the magnitude of Z eq is shown. ω |Z eq | Notes : 1. At low ω , C acts as an open circuit ( ∞ Z eq ). 2. As ω increases, L dominates (increasing to ∞ Z eq ). Department of Electrical and Electronics Engineering - 4 Example : Plot the magnitude of V O from 0 to 1 kHz. 15 Ω 10 ∠ 0 V +- 0.1 H 2.53 mF + V O- Using voltage division, the output V O can be expressed as S C j 1 O V L j R R V ω + ω + = S 2 V 1 LC ) (j CR j CR j + ω + ω ω = 10 1 (2.53)10 ) j (37.95)10 j (37.95)10 j V 4- 2 3 3 O ∠ + ω + ω ω =-- ( Substituting, The magnitude is ( ) ( ) 10 (2.53)10 (37.95)10 (37.95)10 V | 2 4- 2 2 3 3 O ω- + ω ω =-- 1 | Department of Electrical and Electronics Engineering - 5 Plot |V O | by substituting different values of f from 0 to 1 kHz with ω = 2 π f. 2. At high frequencies, the inductor acts as an open circuit so again, V O = 0. We note that: 1. At low frequencies, the capacitor is open so V O = 0. This is an example of a frequency-dependent network and we have shown that its performance is caused by the reactive elements in the circuit. Frequency (Hz) |V O | 10 0 250 500 750 1000 Department of Electrical and Electronics Engineering - 6 The Semilog Plot In the previous example, much of the variation in the low frequencies is difficult to see. A standard technique to get a more meaningful graph is to plot the frequency on a log axis , expanding the low-frequency portions. |V O | 10 10 10 2 10 3 Frequency (Hz) semilog plot 10 Frequency (Hz) |V O | 10 0 250 500 750 1000 Department of Electrical and Electronics Engineering - 7 Natural Frequency - Series RLC Characteristic Equation: LC 1 s L R s 2 = + + LC 1 L 2 R L 2 R s , s 2 2 1- ±- = Roots: Recall the transient response of the series RLC network. R i E +- L C Department of Electrical and Electronics Engineering - 8 where 2 2 o d α- ω = ω L 2 R = α LC 1 o = ω The circuit is underdamped when LC 1 L 2 R 2 < The circuit is undamped when R=0. We get t sin K t cos K i o 2 o 1 t ω + ω = Note:...
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