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1. Decibels, Filters, andBode Plots

2. OBJECTIVES • Develop confidence in the use of logarithms and decibels in the description of power and voltage levels. • Become familiar with the frequency response of high- and low-pass filters. Learn to calculate the cutoff frequency and describe the phase response. • Be able to calculate the cutoff frequencies and sketch the frequency response of a pass-band, stop-band, or double-tuned filter. • Develop skills in interpreting and establishing the Bode response of any filter. • Become aware of the characteristics and operation of a crossover network.

3. INTRODUCTION • The unit decibel (dB), defined by a logarithmic expression, is used throughout the industry to define levels of audio, voltage gain, energy, field strength, and so on.

4. INTRODUCTIONLogarithms • Basic Relationships • Let us first examine the relationship between the variables of the logarithmic function. • The mathematical expression:

5. INTRODUCTIONLogarithms • Some Areas of Application • The following are some of the most common applications of the logarithmic function: • 1. The response of a system can be plotted for a range of values that may otherwise be impossible or unwieldy with a linear scale. • 2. Levels of power, voltage, and the like can be compared without dealing with very large or very small numbers that often cloud the true impact of the difference in magnitudes. • 3. A number of systems respond to outside stimuli in a nonlinear logarithmic manner. • 4. The response of a cascaded or compound system can be rapidly determined using logarithms if the gain of each stage is known on a logarithmic basis.

6. FIG. 21.1 Semilog graph paper. INTRODUCTIONLogarithms

7. FIG. 21.2 Frequency log scale. INTRODUCTIONLogarithms

8. FIG. 21.4 Example 21.1. FIG. 21.3 Finding a value on a log plot. INTRODUCTIONLogarithms

9. PROPERTIES OF LOGARITHMS • There are a few characteristics of logarithms that should be emphasized: • The common or natural logarithm of the number 1 is 0 • The log of any number less than 1 is a negative number • The log of the product of two numbers is the sum of the logs of the numbers • The log of the quotient of two numbers is the log of the numerator minus the log of the denominator • The log of a number taken to a power is equal to the product of the power and the log of the number

10. PROPERTIES OF LOGARITHMSCalculator Functions • Using the TI-89 calculator, the common logarithm of a number is determined by first selecting the CATALOG key and then scrolling to find the common logarithm function. • The time involved in scrolling through the options can be reduced by first selecting the key with the first letter of the desired function—in this case, L, as shown below, to find the common logarithm of the number 80.

11. DECIBELS • Power Gain • Voltage Gain • Human Auditory Response

12. TABLE 21.1 DECIBELS

13. FIG. 21.5 LRAD (Long Range Acoustic Device) 1000X. (Courtesy of the American Technology Corporation.) DECIBELS

14. FIG. 21.6 Defining the relationship between a dB scale referenced to 1 mW, 600Ωand a 3 V rms voltage scale. DECIBELSInstrumentation

15. FILTERS • Any combination of passive (R, L, and C) and/or active (transistors or operational amplifiers) elements designed to select or reject a band of frequencies is called a filter. • In communication systems, filters are used to pass those frequencies containing the desired information and to reject the remaining frequencies.

16. FILTERS • In general, there are two classifications of filters: • Passive filters • Active filters

17. FIG. 21.9 R-C low-pass filter at low frequencies. FIG. 21.8 Low-pass filter. R-C LOW-PASS FILTER

18. FIG. 21.10 R-C low-pass filter at high frequencies. FIG. 21.11 Vo versus frequency for a low-pass R-C filter. R-C LOW-PASS FILTER

19. FIG. 21.12 Normalized plot of Fig. 21.11. R-C LOW-PASS FILTER

20. FIG. 21.13 Angle by which Vo leads Vi. R-C LOW-PASS FILTER

21. FIG. 21.14 Angle by which Vo lags Vi. R-C LOW-PASS FILTER

22. FIG. 21.15 Low-pass R-L filter. FIG. 21.16 Example 21.5. R-C LOW-PASS FILTER

23. FIG. 21.17 Frequency response for the low-pass R-C network in Fig. 21.16. R-C LOW-PASS FILTER

24. FIG. 21.18 Normalized plot of Fig. 21.17. R-C LOW-PASS FILTER

25. FIG. 21.19 High-pass filter. R-C HIGH-PASS FILTER

26. FIG. 21.20 R-C high-pass filter at very high frequencies. FIG. 21.21 R-C high-pass filter at f =0 Hz. R-C HIGH-PASS FILTER

27. FIG. 21.22 Vo versus frequency for a high-pass R-C filter. R-C HIGH-PASS FILTER

28. FIG. 21.23 Normalized plot of Fig. 21.22. R-C HIGH-PASS FILTER

29. FIG. 21.24 Phase-angle response for the high-pass R-C filter. R-C HIGH-PASS FILTER

30. FIG. 21.25 High-pass R-L filter. R-C HIGH-PASS FILTER

31. FIG. 21.26 Normalized plots for a low-pass and a high-pass filter using the same elements. R-C HIGH-PASS FILTER

32. FIG. 21.27 Phase plots for a low-pass and a high-pass filter using the same elements. R-C HIGH-PASS FILTER

33. FIG. 21.28 Series resonant pass-band filter. PASS-BAND FILTERS

34. FIG. 21.29 Parallel resonant pass-band filter. PASS-BAND FILTERS

35. FIG. 21.30 Series resonant pass-band filter for Example 21.7. PASS-BAND FILTERS

36. FIG. 21.31 Pass-band response for the network. PASS-BAND FILTERS

37. FIG. 21.32 Normalized plots for the pass-band filter in Fig. 21.30. PASS-BAND FILTERS

38. FIG. 21.33 Pass-band filter. PASS-BAND FILTERS

39. FIG. 21.34 Pass-band characteristics. PASS-BAND FILTERS

40. FIG. 21.36 Pass-band characteristics for the filter in Fig. 21.35. FIG. 21.35 Pass-band filter. PASS-BAND FILTERS

41. FIG. 21.37 Network of Fig. 21.35 at f =994.72 kHz. PASS-BAND FILTERS

42. BAND-REJECT FILTERS • Since the characteristics of a band-reject filter (also called stop-band or notch filter) are the inverse of the pattern obtained for the band-pass filter, a band-reject filter can be designed by simply applying Kirchhoff’s voltage law to each circuit.

43. FIG. 21.38 Demonstrating how an applied signal of fixed magnitude can be broken down into a pass-band and band-reject response curve. BAND-REJECT FILTERS

44. FIG. 21.39 Band-reject filter using a series resonant circuit. BAND-REJECT FILTERS

45. FIG. 21.40 Band-reject filter using a parallel resonant network. BAND-REJECT FILTERS

46. FIG. 21.41 Band-reject filter. BAND-REJECT FILTERS

47. FIG. 21.42 Band-reject characteristics. BAND-REJECT FILTERS

48. DOUBLE-TUNED FILTER • Some network configurations display both a pass-band and a stop-band characteristic, such as shown in Fig. 21.43. • Such networks are called double-tuned filters.