L09-1 Name ________________________ Date ____________ Partners_______________________________ Lab 9 –AC FILTERS AND RESONANCE OBJECTIVES • • To understand the design of capacitive and inductive filters To understand resonance in circuits driven by AC signals OVERVIEW In a previous lab, you explored the relationship between impedance (the AC equivalent of resistance) and frequency for a resistor, capacitor, and inductor. These relationships are very important to people designing electronic equipment. You can predict many of the basic characteristics of simple AC circuits based on what you have learned in previous labs. Recall that we said that it can be shown that any periodic signal can be represented as a sum of weighted sines and cosines (known as a Fourier series). It can also be shown that the response of a circuit containing resistors, capacitors, and inductors (an “RLC” circuit) to such a signal is simply the sum of the responses of the circuit to each sine and cosine term with the same weights. Recall further that if there is a current of the form (1) I (t ) I max sin t flowing through a circuit containing resistors, capacitors and/or inductors, then the voltage across the circuit will be of the form (2) V t I max Z sin t . Z is called the impedance (and has units of resistance, Ohms) and φ is called the phase shift (and has units of angle, radians). The peak voltage will be given by Vmax I max Z . University of Virginia Physics Department PHYS 2419, Fall 2011 (3) Modified from P. Laws, D. Sokoloff, R. Thornton L09-2 Lab 9 - AC Filters & Resonance Figure 1 shows the relationship between V and I for an example phase shift of +20°. We say that V leads I in the sense that the voltage rises through zero a time t before the current. When the voltage rises through zero after the current, we say that it lags the current. Figure 1 The relationship between and t is given by t 2 or 360 f t T where T is the period and f is the frequency. (4) For a resistor, Z R R and there is no phase shift ( R 0 ). For a capacitor, ZC X C 1 C and C 90 while for an inductor, Z L X L L and L 90 . In other words: VR I max R sin(t ) (5) VC I max X C cos(t ) (6) and VL I max X L cos(t ) (7) XC is called the capacitive reactance and XL is called the inductive reactance. Let us now consider a series combination of a resistor, a capacitor and an inductor shown in Figure 2. To find the impedance and phase shift for this combination we follow the procedure we established before. R V L C Figure 2 University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton Lab 9 - AC Filters & Resonance L09-3 From Kirchhoff’s loop rule we get: V VR VL VC (8) Adding in Kirchhoff’s junction rule and Equations (2), (5), (6), and (7) yields Vmax sin t RLC I max R sin t X L X C cos t Once again using a trigonometric identity1 and equating the coefficients of sin t and cos t , we get Vmax cos RLC I max R and Vmax sin RLC I max X L X C Hence the phase shift is given by X L XC (9) R and the impedance of this series combination of a resistor, an inductor, and a capacitor is given by: tan RLC Z RLC Vmax I max R2 X L X C 2 (10) The magnitudes of the voltages across the components are then VR ,max I max R R Vmax (11) VL ,max I max X L XL Vmax Z RLC (12) VC ,max I max X C XC Vmax Z RLC (13) Z RLC and Explicitly considering the frequency dependence, we see that VR ,max Vmax VL ,max Vmax and VC ,max Vmax 1 R R 2 L 1 C 2 L R L 1 C 2 2 1 C R 2 L 1 C 2 (14) (15) (16) sin( ) sin( ) cos( ) cos( )sin( ) University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton L09-4 Lab 9 - AC Filters & Resonance This system has a lot in common with the forced mechanical oscillator that we studied in the first semester. Recall that the equation of motion was (17) F ma bv kx mx bx kx Similarly, Equation (8) can be written as 1 (18) V Lq Rq q C We see that charge separation plays the role of displacement, current the role of velocity, inductance the role of mass (inertia), capacitance (its inverse, actually) the role of the spring constant, and resistance the role of friction. The driving voltage plays the role of the external force. As we saw in the mechanical case, this electrical system displays the property of resonance. It is clear that when the capacitive and inductive reactances are equal, the impedance is at its minimum value, R . Hence, the current is at a maximum and there is no phase shift between the current and the driving voltage. Denoting the resonant frequency as LC and the common reactance of the capacitor and inductor at resonance as X LC , we see that, at resonance X LC X C LC X L LC so LC 1 LC (19) X LC L C (20) and At resonance the magnitude of the voltage across the capacitor is the same as that across the inductor (they are still 180° out of phase with each other and ±90° out of phase with the voltage across the resistor) and is given by X (21) VC ,max LC VL ,max LC LC Vmax R University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton Lab 9 - AC Filters & Resonance L09-5 In analogy with the mechanical case, we call the ratio of the amplitude of the voltage across the capacitor (which is proportional to q , our “displacement”) at resonance to the driving amplitude the resonant amplification, which we denote as Q, (22) Q VC ,max LC Vmax Hence, Q X LC R L C R (23) Figure 3 (below) shows the voltage across a capacitor (normalized to the driving voltage) as a function of frequency for various values of Q . Figure 3 In this lab you will continue your investigation of the behavior of resistors, capacitors and inductors in the presence of AC signals. In Investigation 1you will explore the relationship between peak current and peak voltage for a series circuit composed of a resistor, inductor, and capacitor. You will also explore the phase difference between the current and the voltage. This circuit is an example of a “resonant circuit”. The phenomenon of resonance is a central concept underlying the tuning of a radio or television to a particular frequency. INVESTIGATION 1: THE SERIES RLC RESONANT (TUNER) CIRCUIT In this investigation, you will use your knowledge of the behavior of resistors, capacitors and inductors in circuits driven by various AC signal frequencies to predict and then observe the behavior of a circuit with a resistor, capacitor, and inductor connected in series. University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton L09-6 Lab 9 - AC Filters & Resonance The RLC series circuit you will study in this investigation exhibits a “resonance” behavior that is useful for many familiar applications such a tuner in a radio receiver. You will need the following materials: • Voltage probes • Multimeter • 510 Ω resistor • test leads • 800 mH inductor • 820 nF capacitor Consider the series RLC circuit shown in Figure 4 (below). [For clarity, we don’t explicitly show the voltage probes.] R V L C Figure 4 Prediction 2-1: At very low signal frequencies (less than 10 Hz), will I R ,max and VR ,max be relatively large, intermediate or small? Explain your reasoning. Prediction 2-2: At very high signal frequencies (well above 3,000 Hz), will the values of I R ,max and VR ,max be relatively large, intermediate or small? Explain your reasoning. University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton Lab 9 - AC Filters & Resonance L09-7 1. On the axes below, draw qualitative graphs of X C vs. frequency and X L vs. frequency. Clearly label each curve. XC and XL Frequency 2. On the axes above (after step 1) draw a curve that qualitatively represents X L X C vs. frequency. Be sure to label it. 3. Recall that the frequency at which Z is a minimum is called the resonant frequency, f LC and that the common reactance of the inductor and the capacitor is X LC . On the axes above, mark and label f LC and X LC . Question 2-1 At f LC will the value of the peak current, I max , in the circuit be a maximum or minimum? What about the value of the peak voltage, VR ,max , across the resistor? Explain. 4. Measure the 510 Ω resistor (you have already measured the inductor and the capacitor): R : __________ 5. Use your measured values to calculate the resonant frequency, the reactance of the capacitor (and the inductor) University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton L09-8 Lab 9 - AC Filters & Resonance at resonance, and the resonant amplification factor. Show your work. [Don’t forget the units!] f LC : __________ X LC : __________ Q : __________ Activity 2-1: The Resonant Frequency of a Series RLC Circuit. 1. Open the experiment file L09A2-1 RLC Filter. 2. Connect the circuit with resistor, capacitor, inductor and signal generator shown in Figure 4. [Use the internal generator.] 3. Adjust the generator to make a 50 Hz signal with amplitude of 2 V. 4. Connect voltage probe VPA across the resistor, VPB across the inductor, and VPC across the capacitor. 5. Use the Smart Tool to determine the peak voltages ( VR ,max , VL ,max , and VC , max ). Enter the data in the first row of Table 2-1. 6. Repeat for the other frequencies in Table 2-1. Table 2-1 f (Hz) VR ,max (V) VL ,max (V) VC , max (V) 50 100 200 400 800 University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton Lab 9 - AC Filters & Resonance L09-9 7. Measure the resonant frequency of the circuit to within a few Hz. To do this, slowly adjust the frequency of the signal generator until the peak voltage across the resistor is maximal. [Use the results from Table 2-1 to help you locate the resonant frequency.] f LC ,exp : __________ Question 2-2: Discuss the agreement between this experimental value for the resonant frequency and your calculated one. 8. Use the Smart Tool to determine the peak voltages at resonance. Vmax : __________ VR ,max : __________ VL ,max : __________ VC , max : __________ Question 2-3: From these voltages, calculate Q and discuss the agreement between this experimental value and your calculated one. University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton L09-10 Lab 9 - AC Filters & Resonance Question 2-4: Calculate your experimental value of X LC and discuss the agreement between this value and your calculated one. Prediction 2-5: What will we get for Q if we short out the resistor? Show your work. 9. Short out the resistor. 10. Measure Q . [You may have to lower the signal voltage to 0.5 V.] Show your work. Explicitly indicate what you had to measure. Q : __________ Question 2-6: Discuss the agreement experimental value and your predicted one. University of Virginia Physics Department PHYS 2419, Fall 2011 between this Modified from P. Laws, D. Sokoloff, R. Thornton Lab 9 - AC Filters & Resonance L09-11 Activity 2-2: Phase in an RLC Circuit In previous labs, you investigated the phase relationship between the current and voltage in an AC circuit composed of a signal generator connected to one of the following circuit elements: a resistor, capacitor, or an inductor. You found that the current and voltage are in phase when the element connected to the signal generator is a resistor, the current leads the voltage with a capacitor, and the current lags the voltage with an inductor. You also discovered that the reactances of capacitors and inductors change in predictable ways as the frequency of the signal changes, while the resistance of a resistor is constant – independent of the signal frequency. When considering relatively high or low signal frequencies in a simple RLC circuit, the circuit element (either capacitor or inductor) with the highest reactance is said to “dominate” because this element determines whether the current lags or leads the voltage. At resonance, the reactances of capacitor and inductor cancel, and do not contribute to the impedance of the circuit. The resistor then is said to dominate the circuit. In this activity, you will explore the phase relationship between the applied voltage (signal generator voltage) and current in an RLC circuit. Consider again our RLC circuit (it is the same as Figure 4). R V L C Figure 5 Question 2-7: Which circuit element (the resistor, inductor, or capacitor) dominates the circuit at frequencies well below the resonant frequency? Explain. University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton L09-12 Lab 9 - AC Filters & Resonance Question 2-8: Which circuit element (the resistor, inductor, or capacitor) dominates the circuit at frequencies well above the resonant frequency? Explain. Question 2-9a: In the circuit in Figure 5, will the current through the resistor always be in phase with the voltage across the resistor, regardless of the frequency? Explain your reasoning. Question 2-9b: If your answer to Question 2-9a was no, then which will lead for frequencies below the resonant frequency (current or voltage)? Which will lead for frequencies above the resonant frequency (current or voltage)? Question 2-10a: In the circuit in Figure 5, will the current through the resistor always be in phase with applied voltage from the signal generator? Explain your reasoning. Question 2-10b: If your answer to Question 2-10a was no, then which will lead for frequencies below the resonant University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton Lab 9 - AC Filters & Resonance L09-13 frequency (current or voltage)? Which will lead for frequencies above the resonant frequency (current or voltage)? 1. Continue to use L09A2-1 RLC Filter. 2. Reconnect the circuit shown in Figure 5. Connect voltage probe VPA across the resistor, VPB across the inductor, and VPC across the capacitor. 3. Start the scope and set the signal generator to a frequency 20 Hz below the resonant frequency you measured in Investigation 2, and set the amplitude of the signal to 2 V. Question 2-11: Which leads – applied voltage, current or neither – when the AC signal frequency is lower than the resonant frequency? Discuss agreement with your prediction. 4. Set the signal generator to a frequency 20 Hz above the resonant frequency. Question 2-12: Which leads – applied voltage, current or neither – when the AC signal frequency is higher than the resonant frequency? Discuss agreement with your prediction. University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton L09-14 Lab 9 - AC Filters & Resonance Question 2-13: At resonance, what is the phase relationship between the current and the applied voltage? 5. Use this result to find the resonant frequency. f LC , phase : __________ Question 2-14: Discuss how this experimental value compares with your calculated one. Question 2-15: How does this experimental value for the resonant frequency compare with the one you determined by looking at the amplitude? Comment on the relative “sensitivities” of the two techniques. University of Virginia Physics Department PHYS 2419, Fall 2011 Modified from P. Laws, D. Sokoloff, R. Thornton