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AC Circuits PH 203 Professor Lee Carkner Lecture 23 Alternating Current Issues Voltage and current vary sinusoidally with time Voltage and current will have a frequency and angular frequency w in radians per second Need to create a more general form of resistance Circuits have natural oscillation frequencies May get resonance Generators We can use induction to produce a continuous source of current Turn a wire loop in a magnetic field with an external source of work to make a generator Produces alternating current Thus the current flows in one direction and then the other Characterized with angular frequency wd Alternating Current e = emax sin wdt As the loop makes one complete rotation (wt goes from 0 to 2p radians) the emf goes from 0, to maximum +, to maximum -, and back to zero again The current through the loop goes one way and then the other, sometimes is weak and sometimes is strong Frequency The number of these cycles made per second is the frequency f = wd/2p We can also refer to the period, the number of seconds per cycle T = 1/f = 2p/wd Note that if wd is similar to the w of the circuit we have resonance and large current amplitudes are possible AC Circuit Elements Resistors (Resistance, R) Capacitors (Reactance, XC) Inductors (Reactance XL) Current and voltage may not be in phase Resistors and AC Resistor connected to an AC generator Voltage through the circuit varies as the emf of the generator does vR = VR sin wdt We can then find the maximum current with Ohm’s Law VR = IRR Resistors and Phase When v is max, i is max iR = IR sin wdt Capacitors and AC As charge builds up on the capacitor, the potential difference across it increases, decreasing the current The capacitor is constantly being charged and discharged In a AC circuit the current will vary with some average rms value that depends on the voltage and the capacitance Capacitive Reactance We define the capacitive reactance to symbolize this “resistance”: XC = 1/(wdC) VC = ICXC Note that the current depends on the frequency At high frequency the capacitor never gets much charge on it Capacitors and Phase When the voltage is zero, the current is a maximum As voltage increases current decreases We say the voltage lags the current by 90 degrees or ¼ cycle The equation for current is then iC = IC sin (wdt + p/2) Capacitor Power Since P = vi, we can see if the power is positive or negative based on the sign of i and v P is positive half the time and negative half the time The capacitor draws energy from and returns energy to the generator in equal measure Inductive Reactance XL = wdL The inductor is most “effective” at high frequency due to the rapid current changes The maximum voltage is again the generator emf and is related to the maximum current by: VL = ILXL Inductors and Phase look at the slope of the current sine wave The induced voltage is zero when the current is a maximum (since that is where the current is not changing) The current at any time is: iL = IL sin (wdt – p/2) Reactance and Frequency Resistor Capacitor Low current at low frequency Inductor Next Time Read 31.9-31.11 Problems: Ch 31, P: 32, 33, 34, 41, 53 For an LC oscillator when the current through the inductor is zero, the charge on the capacitor is, A) B) C) D) E) Maximum Zero ½ maximum p maximum 2p maximum For an LC oscillator the direction of current in the circuit is, A) Always in one direction B) In one direction for one complete cycle and then the other direction for the next complete cycle C) In one direction for ½ of the cycle and then the other direction for the other ½ of the cycle D) Changing direction 2p times per cycle E) Fluctuating unpredictably For an LC oscillator when the energy in the capacitor is ½ maximum the energy in the inductor is, A) B) C) D) E) Maximum Negative maximum Zero ½ maximum 2p maximum