8.02 Spring 2009Experiment 10: Driven RLC CircuitsINTRODUCTIONDriven Circuits: Resonance2. AC/DC Electronics Lab Circuit BoardIn addition, we will connect a voltage probe in parallel with the capacitor (not pictured).GENERALIZED PROCEDUREIn this lab you will measure the behavior of a series RLC circuit, driven sinusoidally by a function generator.Part 1: Driving the RLC Circuit on ResonanceNow the circuit is driven with a sinusoidal voltage and you will adjust to frequency while monitoring plots of I(t) and V(t) as well as V vs. I.Part 2: What’s The Frequency?The circuit is driven with an unknown frequency and you must determine if its above or below resonance.Part 3: What’s That Trace?Current and voltage across the function generator and capacitor are recorded, but you must determine which trace is which.END OF PRE-LAB READINGIN-LAB ACTIVITIESEXPERIMENTAL SETUPMEASUREMENTSPart 1: Driving the RLC Circuit on ResonancePart 2: What’s The Frequency?Part 3: What’s That Trace?MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2009 Experiment 10: Driven RLC Circuits OBJECTIVES 1. To explore the time dependent behavior of driven RLC Circuits 2. To understand the idea of resonance, and to determine the behavior of current and voltage in a driven RLC circuit above, below and at the resonant frequency PRE-LAB READING INTRODUCTION In the last lab, Undriven RLC Circuits, you did the equivalent of getting a push on a swing and then sitting still, waiting for the swing to gradually slow down to a stop. Most children will tell you that although that might be fun, it’s much more fun to get repeated pushes, or, if you have the coordination, to move your body back and forth at the correct rate and drive the swing, making it swing higher and higher. This is an example of resonance in a mechanical system. In this lab we will explore its electrical analog – the RLC (resistor, inductor, capacitor) circuit – and better understand what happens when it is driven above, below and at the resonant frequency. The Details: Oscillations In this lab you will be investigating current and voltages (EMFs) in RLC circuits. Although in the previous experiment these decayed after being given a kick (Fig. 1b), today we will drive the circuit and see continuous oscillations as a function of time (Fig. 1a). 0T 1T 2T-X0X0AmplitudeTime (in Periods) 0T 1T 2T 3T 4T 5T-X0X0AmplitudeTime (in Periods) (a) (b) Figure 1 Oscillating Functions. (a) A purely oscillating function ()0sinxx tωϕ=+ has fixed amplitude x0, angular frequency ω (period T = 2π/ω and frequency f = ω/2π), and phase φ (in this case φ = -0.2π). (b) The amplitude of a damped oscillating function decays exponentially (amplitude envelope indicated by dotted lines) E10-1Driven Circuits: Resonance In the previous lab we charged the capacitor in a series RLC circuit and then “let it go,” allowing the energy to gradually dissipate through the resistor. This time we will instead add a battery that periodically pushed current through the system. Such a battery is called an AC (alternating current) function generator, and the voltage it generates can oscillate with a given amplitude, frequency and shape (in this lab we will use a sine wave). When hooked up to an RLC circuit we get a driven RLC circuit (Fig. 2a) where the current oscillates at the same frequency as, but not necessarily in phase with, the driving voltage. The amplitude of the current depends on the driving frequency, reaching a maximum when the function generator drives at the resonant frequency, just like a swing (Fig. 2b) ωω0xmax I0 (a) (b) Figure 2 Driven RLC Circuit. (a) The circuit (b) The magnitude of the oscillating current I0 reaches a maximum when the circuit is driven at its resonant frequency One Element at a Time In order to understand how this resonance happens in an RLC circuit, its easiest to build up an intuition of how each individual circuit element responds to oscillating currents. A resistor obeys Ohm’s law: V = IR. It doesn’t care whether the current is constant or oscillating – the amplitude of voltage doesn’t depend on the frequency and neither does the phase (the response voltage is always in phase with the current). A capacitor is different. Here if you drive current at a low frequency the capacitor will fill up and have a large voltage across it, whereas if you drive current at a high frequency the capacitor will begin discharging before it has a chance to completely charge, and hence it won’t build up as large a voltage. We see that the voltage is frequency dependent and that the current leads the voltage (with an uncharged capacitor you see the current flow and then the charge/potential on the capacitor build up). Figure 3 Current and Voltage for a Capacitor A capacitor driven with a sinusoidal current will develop a voltage that lags the current by 90º (the voltage peak comes ¼ period later than the current peak). E10-2An inductor is similar to a capacitor but the opposite. The voltage is still frequency dependent but the inductor will have a larger voltage when the frequency is high (it doesn’t like change and high frequency means lots of change). Now the current lags the voltage – if you try to drive a current through an inductor with no current in it, the inductor will immediately put up a fight (create an EMF) and then later allow current to flow. When we put these elements together we will see that at low frequencies the capacitor will “dominate” (it fills up limiting the current) and current will lead whereas at high frequencies the inductor will dominate (it fights the rapid changes) and current will lag. At resonance the frequency is such that these two effects balance and the current will be largest in the circuit. Also at this frequency the current is in phase with the driving voltage (the AC function generator). Resistance, Reactance and Impedance We can make the relationship between the magnitude of the current through a circuit element and magnitude of the voltage drop across it (or EMF generated by it for an inductor) more concrete by introducing the idea of impedance. Impedance (usually denoted by Z) is a generalized resistance, and is composed of two parts – resistance (R) and reactance (X). All of these terms refer to a constant of proportionality between the magnitude of current through and voltage across (EMF
View Full Document
Unlocking...