Experiment 9: AC circuits, Lab Reports of Experimental Physics

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Experiment 9: AC circuits INTRO TO EXPERIMENTAL PHYS- LAB 1493/1494/2699 2 Introduction ● Last week (RC circuit): ● Constant Voltage power source (constant over time) ● This week: ● A new component: the inductor ● Alternating Current (AC) circuits ➢ Time dependent voltage source ● Leads to: ➢ Time dependent currents (alternating currents) ➢ Phase shifts in voltage and currents in components with respect to one another ➢ Resonance PHYS 1493/1494/2699: Exp. 9 – AC circuits 5 Why AC circuits? ● Sensitive to input frequency (i.e. function generator frequency) ● Serve as signal frequency filters: ● High-frequency filters ● Low-frequency filters ● Band-pass filters ● Transformers ● Induction effects - Ability to raise or lower the voltage amplitude. ● Generators and Motors e.g. Radios e.g. Speakers PHYS 1493/1494/2699: Exp. 9 – AC circuits 6 AC circuits: resistors ● AC circuits have an enormous range of applications. Here we cover the most important aspects ● For this lab: ● Consider only sources that vary sinusoidally: ● Simple example: ● Function generator + resistor ● Ohm’s Law: voltage across the resistor is just ~ The voltage across the resistor is then simply: PHYS 1493/1494/2699: Exp. 9 – AC circuits 7 AC circuits: capacitors ● More interesting case: connect a capacitor to the AC voltage source ● Last time we saw that the voltage across a capacitor is given by: ● Therefore, when the current is sinusoidal the voltage is given by: ~ PHYS 1493/1494/2699: Exp. 9 – AC circuits 10 ● Given our expression for VR , the maximum value of the voltage across the resistor is just given by Ohm’s Law: ● The maximum voltage across the capacitor is a function of ω: ● But, the maximum voltage across the inductor is also a function of the driving frequency: Voltage maxima: a closer look Inductive reactance Capacitive reactance Given an oscillating input current the capacitor voltage is higher for small frequencies and lower for high frequencies The inductor voltage is instead higher for large frequencies and lower for small ones PHYS 1493/1494/2699: Exp. 9 – AC circuits 11 Physical explanation: capacitors ● Question: Why does the capacitor resist low-frequency signals more than high-frequency ones? ● Last time: when charging/discharging the capacitor, the current – the rate at which you can charge it – decreases exponentially. It becomes harder and harder to push in more charge as the capacitor fills up. Easy to charge = low reactance (XC) Hard to charge = high reactance (XC) PHYS 1493/1494/2699: Exp. 9 – AC circuits ● Rapidly varying signals (high frequency) quickly charge/discharge capacitor before it fills with charge → low impedance. ● Slowly varying signals (low frequency) charge the capacitor to its limit, slowing down the rate: that is, decreasing the current! ● Now that we have introduced the language of reactances, you can think about the capacitor somehow as a resistor with ω-dependent resistance 12 Physical explanation: capacitors Low reactance (XC) High reactance (XC) PHYS 1493/1494/2699: Exp. 9 – AC circuits 15 RLC circuits: phase shift ● After passing through the three components the voltage will have some phase shift ● Let’s then impose to V(t) to look like: ● Comparing with the equation from the previous slide it must necessarily be: ● And hence the phase shift is: ● The phase shift will depend both on the characteristics of the circuit (R, C, L) and on the frequency of the input signal! PHYS 1493/1494/2699: Exp. 9 – AC circuits 16 RLC circuits: phase shift ● What about the maximum amplitude for the voltage? ● Let take again: ● Let’s now square both equations and add them together: ● The quantity Z is called the impedance of the RLC circuit ● NOTE: the previous equation resembles very closely Ohm’s law for resistors! ● This procedure can actually be generalized introducing the so- called phasor formalism PHYS 1493/1494/2699: Exp. 9 – AC circuits 17 Resonant frequency ● So the whole RLC system has this peculiar frequency dependent “effective resistance”. In particular: ● High-frequencies: killed by the inductor ● Low-frequencies: killed by the capacitor ● We therefore expect to have a particular frequency (ω0) in the middle range that goes through the system almost untouched ● For a given input voltage, the current in the circuit is maximum when Z is minimum i.e. when XL = XC. ● The resonant frequency is given by: High LowResonant L C PHYS 1493/1494/2699: Exp. 9 – AC circuits 20 Main goals ● Resonance of RLC circuit: ● Measure the resonant frequencies and FWHM for three known circuits ● Compute the unknown inductance of a copper coil by finding the resonant frequency of the whole system ● Observe the phase shift, φ, between the driving signal and the three components (R, L and C) of the circuit ● Compare with expected value PHYS 1493/1494/2699: Exp. 9 – AC circuits 21 Experimental setup CapacitorInductor Resistor with variable resistance PHYS 1493/1494/2699: Exp. 9 – AC circuits 22 Experimental setup ● Recommendations: ● Set the function generator peak-to-peak voltage to 20 V, the maximum allowed. − There is a 0-2V / 0-20V selector button in addition to the voltage knob. ● Make sure the oscilloscope is set to trigger on channel 1, the function generator signal. You can do this by pressing the TRIGGER button and checking in the window menu that CH 1 is selected. ● Use the MEASURE tools to observe peak-peak amplitudes, signal periods, and signal frequencies. Let the scope do the work for you! ● Make sure that both peaks are in the viewable range of the scope! PHYS 1493/1494/2699: Exp. 9 – AC circuits 25 Phase shift measurement ● Replace the copper ring with the known capacitor again and set R = 30 Ω ● Locate the resonant frequency and for 5 values at, above and below it measure the phase shift across the resistor ● Now increase the frequency well above the resonant one. This makes the inductor much more important than the capacitor. Measure φ ● Now make the capacitor more important by going way below the resonant frequency. Measure φ again Compare the results obtained with the expected ones PHYS 1493/1494/2699: Exp. 9 – AC circuits 26 Tips ● Don’t get confused! The frequency reported by the oscilloscope (in the MEASURE mode) is f. It is related to what we called frequency so far by ω = 2π f. ● Once you manage to locate the maximum of the Vpp curve and you took 20 points above and below it, try to take more points right around you maximum. This will reduce your uncertainty on ω0 ● When plotting Vpp vs ω remember to take enough data to measure the FWHM. When R is large, the peak is very broad. Keep taking data until you pass half height ● The resistor is old but if you look carefully on each knob there is a label telling you how many Ω correspond to that knob. PHYS 1493/1494/2699: Exp. 9 – AC circuits