Pre-Lab Read through this entire lab. Perform all of your calculations (calculated values) prior to making the required circuit measurements. You may need to measure circuit component values to obtain your calculated values. All calculations should be written on a separate piece of paper (or in your lab notebook). They should be legible and written so that someone else can clearly understand your thought process. This is to demonstrate your understanding of the material, as well as aid in the troubleshooting process. Introduction Filtering: Last term, we explored circuit behavior for AC circuits and saw that frequency plays a key role in how a circuit behaves. For example, to very low frequencies, a capacitor looks like an open circuit and an inductor looks like a short circuit. To very high frequencies, a capacitor looks like a short circuit and an inductor looks like an open circuit. To all frequencies, a resistor always behaves the same. This lab will deal with low-pass filters (LPFs). Low-pass filters are used to remove highfrequency content, while keeping the low-frequency content. A few uses for LPFs are listed below: Music filtering Removing fast changes in power supply voltage (spikes) to protect equipment Preparing an analog voltage signal for analog-to-digital conversion (they may be called anti-aliasing filters in this context) Removing unwanted harmonics before transmitting an RF signal A circuit will behave differently depending on what frequency our circuit is at. Some circuits are meant to operate with many different frequencies simultaneously. Music may contain any or all frequencies audible to the human ear (20Hz to roughly 20kHz). Music is often represented as a time-varying voltage containing all of these frequencies. If a circuit is built to supply a subwoofer, a speaker meant to play only low frequencies, you would likely want to get rid of the higher frequencies before supplying the subwoofer with your voltage signal. To do this, you can build a circuit to filter out the high frequencies while keeping the low frequencies, a low-pass filter. 2014 Dan Kruger 1
A low-pass filter can be built using only two circuit elements. In Figure 1 you will see a series RC circuit. V in R V R V in R V R C V C H L X V L Figure 1 Figure 2 The output voltage for the LPF in Figure 1 will be the voltage across the capacitor. At very low frequencies, the capacitor looks like an open circuit with infinite impedance, thus taking all of the voltage in the voltage divider. So, low frequencies show up in the output voltage virtually unchanged. At high frequencies, the capacitor looks like a short circuit, and the resulting voltage on the output is very low. Figure 2 shows a series RL circuit. If you wanted to make an LPF out of this circuit, should your output voltage be across the resistor or the inductor? The band of frequencies you would like to keep is called the passband, as they will be allowed to pass through the circuit. The band of frequencies that will be rejected will be called the stopband. These frequencies will still pass through the circuit, but they will be attenuated, or reduced in amplitude. For a low-pass filter, the boundary between the passband and the stopband is called the cutoff frequency (f c ). Ideally, the transition between the passband and stopband would be immediate, but for an actual circuit, this cannot be achieved. V out Ideal LPF Response V out Actual LPF Response V max.707v max Passband Stop band Passband Stop band f c f f c f Figure 3 Observe the difference between ideal and actual low-pass filter (LPF) responses in figure 1. These are plots of output voltage versus circuit frequency. The actual response shown is the shape for the output voltage of a series RC circuit. A response closer to that of the ideal response is possible with more complex circuitry. 2014 Dan Kruger 2
The cutoff frequency is defined to be the frequency at which the circuit s output is at half of its 2 V maximum power. Since power is proportional to the square of voltage P =, the output R 1 voltage is at 0.707 of its maximum value at the half-power, or cutoff frequency. Its 2 V out maximum value is the circuit s input voltage. So when plotting circuit gain AV =, you Vin will see that the gain is at about 0.707 at the cutoff frequency. For the RC circuit in Figure 1, where the output voltage is across the capacitor, the cutoff frequency can be found with this equation. f c = 1 2π RC Hz Eqn 1 Bode Plot (Magnitude): The Bode plot is a straight-line approximation of a filter s output voltage or voltage gain. Its y- axis is the circuit s voltage gain in decibels (db). The x-axis represents circuit frequency and it 1 has a logarithmic scale. Since the cutoff frequency happens where the circuit s gain is at 2, 1 its location is 20 log 3dB 2 shown in Figure 4. 0dB A VdB on the Bode plot s y-axis. A Bode plot of an RC circuit is actual f c -3dB Bode approximation 10f c f (Hz), log-scale Figure 4 passband stopband -20dB The Bode plot is made of the solid lines and the actual response is the dotted line. Notice the Bode plot is a very close approximation to the actual response, and is more accurate the further you get from f c, with a maximum error of 3dB occurring at f c. Using db for the y-axis and a logscale for the x-axis, this quick and easy straight line approximation can be quickly sketched. In the passband, the circuit s gain is approximately 1. This has a decibel value of 0dB, so the passband is sketched as a straight line at 0dB. Notice that the stopband frequencies get 2014 Dan Kruger 3
attenuated further and further down as frequency increases. This gain has a slope of -20dB per decade. One decade means a factor of ten. If f c = 5kHz, then the Bode approximation for an RC LPF would have the following values: f < 5kHz: 0dB f = 5kHz: 0dB f = 50kHz: -20dB f = 500kHz: -40dB f = 5MHz: -60dB Example 1 Let s say we want to find the Bode approximation for output gain of this same filter (f c = 5kHz) at f = 100kHz. This is not an integer number of decades (factors of ten) above f c. We must answer this question: how many decades is f = 100kHz above f c = 5kHz? f 100k log = log = 1.301 decades fc 5k Since we know there are 20dB of attenuation per decade above f c, the Bode approximation of circuit gain at f = 100kHz is: 20dB ( 1.301 decades) = 26.02dB decades Phase Response: Filters are usually built to manipulate output voltage magnitude; however they also alter the phase of the output voltage as well. For the series RC LPF, the phase of V out is shifted by 0 at very low frequencies and by -90 at high frequencies. At f c, the phase of V out is equal to -45. Required Equipment: Oscilloscope (Tektronix TDS 2002C) Function Generator R 1 = 1kΩ C 1 =?nf, C 2 = 100nF Fender Mini-Deluxe amplifier 6.35mm (0.25in) audio phone connector-to-bnc connector 3.5mm (0.25in) audio phone connector-to-breadboard cable 2014 Dan Kruger 4
Part A: RC LPF Basics For this experiment, we will want to design a series RC LPF using a 1kΩ resistor and we will use Excel to plot output voltage versus frequency on a log scale. 1) Assemble the circuit pictured in Figure 5. Your function generator should produce a 10V pp sine wave with no DC offset. R 1kΩ Figure 5 V in 10 0 V pp C V out 2) A cutoff frequency of 3,386 Hz is desired. Based on this information, solve for your capacitor value and place the capacitor with the closest value to your result into your circuit. 3) Create an Excel table like the one in Figure 6. Figure 6 f (Hz) V out (V pp ) θ ( ) A V (db) 10 9.92-1.08-0.0698 339 9.92-6.34-0.0698 1,000 9.94-17.30-0.0523 2,400 8-34.50-1.9382 3,100 7.07-41.90-3.0116 5,500 5-58.30-6.0206 10,000 3.12-70.80-10.117 33,860 1.02 84.50-19.828 338,600 0.0672-89.10-43.453 4) Hook up CH 1 of your oscilloscope across your function generator. Adjust your function generator so that your scope measures as close to 10V pp as possible. Hook CH 2 to measure V out. 5) Fill in the first 3 columns of your data table with the appropriate measurements. For the rows in which frequency is specified, tune your function generator to that frequency and measure V out s magnitude (peak-to-peak) and phase. For the rows in which V out s magnitude is specified, adjust your frequency until your scope reads V out s voltage to be as close as possible to the specified values, then record your frequency and phase. Remember that for the best possible accuracy, you want to use as much as your scope s screen as you can without your waveform going past the top or bottom of the screen. You 2014 Dan Kruger 5
should be adjusting your vertical and horizontal scales accordingly. For the measurement at f = 338,600 Hz, your voltage will be a low value and you will see a noisy waveform, made evident by the thick lines. The thickness of the line is higher-frequency noise. We are only interested in the underlying sinusoid. Use your eyes and your cursors to measure the peak-to-peak voltage of the sinusoid without including its noise. The measure function will only give you measurements that include noise. 6) Fill in the 4 th V out column with decibel value of your circuit gain, that is AVdB = 20log. Vin 7) Make an Excel scatter plot of A VdB versus frequency using your data. Change your x-axis to a logarithmic scale: Right click your plot s x-axis. Make sure that just the x-axis is highlighted. Select Format Axis Under Axis Options, check the Logarithmic Scale box. It should have a base 10 filled in. Hit OK. Your axis should resemble the actual response in figure 4. 8) Make an Excel scatter plot of the phase angle versus frequency using your data. Change your x-axis to a logarithmic scale. 9) Use percent error to compare your measured cutoff frequency to your calculated value. 10) Using your db value for gain at f = 338.6kHz, how many db of attenuation per decade of frequency did you measure? Part B: RC LPF Applied In this part of the experiment, we will have an unwanted voltage signal at 10kHz applied to some music und use the RC LPF from part A to attenuate it, and observe the consequences of doing this. 1) Obtain a Fender Mini-Deluxe amplifier along with its power supply. Plug it in. Turn the Tone and Drive knobs to their middle position. Adjust the volume to a low and reasonable level, and turn the unit off. 2) Use the 3.5mm (0.25in) audio phone connector-to-breadboard cable to connect the computer s headphone jack to the breadboard. Use the 3.5mm (0.25in) audio phone connector-to-breadboard cable to connect Input jack of the amplifier to the computer headphone jack via the breadboard. We will be using music from the computer as the voltage source that we will be filtering. 2014 Dan Kruger 6
One side should be connected to ground (the lead soldered to the bare wire) and the other side should either be connected to the left or right channel. It does not matter which one. Play some music on the computer. Ideally, this should be a song that has persistent highfrequency content. It should be appropriate for class. One such song can be found here: http://www.youtube.com/watch?v=y1d3a5edjis Turn on your amplifier and adjust the volume to a reasonable level (level 2, or so). If the song sounds distorted, you may want to turn down the volume on the computer until the distortion goes away. Leave the amplifier off when you do not need to hear the music. 3) After making sure your amplifier is playing music satisfactorily, add your noise source. Tune your function generator to generate a sinusoid at 10kHz with amplitude of 2.0V pp with no DC offset. This will be the annoying sound that we will try to attenuate with our RC filter. Hook only the positive end of the function generator to your breadboard at the same node at which your music voltage source s positive lead is. Leave the negative end of the function generator floating. Briefly turn on your amplifier. You should hear both the music and the annoying 10kHz tone. This 10kHz tone represents something that might be feedback, or some other noise that s imbedded in a recording. We re trying to simulate a situation where the signal (music) and the noise (10kHz) tone are already combined. We re pretending we cannot simply disconnect a wire to get rid of it. 4) Connect CH 1 of the scope so that it is measuring the computer headphone jack s (our voltage source s) voltage, along with the noise created by the function generator. Put your scope into FFT mode using the Math button. 5) Using Example 1, predict how much attenuation (db) a 10kHz signal will suffer going through the circuit from Part A that had a cutoff frequency of 3,386Hz. 6) Adjust your vertical scale to 10dB and your horizontal scale to 1.25kHz. You should see a spike of energy at 10kHz. Use your cursor to measure the magnitude of the noise in db. 7) Now, hook up your RC LPF so that you are low-pass-filtering the voltage source. Your output of the filter will be across the capacitor. Your amplifier and CH 1 of the scope should be hooked up across your capacitor. The function generator s positive lead should remain at the voltage source s positive node. 8) Use the cursor to measure the magnitude of the 10kHz sinusoid in db. Measure many db this noise attenuated by your LPF? You will need to compare your filtered 10kHz signal magnitude to that of an unfiltered signal. Can you hear the difference between the two? 2014 Dan Kruger 7
9) Calculate what f c would be if your cap was 100nF. Repeat steps B5-B8 using a capacitor of value 100nF. 10) (optional) Hook up a 220nF capacitor in parallel with your existing capacitor and observe the resulting sound. 11) What did you do to your filter s cutoff frequency by increasing your capacitance? What effect did this have on your noise? What affect did this have on your music? 2014 Dan Kruger 8