Week 13 Day 1 AC Series RLC Circuit Z Z Z Z R i L 1 C arctan Resonance Driven Series RLC Circuit Resonance occurs when impedances match The current amplitude has the maximum values The phase constant is zero Resonance So the two voltage drops sum to zero The current is exactly in phase with the voltage source 0 Concept Question RLC Circuit with Light Bulb At resonance the voltage across the capacitor is equal in magnitude to the voltage across the induction by phase shifting by 180 Correct answer 4 As I slide the core into the solenoid the light bulb changes brightness I am driving the circuit though resonance by continuously decreasing the natural frequency of the oscillations in the circuit Correct answer 1 In plots that show driving voltage V and current I in a driven RLC circuit you know which is leading by looking at which curve peaks rst Concept Question Leading or Lagging Group Problem Analyzing Circuit Elements The circuit shown contains an AC generator a resistor with resistance R 1 and a black box which contains either an inductor or a capacitor or both We rst measure the current in the circuit at an angular frequency 1 1 radians sec and nd that it is leads the driving emf by exactly 4 radians We then measure the current in the circuit at an angular frequency 2 2 radians sec and nd that it lags the driving emf by exactly 4 radians Note 4 radians 45 tan 4 1 a What is inside the black box We know that the black box must contain both L and C This is because the current sometimes leads and sometimes lags b What is are the value s of the inductance or and capacitance Electrical Power c Will the circuit resonate If so at what angular frequency Power is change in energy per unit time P I V Time Dependence and Averaging The voltage and current are both time dependent We have both instantaneous and time averaged power Examples to remember sin t 0 and sin t Average Power Resistor Power Voltage Source P I I cos t R I R cos t I R De ne root mean squared RMS current and voltage by When the current amplitude has a maximum value The full width at half power maximum occurs when I Resonance Width Series RLC Width for Narrow Resonance Narrow width approximate Amplitude and phase When At time averaged power Equal one half maximum power at resonance Capacitive Case 1 C L The power delivered to a circuit by the voltage source is given by P t Re I Re V I V cos t cos t I V cos t cos sin t sin cos t I V cos t cos sin t cos t sin The time average is P t I V cos t cos I V sin t cos t sin Therefore P I V cos Power dissipation is maximum at resonance P V V R At very low frequencies capacitor dominates capacitor charges reducing the amplitude of the current phase shift is negative current leads At very high frequencies inductor dominates current tries to change but inductor blocks it and so amplitude of the current is small phase shift is positive current lags voltage voltage Acs like DC circuit as approaches zero Inductive Case 1 C L Concept Question Which Dominates Concept Question Residential Transformer Correct answer 1 The circuit is dominated by inductance We know this because the current lags the voltage so 1 C L Correct answer 1 The house needs a lower voltage so we step down to the house meaning there are fewer turns on the house side
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