Physics 241 Lab – RLC Radios http://bohr.physics.arizona.edu/~leone/ua/ua_spring_2010/phys241lab.html Name:____________________________ Section 1: 1. Begin today by reviewing the experimental procedure for finding C, L and resonance. This may help you to do well on your lab practical. (Be sure to sign up for your lab practical time slot today.) 1.1. Reminder of how to measure an unknown capacitance C with an RC circuit: The component voltage amplitudes of an RC circuit driven sinusoidally with angular frequency drive are given by VC,amplitudeCZVsource,amplitude and VR,amplitudeRZVsource,amplitude. These component voltage amplitudes are equal when CR. Substituting R C1driveC and solving for C gives C 1R 2 fdrive. Remember this is only true when the voltage amplitude across the resistor is equal to the voltage amplitude across the capacitor. Therefore, to find an unknown capacitance C, measure R and find the driving frequency where VC,amplitude VR,amplitude. (No question.) 1.2. Reminder of how to measure an unknown inductance L with an RL circuit: In an RL circuit driven sinusoidally with angular frequency drive, VL,amplitudeLZVsource,amplitude and VR,amplitudeRZVsource,amplitude. These component voltage amplitudes are equal when LR. Substituting RLdriveL and solving for L gives L R2 fdrive. Remember this is only true when the voltage amplitude across the resistor is equal to the voltage amplitude across the inductor. Therefore, to find an unknown inductance L, measure R and find the driving frequency where VL,amplitude VR,amplitude. (No question.) 1.3. Reminder of how to measure the resonant frequency resonance of an RLC circuit: In a sinusoidally driven RLC circuit, there is a driving frequency at which the current in the resistor is maximized (i.e. absorbing the most power from the oscillating source). This happens when the total circuit impedance Z R2LC2 is minimized. This occurs when LC2 0. The driving frequency at which LC0 occurs is called the resonant frequency. This can be found by setting C equal to L so that 1driveCdriveL. Solving gives driveresonance1L C. (No question.)1.4. A plot of current vs. driving frequency in an RLC circuit has a maximum at the resonant frequency. This is useful because an RLC circuit can be tuned to a specific resonant frequency by adjusting its C or L. If you lower the resistance of an RLC circuit and retake your measurements, the resonant frequency won’t change (since L and C don’t change), but you will measure a higher quality factor: The smaller the resistance, the sharper this peak gets. This is useful because an RLC circuit with low resistance only “reacts to” frequencies very near resonance while ignoring other driving frequencies. A radio can be made using a low-resistance RLC circuit that responds only to a specific radio frequency electromagnetic wave. How could you make the peak as sharp as possible in an RLC circuit for use in a radio? Take the resistor out so that only the tiny resistance of metal wires is present. (No question.) 1.5. Reminder of how to observe fresonance using an oscilloscope: The most accurate way to find fresonance is to utilize the fact that at resonance, VR(t) and Vsource(t) are exactly in phase with each other with equal amplitudes. You should place each of these voltages on your oscilloscope channels and examine an XY formatted display. The resonance frequency is easily found because you will see an ellipse when VR(t) and Vsource(t) are out of phase and a diagonal line when they are in phase. You see a straight line when they are in phase because both voltages must reach zero simultaneously. (No question.) Section 2: Modulating High Frequency Waves with Low Frequency Waves 2.1. Imagine that you want to transmit the following sound wave from one solenoid (the transmitter) to another solenoid (the receiver). The two solenoids are not connected in any way so that the oscillating magnetic field inside one solenoid must be made to oscillate within the other solenoid to utilize Faraday’s Law.Unfortunately, this wave is alternating much too slowly to induce a large voltage in the receiving solenoid. Remember the equation for mutual inductance, Vinducedin circuit 2dIcircuit 1dtM1 to 2, where M is a constant that describes how much the solenoids overlap. If the current doesn’t oscillate rapidly enough, then Vinducedin circuit 2 is very small. “Gee, I wish this wave oscillated more quickly to cause a bigger induced voltage in the receiving solenoid,” you might say. But then it wouldn’t be the same sound pitch that you wanted to hear in the first place! Still, it sounds like something you would say. (No question.) 2.2. Next examine a wave that oscillates quickly, radio or slightly sub-radio frequency for example. This isn’t the frequency you want to hear (you are not able to!), but it does oscillate so quickly as to create a large induced voltage in the receiving solenoid. I.e., it oscillates quickly enough to be transmitted into the receiving circuit through the mutual inductance of the “transformer” (overlapping solenoids). 2.3. The solution is to combine the two waves by multiplying them together. This modulated wave has the properties of both waves: it carries information about the audio frequency component and it oscillates quickly enough to generate a highly induced voltage in the receiver circuit.The modulating wave (or envelope wave) is the low frequency oscillation while the high frequency oscillation is often called the carrier wave. 2.4. Why are we interested in using a low frequency envelope wave in today’s lab? Your answer: 2.5. Why do we need to use a rapidly oscillating carrier wave in today’s lab? Your answer: 2.6. In today’s lab, we would like to transmit a modulated sound wave transmitted by one solenoid into another “receiving” solenoid. We will use a capacitor in the second “receiving” circuit to make an RLC “receiving” circuit. By changing the capacitor of the “receiving” circuit, we can adjust its resonant frequency. Therefore, we will be able to “tune” our “receiver” to a particular radio frequency. Now you would like to listen to your transmitted wave. But there is a huge
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