PHY2049ch31C%283-22-10%29

PHY2049ch31C%283-22-10%29 - Alternating current (AC)...

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Alternating current (AC) circuits (Chapt. 31, continued) t v i T 2T V C –V C v, i I –I
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Last class, in preparation for understanding the behavior of the driven RLC oscillator ( continuously powered by a sinusoidally oscillating voltage): md (t) sin t ξ= ξ ω We developed the relation between voltage and current amplitudes in the phasor and time domains. ω d ξ (t) , i(t) ξ (t), i(t) t φ I ξ m Phasor domain Time domain
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The voltage and current amplitudes are rotating phasors (with angular frequency ω d ) while their projections on the vertical axis are the time dependent instantaneous values: md (t) sin t ξ ω ω d ξ (t) , i(t) ξ (t), i(t) t φ I ξ m d i(t) Isin( t ) = ω− φ Before considering the full, driven RLC circuit we first considered the individual components driven by this AC source.
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We found, for the sinusoidally driven individual circuit elements , Circuit Element Resistance, or Reactance Phase of current I Phase constant φ Amplitude Relation Resistor (R) In phase with V R 0 o Capacitor (C) Leads V C by 90 o –90 o Inductor (L) Lags V L by 90 o +90 o C d 1 X C = ω Ld XL = ω RR VI R = CC C X = LL L X = R These phases differences between the voltages across the element and the currents through them are a property of the element and also hold for them in any more complex circuit .
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We apply the AC emf (a voltage), md sin t ξ=ξ ω This will the generate an AC current, d iI s i n (t ) φ Our goal is to determine the resulting current amplitude , I and its phase , φ ? This puts us in a position to consider the driven series RLC circuit.
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The phasor for the emf is a rotating vector of magnitude , rotating with angular velocity ω d , so that its angle with the horizontal axis is at each instant in time, . d t ω The instantaneous current , which must be the same everywhere in the series circuit is the projection of I on the vertical axis. What is not the same throughout the circuit is the voltage across each of the elements R , L and C . We can plot the phasor for the voltage across each element on a voltage phasor plot. The current phasor of (presently) unknown magnitude, I , “follows” with a phase difference from it of φ (also presently unknown). m ξ d t ω −φ I m ξ d t ω φ m ξ
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For the resistor , because the current through it is in phase with the voltage v R across it, the resistor’s voltage phasor lies, parallel to the circuit’s current phasor as the two rotate together with the same ω d , d t ω −φ I i d t ω− φ R v d t ω I R V L V L v The current through an inductor lags the voltage across it by 90 o . So the inductor’s voltage phasor is ahead of the current, I , and therefore the resistor’s voltage phasor, by 90 o .
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d t ω −φ I C V C v The current through a capacitor leads the voltage across it by 90 o , so the capacitor’s voltage phasor lies 90 o behind the current I (and the resistor’s voltage phasor).
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This note was uploaded on 05/17/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.

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PHY2049ch31C%283-22-10%29 - Alternating current (AC)...

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