Physics 241 Lab: RLC Circuit – AC Source http://bohr.physics.arizona.edu/~leone/ua/ua_spring_2010/phys241lab.html Name:____________________________ Section 1: The RLC Circuit 1.1. In a circuit where an inductor, resistor and capacitor (RLC) are connected in series and driven by a sinusoidal voltage source, the properties of the RC circuit and RL circuit that you studied previously combine in a straightforward manner. Let’s summarize the results for the LRC circuit that you should already suspect. The voltage across each component oscillates at the same frequency as the driving frequency of the source, drive. The properties of the inductor and capacitor are frequency dependent, which makes the circuit respond differently to different driving frequencies. The resistor voltage is chosen as the voltage reference because it is the only Ohmic device and therefore can be used to determine the current in the circuit: V(t)resistorRZVsourceamplitudesindrivet and therefore I(t) 1ZVsourceamplitudesindrivet, where Z [Ohm] is the total circuit impedance, Z R2LC2. Also, C [Ohm] is the capacitive reactance C1driveC and L [Ohm] is the inductive reactance LdriveL. Is the current in the circuit increased or decreased when the difference between the capacitive reactance and inductive reactance is increased? Explain your answer. Note that LC2CL2. Your answer and explanation: 1.2. The driving source voltage is given with a phase shift with respect to the resistor: V(t)sourceVsourceamplitudesindrivetsource, where the source phase shift is given by source tan1LCR. This shows that the source voltage will be in phase with the resistor voltage when LC because tan100. It also shows that if LC, then source 0 (the source voltage leads the resistor voltage), and if LC, then source0 (the source voltage lags the resistor voltage). Is it possible for the source voltage to ever be 180o out of phase with the resistor voltage? Use your graphing calculator to graph y=arctan(x) and find when y = or –. Your answer and reasoning:1.3. The capacitor voltage is phase shifted to lag the resistor voltage by 90o: V(t)capacitorCZVsourceamplitudesindrivet 2. The inductor voltage is phase shifted to lead the resistor voltage by 90o: V(t)inductorLZVsourceamplitudesindrivet 2. Sometimes these four boxed (important) voltage equations are rewritten using the fact that IamplitudeVsource amplitudeZ: V(t)resistorRIamplitudesindrivet V(t)sourceVsourceamplitudesindrivetsource (no change) V(t)capacitorC Iamplitudesindrivet 2 V(t)inductorL Iamplitudesindrivet 2 If R = 10 , L = 10 H, C = 10 F, Vsource amplitude = 10 volts and fdrive = 10 Hz, find the voltage amplitudes of the inductor and capacitor. Your work and answers: 1.4. It is useful to examine an example graph showing the relationships between each of the component’s voltages: In this example, C is larger than L so that Vcapacitor amplitude is larger than Vinductor amplitude. Note that Vresistor amplitude is larger than the other two voltage amplitudes, which indicates that R is larger than C and L. This need not be the case since you could always choose to use a smaller resistor in your circuit. Since the capacitor voltage lags the resistor voltage by /2 while the inductor voltage leads the resistor voltage by /2, the capacitor and inductor voltages themselves are 180o out of phase (/2 + /2 = ).Conservation of energy indicates that the sum of the voltages of the three components at any instant of time must equal the voltage of the source at that time. Therefore, if you add the three graphed voltages, you obtain the source voltage: Vsource(t)VL(t)VR(t)VC(t). The sum of the three component voltage functions in the previous graph is shown below with a dotted line. Notice that in this example the capacitor voltage amplitude is larger than the inductor voltage amplitude. This causes the source voltage to reach it’s amplitude a little after the resistor. Recall that source tan1LCR and since C is larger than L, you obtain source < 0. This means that the source voltage lags the resistor voltage. At driving frequencies where L > C, you find that VL > VC, and this causes the source voltage to lead the resistor voltage. Finally, imagine that you are able to adjust the source frequency while leaving Vsource,amplitude constant. Since IamplitudeVsource amplitudeZ, the current in the resistor can be maximized by minimizing Z. Remember that Z depends on drive since both C and L each depend on drive. Since Z R2LC2, Z is minimized when C is equal to L so that L – C, = 0. This gives Zminimum = R. The driving frequency at which L – C, = 0 occurs is called the resonant frequency. This can be found by setting C equal to L so that 1driveCdriveL. Some algebra (that should appear in your lab report) gives driveresonance1L C. 1.5. Imagine that you have an RLC circuit being driven sinusoidally at resonance. Assume you have placed the inductor voltage and capacitor voltage on the two channels of your oscilloscope (correctly inverting one of the channels). Make a quick sketch of what you should see on the oscilloscope screen. Think carefully about what the voltage amplitudes should be for these two components at resonance. Your quick sketch below:Section 2: Let’s practice the math for an example of a sinusoidally driven series RLC circuit. 2.1. Calculate each of the basic RLC circuit parameters in SI units though not necessarily in the order given.. Write the numerical value with correct SI units for each listed parameter: C L Z VR,amplitude VC,amplitude VL,amplitudesource Iamplitude 2.2. Next write equations using the numerical results from 2.1 to describe the time-dependent behavior of each component of
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