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LECTURE 35 AC CIRCUITS (RLC & LC) AND

Instructor: Kazumi Tolich Lecture 35

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¨ Reading chapter 23-12. ¤ RLC circuits ¤ LC circuits ¤ Resonance RLC circuit

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¨ Consider an RLC circuit: a circuit with a resistor with a resistance R, a with a C, an with an of L connected in series with an ac generator oscillating at ω and maximum of Vmax. ¨ diagram for an RLC circuit:

¤ Current phasor with a magnitude Imax.

¤ Resistor-voltage phasor with a magnitude ImaxR. The resistor-voltage phasor is in phase with the current phasor.

¤ Capacitor-voltage phasor with a magnitude ImaxXC. The capacitor-voltage phasor lags current phasor by 90°.

¤ Inductor-voltage phasor with a magnitude ImaxXL. The inductor-voltage phasor leads current phasor by 90°. ¤ The total voltage phasor is the vector sum of the resistor-voltage, capacitor-voltage, and inductor-voltage . Total reactance and impedance

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¨ The magnitude of the phasor representing the vector sum of the inductor-voltage and capacitor-voltage phasors is Imax|XL – XC|, where XL – XC is the total reactance. ¨ The impedance, Z (in Ω) for an RLC circuit is given by

1 � = � + � − � = � + �� − ��

In the case where XL > XC Phase angle ϕ for an RLC circuit

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¨ The phase angle for an RLC circuit is given by

I X X max ( L − C ) X L − XC tanφ = = I R R max

¨ The power factor is given by

R cosφ = Z Clicker question: 1

6 Example: 1

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¨ The ac generator has an rms voltage of

Vrms = 6.00 V and a frequency of f = 30.0 kHz. The inductance of the circuit is L = 0.300 mH, the capacitance is C = 0.100 µF, and the resistance is R = 2.50 Ω. Calculate the rms voltage across the followings. a) The resistor b) The inductor c) The capacitor Resonance of a RLC circuit

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¨ When total reactance � − � is zero: ¤ Impedance, � = � + � − � = �, the minimum allowed. ¤ � is the maximum since � = . ¤ The phase constant ϕ = 0 since tan � = = 0. ¤ The current is in phase with the total voltage. ¤ The power delivered is the maximum since power factor is cos� = cos0 = 1. ¤ The circuit is at resonance, with a resonant frequency given by 1 ω0 = LC Z vs. ω and I vs. ω

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¨ In an RLC circuit with an ac power source, the impedance is the minimum at the resonant frequency.

¨ The current peaks at the resonant frequency.

¨ The smaller the resistance, the larger the resonant current.

Resonance curves Demo: 1

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¨ Driven RLC circuit ¤ Demonstration of resonance in an RLC circuit receivers

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¨ Resonance circuits are used in radio receivers.

¨ Each radio station broadcasts at a particular frequency of radio waves.

¨ You vary the capacitance or inductance of your circuit to match the natural frequency of it to the incoming radio waves.

¨ If the resistance of the receiver circuit is low, the current due to other stations is negligible. Example: 2

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¨ An AM radio picks up a fres = 1000 kHz signal with a maximum voltage of Vmax = 5.0 mV. The tuning circuit consists of an inductor with a inductance of L = 60 µH in series with a . The inductor coil has a resistance of R = 0.25 Ω, and the resistance of the rest of the circuit is negligible. a) To what capacitance should the capacitor be tuned to listen to this radio station? b) What is the maximum current through the circuit at resonance? c) A stronger station at f’ = 1050 kHz produces a maximum antenna signal of Vmax’ = 10 mV. What is the current in the radio at this frequency when the station is tuned to 1000 kHz. LC circuits

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¨ Consider an LC circuit where a capacitor with a capacitance C is initially charged with Q and connected to an inductor with an inductance of L.

¨ The current oscillates in the circuit.

¨ The total energy in the circuit is conserved, but oscillated between in the within the capacitor and in within the inductor.

¨ The system will oscillate indefinitely with a natural frequency, ω0. 1 ω0 = 2π f = LC Demo: 2

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¨ LC oscillation ¤ Demonstration of oscillation in an LC circuit Demo: 3

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¨ ¤ Produces 0.5 million at 350 kHz ¤ Demonstration of resonance

Spark gap

L L’ Lp L Cs 110Vac Cp s ZAP! 110V to 15kV ac ac 1 1 transformer ω0 = = L C L C p p s s Example: 3

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¨ You have an inductor with an inductance of L = 10 mH. What capacitor should you use with it in series to make an oscillator with a frequency of f = 920 kHz? (This frequency is near the center of the AM radio band.)