Practical RF System Design

7 and σ 2Fcas5 = 10−Fcas5/5 dB   10Fcas3/5 dBσ 2F cas3 +10 ( F5− 4∑ j=1 Gj )/ 5 dB σ 2F5   , (10) where Fcasj is F for the cascade through stage j . Equation (10) is the same as Eq. (8). We can also write this as σ 2Fcas,n = 10−Fcas,n/5 dB   10Fcas(n−1)/5 dBσ 2F cas(n−1) +10 ( Fcas,n− n−1∑ j=1 Gj )/ 5 dB σ 2Fn   , (3.31) depending upon σ 2Fn being zero for all interconnects. Then every other application of Eq. (3.31), those that apply to interconnects, will produce a change in cascade noise figure variance only because of the change in cascade noise figure. APPENDIX X CROSSOVER SPURS Table X.1 is a list of all crossover spurs for the range of m and n shown in its upper left. The ratios of RF to IF and of RF to LO, which apply to the two types of normalizations that we have considered, are listed for the three desired-signal TABLE X.1 Crossover Spurs desired m(LO) = 1 desired n(RF) = −1 RF/IF m n RF/LO 0.2 0 5 0.1666667 0.25 0 4 0.2 0.3333333 0 3 0.25 0.3333333 2 −5 0.25 0.5 −1 5 0.3333333 0.5 0 2 0.3333333 0.5 2 −4 0.3333333 0.6666667 −1 4 0.4 1 −2 5 0.5 1 −1 3 0.5 1 0 1 0.5 1 2 −3 0.5 1 3 −5 0.5 1.5 −2 4 0.6 2 −3 5 0.6666667 2 −1 2 0.6666667 2 3 −4 0.6666667 3 −2 3 0.75 3 4 −5 0.75 4 −3 4 0.8 5 −4 5 0.8333333 spur set max n = 5 giving max m 10 10 359 Practical RF System Design. William F. Egan Copyright  2003 John Wiley & Sons, Inc. ISBN: 0-471-20023-9 360 APPENDIX X CROSSOVER SPURS TABLE X.1 Crossover Spurs (continued ) desired m(LO) = 1 desired m(LO) = −1 desired n(RF) = 1 desired n(RF) = 1 RF/IF m n RF/LO RF/IF m n RF/LO 0.1428571 2 −5 0.1666667 1.1 10 0 11 0.1666667 2 −4 0.2 1.1111111 9 0 10 0.2 0 5 0.25 1.125 −10 2 9 0.2 2 −3 0.25 1.125 8 0 9 0.25 0 4 0.3333333 1.1428571 −9 2 8 0.25 2 −2 0.3333333 1.1428571 7 0 8 0.25 3 −5 0.3333333 1.1666667 −8 2 7 0.2857143 3 −4 0.4 1.1666667 6 0 7 0.3333333 −1 5 0.5 1.2 −7 2 6 0.3333333 0 3 0.5 1.2 5 0 6 0.3333333 2 −1 0.5 1.2222222 10 −1 5.5 0.3333333 3 −3 0.5 1.25 −6 2 5 0.3333333 4 −5 0.5 1.25 4 0 5 0.375 4 −4 0.6 1.25 9 −1 5 0.4 −1 4 0.6666667 1.2857143 −10 3 4.5 0.4 3 −2 0.6666667 1.2857143 8 −1 4.5 0.4 5 −5 0.6666667 1.3333333 −9 3 4 0.4285714 −2 5 0.75 1.3333333 −5 2 4 0.4285714 4 −3 0.75 1.3333333 3 0 4 0.4444444 5 −4 0.8 1.3333333 7 −1 4 0.4545455 6 −5 0.8333333 1.375 10 −2 3.6666667 0.5 −3 5 1 1.4 −8 3 3.5 0.5 −2 4 1 1.4 6 −1 3.5 0.5 −1 3 1 1.4285714 9 −2 3.3333333 0.5 0 2 1 1.5 −10 4 3 0.5 2 0 1 1.5 −7 3 3 0.5 3 −1 1 1.5 −4 2 3 0.5 4 −2 1 1.5 2 0 3 0.5 5 −3 1 1.5 5 −1 3 0.5 6 −4 1 1.5 8 −2 3 0.5 7 −5 1 1.5714286 10 −3 2.75 0.5384615 8 −5 1.1666667 1.6 −9 4 2.6666667 0.5454545 7 −4 1.2 1.6 7 −2 2.6666667 0.5555556 −4 5 1.25 1.6666667 −6 3 2.5 0.5555556 6 −3 1.25 1.6666667 4 −1 2.5 0.5714286 −3 4 1.3333333 1.6666667 9 −3 2.5 0.5714286 5 −2 1.3333333 1.75 −8 4 2.3333333 0.5714286 9 −5 1.3333333 1.75 6 −2 2.3333333 0.5833333 8 −4 1.4 1.8 −10 5 2.25 0.6 −5 5 1.5 1.8 8 −3 2.25 0.6 −2 3 1.5 1.8333333 10 −4 2.2 0.6 4 −1 1.5 2 −9 5 2 0.6 7 −3 1.5 2 −7 4 2 0.6 10 −5 1.5 2 −5 3 2 0.6153846 9 −4 1.6 2 −3 2 2 0.625 −4 4 1.6666667 2 1 0 2 0.625 6 −2 1.6666667 2 3 −1 2 0.6363636 −6 5 1.75 2 5 −2 2 0.6363636 8 −3 1.75 2 7 −3 2 0.6428571 10 −4 1.8 2 9 −4 2 0.6666667 −7 5 2 2.2 10 −5 1.8333333 spur spur 361 TABLE X.1 Crossover Spurs (continued ) desired m(LO) = 1 desired m(LO) = −1 desired n(RF) = 1 desired n(RF) = 1 RF/IF m n RF/LO RF/IF m n RF/LO spur spur 0.6666667 −7 5 2 2.2 10 −5 1.8333333 0.6666667 −5 4 2 2.25 8 −4 1.8 0.6666667 −3 3 2 2.3333333 −8 5 1.75 0.6666667 −1 2 2 2.3333333 6 −3 1.75 0.6666667 3 0 2 2.5 −6 4 1.6666667 0.6666667 5 −1 2 2.5 4 −2 1.6666667 0.6666667 7 −2 2 2.5 9 −5 1.6666667 0.6666667 9 −3 2 2.6666667 7 −4 1.6 0.6923077 −8 5 2.25 3 −7 5 1.5 0.6923077 10 −3 2.25 3 −4 3 1.5 0.7 −6 4 2.3333333 3 2 −1 1.5 0.7 8 −2 2.3333333 3 5 −3 1.5 0.7142857 −9 5 2.5 3 8 −5 1.5 0.7142857 −4 3 2.5 3.5 6 −4 1.4 0.7142857 6 −1 2.5 4 −5 4 1.3333333 0.7272727 −7 4 2.6666667 4 3 −2 1.3333333 0.7272727 9 −2 2.6666667 4 7 −5 1.3333333 0.7333333 −10 5 2.75 5 −6 5 1.25 0.75 −8 4 3 5 4 −3 1.25 0.75 −5 3 3 6 5 −4 1.2 0.75 −2 2 3 7 6 −5 1.1666667 0.75 4 0 3 0.75 7 −1 3 0.75 10 −2 3 0.7692308 −9 4 3.3333333 0.7777778 −6 3 3.5 0.7777778 8 −1 3.5 0.7857143 −10 4 3.6666667 0.8 −7 3 4 0.8 −3 2 4 0.8 5 0 4 0.8 9 −1 4 0.8181818 −8 3 4.5 0.8181818 10 −1 4.5 0.8333333 −9 3 5 0.8333333 −4 2 5 0.8333333 6 0 5 0.8461538 −10 3 5.5 0.8571429 −5 2 6 0.8571429 7 0 6 0.875 −6 2 7 0.875 8 0 7 0.8888889 −7 2 8 0.8888889 9 0 8 0.9 −8 2 9 0.9 10 0 9 0.9090909 −9 2 10 0.9166667 −10 2 11 curves, with m and n (±1) for them shown at the top. The spurs have been sorted from lowest to highest frequency ratios or vise versa. Crossovers at zero or infinity are not shown. Table X.1 has three major divisions, according to the sign of m and n that applies to the desired response. Within the divisions for the 1 × 1 or 1 × −1 362 APPENDIX X CROSSOVER SPURS desired responses, the values of m and n for the crossover spurs, on any given line, are the same for RF/IF as for RF/LO. That is because, as the RF increases along these curves (Fig. 7.28), both RF/IF and RF/LO increase. Therefore, the sequence in which they cross spurs is the same. However, within the third segment, that for the −1 × 1 desired response, RF/IF increases with increasing RF but RF/LO decreases. For this reason, the ratio RF/LO has been sorted by decreasing value in this last segment so that both ratios on a line refer to the same m and n values. APPENDIX Z NONSTANDARD MODULES Here we treat unilateral modules that are specified by their input and output impedances and by their transducer gains or their maximum available gains (Appendix G). Figure Z.1 shows two such modules, each represented by its input and output impedances (Z11) and (Z22) and a voltage generator that depends on the voltage across the input resistance (thus on the square root of the input power). We will see how to compute the gain of a cascade of such modules, using a spreadsheet as an aid, and how to find the S parameters for such modules and cascades. Z.1 GAIN OF CASCADE OF MODULES RELATIVE TO TESTED GAIN What is the gain of a cascade of modules that interfaces with impedances that are different than those used in obtaining their gains when they were tested with matched loads (maximum available gain), assuming negligible reverse transmis- sion (Z12, S12 = 0)? The ratio of the voltage across the real part of a driven load to the voltage across the real part of the module’s input is1 aj
= e(j+1) ej = a′j R11(j+1) Z22j + Z11(j+1) . (1) The voltage at the cascade source is e1. The load Z11(j+1) is the input to the next stage except that Z11(N+1) is the load for a cascade of N modules. 363 Practical RF System Design. William F. Egan Copyright  2003 John Wiley & Sons, Inc. ISBN: 0-471-20023-9 364 APPENDIX Z NONSTANDARD MODULES vj jX11j jX22j jX11j +1 jX22j +1R22j +1 R11j +1 Z11j +1 Z11j R22j Z22j R11j ej a ′j ej a ′j+1 ej +1 ej+1 vj +1 Fig. Z.1 The j th module in the cascade. During module characterization (test), when the load is matched to (i.e., the complex conjugate of) the module’s output impedance, Eq. (1) becomes aTj
= eT (j+1) eTj = a′j R22j R22j + R22j = a′j 2 . (2) (The imaginary parts in the denominator cancel since the impedances are conjugate.) Since the gain is defined with respect to the voltage across R11, the input match during test is not significant here. From Eq. (2) we can obtain the internal parameter a′ in terms of the tested transfer function aT . Substituting a′ from Eq. (2) into Eq. (1), we obtain the transfer function in the cascade relative to the tested transfer function for the module: aj = aTj 2R11(j+1) Z22j + Z11(j+1) . (3) The voltage transfer function for a cascade of N modules is then acas =   N∏ j=1 aTj     N∏ j=1 2R11(j+1) Z22j + Z11(j+1)   . (4) Note that the first product is the transfer function the cascade would have if not for the differing impedances between test and use, and the second product is the modification due to the differing impedances. The (actual) power gain gact,j is the ratio of power absorbed in the load (R11,j+1) to the power dissipated by the module input resistance (R11,j ). Due to the impedance matches at input and output, the power gain during test is the maximum available power gain gmaT : gmaTj = ∣∣∣∣∣ e2T (j+1)/R22j e2Tj /R11j ∣∣∣∣∣ = |amaTj |2 R11j R22j , (5) GAIN OF CASCADE OF MODULES RELATIVE TO TESTED GAIN 365 but the power gain of a module in the cascade is gact,j = ∣∣∣∣∣ e2j+1/R11(j+1) e2j /R11j ∣∣∣∣∣ (6) = |aj |2 R11j R11(j+1) = |amaTj |2 4R211(j+1) |Z22j + Z11(j+1)|2 R11j R11(j+1) (7) = gmaTj R22j R11j 4R211(j+1) |Z22j + Z11(j+1)|2 R11j R11(j+1) = gmaTj 4R22jR11(j+1)|Z22j + Z11(j+1)|2 . (8) Here Eqs. (3) and (5) have been used. From this we obtain the actual power gain (Appendix G) for a cascade of N modules: gact,cas = N∏ j=1 gact,j =   N∏ j=1 gmaTj     N∏ j=1 4R22jR11(j+1) |Z22j + Z11(j+1)|2   . (9) Here the subscript N + 1 refers to the load and gmaTj is the maximum available power gain measured when the module was characterized with a matched source and load. The actual power gain is the ratio of power delivered to the load to the power absorbed at the input of the cascade. Example Z.1 Cascade Gain, Nonstandard Modules Figure Z.2 shows a spreadsheet that executes Eq. (9). The input and output impedances and the power gain in test (gmaT i) are listed for each module in lines 5–8 with the cascade’s load A B C D E F 1 Maximum 2 Available 3 Power Gain 4 in test R11 X11 R22 X22 5 Module 1 6.00 dB 200.00 Ω 100.00 Ω 300.00 Ω 150.00 Ω 6 Module 2 9.00 dB 1500.00 Ω −250.00 Ω 1200.00 Ω −200.00 Ω 7 Module 3 4.50 dB 1000.00 Ω 200.00 Ω 500.00 Ω 200.00 Ω 8 Module 4 22.00 dB 250.00 Ω 45.00 Ω 55.00 Ω 10.00 Ω 9 Load 100.00 Ω 25.00 Ω 10 11 Module CUMULATIVE 12 Module 1 3.41 dB 3.41 dB 13 Module 2 8.94 dB 12.36 dB 14 Module 3 3.53 dB 15.88 dB 15 Module 4 21.38 dB 37.27 dB actual gain in use Fig. Z.2 Spreadsheet for cascade of nonstandard modules. 366 APPENDIX Z NONSTANDARD MODULES given in line 9. The actual gain of each module in the cascade [Eq. (8)] is given in cells B12–B15 with the cumulative gain in cells C12–C15. Note that the sum of the tested gains is 41.5 dB, whereas the cascade gain is only 37.27 dB. Z.2 FINDING MAXIMUM AVAILABLE GAIN OF A MODULE We can obtain the value of gmaTj from test data that was not obtained with matched source and load but rather in a transducer-gain test, with a signal gen- erator and power meter (Fig. Z.3). Using Eqs. (6) and (8), we write gmaTj = ∣∣∣∣∣ e2j+1/R11(j+1) e2j /R11j ∣∣∣∣∣ |Z22j + Z11(j+1)|2 4R22jR11(j+1) . (10) Here, Z11(j+1), including R11(j+1), is the test load. Assume that the power meter presents a real impedance equal to that of the connecting cable. Then Z11(j+1) = R11(j+1). We recognize the first ratio as the ratio of power absorbed in the meter (assuming negligible cable loss) to that absorbed in the module, so we can write gmaTj = po,j+1 po,j − pi,j |Z22j + R11(j+1)|2 4R22jR11(j+1) = po,j+1 po,j − pi,j |1 + Z22j /R11(j+1)|2 4R22j /R11(j+1) . (11) If we do not have a value for pi,j , we can relate it to po,j and to the transducer gain of the module, gtj , by gmaTj = po,j+1 po,j (1 − |S11j |2) |1 + Z22j /R11(j+1)|2 4R22j /R11(j+1) (12) = gtj |1 + Z11j /R0j | 2 4R11j /R0j |1 + Z22j /R11(j+1)|2 4R22j /R11(j+1) . (13) Cable impedance = R22j +1Measure forward and reverse power poj pij Z11j R22j R11j+1 Z22j R11j jX11j jX22j ej a ′j ej Directional coupler Power meter poj +1 ej+1 Fig. Z.3 Testing module for available gain. EQUIVALENT S PARAMETERS 367 Here R0j is the characteristic impedance of the input cable (at port j ) during test, and S11j can be obtained in terms of impedances from Eq. (14) below. Z.3 INTERCONNECTS Interconnect impedances may be included as part of the input impedance Z11,j+1 of the following module or the output impedance Z22j of the preceding module. When a true transmission line is used, Section F.4 may be helpful in translating the input impedance of the following module to a value at the output of the preceding module. If the line is made part of the following module and is lossless, whatever power is absorbed into the combined input structure must be absorbed in R11,j+1 of the following module. This power is part of the power gain equation [the numerator in Eq. (6)]. For use with voltage gain [Eq. (1)], e2j+1 can be obtained from the power by multiplying the power by R11,j+1. Z.4 EQUIVALENT S PARAMETERS Here we will consider how to convert the description of a module in terms of nonstandard impedances into a description using S parameters for standard impedances. If the interfaces in a cascade are matched to various resistive val- ues, the modules on either side of an interface being each matched to the same resistance with specified deviation therefrom, such a cascade can be treated as a standard cascade. The variation of standard impedance (e.g., 75 , 120 , etc.) from interface to interface does not invalidate that method. However, when we convert to an S-parameter description at an impedance significantly different than the actual interface impedance and apply the methods of Section 2.3, we may be throwing away significant information and, as a result, generating unnecessarily large uncertainties in overall performance. In other words, if the impedances of modules in a cascade are known in detail, rather than by their allowed devia- tion from a standard impedance, converting them to the latter type of description throws away useful information. It may be better to compute cascade gain as in Section Z.1 and then possibly describe the overall cascade by S parameters, as we will do here. Figure Z.4 shows the module as it is during characterization. (Note that only vin,j , vout,j , and vout,j+1 are normalized variables here.) From this figure we can see (Section F.2) S11j = Z11j − R0j Z11j + R0j , (14) S22j = Z22j − R0,j+1 Z22j + R0,j+1 , (15) S21j = vout,(j+1) vin,j = v˜out,(j+1) v˜in,j √ R0j R0,j+1 , (16) 368 APPENDIX Z NONSTANDARD MODULES vin, j vout, j+1 vj ej vout, j jX11j Z11j R22j Z22j R11j jX22j a'j ej R0 Fig. Z.4 The j th module in test. = R0,j+1 Z22j + R0,j+1 a ′ j ej v˜in,j √ R0j R0,j+1 (17) S12j = 0. (18) To put S21j in a usable form, we write ej in terms of the waves used in defining S parameters by observing Z11j R11j ej = v˜in,j + v˜out,j = v˜in,j (1 + S11j ). (19) We then substitute ej from (19) and a′j from (2) into (17) to obtain S21j = √ R0jR0,j+1 Z22j + R0,j+1 2aTj R11j Z11j (1 + S11j ). (20) Then, substituting for S11j from Eq. (14), we obtain S21j = 4aTj √ R0jR0,j+1R11j (Z11j + R0j )(Z22j + R0,j+1) . (21) This expression, along with those of Eqs. (14), (15), and (18), allow S parameters to be written in terms of nonstandard-module parameters. The second page of the workbook containing Fig. Z.2 shows how these con- versions can be made using a spreadsheet. The spreadsheet is written for the usual case where R0j = R0,j+1 = R0. It uses the module parameters from the Fig. Z.2 spreadsheet and only the phase of aTj must be added. Z.5 S PARAMETERS FOR CASCADE OF NONSTANDARD MODULES We determine the S parameters for a cascade of nonstandard modules so we can use that cascade as an element in a cascade with standard modules having ENDNOTE 369 R0 interfaces. Because the modules are unilateral, the input impedance of the cascade is that of the first module from Eq. (14): S11 = Z111 − R0 Z111 + R0 , (22) and the output impedance is that of the last module in the cascade, S22 = Z22N − R0 Z22N + R0 , (23) where the last subscript refers to the number of the module in the cascade of nonstandard modules. The forward transfer ratio is (Fig. Z.1) S21 = vo(N+1) vo1 = v1 vo1 e1 v1 vo(N+1)T e1 (24) = (1 + S111)R111 Z111 acas|Z11(N+1)=R0 (25) = 2 R111 Z111 + R0 acas|Z11(N+1)=R0 , (26) where aN |Z11(N+1)=R0 is given by Eq. (4) with Z11(N+1) = R11(N+1) = R0 (i.e., with the nonstandard cascade properly terminated). [Equation (22) was used in obtaining Eq. (26)]. Due to our assumption that S12j ≈ 0 for these modules, S12 = 0 for the cas- cade also. Thus the cascade meets the unilaterality requirement for modules in Section 2.3. Note, from Eqs. (22) and (23), that it is easy to determine variations in the reflection coefficients of the cascade from the variations in the individual mod- ules. Also, from Eqs. (26) and (4), the effect of variations in aTj on S21 can be easily determined. Sensitivity analysis may be helpful in determining the effects of the various impedances in Eq. (4) on overall gain if that becomes important. 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INDEX 180◦ hybrid, 152 90◦ hybrid, 150 ABCD parameters, 45 acceptance band in spur plot, 279 actual gain, 315 additive noise, effect of loop on, 262 AGC, 84 Allan Variance, 271 AM radio, Example 7.13, 211 suppression, 225 transfer from LO, 225 amplifiers, combining 180◦hybrid, 152 90◦hybrid, 150 anomalous IMs, 115 appendixes, use of, 3 architectures that improve linearity, 149 Ch. 6 Summary, 163 asymmetry, filter, 286 attenuator noise factor, 55 available gain, 313 average gain, 19 band conversion Example 7.3, 182 Example 7.7, 193 bandwidth noise, 48, 246 with feedforward, 161 bilateral and unilateral modules combined, 28 modules in cascade, 24 binomial coefficient, 345 cable gain, 18 Example 2.1, 21 maximum, 19 minimum, 19 calculator frequency conversion, 170 receiver, 289 synthesizer, 291 carrier recovery loop, effect on phase noise, 260 cascade analysis of combiner tree, 157 optimizing, 139 phase shift of, 27 processing phase noise in, 252 standard, 16 CATV Second-Order IMs in, 134 Third-Order IMs in, 136 changes, parameter, on spreadsheet, 43 changing standard impedance, Appendix I, 321 circle, constant noise, 61 379 Practical RF System Design. William F. Egan Copyright  2003 John Wiley & Sons, Inc. ISBN: 0-471-20023-9 380 INDEX class B, 154 combiner tree, 156 cascade analysis, 157 combining amplifiers 180◦ hybrid, 152 harmonics, 153 intermods, 153 90◦ hybrid, 150 harmonics, 151 intermods, 151 parallel, 149 parameters on one spreadsheet, 139 SWRs, 306 complex Z0, 45 composite distortion, 133 second-order, 133 triple beat, 133 S parameters, Appendix S, 349 compression from third-order response, 102 in a cascade, Example 4.6, 119 in mixer, 166 constant-noise circle, 61 contaminating signal, 219 in nonlinearities, summary of Ch. 8, 243 LO, 228 summary, 236 contents, vii control(s) gain, 84 level, 86 using on spreadsheets, 147 conversion arithmetic, Appendix C, 289 direct, 195 double, 202 frequency in receivers, 167 in synthesizers and exciters, 170 loss, 166 coupler, directional, 159 cross modulation, 102 crossover spur, 170, 182 Appendix X, 359 crystal video receiver, 129 with preamplification, 129 CSO, 133 CTB, 133 data clock, transfer of phase noise from, 256 effect of phase noise on, 258 errors, 247 effect of phase noise on, 263 DC term, 93 from noise, 126 decibel, 303 decomposition of single sideband, 220 density noise, 126 phase-power spectral, 246 power spectral, 125 single-sideband, 246 desensitization, 102 receiver, 249 design process, 1 detection, 93 difference-frequency term, 93 diplexer, 67, 90 direct conversion, 195 directional coupler, 159 distortion, composite second-order, 133 triple beat, 133 double conversion, 202 downconversion limitation due to two-by-twos, 206 multi-band, Example 7.15, 212 dynamic range, 137 other limitations, 139 spurious free, 137 effective power gain, 19 end elements in cascade, 26 end notes, 5 enhancements, spreadsheet, 146 error probability, 258 errors, data, 247 evanescent fields, 45 even-order terms, other, 97 Example 2.1 Cable Gain, 21 Example 2.2 Effect of Mismatch, 22 Example 2.3 Cascade Calculations, 27 Example 2.4 Composite from Bilateral and Unilateral Modules, 30 Example 2.5 Attenuator in Cascade, 35 Example 2.6 Sensitivities Using Spreadsheet, 42 Example 2.7 Changes Using Spreadsheet, 43 Example 3.1 Cascade Noise Figure, 51 Example 3.2 Noise Figure to Meet System Requirement, 52 Example 3.3 Cascade Noise Factor, 56 Example 3.4 Effect of Image Noise, Simple Front End, 69 Example 3.5 Spreadsheet with Image Noise, Broadband System, 70 INDEX 381 Example 3.6 Parameters Differing at Image Frequency, 72 Example 3.7 (NF) Combined with Interconnects ..., 74 Example 3.8 Noise Factor in Mixed Cascade, 80 Example 3.9 AGC, Gain Determines Input, 84 Example 3.10 AGC, Input Determines Gain, 85 Example 3.11 Level Control, Open-Loop, 87 Example 3.12 Level Control, Closed-Loop, 88 Example 4.1 Second Harmonic, 94 Example 4.2 Third-Order IM, 100 Example 4.3 Computing IMs of a Cascade, 111 Example 4.4 . . . IMs That Do Not Add, 111 Example 4.5 Coherent and Noncoherent Addition, 115 Example 4.6 Compression in the Cascade, 119 Example 5.1 NPR, 132 Example 5.2 CSO and CTB, 136 Example 5.3 ISFDR, 138 Example 5.4 Combined Parameters, Standard Cascade, 141 Example 5.5 Combined Parameters, Less Ideal Cascade, 141 Example 5.6 Simplified Combined Spreadsheet, 143 Example 5.7 Optimization, 143 Example 7.1 Spur Levels, 175 Example 7.2 Mixer IM, 177 Example 7.3 band conversion, 182 Example 7.4 Relative Level of LO Leakage, 184 Example 7.5 Conversion to a Single IF, 186 Example 7.6 Conversion to an IF Range, 192 Example 7.7 Band Converters, 193 Example 7.8 Zero IF, 195 Example 7.9 Filter Requirements Table, 197 Example 7.10 Plotting the Filter Requirements, 200 Example 7.11 IF Filter, 201 Example 7.12 Limitation Due to 2×2 Spurs, 206 Example 7.13 AM Radio, 211 Example 7.14 Switched Preselector, 211 Example 7.15 Multi-Band Downconverter, 212 Example 7.16 Design Aid for Switched Preselectors, 212 Example 8.1 Sideband Transfer from LO, 227 Example 8.2 FM Contaminant Transferred from LO to IF, 230 Example 8.3 SSB Contaminant on Verge of Transfer, 230 Example 8.4 SSB Contaminant Not Transferred, 232 Example 8.5 SSB Contaminant on Verge at Other End, 232 Example 8.6 SSB Contaminant Not Transferred at Other End, 232 Example 8.7 LO Contaminant Converted into IF, 234 Example 8.8 LO contaminant leaking into IF, 234 Example 8.9 LO Contaminant Equivalent Sideband Leaking into IF, 235 Example 8.10 Mixer Noise Factor Increase Due to LO Noise, 237 Example 8.11 Noise with High-Ratio Up Conversion, 239 Example 8.12 Frequency Divider Spectrums In and Out, 241 Example 8.13 Frequency Multiplier Spectrum In and Out, 243 Example 9.1 Desensitization, 249 Example 9.2 Contribution of Phase Noise to Data Errors, 263 example of frequency conversion, Appendix E, 293 Example Z.1 Cascade Gain, Nonstandard Modules, 365 expected value of noise figure, 58 feedback, 158 feedforward, 159 and bandwidth, 161 harmonics with, 160 intermods with, 160 files, getting from Wiley site, xix filter asymmetry, 286 filtering of phase noise by, 254 IF, 168 requirements, 200 image rejection, 66 requirements table, Example 7.9, 197 RF, 168 requirements, 197 shape factor, 184, 196 filtering of LO noise, 238 of noise by filters, 254 by PLLs, 253 flicker noise, 48 FM, transfer from LO, 226 footnotes, 5 formulas for IMs and harmonics, Appendix H, 317 382 INDEX frequency bands in spur plot, Appendix B, 279 conversion, 165 calculator, 170 design method, 170 effect on IM addition, 111 example of, 171 Appendix E, 293 higher values of m, 209 in feedback path, 217 in receivers, 167 in synthesizers and exciters, 170 operating regions, 203 frequency dividers effect on contaminants, 240 internal noise, 242 sampling in, 241 frequency multipliers, effect on contaminants, 242 frequency, functions of, 7 gain, 7 actual, 315 available, 313 ratio to transducer gain, 64 average, 19 cable, 18 variance, 22 cascade of nonstandard modules, 365 controls, 84 effective, 19 insertion, 315 maximum available, 313 nonstandard module, 366 mean cable, 20 module, 15 nonstandard relative to tested, 363 round-trip, 18 simple, 8 summary of Ch. 2, 43 tolerance, 8 transducer, 314 ratio to available gain, 64 types of power, Appendix G, 313 variance of a cascade, 25 variation due to SWR, 21 with parallel combining, 155 getting files, xix glossary, xxi hard limiting, 223 harmonic formulas for, Appendix H, 317 second, 93 third, 100 with feedforward, 160 heterodyning, 165 homodyne, 195 HRC CATV system, 133 hybrid 180◦, 152 90◦, 150 i (direction of propagation), 9 IF Filter Requirements, 200 Example 7.11, 201 mixer output, 165 range, conversion to, Example 7.6, 192 IIP, 94 image frequency, parameters differing at, 72 noise, 65, 67 standard cascade, 74 rejection filter, 66 impedance match, 8 with hybrid, 151 nonstandard, 8 transformations in cables, 310 IMs adding coherently, 106 randomly, 108 anomalous, 115 formulas for, Appendix H, 317 in cascade, spreadsheet for, 111 in mixers, Appendix P, 345 measuring, 116 relative phases at modules, Table 4.1, 108 second order, 93 that do not add, 109 third-order terms at input frequency, Appendix T, 353 two-signal in mixer, 176 with feedforward, 160 in (direction of propagation), 9 insertion gain, 315 instantaneous SFDR, 137 integration of phase noise, 258 limits for, 252 intercept point effect of mismatch on, 110 second order, 93 third-order, 99 interconnect in nonstandard cascade, 367 noise factor, 56, 334 INDEX 383 with mismatch, 335 reflection at, 39 transmission line, 16 intermediate frequency, 165. See IF intermods. See IMs intermodulation of noise, 123 internal spur, 168 introduction, 1 IRC CATV system, 133 isolation, mixer, 167 jitter, 248, 269 Johnson noise, 48 leakage LO-to-IF on spur plot, 184 LO-to-RF on spur plot, 184 level control, 86 limiting distortion in, 225 hard, 223 soft, 223 limits of integration in computing phase variance, 259 linearity, architectures that improve, 149 literature, use of technical, 5 LO components, mixing between, 228 contamination, effect on noise figure, 236 filtering, 238 mixer input, 165 summary of troublesome frequency ranges in, 236 transfer from, of AM, 225 FM, 226 phase noise, 255 single sideband, 226 troublesome frequency ranges in, 228 load, power delivered to, 23 Appendix L, 325 local oscillator, 165. See LO look-up tables, 146 lossy interconnections, 32 m, high values in frequency conversion, 209 matching impedance with hybrid, 151 matrix multiplication, Appendix M, 327 maximum available gain, 313 finding for a nonstandard module, 366 maximum SWR from multiple reflections, 306 mean cable gain, 20 measurement of IMs, 116 NPR, 131 S parameter, 10 T parameter, 13 minimum SWR, multiple reflections, 306 mismatch, effect of, 16 Example 2.2, 22 on intercept points, 110 on interconnect noise factor, 335 mixer, 165 doubly balanced, 166 IF output, 165 IMs in, Appendix P, 345 LO input, 165 noise factor due to LO contamination, 236 effective, 66 parameters, 166 RF input, 165 singly balanced, 166 terminations, 174 transfer from LO, 225 mixing between LO components, 228 modules bilateral in cascade, 24 nonstandard, Appendix Z, 363 unilateral, 8 multiband downconverter, Example 7.15, 212 noise additive, effect of loop on, 262 and nonlinearity, 123 summary of Ch. 5, 147 bandwidth, 48, 49, 246 density, 49, 126 effect of loop on additive, 262 factor, 47. See also noise figure attenuator, 55 cascade, 50 module contribution, 50 effect from LO contamination, 236 effect of source impedance on, 341 equivalent for voltage amplifier, 79 impedance dependent, 59 representation, 59 implication re phase noise, 255 mixer, due to LO contamination, 236 of interconnect, 56, 334 of mixer, effective, 66 Op-Amp calculations, Appendix A, 273 parallel combining, 156 single-sideband, 66 standard, 54, 331 relation to theoretical, 62 384 INDEX standard and theoretical, Appendix N, 329 summary of Ch. 3, 88 summary of relationships, 53 theoretical, 54, 329 relation to standard, 62 using, 64 summary, 65 two-element cascade, 51 voltage amplifier, 74 with unilateral modules, 79 with extreme mismatch, 74 figure, 47. See also noise factor expected value, 58 sensitivity, 79 sensitivity, Appendix V, 355 spot, 49 variance, 58 variance, Appendix V, 355 flicker, 48 image, 65, 67 intermodulation of, 123 Johnson, 48 phase, 245 power ratio, 131 products DC term, 126 second-order, 125 third order, 130 sidebands, oscillator, 238 source, isolated, 59 temperature, 47, 48 cascade, 51 in operational environment, 52 system, 51 with operational source, 52 thermal, 48 nonideal effects in parallel combining, 162 other, 121 nonlinear products frequency dependence, 102 general cascade, 105 in cascade, 102, 103 relationship between, 102 two-module cascade, 104 nonlinearity and noise, 123 in signal path, 91 summary of Ch. 4, 121 representing, 91 nonstandard impedances, 40 interface impedance, 8 modules, Appendix Z, 363 normalized waves, 11 notes, end, 5 NPR, 131 Measurement, 131 o (direction of propagation), 9 odd-order terms, other, 101 OIP2, 93 Op-Amp noise factor, Appendix A, 273 operating regions, frequency conversion, 203 optimizing cascades, 139 organization of the book, 2 oscillator noise sidebands, 238 phase noise representations, 252 out (direction of propagation), 9 parallel combining, 149 gain with, 155 noise factor, 156 nonideal performance in, 162 parameters mixer characterized by, 166 range in composite modules, 39 S, 9 scattering, 9 T, 12 two-port, 9 passband in spur plot, 279 performance, deviations from desired, 4 phase noise, 245 adverse effects, 247 effect of carrier recovery loop on, 260 effect on data, 258 effect on data errors, 263 implication of noise figure, 255 integration of, 258 low frequency, 268 measures of, 269 oscillator spectrum, 250 sources of, 250 transfer from LO, 255 power spectral density, 246 shift of a cascade, 27 variance, limits of integration, 259 variation, 24 PLLs, filtering of phase noise by, 253 plotting filter requirements, Example 7.10, 200 power delivered to load, 23 Appendix L, 325 gain, 8 ratio for two noise factors, 342 INDEX 385 in a traveling wave, 12 spectral density, 125 PPSD, 246 predictable spur levels, power range for, 177 preface, xvii processing phase noise in a cascade, 252 propagation direction subscript, 9 PSD, 125 push–pull, simple, 154 random-walk FM, 251 range of parameters in composite modules, 39 ratio of power gains, standard and theoretical noise factors, 342 receiver calculator, 289 crystal video, 129 desensitization, 249 references, 5, 371 reflection coefficient, 8, 304 reflections at interconnects, 39 other, 82 relative sideband amplitude, 245 density, 246 response, standard cascade, 25 return loss, 8 RF mixer input, 165 round-trip gain, 18 S parameters, 9 composite, Appendix S, 349 measurement, 10 nonstandard cascade, 368 module, 367 relative to T parameters, 13 sampling in frequency dividers, 241 scattering parameters, 9 second harmonic, Example 4.1, 94 second-order IMs in CATV, 134 products mathematical representation, 95 of noise, 125 terms, 92 sensitivity noise figure, 79 Appendix V, 355 using, 82 use of to find variations, 40 severe nonlinearities, 219 SFDR, instantaneous, 137 shape factor definitions, 197 filter, 184, 196 sideband amplitude, relative, 245 density, relative, 246 signal with noise, effect of, 128 simple push–pull, 154 simulation, 3 single frequency spur, 168 IF, conversion to, Example 7.5, 186 sideband decomposition of, 220 density, 246 transfer from LO, 226 Smith Chart, 310 soft limiting, 223 source impedance, effect on noise factor, 341 source resistance, effect on Op-Amp noise factor, 274 specifications, creating and using, 1 spectrum, oscillator, 250 spot noise figure, 49 spreadsheet enhancements, 146 getting from Wiley site, xix spur plot, 180 use of, 2, 3 spur(ious), 168 crossover, 170, 182 Appendix X, 359 free dynamic range, 137 internal, 168 level chart, 168 in DBM from balance parameters, 217 in mixer, 168, 171 dependence on signal strength, 171 estimating, 173 m-by-n, 168 plot IF reference, 186 LO reference, 180 normalized to LO, 184 representation of bands, Appendix B, 279 spreadsheet, 180 single-frequency, 168 standard cascade, 16 overall response, 25 CATV system, 133 impedance, 8 changing, Appendix I, 321 386 INDEX noise factor, 331 state variables, standard cascade, 18 sum-frequency term, 93 superheterodyne, 165 switched preselector design aid for, Example 7.16, 212 Example 7.14, 211 SWR(s), 8, 304 combining, 306 maximum sum, 306 minimum sum, 306 variation in, 38 symbols, list of, xxi synthesis calculator, 291 system design process, 1 T matrices, 14 multiplying, 14 T parameter(s), 12 measurement, 13 other definitions, 45 relative to S parameters, 13 restrictions on, 14 Table 3.1 Summary of Noise Relationships, 53 Table 4.1 Phases of Close (in frequency) Signals and IMs Formed at Two Different Locations, 108 Table 5.1 Effects of Redistributing Amplifiers, 146 Table 7.1 Ratio (r) of Largest IM to Mixer Spur, 176 Table 7.2 Values for Fig. 7.38, 215 Table 8.1 Characteristics of Troublesome Ranges in LO with Attenuation from LO to IF shown for SSB contaminant, 236 Table A.1 Op Amp Noise Factors for Various Parameter Values, 277 Table P.1 Binomial Coefficients, 346 Table S.1 S Parameters for Composite of Two Modules, 352 Table X.1 Crossover Spurs, 359 Taylor series, 91 technical literature, use of, 5 terminations, mixer, 174 terms, list of, xxi test, 3 theoretical noise factor, 329 thermal noise, 48 third-order IMs in CATV, 136 IP, 99 Example 4.2, 100 products mathematical representation, 100 of noise, 130 terms, 97 time dependence, 92 tolerance, gain, 8 transducer gain, 314 transfer of phase noise from data clock, 256 transformation of impedance by cable, 310 transmission line interconnection, 16 triplexer, 67, 90 troublesome frequency ranges in the LO, 228 summary, 236 two-by-twos, 206 two-port parameters, 9 two-signal IMs in mixer, 176 unilateral modules, 8 bilateral modules becoming effectively, 33 combined with, 28 simplification with, 15 variance cable gain, 22 noise figure, 58 Appendix V, 355 variation in SWRs, 38 phase, due to reflection, 24 waves, normalized, 11 Z0, imaginary component, 45 zero IF conversion to, 195 Example 7.8, 195