University of Rochester Department of Electrical and Computer Engineering ECE113 Lab. #7 Higher-order filter & amplifier designs March, 2012

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1 University of Rochester Department of Electrical and Computer Engineering ECE113 Lab. #7 Higherorder filter & amplifier designs March, 2012 Writeups, due one week after the lab is performed, should provide a detailed description of the laboratory exercise with diagrams of all circuits, a description of your procedures, loglog plots of gain versus frequency, and a separate abstract. Late work is not accepted. I. Background The attached pages describe dual sets of active amplifier/filter circuits, one 2nd order and the other 3 rd order. Enough information is provided on these pages to allow you to design lowpass and highpass filters with prescribed frequency breakpoints. The design methodology is based on frequency and amplitude scaling to determine the component values for your chosen filter. Your design is then to be tested in the laboratory. II. Preparation for laboratory exercise Read this entire assignment first, then with your partner select one (1) of the four circuits listed below and create a design according to the specifications. a) 2 nd order LP ampl. with G LP = 5: critical 10 khz b) 2 nd order HP ampl. with G HP = 5: critical 5 khz c) 3 rd order LP filter (G LP = 1): critical 20 khz d) 3 rd order HP filter (G HP = 1): critical 5 khz Start with a Bode diagram of the desired circuit. To determine the values of the components, especially for the 3 rd order filters, you can use frequency and amplitude scaling to great advantage. You will need to use a trialanderror approach to calculate some of your component values. For any of the circuits selected, restrict resistor values to the range of 1 to 20 kω. NOTE: There is no single, unique solution for any of these designs, though some choices will be better than others. Before you come to the lab, prepare a onepage (singlesided) design sheet. This design sheet must clearly show your circuit, the design methodology, and a parts list. The amplitude & phase Bode plots used in the design should be attached. Bring two copies of these sheets to the lab and hand one of them to the TA before you start the lab. Loglog paper for use in your Bode diagram will be available in the plastic tray outside my office and in the lab.

2 pg 2 of 5 III. Experimental procedure A. Assemble your circuit. After doublechecking your connections, test it thoroughly across a frequency range appropriate to the critical frequency specified for the design you selected. Measure both the amplitude and the phase of the transfer function. Be smart about the frequency values chosen for your measurement: concentrate the data collected close to the breakpoint, that is, where H(jω) and H(jω) change most rapidly. The smart way to record date is to plot it directly on the loglog paper. B. If the performance of your circuit is far out of specs, look closely at the design to see if you have made some error in either the calculations or the design itself. Make appropriate corrections, and repeat the test measurements. The thirdorder filters can be made to work properly if you lay out the circuit carefully, use star grounding, and choose component values correctly. Do not use electrolytic capacitors as filter elements. Feel free to use any opamp available in the lab; 741 s are not good choices for 3 rd order filters. IV. Writeup In your writeup, summarize the design procedure and include a copy of the onepage design sheet and Bode plot. Describe and analyze the performance of your circuit, comparing the actual measured break frequency, behavior of the filter in the pass band, and rolloff in db/decade. Include carefully prepared plots of your data with the Bode diagrams superimposed. Summarize any data analysis or theoretical calculations relevant to your work. In the last section of the report, discuss any shortcomings of your circuit and consider explanations for such discrepancies. Do not forget to include a separate abstract with your writeup.

3 pg 3 of 5 L o w g a i n 2 n d o r d e r a m p l i f i e r s Lowgain 2ndorder LP and HP amplifiers can be designed from a common circuit topology using two capacitors, four resistors, and one opamp. Lowpass (LP) amplifier The critical break frequency of this LP amplifier is! h = 1/ C 1 C 2, while the lowfrequency voltage gain is G LP = 1 R b /R a. Subject to the assumption that the openloop gain of the opamp is very high, the transfer function of this LP filter is: R a R b Highpass (HP) amplifier G LP H(s) = s 2 C 1 C 2 s [C 2 ( ) C 1 R b / R a ]1 The critical break frequency of this HP amplifier is! l = 1/ C 1 C 2 and the highfrequency gain is G HP = 1 R b /R a. Subject to the assumption that the openloop gain of the opamp is very high, the transfer function of this HP filter is: R a R b H(s) = G H P s 2 C 1 C 2 s 2 C 1 C 2 s [ (C 1 C 2 ) C 2 R b / R a ]1 You can base your design methodology for either of these amplifiers on the given analytical expressions for H(s) as follows: (i) identify the damping factor ζ within H; (ii) set ω break = 1 rad/sec, = = R a = 1 Ω, and then use G plus a reasonable guess for the value of ζ to determine R b, C 1, and C 2 ; (iii) employ frequency & amplitude scaling to move the break frequency where you want it and to put the resistors and capacitor values into practical ranges. NOTE: You must use trial and error methods to determine your capacitor values. In either case, make sure the coefficient of s in the denominator is positive! If it goes negative, the poles move to the righthalf plane and instability results.

4 pg 4 of 5 U n i t y g a i n 3 r d o r d e r f i l t e r s 3rdorder LP and HP filters can be designed to give unity gain in their pass bands and very steep rolloff (60 db/decade) in their stop bands. The transfer function H(s) of a 3rdorder filter is algebraically hard to manipulate. Thus, it is common to start with given component values (e.g., derived from the maximally flat condition, H = 3 db with a breakpoint at ω break = 1 rad/sec). To create a ckt to meet practical specifications, the component values are frequencyscaled to achieve the desired corner frequency and amplitudescaled to put components into practical ranges. LP filter The 3rdorder LP filter ckt uses 3 resistors & 3 capacitors, as shown at right. Starting point values giving the maximally flat condition with ω h = 1 rad/sec are: R 3 C3 = = R 3 = 1 Ω, C 1 = F, C 2 = 3.57 F, & C 3 = 1.39 F. HP filter The 3rdorder HP filter exchanges the 3 resistors and 3 capacitors, as shown at right. C3 Starting point values giving the maximally flat condition & a corner freq. ω l = 1 rad/sec are: R 3 C 1 = C 2 = C 3 = 1.00 F, = 4.94 Ω, = Ω, & R 3 = Ω.

5 pg 5 of 5 If you are ambitious and enjoy algebra, you can attempt to use the mesh equations to solve for the transfer function H(s). If you succeed, you can study the influence of component values on frequency response. This is not necessary for completion of the assignment but, if you succeed in obtaining H(s), let me know and I will plot it for you. Note that 3 rd order filters exhibit rather strong sensitivity to component values, indicative of the everpresent tradeoff between component cost and performance.