Lab E5: Filters and Complex Impedance


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1 E5.1 Lab E5: Filters and Complex Impedance Note: It is strongly recommended that you complete lab E4: Capacitors and the RC Circuit before performing this experiment. Introduction Ohm s law, a well known experimental fact, states that the current through a circuit is related to the overall voltage drop and the resistance of the components by the relation (1) I = This can be further generalized to become (2) I = where Z represents the complex impedance, which has both real and imaginary components. Impedance is analogous to direct current (DC) resistance, but in addition to the real resistance also includes an imaginary reactance term due to the oscillatory effects of, for instance, charging and discharging a capacitor that come into play when we switch to using an oscillating alternating current (AC) source to power our circuit. (Recall that the simplest form of a capacitor is two parallel plates with a gap inbetween them, creating time dependence as observed in the E4: Capacitors lab. In that lab we looked at a single discharging cycle of the capacitor to determine the time constant of the circuit. Here we apply an oscillating voltage that causes the capacitor to constantly charge and discharge.) The imaginary nature of the reactance gives phase to the impedance, indicating that the current is out of phase with the voltage across that component. Another common passive component, the inductor, creates a time dependence of its own. An inductor is essentially just a coil of wire; when current flows through, a magnetic field is created in the coil. If the input voltage changes, the inductor creates a voltage across itself opposing that change (you may remember Faraday s Law of Inductance and Lenz Law from Physics 1120). The voltage drop across the inductor is given by (3) V = L( "() " ) where L, known as the inductance, is determined by the geometry and number of coils in the loop, and i(t) is the time dependent current through the inductor. We have switched to the lower case i in order to indicate time dependence, a common convention. However, from this point forward in the lab i will denote the imaginary number, i = 1. Notice that there must be a time dependence in the current for any voltage drop to occur for this reason inductors don t factor into DC circuit analysis, where the input voltage is constant in time. The same is true for capacitors. The complex impedances of the capacitor and inductor are given by (4) Z = = "# " (impedance of a capacitor)
2 E5.2 and (5) Z = iωl. (impedance of an inductor) where (6) ω = 2πf The impedances are purely imaginary and depend on frequency, indicating that they are only relevant when the voltage changes with time. In contrast, the impedance of a resistor is purely real, where (7) Z = R. (impedance of a resistor) This is to be expected, since the resistor creates a voltage drop even in a DC circuit with a time independent voltage. These complex impedances add in the same way the resistances add; recall that in series this means that (8) Z = Z + Z + Z + while in parallel: (9) Z = A common method of decreasing the voltage sent to a given part of your circuit is to create a voltage divider (pictured below). We can analyze the divider by using Ohm s law: Figure 1: Voltage Divider (10) I = " "#$ (Think about it: where is I measured? Does it matter?) (11) Z "#$ = Z + Z
3 E5.3 V out, the voltage we re trying to find, is the voltage dropped across the second impedance, so we use Ohm s law once again to find that (12) V "# = IZ = V ". When we use resistors for Z 1 and Z 2, the divider simply reduces the voltage output independent of the signal frequency, even if we are using an alternating current (AC) input. Figure 2: CR highpass filter When we use an inductor or a capacitor, however, we find (through a little algebra) interesting frequency dependence. Given the circuit above, we see that: (13) I = " = " "#$ = " " " (14) V "# = IZ = IR = " " R (15) "# " = "# " "# " = Note: the * indicates the complex conjugate. (16) V "# = "#$ "#$ V " This circuit is called a CR highpass filter. You can see that, for large frequencies f, Equation (15) approaches unity, whereas for small f it approaches 0. If we switch the positions of the resistor and capacitor above we get a similar result, (17) V "# = "#$ V " which is known as an RC lowpass filter. For these circuits we define the cutoff frequency to be the point at which the ratio
4 E5.4 (18) "# " =.707 The cutoff frequency in either case is therefore (19) f c = If we were to construct a loglog scale plot for this ratio as a function of frequency (using R = 4700 Ω and C = 33 nf for example) we would get " for the first circuit and for the second. The cutoff frequency (~1026 Hz) is marked for each case. As we can see the two circuits block out low and high frequency inputs respectively.
5 E5.5 Similar filters can be constructed using resistors and inductors, although with slightly different time dependence (see prelab questions). Procedure Part 1. CR High Pass Filter In this part we will analyze a simple high pass filter and plot attenuation (the ratio of V in to V out ) vs. frequency over a broad range. The circuit is built for you at the lab station and the components are labeled with their actual values. Determine the cutoff frequency f c for the CR circuit (figure 2) at your table. You will use this calculated cutoff frequency in your measurements. The values should be about 33 nf and 4.7 kω, but the actual values can vary so use the values labeled on the circuit box. To connect the signal generator to your circuit you will use a BNC cable. These cables carry your signal internally but also have a grounded shell (not connected to the inner wire). Connect a BNC cable to the Input terminal on the circuit box. Connect the other end to the function generator using a BNC Tsplitter. Use another BNC cable to connect the other end of the Tsplitter and Channel 1 on the Oscilloscope. Take another BNC cable and attach it from the output TTL on the function generator to the EXT Trig slot on your scope. This will be used to trigger the measurement. Triggering tells the scope when to take measurements in order to get a consistent signal, as opposed to taking a different signal with each sweep which shows up as a signal that appears to move across your screen. Turn on the function generator making sure channel 1 is onscreen on your oscilloscope and that the oscilloscope is set to trigger off of the channel connected to Output TTL on the function generator.
6 E5.6 Figure 3: Wiring Setup (Figure 2 gives the circuit diagram) Connect another BNC to the circuit Output and into channel 2 on the oscilloscope. Write down the input voltage, with error, as measured on your scope, using the vertical scale divisions knob to make sure the entire signal is on screen while also filling it as much as possible. This allows you to take the most accurate measurements possible when collecting data for the remainder of the lab always be sure to use this technique. Any time you adjust the input frequency you will need to check your waveforms so that they fill the screen on the oscilloscope again. Make voltage measurements at values of.001*(f c ),.01*(f c ),.1*(f c ),.5*(f c ), f c, 2*(f c ), 10*(f c ), 100*(f c ), and 1000*(f c ). Where f c is the cutoff frequency that you calculated earlier. Record the exact frequency off of your function generator. For very low frequencies, such as.001 and.01 f c, you may not be able to trigger your signal. If this is the case, use the Run/Stop button on your oscilloscope, allowing you to look at just a single measurement. Divide your measurements by V in, and create a data table including error for each result. Plot your data and the theory curve on a loglog scale plot in Mathematica, and comment on any discrepancy. Part 2. Integrator and Differentiator Circuits Using the high pass filter you constructed in the previous part of this lab, look at the input and output signals for various input waveforms (i.e. square wave, triangle wave, sine wave, etc.) found on the right side of the function generator. Keep the frequency of the signals well below your calculated f c (in the 0.01 f c f c range). What do you notice about the waveform of the output signal as compared to the input (Hint: see section title)? Switch the resistor and capacitor to create a low pass filter, and change your input frequency to well above the circuit s f c. What do you notice now? Which of these two circuits
7 E5.7 would you call an integrator? Which a differentiator? Explain. State what the integral and derivative waveforms of each of the three inputs mentioned above are. Do the circuits make a good approximation of differentiation and integration? Hint: think carefully about what the integral of a triangle waveform should look like. It is NOT a sine wave. Part 3. Signal Processing and the ECG One of the most important applications of filters is to reduce electronic noise in signals from measurement devices. When measuring a signal of a certain frequency you are nearly certain to have your measurement distorted by a multitude of other signals over a range of other frequencies. If your signal amplitude is low enough your measurement can be drowned out completely The goal of signal processing is to reduce this noise as much as possible so that you only look at the data you are interested in. By attenuating signals far from the cutoff frequency in one direction while allowing signals on the opposite side to pass through, RC and RL low and high pass filters used in combination are good candidates for this task. The electrocardiogram (ECG) is commonly used in hospitals to monitor patients heart beats and detect heart problems as a diagnostic tool. It works by amplifying tiny voltages present in the skin caused by polarization and depolarization of the heart. The human heart has four chambers: two atria, where blood enters the heart, and two ventricles, which pump the blood out. During the heart s cycle the atria and then the ventricles contract then relax in succession, pushing blood through the body. These contractions are caused by an electrical signal proliferated by the sinoatrial node, otherwise known as the pacemaker due to the fact that the cells spontaneously and rhythmically depolarize of their own accord. This electrical signal creates a very small net voltage difference across the body due to the asymmetric positioning of the transduction cells on the heart. This tiny signal can be read off the skin on opposite sides of the body and amplified to give doctors an idea of how a heart is functioning. The largest peak in an ECG readout, known as the QRS complex, represents ventricular contraction. Depending on the heart of the person in question, the conductivity of their skin, and the positions at which the signals are measured, the voltage difference across the body at this point will be around 13 mv. This presents a challenge, since the largest source of noise in most of these circuits, which comes from power lines, is large enough to completely obscure the signal and will be amplified during the amplification phase of the circuits. Luckily, power lines operate at 60 Hz, whereas the human heart (which beats only around 100 times per minute) operates at a frequency between 1 and 2 Hz. Almost all of the circuitry that goes into an ECG is aimed at eliminating this noise by applying various filters to allow visualization of the heart cycle. In this part of the lab you will compute the actual maximum voltage difference created at your hands by your heart. ECG circuits first amplify the signal before filtering it, since otherwise peaks would be too small to resolve. To determine your heart voltage you will thus need to know the degree to which the ECG you are using amplifies the signal. Make sure the power cord on the ECG box is plugged in, then connect the OUTPUT jack to channel 1 of your oscilloscope using a BNC cable. Turn on the circuit, set the mode to CALIBRATE, and switch three to Vin. Using a DMM (digital multimeter) with a BNC cable attachment, measure the input voltage to the calibration pulse and it s error by connecting the DMM to the LEFT/Vin socket. Pressing the CALIBRATING PULSE button will send a signal into the circuit at this voltage. Remove the
8 E5.8 DMM and flip switch three to OPERATE. On the oscilloscope, adjust the time scale to around 250 ms/division and, using the calibrate button, adjust the voltage divisions until the pulses take up most of the oscilloscope screen. Measure the output calibration voltage on the oscilloscope with error (the error on any oscilloscope measurement is + one small division note that making the large divisions as small as possible minimizes error, which is why it is best to take up as much of the screen with the pulses as you can without clipping the signal). Use the input and output calibration voltages to determine the amplification factor of the circuit and its error. Next attach the handle bar BNC cables to the LEFT/Vin and RIGHT sockets on the box, and switch the mode to OPERATE. In this mode the box will pick up signals from the handlebars, amplify and filter the signal, then output what remains. To get an accurate reading, rest your hands gently on the bars and stay as still and relaxed as you can muscle contractions also create transient voltages that will distort your signal. If you don t see your heartbeat right off the bat don t worry sometimes it can take seconds to resolve to a steady pattern. When you see your heartbeat, pick a typical cycle and measure the peak to peak voltage of the QRS complex with error. Using the amplification factor and the measured output voltage for your heartbeat, calculate the voltage running across your hands whenever your ventricles contract. Neat Guiding Questions 1) What is the function in Mathematica used to create a loglog scale plot of discrete data points? (1 point) 2) When plotting the theory curve on a loglog scale, be careful not to begin your plot range at 0. Why is this necessary? (1 point) 3) For an EKG monitor, one problem in signal amplification is power line noise which comes in at around 60 Hz (the human heart beats at around 1 Hz). If you wanted to eliminate this noise using an RC filter with a 1 µf capacitor, what valued resistor would you choose? Draw a schematic of the filter. (2 points) 4) Derive the ratio "# " for the RL filter shown below, and determine the cutoff frequency. Hint: see equations Your final answer should be "# " = /"#. (3 points) Figure 4: RL lowpass filter 5) Show that the units on the right hand side of the ratio above cancel to give a unitless quantity. (1 point) 6) Filters have many real world applications. Write down a couple ideas you have for possible uses. (1 point) 7) By combining a resistor, an inductor, and a capacitor all into one RLC circuit, as shown below, it is possible to filter out signals above and below a certain frequency. This is
9 E5.9 known as a bandpass filter, since only a narrow band of signals is allowed through. The width of the band is determined by the Q (for quality) factor of the circuit, which we won t discuss in depth. Give the values for Z 1 and Z 2 in the circuit below. (1 point) Figure 5: RLC bandpass filter
10 E5.10 Prelab assignment for E5: Filters and Complex Impedance. This prelab is designed to focus your attention on what is important in the lab before you start the experiment and give you a leg up on writing your report. Use this template when you write your report. 1. Carefully read the lab instructions for E5 2. Using Mathematica, set up a template for your lab report. This will include: I. Header information such as title, author, lab partner, and lab section: (to do this, In Mathematica, go to File, Print Settings, Header and Footer from this dialog box you can enter all of the required information.) Mathematica automatically enters the file name in the right side of the header. Leave this alone. II. Section Headings including Title, Summary of Experiment, Data and Calculations, Discussion of Uncertainty, and specific section headings for each part of the experiment. (All of the Physics 1140 lab manuals include multiple parts.) 3. Write a Summary of the experiment. What are you going to measure and what data will you collect to make the measurement? (For example, in part one of M1, you will measure the length of a pendulum along with the period of the pendulum to determine the acceleration due to gravity.) 4. For each part of the lab do the following: A.) For part one: Suppose you place a 3600 Ohm resistor and a 23nF capacitor in the High Pass circuit as described above. Calculate the cutoff frequency and create a theory plot of the output voltage (V out ) versus the frequency. Use equation (16) for inspiration and assume a 5V input voltage. B.) For Part two: you are asked to determine whether the high pass filter is an integrator or differentiator, and then again determine the same thing for the low pass filter. For this part, draw what you would expect to see as a graph for the differentiator of the triangle wave. Draw the same for the square wave. Don t worry about doing this in Mathematica, just draw it by hand. C.) For Part three: our ECG uses electrodes that you place your hands on. In two or three sentences describe how the ECG can detect your heartbeat. Also, how do high and low pass filters hone in on your heart beat?
11 E5.11 Remember, all plots you make in a prelab or lab report should include the following: I. Plot your theory or expectation as a line. (In Mathematica, you should define a function and plot it using the command Plot ) Reference the Mathematica tutorial to do this. II. Label your axes, in English (not just symbols) and with units. III. Include a brief caption (namely, a text statement of what the plot shows.) IV. Set the x and y range of the plot to be close to what you expect for your data. For example: in M1 your longest length pendulum should be no more than 130 cm in length. 5. Write a Discussion of Uncertainty. What are the major sources of uncertainty in the experiment and how will you account for them? 6. Turn in a printout of your Mathematica document that includes 25 above. This document should be no more than 2 pages long.
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