Filters & Resonance PHYSICS 2B03 D. FITZGREEN November 16, 2022 1 Objective Students will use an inductor, capacitor and resistor to make a bandpass filter and measure its resonant frequency band. 2 Introduction 2.1 Inductors Inductors are very similar to capacitors in that they transform voltage into energy; a capacitor does this by generating an electric field, while an inductor does this with a magnetic field. An inductor is basically a solenoid (wire wrapped along a tube). The inductance of a solenoid is L - ANZA (1) where L is the inductance, & is the magnetic permeability of free space, NV is the number of turns in the solenoid, A is the cross-sectional area of the solenoid, and / is the length of the solenoid. You won't have to use that equation in the lab, but it's good to know that it exists. While capacitors allow a quickly-changing voltage to pass through uninhibited, inductors allow slowly- changing or DC voltages through. When there is a change in the current going through the inductor, an electromotive force is induced that opposes the change in current that created it. Another way to say that is that an inductors impedance increases with frequency. Inductors are used to prevent sudden changes in circuits, like preventing radio frequency and electromagnetic interference from entering sensitive circuits. 2.2 RL Filters We can make filters just like we did with capacitors and determine that the time constant is L/ R. When you are building your circuits, pay attention to the order of the components and how that differs from your capacitive filters. Figure 1 shows a high and low pass filter using an inductor. Should look pretty familiar to the capacitor stuff you did in the last lab. 2.3 RLC Filters When we combine all three of the components we've been studying so far, we get an RLC filter, as shown in Figure 2. The inductor will filter out high-frequency signals while the capacitor will filter low-frequency signals. If we choose our component values carefully, then we can create a filter that will only allow aMcMaster University Physics 2803 Filters & Resonance A. Low-Pass Filter B. High-Pass Filter R 160mH 16000 Ssoon You R 3 2 30M V. -Jde Point 1 -20 on [dB 1. = 497 HI 40 -60 -BO .80 10 100 100% 100 10% 100 Frequency [HE) Figure 1: (A) Low and (B) high pass filters using an inductor. Figure from Practical Electronics for Inventors. V1 L1 C1 R1 sine 1 kHz Figure 2: An RLC circuit that acts as a bandpass filter. The inductor and capactiro remove high and low frequencies, respectively. narrow band of frequencies through the circuit. We say that the circuit has a resonant frequency, and other frequencies are attenuated. We call this a bandpass filter. The resonant frequency is fr = = 27VLC (2) When dealing with resonances, the width of the resonance (in frequency space) is just as important as the resonant frequency itself. We use the same definitions for cut-off frequency to define the band of allowed frequencies in the circuit. The 'Q' value of the circuit is used as a measure of the relative loss in the circuit: fr Q = Af' (3) where fr is the resonant frequency and Af is the difference between the high and low cut-off frequencies. 3 Experimental Apparatus The circuit components that you will use in your inductor/ filter circuits are shown in Figure 3. Note that there are several values of resistor: the green ones are approximately 200 0, the beige one is 2.2 kf, and theMcMaster University Physics 2803 Filters & Resonance blue one is 22 kf. There is only one value of capacitor (0.01 #F) and inductor (which you will measure in units of Henries). Figure 3: Circuit components used to study inductors and filters. 4 Procedure 4.1 Inductors as Filters (a) (b) V1 R1 V2 L1 square square 1 kHz 1 kHz MM L1 R1 VM1 VM1 N/A N/A Figure 4: Circuit components used to study inductors and filters. 1. Build the circuit shown in Figure 4(a) with a 200 Ohm resistor. 2. Use a square wave from the PASCO interface at 1000 Hz.McMaster University Physics 2B0 Filters & Resonance 3. Use the oscilloscope to measure the voltage across the inductor. You should see a familiar sight: A square wave with the DC/low-frequency part of the signal filtered out. 4. Use the same technique as Lab 3 to measure the time constant (the time between the max amplitude and 1/e of the max amplitude). Use that measured time constant to estimate the inductance you are using. 5. Remove the 200 0 resistor and replace it with a 2.2 k resistor. Repeat your calculation of the time constant and he inductance. How do your values compare to eachother? Swap in the 22 kf resistor. Do you still even see a signal? 6. Swap the position of the resistor and inductor (as shown in Figure 4(b)) and set up the oscilloscope to measure across the resistor. Keep using the 22 k$2 resistor. Comment on what you're seeing 7. Swap in the 2.2 kf, then 200 Ohm resistors and comment on how the filter behaviour changes. Are you seeing underdamped, overdamped behaviour? It may be critically damped but it's unlikely that we have exactly the right component values for that. 8. With the 200 Ohm resistor in place, change the frequency until you find the cut-off frequency for this low-pass filter. 9. Just for fun, set the frequency to 1 Hz and hold the compass near the hole through the inductor. What do you notice? 4.2 RLC Bandpass Filter 1. Build the circuit shown in Figure 2. Set the oscilloscope to measure across the resistor. 2. Vary the frequency from the PASCO controller to find the resonant frequency. Take about 5 amplitude measurements on either side of the resonant frequency. Make sure you measure the upper and lower cut-off frequencies. 3. Check that your measured resonant frequency matches your calculation from Equation (2). 4. Calculate the Q of your circuit from Equation (3). 5 Prelab Assignment 1. Use Circuitlab to build the circuit shown in Figure 5. 2. Run the time-domain simulation for 0.01 seconds with a time-step of 0.0001 s. Measure the voltage across the resistor. 3. Save a screenshot of your resulting plot. Change one of the values in your circuit by about 50% and rerun the simulation. Submit a screenshot of your two simulations and comment on the differences.McMaster University Physics 2803 Filters & Resonance V1 L1 sine 1 JH C1 1 KHz 1 UF R2 100 Q Figure 5: An RLC circuit to simulate in Circuitlab. Measure the voltage output across the resistor. 6 Experimental Results to Submit 1. State your measurement of the inductor. 2. Show a plot of your amplitude measurements as a function of frequency in the RLC circuit. 3. State your measured resonant frequency and compare it to your calculations. 4. State your measured @ for the RLC circuit. 5Lab 4 data 4.1 4. 110 microseconds is the time constant a. 0.00011seconds = L/200Ohms; L = 0.022 Henries 5. 12 microseconds is time constant when resistor is 2.2k a. 0.000012seconds = L/2200Ohms; L = 0.0264 Henries 3.2 microseconds is time constant when resistor is 22k, we saw signal a. 0.0000032 * 220000OhmAS = 0.0704 Henries (don't include thist 6. now it looks like proper square waves (low pass filter looking head ass) 7. 2.2 kOhms: underdamped a. 200 Ohms: overdamped 8. when frequency is 1000Hz (200 Ohms), amplitude is 880mV a. when frequency is 4800Hz (200 Ohms), amplitude is 440mV (cutoff freq)4.2 calculated fr: 1/(2pi(0.0242*1e-8) ^0.5) = 10231 10494.4 Hz frequency (hz) [PK to PK]/2 (mV) 8000 245 8500 295 9000 375 9500 490 10000 660 10231 (using average of both 755 inductance) 10650 (measured) 875 11000 815 11500 655 12000 495 12500 405 13000 340 high cutoff: 12350 Hz (Pk to PK 840mV) low cutoff: 9300Hz (Pk to PK 860mV) Q = 10650/(12350 - 9300) = 3.328 Lab 4Lab 4 1. State your measurement of the inductor. . 110 microseconds is the time constant when resistor is 200 Ohms . 0.00011seconds * 200Ohms = 0.022 H . 12 microseconds is time constant when resistor is 2.2k . 0.000012seconds * 2200Ohms = 0.0264 H (0.022+0.0264)/2 = 0.0242 H (average inductance) 2. Show a plot of your amplitude measurements as a function of frequency in the RLC circuit. 1000 900 800 700 600 Voltage (mV) 500 400 . . 300 200 100 0 0 2000 4000 6000 8000 10000 12000 14000 Frequency (Hz) Figure 1: Amplitude of the voltage across resistor as a function of frequency. Both the measured and calculated resonant frequencies are included.3. State your measured resonant frequency and compare it to your calculations. Measured resonant frequency is 10650 Hz. Calculated resonance frequency: 1/(2pi(0.0242*1e8)"0.5) = 10231 Hz 4. State your measured Q for the RLC circuit. high cutoff: 12350 Hz low cutoff: 9300Hz Q 2 fr/ (high cutoff low cutoff) Q (with calculated fr) = 10231/(12350 9300) = 3.354 Q (with measured fr) = 10650/(12350 9300) = 3.492