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RLC Circuits Physics 102 Professor Lee Carkner Lecture 25

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Three AC Circuits V max = 10 V, f = 1Hz, R = 10 V rms = 0.707 V max = (0.707)(10) = R = I rms = V rms /R = I max = I rms /0.707 = Phase Shift = When V = 0, I = V max = 10 V, f = 1Hz, C = 10 F V rms = 0.707 V max = (0.707)(10) = X C = 1/(2 fC) = 1/[(2)( )(1)(10)] = I rms = V rms /X C = I max = I rms /0.707 = Phase Shift = When V = 0, I = I max =

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Three AC Circuits V max = 10 V, f = 1Hz, L = 10 H V rms = 0.707 V max = (0.707)(10) = X L = 2 fL = (2)( )(1)(10) = I rms = V rms /X L = I max = I rms /0.707 = Phase Shift = When V = 0, I = I max =

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For capacitor, V lags IFor inductor, V leads I

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RLC Circuits Z = (R 2 + (X L - X C ) 2 ) ½ The voltage through any one circuit element depends only on its value of R, X C or X L however

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RLC Circuit

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RLC Phase The phase angle can be related to the vector sum of the voltages Called the power factor

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RLC Phase Shift Also: tan = (X L - X C )/R The arctan of a positive number is positive so: Inductance dominates The arctan of a negative number is negative so: Capacitance dominates The arctan of zero is zero so: Resistor dominates

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Frequency Dependence The properties of an RLC circuit depend not just on the circuit elements and voltage but also on the frequency of the generator Frequency affects inductors and capacitors exactly backwards High f means capacitors never build up much charge and so have little effect

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High and Low f For “normal” 60 Hz household current both X L and X C can be significant For high f the inductor acts like a very large resistor and the capacitor acts like a resistance-less wire At low f, the inductor acts like a resistance- less wire and the capacitor acts like a very large resistor

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High and Low Frequency

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Today’s PAL a) How would you change V rms, R, C and to increase the rms current through a RC circuit? b) How would you change V rms, R, L and to increase the rms current through a RL circuit? c) How would you change V rms, R, and to increase the current through an RLC circuit? d) What specific relationship between L and C would produce the maximum current through a RLC circuit?

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LC Circuit The capacitor discharges as a current through the inductor This plate then discharges backwards through the inductor Like a mass on a swing

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LC Resonance

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Oscillation Frequency Since they are connected in parallel they must each have the same voltage IX C = IX L = 1/(LC) ½ This is the natural frequency of the LC circuit

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Natural Frequency Example: a swing If you push the swing at all different random times it won’t If you connect it to an AC generator with the same frequency it will have a large current

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Resonance Will happen when Z is a minimum Z = (R 2 + (X L - X C ) 2 ) ½ This will happen when = 1/(LC) ½ Frequencies near the natural one will produce large current

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Impedance and Resonance

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Resonance Frequency

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Resistance and Resonance Note that the current still depends on the resistance So at resonance, the capacitor and inductor cancel out Peak becomes shorter and also broader

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Next Time Read 22.1-22.4, 22.7 Homework, Ch 21, P 71, Ch 22, P 3, 7, 8

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