1. Attenuators

2. Attenuators are simple but very important instruments. Unlike an amplifier, which is ordinarily used to increase a signal level by a given amount, the attenuator is used to reduce the signal level by a given amount.

3. The use of attenuators has become so widespread that a study of their design and use is important in the study of electronic instruments. Attenuators may be constructed in many ways. We will confine our discussion to lumped- resistance attenuators.

4. The L-type Attenuator One of the simplest types of attenuators is the L type or the ordinary voltage divider. The voltage gain of this network is the out­put voltage divided by the input voltage.

5. We can find the formula for gain in the following manner: (1)

6. Equation (1) should be familiar. It tells us that the voltage gain of a simple voltage divider is equal to R2 divided by the sum of R1 and R2.

7. For instance, if R1 equals 9 kilo-ohms and R2 equals 1 kilo-ohm the voltage gain equals one-tent. As usual, to find the voltage gain expressed in decibels, we take 20 times the base 10 logarithm of A. In this case, for A equals 0.1, Adb equals -20 db.

8. Since attenuators always reduce the signal level, the value of A is always less than unity, and Adb is therefore always negative. If we like, we can use the reciprocal of A and thereby avoid values that are less than unity. In other words, let us define the attenuation as: (2)

9. For the voltage divider with R1 equal to 9 kilo-ohms and R2 equal to 1 kilo-ohm, we now say that the attenuation a is 10, or adb is 20 db. The distinction between A and a is merely a detail, but it should be under­stood, since both terms are commonly used in practice.

10. The Characteristic Resistance of Symmetrical Attenuators Usually, the term attenuator refers to a device that not only introduces a precise amount of attenuation but also provides an impedance match on the input and output terminals.

11. For instance, an attenuator has been inserted between a source and a load. If it has been properly designed, the input resistance will be 50ohms, thereby matching the source resistance. Further, at the output terminals, the Thevenin resistance looking back is 50ohms, so that the load is matched. There are many systems where impedance matching is very important.

12. In fact, in any system using transmission lines a great deal of trouble maybe encountered unless all devices are impedance-matched. fig.2 fig.3

13. This is especially true in those situations where the wavelength associated with the signal becomes comparable to the length of the transmission line. Telephone, television, and microwave systems are example situations where impedance matching is normally used.

14. We will now be concerned with attenuators that have been designed to provide an impedance match on both the input and output side. Further, we will confine our attention to a practical class attenuators known as unbalanced symmetrical attenuators.

15. The two most basic forms in this class are shown in Fig. 7-3 (a) and (b). Note that these are symmetrical about vertical center line. The word "unbalanced" refers to the fact that there is a common connection from input to output (the lower wire). One extremely important property of symmetrical attenuators is what is commonly referred to as the characteristic (or image) resistance

16. Consider the situation in Fig. 7-4. A variable-load resistor has been connected to the output of the attenuator. fig.4

17. As we look into the attenuator, we see a value of input resistance that depends upon the value of load resistance. Suppose that for RL = 100 ohms, Rin = 60 ohms. As we change RL to 70 ohms, we might observe that Rin changes to 55 ohms. As we vary RL again to a new value of 50 ohms, we might have an Rin equal to 50 ohms. Continuing in this way, we would find that Rin takes on different values ­for each value of RL

18. In general, the characteristic resistance of an attenuator is that value of load on an attenuator which produces the same value of input resistance. Every attenuator has a characteristic resistance. Under normal circumstances, attenuators should always be loaded in their characteristic resistances. If this is done, the resistance value is maintained throughout the system in which the attenuator is used.

19. For instance, if a system uses 50-ohm load resistors and 50-ohm source resistances, using an attenuator with a characteristic resistance of 50 ohms between a load and a source will match the impedance on the source and load sides of the attenuator.

20. A useful formula for the characteristic resistance Ro of an attenuator is (3) where : is the input resistance of the attenuator with the output terminals shorted. : is the input resistance of the attenuator with the output terminals open. This equation can be derived by use of z para­ meters. We will accept it without derivation.

21. Symmetrical T Analysis Formulas In this section we wish to find a pair of formulas that will be useful when­ever we have to deal with the symmetrical T attenuator. In order to simplify the formulas it is helpful to define the ratio of R2 to RI as follows: (4) or R2 = mR1 (5)

22. With this definition for m, the attenuator may be relabeled fig. 5

23. First, let us find Ro (6) After a few simplifying algebra steps we can obtain : Ro = R1 (7)

24. The second formula that will be useful in our work is a formula for the amount of attenuation that takes place from the input to the output terminals of an attenuator loaded in its characteristic resistance.

25. We require a formula for, that is, the attenuation. There are several approaches that can be used in finding such an expression. We will merely note that after the required algebra, it can be shown that (8) fig. 6

26. Let us summarize. In Eqs. (7-7) and (7-8) we have the analysis formulas for a symmetrical T attenuator. Given the values of R 1 and R 2 (such as on a schematic), we can find m by taking the ratio of R 2 to R 1. Then, sub­stituting into Eqs. (7-7) and (7-8) we can easily find the characteristic resistance and the attenuation for the T attenuator.

27. EXAMPLE 5: Compute the characteristic resistance and the attenuation of a symmetrical T attenuator which has R 1 = 409 Ω and R 2 = 101 Ω.

28. SOLUTION:

29. Thus, the given attenuator should be used with a 500-ohm load and source resistance. If this is done, the attenuation will be equal to 10, which is equivalent to 20 db.

30. EXAMPLE 6: If the attenuator of Example 5 has all resistances reduced by a factor of ten, what is the value of R o and a for the new attenuator?

31. SOLUTION: We note that now R 1 = 40.9 and R 2 = 10.1 ohms. Therefore, m still equals 0.247, as in Example 5. From Eq. (8), the attenuation still equals 10. Next, we observe from Eq. (7) that R, is directly proportional to R 1. The value of m is still the same. Therefore, R o equals one-tenth of 500, or 50 ohms. Thus, by reducing all resistances by a factor of ten, we have reduced the characteristic resistance by ten, but the attenuation has remained the same.

32. 5- Symmetrical T Design Formulas: In this section we develop design formulas for the symmetrical T attenu­ator. Equations (7) and (8) tell us how to find R o and a, given the value of R 1 and R 2. If, on the other hand, we wish to design a T attenu­ator, ice will be given the value of R o and a and will need to find R 1 and R 2. We can find formulas for R 1 and R 2 by solving Eqs. (7) and (8) simultaneously.

33. 5- Symmetrical T Design Formulas: If this is done, we obtain: (9) (10) These are very useful formulas for designing T attenuators. For instance, if the characteristic resistance is to be 50 ohms, and if the attenuation is to be 10,

34. 5- Symmetrical T Design Formulas: we would obtain From Eqs. (9) and (10) it should be clear that R 1 and R 2 are func­tions of a and R o. Note especially that R 1 and R 2 are directly proportional to R o. If we like, we can generate a table of design values for future reference. In order to simplify the table we may choose R o equal to 50 ohms.

35. 5- Symmetrical T Design Formulas: With R o equal to 50 ohms, R 1 and R 2 become functions of a only. To make a table we can select values of a and calculate the corresponding values of R 1 and R 2, as shown in Table 1. Note that for any characteristic resistance not equal to 50 ohms, we need only multiply each value for R 1 and R 2 by R o /50. For instance, if we need a 500-ohm 20-db attenuator, according to our table, the value of R 1 and R 2 should be 40.9 and 10.1 ohms, respectively, for a 50-ohm attenuator.

36. 5- Symmetrical T Design Formulas: For a 500-ohm attenuator we need only multiply each R 1 and R 2 value by, or 10. Thus, we obtain R 1 = 409 and R 2 = 101 ohms for a 20-db 500-ohm attenuator.

37. EXAMPLE 7 : Design a 12-db 50-ohm T attenuator. SOLUTION: From Table 1 we find R 1 = 29.9 and R 2 = 26.8 ohms. Since the attenuator resistance is to be 50 ohms, these values are used as they are.

38. Table 7-1 Symmetrical T Design, R o = 50 D db R1R1 R2R2 12.88433 25.73215 38.55142 411.3105 616.666.9 821.547.3 102635.1 1229.926.8 1636.316.3 2040.910.1 2444.16.34 2846.33.99 3046.93.17 3548.31.78 40491.00

39. EXAMPLE 8: Design a 20-db 600-ohm T attenuator. SOLUTION: From Table 1 we find R 1 = 40.9 and R 2 = 10.1 ohms. Since the characteristic resistance is to be 600 ohms, we must multiply R 1 and R 2 by, or 12. If we do this, we obtain R 1 = 490 and R 2 = 121 ohms.

40. 6- Cascading T Sections: Up to this time we have dealt only in basic T attenuators, that is, a single T composed of three resistors. Basic T sections may be cascaded, as shown in Fig. 9. Fig. 9 Cascading symmetrical T sections.

41. 6- Cascading T Sections: Note that the resistance level is preserved as we move from load to source. That is, starting from the load end, we see that the input resistance of the second attenuator is R o. This means that the first attenuator is also loaded correctly in its characteristic resistance. As a result, the input resistance of the first attenuator is also equal to R o. This useful property of attenuators allows us to cascade as many attenu­ators as we like. The R o resistance is preserved throughout the entire sys­ tem, with the result that a perfect impedance match occurs at all input and output terminals.

42. 6- Cascading T Sections: There is a good reason for wanting to cascade attenuator sections. From Table 1 note that as we approach higher values of attenuation, R 2 becomes very small. Because of this, values of attenuation above 40 db will require R 2 values that are impracticably small. Therefore, in the construction of, say, a 90-db attenuator, we can cascade three 30-db sec­tions.

43. 6- Cascading T Sections: For instance, if we wish a 90-db 50-ohm attenuator, we obtain R 1 = 46.9 ohms and R 2 = 3.17 ohms for each 30-db section. After cascading the 30-db sections, we have the circuit shown in Fig. 10a. The 46.9-ohm resistances can be lumped into a single 93.3-ohm resistor, Fig. 10 (a) Three-section T attenuator. (b) Combining resistances.

44. 6- Cascading T Sections: as shown in Fig. 10b. This circuit has characteristic resistance of 50 ohms and an attenuation of 90 db. EXAMPLE 9: Design a 50-db 500-ohm attenuator using three basic T section.

45. 6- Cascading T Sections: SOLUTION Arbitrarily, we can choose to use two 20-db sections and one 10-db section. For the 20-db section we obtain from Table 1 R 1 = 40.9andR 2 = 10.1 ohms for R o = 50 ohms or R 1 = 409 andR 2 = 101 ohm for R o = 500 ohms

46. For the 10-db section we obtain R 1 = 26andR 2 = 35.1 ohmsfor R o = 50 ohms or R 1 = 260andR 2 = 351 ohmsfor R o = 500 ohms The final design is shown in Fig. 11 Fig. 11 Example 9.

47. 7- The Symmetrical Attenuator The section of Fig. 12a is as basic as the T section. Occasionally, the section may be preferred to the T section. By defining m as the ratiodefining Fig. 12

48. of R 2 to R 1, we can reliable the values as shown in Fig. 12b. Proceeding as we did for the T section, we can find the following formulas for analysis and design: Analysis (11) (12)

49. Design (13) As with the T section, we can generate a table for future reference. Table 2 gives the design values of R 1 and R 2 for a characteristic resist­ance of 50 ohms. For any other value of R o, we need only multiply R 1 and R 2 by R o /50.

50. Design, R o = 50 D db R1R1 R2R2 15.77870 211.6436 317.6292 423.8221 637.4150 852.8116 1071.296.2 1293.283.5 1615468.8 2024861.1 2439556.7 3079053.3 40250051 Table 2 Symmetrical

51. EXAMBLE 10: Design a 20-db 300-ohm attenuator. SOLUTION : From Table 2 we see that R 1 = 2-18 and R 2 = 61.1 ohms for an R o of 50 ohms. For an R o of 300 ohms we must multiply by 300/50, or 6. Hence, R 1 = 1488 and R 2 = 366.6 ohms.

52. 7-The Bridged T Attenuator The last specific type of attenuator we wish to study is the bridged T attenuator. The circuit for a bridged T is shown in Fig. 13. The reason Fig. 13 Bridged T Attenuator

53. for the name bridged T should be clear. To an ordinary T attenuator is added a resistor that bridges from the input to the output. The analysis and design formulas for this important attenuator are as follows: Analysis (15) (16)

54. Design (17) (18) Note carefully that tote design formulas indicate that R 1 is not a function of a. Only R 2 and R 3 depend upon a. This is a very important property, as will be discussed in the next section.

55. EXAMPLE 11 Design a 20-db 50-ohm bridged T attenuator. SOLUTION R 1 = 50 ohm

56. 9 -Variable Attenuators Commercial instrument attenuators are often of the variable type to allow selection of different values of attenuation, while maintaining the resistance at the value of R o. One way of building a continuously variable attenuator is by using ganged rheostats for the resistors in the attenuator. In either the T or the attenuator all three resistances must be varied in accordance with the design equations. It is actually much easier to use a bridged T attenuator. Recall that the R 1 value does not depend upon the amount of attenuation. R 1 may be fixed at the value of characteristic resistance that is desired.

57. 9 -Variable Attenuators We need only vary R 2 and R 3, as shown in Fig. 14. Note that R 2 and R 3 are ganged together. In order to work properly, Fig. 14 Continuously variable attenuator

58. 9 -Variable Attenuators these rheostats must track according to Eqs. (17) and (18). In these equations we note that R 2 is inversely proportional to a, whereas R 3 is directly proportional to a. A study of these equations shows that linear rheostats will not track properly. However, logarithmic rheostats can be ganged together, so that Eqs. (17) and (18) are satisfied. The result is then a continuously variable attenuator whose characteristic resistance remains constant.

59. 9 -Variable Attenuators Of course, the tracking of R 2 and R 3 cannot be made perfect. Some deviation in the value of R o is to be expected. However, the variable bridged T attenuator provides a reasonably accurate value of R o over its range of adjustment. If more precise values of R o and a are required, the usual procedure is to build a step attenuator, which is an attenuator varied in discrete steps. For instance, a step attenuator might be designed to cover from 0 to 10 db in steps of 1 db. We would then be able to select 0, 1, 2, 3, or any whole number of decibels up to 10 db.

60. 9 -Variable Attenuators One of the most popular ways of building a step attenuator is by building a number of simple T attenuators with different values of attenu­ation. By means of a proper switching arrangement, various combinations can be cascaded to produce the desired amount of attenuation. For example, suppose we desire to build a 0- to 10-db attenuator with 1-db steps. We could build four basic attenuators with values of 1, 2, 3, and 4 db.

61. 9 -Variable Attenuators There are several switching arrangements that can be used to switch in any attenuation from 1 to 10 db, simply by using combinations of the four basic attenuators. To obtain 6 db, for example, the 2- and 4-db attenuators must be cascaded. To obtain 9 db, we must connect the 2-, 3-, and 4-db attenuators (see Prob. 16).

62. 9 -Variable Attenuators A 100-db attenuator with 1-db steps can be made by using two step attenuators in cascade. The first unit would be a 0- to 10-db step attenu­ator with 1-db steps. In cascade with this, we would use a 0- to 90-db at­tenuator with 10-db steps. Thus, any whole number of decibels between 0 and 100 db can be selected.