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CourseNotes.49

CourseNotes.49 - m r R 2 ± ω d L-1 ω d C ² 2(48 We can...

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12.4 Phasors and the RLC Circuit Recall from last semester that a phasor was a vector rotating the the xy plane at a given angular frequency. take our RLC circuit and project the potential of each of the components onto this phasor diagram, e find an easy way to calculate the amplitude of the final current amplitude. The current and its phase will be the same throughout the circuit, so we must project our phasors with the proper angular shifts derived above. Resistor: Current and voltage are in phase. Capacitor: The voltage follows the current (and hence the voltage of the resistor) by 90 . Inductor: The voltage leads the current (and hence the voltage of the resistor) by 90 . To find the final voltage in the circuit, we simply need to add the lengths of all of these vector vectorially. The amplitude is given by E 2 m = ( IR ) 2 + ( IX L - IX C ) 2 I = E m p R 2 + ( X L - X 2 C ) Or, plugging in for X L and X C , the amplitude of the current in the circuit is given by
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Unformatted text preview: m r R 2 + ± ω d L-1 ω d C ² 2 (48) We can also ﬁnd the overall phase shift between the current and the potential using the standard methods of vector analysis. tan φ = X L-X C R If we plot this current amplitude versus the natural frequency, we see the phenomena of reso-nance emerge (see ﬁgure 35). By looking at the plot we can clearly see that the largest amplitude of the current is induced when the circuit is driven precisely at its resonance frequency. ω d = ω = 1 √ LC Figure 35: Resonance in an RLC circuit. 12.5 Problems Problem 31.21 In an oscillating LC circuit, L = 3 mH and C = 2 . 7 μF . At t = 0 the charge on the capacitor is zero and the current is 2 A . (a) What is the maximum charge that will appear on the capacitor? (b) At what earliest time t > 0 is the rate at which energy is stored in the capacitor greatest, and (c) what is that greatest rate? 49...
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