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EE 311: Electrical Engineering Junior Lab Active Filter Design (Sallen-Key Filter) Objective The purpose of this experiment is to design a set of second-order Sallen-Key active filters and to investigate their performance characteristics. Figure 1. Second order Sallen-Key band-pass filter. Background Theory Filters are found in various types of electronic equipment and perform various functions. For example, they are used in power supplies to attenuate undesirable ripple, in audio circuits for bass and treble control, and in signal processing applications where they are often used to band limit a signal before it is sampled. There are four basic types of filters: high-pass, low-pass, band-pass and band-reject or notch. All filters fall in one of two categories: passive or active. Passive filters consist of only passive elements, i.e., resistors, inductors and capacitors. On the other hand, active filters consist of passive elements along with active devices, such as transistors or op-amps. It should be noted, that one can not take the output of a passive filter and amplify it using an op-amp or transistor to produce an active filter! Typically, op-amps are chosen over transistors in active filters to take advantage of their high performance characteristics and minimal cost. The question one needs to ask is why bother with active filtering? The answer is really quite simple: an active filter uses combinations of op-amps, resistors and capacitors to obtain a response equal to or better than conventional passive filters. For example, in order to obtain a sharp response with a passive filter, we would need to cascade several passive stages. Each cascaded stage, however, loads the previous stages. This loading attenuates the desired part of the signal, i.e., frequencies within the filter pass-band, as well as unwanted frequency content within signal. This problem is commonly known as insertion loss. Active filtering practically eliminates insertion loss due to the high input impedance and the low output impedance of an op-amp. Furthermore, with active filtering, we can attenuate unwanted frequencies while amplifying desired frequencies! Two other advantages of active filters include simple design and ease of tuning. Lastly, active filters usually do not require the use of inductors, which are typically bulky, costly and depart further from ideal models compared to capacitors. Since the first designs in this experiment investigate the performance characteristics of a band-pass filter, we will need to define some important concepts related to band-pass filters. Refer to Figure 2 to obtain some physical intuition to the following terms. Center Frequency The center frequency, which is sometimes called the resonant frequency, is given by f o . In second order circuits and some higher order circuits, the center frequency is easy to determine since it is the frequency at which the maximum gain of the filter occurs. Theoretically, the center frequency is the geometric mean of the two half-power frequencies, f l and f h . That is, fo = fl ⋅ fh . Lower and Upper Cutoff Frequencies The lower and upper cutoff (or half-power) frequencies are given by f l and f h , respectively. The lower cutoff frequency is the lower frequency at which the gain is 3dB less than the gain at the center frequency. Similarly, the upper cutoff frequency is the upper frequency at which the gain is 3dB less than the gain at the center frequency. Maximum Gain The maximum gain of a filter is given by Ho . It is the ratio of V o to V i at the filter's center or resonant frequency. Decibels (dB) are often used as a relative measure of filter gain in the lab, where it is common practice to uncalibrate an instrumentation channel so that the output voltage reads 0 db at the point of maximum gain. If you do this, you still need to record the actual, i.e., unscaled, gain for a given frequency. Ho HO Figure 2. Typical frequency response for a band-pass filter with peak gain of 1.0. Passband The passband is the frequency range for the part of the signal that is not attenuated, i.e., the gain is within 3dB of the maximum gain. Hence, the passband frequency range lies between the lower and upper cutoff frequencies, f l and f h . Bandwidth The bandwidth, β , is closely related to the passband, being the difference between the upper and lower filter cutoff frequencies, β = f h − f l . Quality Factor The quality factor, Q , is a dimensionless figure of merit used to measure the selectivity of a filter and is expressed as the ratio of the filter's center frequency to bandwidth, Q = f o / β . For example, given a filter with a fixed center frequency, decreasing the filter's bandwidth (i.e., increasing its sharpness) increases Q . Figure 3. Non-inverting amplifier subcircuit. Sallen-Key Band-pass Filter In this experiment, we will investigate active filtering by the aid of a popular filter called a Sallen-Key band-pass filter, which was named after its inventors. This filter is shown in Figure 1. To fully understand this configuration, we must obtain the transfer function, T ( s ) , of the filter. We can simplify our analysis by first considering the basic noninverting op-amp configuration shown in Figure 3. Recall that the gain, K , of this circuit is given by K= Vo Rf = 1+ Vx Ri (1) Note that the Sallen-Key filter under investigation has the same non-inverting op-amp stage incorporated in its design, i.e., in the boxed region of Figure 1. We can use this observation to simplify the derivation of the transfer function of the Sallen-Key filter. It can be shown that the transfer function of the filter is Ks Vo R1C1 T (s) = = 1 1 1 1− K R1 + R2 Vi s 2 + + + + s + R1C1 R3C1 R3C 2 R2 C1 R1 R2 R3C1C 2 (2) Observe that the transfer function shown in equation (2) is quite complex. The analysis can be greatly simplified, however, if all frequency-determining capacitors are set equal to each other, as well as all frequency-determining resistors. Under these conditions, the filter of Figure 1 is sometimes referred to as Equal Component Sallen-Key Filter. We can take advantage of this trick by letting R1 = R 2 = R3 = R and C1 = C 2 = C 2 ( 3) The original transfer function of equation (2) then simplifies to Ks RC T (s) = 2 3 K − s + 1 s 2 + RC RC ( 4) A second order band-pass transfer function can written in the following standard form T (s) = ωo Ho s Q (5) ωο s + s + ω o2 Q 2 Comparing coefficients of (4) and (5), we see that 1 RC 1 Q= 3− K K Ho = 3− K ωο = ( 6) ( 7) (8 ) where ωο, Q , and Ho have been defined previously. From equations (7) and (8), we notice that Q and not independent, being determined by the op-amp gain K of equation (1). In your Pre-lab, you will be asked to verify that the poles of the transfer function given in equation (4) are determined by 2 3− K 3− K 1 − ± − 4 RC RC RC s1, 2 = 2 2 ( 9) Ho are In order to ensure stability of the filter, we must ensure that the poles of the transfer function lie in the left-half of the complex s -plane, or ℜ( s1, 2 ) < 0 . Thus, we must ensure that the gain of the op-amp is less than 3, i.e., K < 3. One final topic to be addressed is the issue of frequency scaling. Frequency scaling is a method of denormalizing a filter by changing its frequency. This method is extremely useful once one has designed a filter with a satisfactory response (i.e., ωο, Q , and Ho ) and then merely wants to change, for example, the center frequency. To increase the center frequency of a filter without affecting any of its other characteristics (i.e., Q and Ho ), we can simply divide all frequency determining capacitors or divide all frequency determining resistors by the desired scaling factor. As an example, to triple the center frequency, divide all capacitor values by 3 or divide all resistor values by 3. We now have all the tools in our possession to design our own Sallen-Key band-pass filter. Typically, we are given the center frequency and bandwidth of the desired filter. In this lab, we can use equations (1), (6), (7) and (8) to solve for the proper resistor and capacitor values. Note: For your design, let R1 = R 2 = R3 = R i = 2k Ω 2 Sallen-Key Low-pass Filters Sallen-Key filters, like many other active filters, have low-pass, high-pass, and band-reject filter implementations, as well as band-pass implementations. An ideal low-pass filter passes all frequencies from zero up to the corner frequency f o , and blocks all frequencies above this value. In actual filters, there is a transition region between the passband and the stopband. The standard form for a low-pass filter is T ( s) = Hoω 2o ωο s 2 + s + ω 2o Q (10) where ωo = 2 πf o is the corner frequency of the low pass filter and Q is the quality factor. The frequency response of the low-pass filter is not as straightforward to analyze as that of the band-pass filter, however. For values of Q less than 0.5, the poles of the transfer function are real. For values of Q greater than 0.5, the poles are complex. For Q >0.707, the frequency response peaks above Ho at ωo . This peak can be quite large for large values of Q , while Q =0.707 produces a maximally flat response, i.e., the sharpest fall-off near ωo without any peaks larger than Ho . Figure 4. Sallen-Key low-pass filter. Figure 4 shows a potential Sallen-Key low-pass filter. Note that this filter contains the same non-inverting amplifier sub-circuit (i.e., the boxed region) analyzed in the band-pass filter development. Preliminary Report Questions 1. Show that the poles of the band-pass filter transfer function, specified in equation (4), are given by equation (9), and verify that the filter is stable if and only if K < 3. 2. Design a Sallen-Key band-pass filter which has a center frequency, f o , of 1600Hz and a bandwidth, β , of 640Hz. (Use the resistor values specified above in the Background Theory.) 3. Calculate the quality factor, Q , and maximum gain, Ho , for the filter designed in the previous question. 4. Assuming the same center frequency, repeat steps 2 and 3 for a filter bandwidth of 320Hz and 160 Hz. 5. Derive the transfer function for the low-pass filter shown in Figure 4. From this, develop expressions for ωο, Q, and Ho in terms of the resistances and capacitances. Choose convenient relationships between the 2 resistors and between the 2 capacitors to simplify the transfer function analysis. Describe the meaning of ωο, Q, and Ho for the low pass filter. 6. Design low-pass filters with a corner frequency of 4800 hertz and Q values of 0.707 and 5.0. 7. Prepare schematic diagrams for the filters designed above, with appropriate labels so that your lab work can be completed efficiently. Label all components, number all pins and indicate how the wave analyzer, oscilloscope and dc power supply should be connected to test the performance of your circuit. 8. Calculate the maximum allowable peak-to-peak input voltage for each of your designs. Assume that the maximum output voltage swing of the LM741 is ±10 volts when using a supply voltage of ±12 volts. 9. Use PSpice to obtain the magnitude and phase frequency response plots for the designed low-pass filters. You must also include a netlist, i.e., device, device value, + node and - node, of the components used by PSPICE to produced the plots for each filter. 10. Analyze how a signal source with high output impedance will affect the performance of your filter circuits. How can you minimize these effects? (You can use your PSpice models or the above analysis to predict the effects). 11. *Use the Sallen-Key topology to implement a second-order low-pass Bessel filter with the same corner frequency as in step 6. Use Pspice to simulate your Bessel filter design and compare its performance to the lowpass filters of step 9. Show all work in this process. How does the Bessel response differ from the responses of the previously designed low-pass filters? * Bonus problem for extra credit. Procedure A few notes on data taking: The data in this experiment will be taken manually. We will use the HP 3581A wave analyzer to provide the input voltage and to measure the output voltage. The HP 3581A has a fairly high output impedance, so use either a voltage divider or an op-amp buffer (e.g., voltage follower) to reduce the effect of this source impedance on your filter circuits. The most important data set for the band-pass filter is the center frequency and gain at center frequency. The next most important points are the upper and lower 3 db points. Outside the passband, it is sufficient to measure the change in db per decade or octave for this simple, singly tuned filter. 1. 2. Construct your first band-pass filter and verify that it is working properly. Use the decade resistor box for R F to facilitate changing Q later on. Connect both input voltage and output voltage to the oscilloscope so that you can observe that the waveshapes are correct. Take the input voltage from the wave analyzer, setting this voltage at a suitable level. Measure the variation in input voltage as the frequency is swept through the filter center frequency. Determine if this variation in input voltage is significant. If Step 1 shows significant change in the input voltage, construct either a voltage follower circuit or a 100Ω − 5Ω voltage divider on the proto-board and connect the output signal on the back of the wave analyzer to the input of the voltage divider. This output signal is a sine wave of the frequency being measured by the meter. The amplitude is adjustable by the knob on the back of the meter. 3. Set the wave analyzer to the manual mode. Adjust the wave analyzer resolution bandwidth and frequency span so that the wave analyzer ADJUST light remains off (you may need to change the reset frequency and sweep time as well). Also adjust the input sensitivity on the wave analyzer so that the needle obtains a full-scale deflection at the resonant frequency. 4. Using the manual sweep mode and the vernier frequency control dial on the mode switch, adjust the frequency to the resonant frequency of the filter. The resonant frequency occurs when the needle on the wave analyzer obtains its maximum deflection. Measure the absolute gain (volts per volt) at this frequency. Now, go to the dB scale and adjust the input sensitivity so that a 0 dB reading, relative gain, is obtained at the resonant frequency and measure the upper and lower 3-db points. Use this data to determine the filter Q value. 5. Go to a frequency somewhat below the lower 3-dB point and measure the relative gain. Go down an octave in frequency, and measure the gain again. Does this correspond to the expected change per octave? Repeat for a decade change in frequency. Repeat both measurements above the upper 3-dB point. Record data and comment. Use these 7 data points to generate a plot of the filter response (Matlab is available for doing this). 6. Construct the two other band-pass filters described in the Pre-lab. Measure each circuit's frequency response, then determine gain, center frequency, and Q factor. Note: The last filter may be unstable. Adjust R F and observe both stable and unstable operation. Comment on the two cases. Note the values for R F involved, then record data for the stable case. 7. Construct the two low-pass filters described in the Pre-lab. Using the manual sweep mode, determine low frequency gain, maximum gain, and corner frequency, i.e., 3-dB frequency. Plot the frequency response for the two cases. Final Report Questions • Prepare a table that presents the theoretical and experimental β, Ho, Qs, and ωo for each of the band-pass filters. Your table should also include the percent error between the theoretical and experimental results for each of the preceding entries. Explain any discrepancies between the theoretical and experimental results. • Discuss the stability of the third band-pass filter investigated. • Discuss the low-pass filter experimental results and compare them with expected performance. • Explain why the voltage divider or voltage follower at the input of the filter was necessary in terms of filter performance. Equipment List Proto-board 2- 741 op amps 100 ohm, 5 ohm resistors for voltage divider decade resistor (for R f ) capacitors and resistors determined in Pre-lab Last Revised: 9/7/02 jjc