OP AMP NOISE FACTOR CALCULATIONS

Transcription

1 Practical RF System Design. William F. Egan Copyright 2003 John Wiley & Sons, Inc. ISBN: APPENDIX A OP AMP NOISE FACTOR CALCULATIONS This appendix details the effects of certain changes in the representation of the circuit discussed in the Example 3.8 (Section 3.12) and shown in Fig Results are discussed in Section A.1 INVARIANCE WHEN INPUT RESISTOR IS REDISTRIBUTED The cascade is not changed if we consider part of the input resistor to op amp 2 or 3 to be part of the output resistance of the previous stage. This is just a matter of redrawing the boundaries between stages. Therefore, the cascade noise figure should be unaffected. To verify this, Fig. A.1 is another spreadsheet representing Fig except that 1 k of the input resistors of op amps 1 and 2 are moved to the previous stages. The last three stages, so considered, are shown in Fig. A.2. The output resistances R 22 (k-) seen by the last two stages now increase by 1 k, and the noise figure of each stage is changed because of the new values of R 22 (k-) and of gain and of the resistor noises that change with the resistor configuration. The change in noise factor for Op Amp 1 is fairly easily computed, being due only to the addition of the noise of a 1-k output resistor. Section A.3 gives a model for the other two op amps that is used to compute their noise factors before and after the change. Cells G48 H50 in Fig. A.1 contain the ratios of the cascade noise factors in Fig. A.1 to those in Fig and the overall noise factor can be seen to be the same, even though the parameters of the various amplifiers change considerably. (Compare cells F13 F15 on the two spreadsheets.) 273

2 274 APPENDIX A OP AMP NOISE FACTOR CALCULATIONS A B C D E F G H 12 Filter 7.0 db 0.3 db R 0 R 22k 1/g c k a k 13 Op Amp db 2000 Ω 2000 Ω db Op Amp db 2000 Ω 1020 Ω db Op Amp db 2000 Ω 1020 Ω db DERIVED (italics above are derived also) 17 NF using mean NFs (see Note*) 18 mean max min ± mean G max G min G 28 Op Amp db db db 0.00 db 6.53 db 6.53 db 6.53 db 29 Op Amp db 0.09 db 0.09 db 0.00 db 8.52 db 8.52 db 8.52 db 30 Op Amp db db db 0.00 db 6.30 db 6.30 db 6.30 db at output of mean max min ± mean G max G min G 42 Filter 7.43 db db 0.09 db 7.34 db 3.09 db 2.47 db db 43 Op Amp db db db 7.34 db 4.27 db 2.74 db db 44 Op Amp db db db 7.34 db 4.38 db 2.77 db db 45 Op Amp db db db 7.34 db 4.44 db 2.79 db db CUMULATIVE NF using mean NFs 48 Op Amp db 49 Op Amp db 50 Op Amp db 51 *Note: Cable NF depends on SWR, which is assumed to be fixed. Fig. A.1 Alternate spreadsheet for Fig Here each of the last two amplifiers is partitioned in the middle of its input resistor. Missing lines are identical to those with the same number in Fig Op Amp 1 Op Amp 2 Op Amp 3 2 kω 3 kω 1 kω 1 kω 2 kω 1 kω 1 kω 20 kω 1 kω Fig. A.2 Last stages with resistors reassigned. A.2 EFFECT OF CHANGE IN SOURCE RESISTANCES We have used 20 as the output resistance of the op amps. What is the effect of an inaccurate value for this output resistance? From Eq. (3.73), we find that the contribution of each stage to total noise factor is proportional to R 22(k-). However, Eq. (3.66) shows that f for each stage is inversely proportional to R 22(k-) so the dependencies cancel except for the 1 in Eq. (3.73). This represents subtraction of the noise attributed to the source, so it will not appear both as part of the noise of the preceding stage and of the source

3 EFFECT OF CHANGE IN SOURCE RESISTANCES 275 for the stage in question. Thus an error in the value of R 22 (k-) causes some error in overall noise factor due to the noise difference between the correct and erroneous values at that point in the circuit. In addition, R 22 (k-) has some influence on the preceding gain through c k, but these values will often be close to unity for circuits with output impedance that are small compared to the driven impedances. Figure A.3 is like Fig except the output resistances R 22(k-) of the last three op amps are changed from 20 to 40, and noise factors have been recomputed for those values. We can see, from cell H58, that this 2:1 change in assumed output impedance causes only db change in overall noise figure in this particular cascade, even though the noise figure of the last two amplifiers have changed nearly 3 db. Figure A.4 shows that a change of 10% in source impedance for the first op amp (cell E13) results in only a 0.03-dB change in overall noise figure for this example. A B C D E F G H 12 Filter 7.0 db 0.3 db R 0 R 22k- 1/g Ck Ak 13 Op Amp db 2000 Ω 2000 Ω db Op Amp db 2000 Ω 40 Ω db Op Amp db 2000 Ω 40 Ω db Op Amp 1 1/g db 17 Op Amp 2 changes: db 18 Op Amp db mean max min ± mean G max G min G 30 Filter 7.00 db 6.70 db 7.30 db 0.30 db 7.00 db 6.70 db 7.30 db 31 Op Amp db db db 0.00 db 6.50 db 6.50 db db 32 Op Amp db 0.17 db 0.17 db 0.00 db 8.57 db 8.57 db db 33 Op Amp db db db 0.00 db 6.82 db 6.82 db db Op Amp db 37 Op Amp db 38 Op Amp db at output of mean max min ± mean G max G min G 50 Filter 7.43 db db 0.09 db 7.34 db 3.09 db 2.47 db db 51 Op Amp db db db 7.34 db 4.26 db 2.74 db db 52 Op Amp db db db 7.34 db 4.37 db 2.77 db db 53 Op Amp db db db 7.34 db 4.44 db 2.79 db db DERIVED (italics above are derived also) NF using mean NFs (see Note*) CUMULATIVE NF using mean NFs 56 Op Amp db 57 Op Amp db 58 Op Amp db 59 *Note: Cable NF depends on SWR, which is assumed to be fixed. Fig. A.3 Effect of assumed output resistance. Modification of Fig where output resistances of Op Amps 1 through 3 change from 20 to 40. Missing lines are as in Fig

4 276 APPENDIX A OP AMP NOISE FACTOR CALCULATIONS A B C D E F G H 12 Filter 7.0 db 0.3 db R 0 R 22k- 1/g Ck Ak 13 Op Amp db 2200 Ω 2200 Ω db Op Amp db 2200 Ω 20 Ω db Op Amp db 2200 Ω 20 Ω db mean max min ± mean G max G min G 27 Filter 7.00 db 6.70 db 7.30 db 0.30 db 7.00 db 6.70 db 7.30 db 28 Op Amp db db db 0.00 db 6.50 db 6.50 db db 29 Op Amp db 0.09 db 0.09 db 0.00 db 8.17 db 8.17 db db 30 Op Amp db db db 0.00 db 6.45 db 6.45 db db Op Amp db 34 Op Amp db 35 Op Amp db at output of mean max CUMULATIVE min ± mean G NF using mean NFs max G min G 47 Filter 7.43 db db 0.09 db 7.34 db 3.09 db 2.47 db db 48 Op Amp db db db 7.34 db 4.26 db 2.74 db db 49 Op Amp db db db 7.34 db 4.36 db 2.77 db db 50 Op Amp db db db 7.34 db 4.42 db 2.78 db db DERIVED (italics above are derived also) NF using mean NFs (see Note*) db db db 56 *Note: Cable NF depends on SWR, which is assumed to be fixed. Fig. A.4 Effect of 10% change in cascade source resistance. Modification of Fig Missing lines are as in Fig A.3 MODEL Refer to Fig. A.5. The mean-square equivalent input noise voltage due to the resistors is [ ( ) ] er 2 = 4kT Rs R 2 in 0B R s R in (R f R out ), (1) where the last factor is the reciprocal of the op-amp gain (squared) and refers the output noise to the input. The term R out is the output resistance of the op amp, as reduced by the gain of its feedback loop. Its value will be frequency dependent because the op-amp gain is frequency dependent. The mean-square input voltage that produces the same current through R f as does i n is e 2 i = i 2 n (R s R in ) 2. (2) R f

5 MODEL 277 R f R s R in i n v n R out Fig. A.5 Op amp with noise sources. The mean-square input voltage that produces the same output as does v n is ev 2 = (1 a) 2 v2 n a 2 = v 2 n ( 1 1 a ) 2, (3) where a is the voltage gain of the op amp, R f /(R s R in ).Herev n multiplied by the gain from the noninverting input (i.e., 1 a) equals e n multiplied by a, the gain from the inverting input. Adding these three equivalent mean-square input voltages and dividing by the voltage equivalent to the available power from the source resistor, we obtain the TABLE A.1 Op Amp Noise Factors for Various Parameter Values A B C D E F 2 Nt E-21 W 3 Rs 1,020 1, Rin 1,000 1,000 2,000 2,000 5 Rf 20,000 2,000 20,000 2,000 6 a In 4.00E-12 A 8 Vn 4.00E-09 V 9 er term from R s (less Rout) ev term From Vn ei term From In f without Rout Rout f with Rout NF db db db db Rout f with Rout NF db db db db

6 278 APPENDIX A OP AMP NOISE FACTOR CALCULATIONS noise factor (Steffes, 1998): f = e2 R e2 i e2 v 4kT 0 BR s (4) ( = 1 R )( in 1 R s R in R ) out R s R in e2 i e2 v R s R f R f R f 4kT 0 BR s (5) ( = 1 R in R s )( 1 1 a R out R f a ) i2 n (R s R in ) 2 vn 2 (1 1/a)2. (6) 4kT 0 BR s By adding the mean-square voltage due to the noise current and the noise voltage sources, we are assuming their independence. (We can do that for an example, but the issue can be important in practice where correlation may have to be taken into account.) The noise factor for various values of these variables is shown in Table A.1. Note, from Eq. (6) and from the table, how high gain improves the noise figure. It would seem to be better to get all the gain in one op amp, rather than two as is done in Fig There could be other requirements, however, such as a specific gain required at an intermediate output after Op Amp 2, or wide bandwidth (which is adversely affected by high closed-loop gain in a classic op amp), or the desire to study the effects of changes in Op Amp 3 in an example.