Frequency-Dependent Distortion Mechanism in a Broadband Ampliﬁer Jodi Steel, Anthony Parker Electronics Department, Macquarie University, Australia jodis, email@example.com March 25, 1999 Abstract the use of equalization and error correction schemes to reduce the impacts of system im- pairments and is therefore likely to result in Investigation has been undertaken to as- ineﬃcient digital networks. While simulation sess the feasibility of black-box models for models that require the inclusion of a mea- broadband cable ampliﬁers. Frequency- sured system frequency response are a very dependence in the distortion behaviour of useful tool in the understanding of digital sys- these ampliﬁers is not accounted for in tem behaviour, they do not readily enable the earlier-proposed two-block models. A new design or upgrading of networks. model with inherent frequency dependence in its distortion behaviour is presented. The aim of the work has been to develop broadband element models, which can be de- termined from “black box” measurements, 1 Introduction for use in simulation for design of broadband cable networks. The complexity of the de- sign and behaviour of elements such as ca- Recognizing the need for a new approach to ble ampliﬁers and their commercial sensitiv- the design of cable networks with the ad- ity renders the simulation of such elements by vent of digital services, work began in 1994 circuit-level models impractical. To be most to study the characteristics of digital net- useful to network designers, a model should works and the competitive environment in be able to be ﬁtted using a small number of which they would operate. With both high simple measurements of a network element, costs and high risks involved in establish- without the need for highly-specialized and ing new networks or upgrading existing net- expensive equipment. works to carry broadband digital services, ac- curate simulation of the networks during de- Particular focus has been placed on the char- sign can assist in technical risk identiﬁcation acterization of distortion behaviour of broad- and management. band cable ampliﬁers for low signal levels; that is, below levels where clipping or eﬀects To facilitate the design of digital networks of order greater than three are signiﬁcant. using simulation, models of cable network el- Early work on this project was reported in ements – cables, ampliﬁers, passive devices – , which described a block-model approach are required. The link-budget approach used to the characterization and modeling of cable to design analog networks does not consider ampliﬁers and other network elements. only two or three blocks in total) in se- ries. In the second case, the distortion and Subsequent work has revealed greater com- frequency-dependence are inseparable. For plexity in the distortion behaviour of a repre- clarity, we have termed these cases “wide- sentative broadband cable ampliﬁer than can band” and “broadband,” respectively. be readily accommodated in the block model. This paper presents the results of the char- Table 1 summarizes the classiﬁcation of dis- acterization work and proposes a new model tortion behaviour from a modeling perspec- to represent the observed behaviour. tive. As discussed above, narrowband net- works are a simple modeling case where both the frequency response and any frequency de- pendence in the distortion behaviour can be 2 Distortion classiﬁcation ignored. For completeness, a fourth category, which we have named “bandpass,” has been The importance of distortion behaviour in a included for cases where an ampliﬁer may given network depends on the relative band- have a frequency response that is essentially width of that network, which determines the ﬂat over the band of interest and yet still dis- type and placement of in-band distortion plays frequency-dependence in its distortion products. behaviour. In narrowband networks, only odd-order dif- ference products fall in-band; only third- 3 Characterization results order diﬀerence products are normally con- sidered signiﬁcant. Accordingly, only the distortion products associated with adjacent Hamilton and Stoneback  performed dis- tones will aﬀect a given carrier of interest. tortion characterization on several ampliﬁers Since the variations in network or ampli- using 77 or more tones. Assuming that third- ﬁer frequency responses are typically small order intermodulation beats would be in- or even negligible over a narrow frequency signiﬁcant with respect to triple beats, they range, little or no variation is present in the calculated a measure they termed Individ- narrowband distortion levels. In other words, ual Triple Beat (ITB) by dividing the mea- the distortion products do not display fre- sured CTB (composite triple beat) by the ex- quency dependence. In other networks with pected number of triple beats for a particu- broader frequency ranges, the levels of distor- lar frequency. They proposed that the level tion produced by a given pair of input tones of a triple beat depended on the frequency are more likely to display frequency depen- at which the beat fell and not on the fre- dence, according to the frequencies of the in- quency of the generating tones. Their results put tones, the frequencies of the distortion showed a strong frequency-dependence in the products, or both. ITB levels and indicated that the triple beats did indeed depend only on the frequency at The distortion frequency dependence may be which the beat fell. of two types. In the ﬁrst, the distortion mechanism is able to be separated from the Measurements of discrete distortion products frequency dependence, leaving a frequency- were taken for this work using a standard independent nonlinearity and a linear fre- two-tone test method, even though third- quency response. This is the basis for am- order triple-beat products (ie of type A±B ± pliﬁer block models, which use a combina- C) are more numerous than third-order in- tion of ﬁlter and nonlinear blocks (usually termodulation products of the 2A ± B type. Ignore Ignore distortion Class frequency response frequency dependence Model comments Narrowband YES YES Simple Wideband NO YES Block model possible, must in- clude ﬁlter Broadband NO NO Simple serial block model not possible, new model under in- vestigation Bandpass YES NO Tunable narrow-band version of broadband model Table 1: Classiﬁcation of distortion behaviour The two-tone test method is consistent with gain tilt was used to ensure any frequency- the aim to ﬁt models to simple measure- dependence observed was inherent to the am- ments without the need for expensive and pliﬁer. For a given pair of input tones and specialist equipment. For measurements us- for products of the same order, distortion ing more than two input tones, the equip- product output levels did increase as the fre- ment and setup complexity increases signiﬁ- quency of the output products increased, ex- cantly unless specialist equipment such as a cept for those products which fell outside the multiple-tone matrix generator is used. Both ampliﬁer’s passband and were therefore at- second- and third-order distortion products tenuated. However, when results from the can be characterized using two-tone measure- six diﬀerent pairs of input tones were com- ments and triple-beat product behaviour is bined, the distortion products showed only a expected to be predictable from those results. general trend of increasing level with increas- ing output frequency – there is also some de- Early characterization work (reported in ) pendence on the frequency of the generating using only one pair of input tones and for tones. three diﬀerent ampliﬁer tilt levels (includ- ing no tilt) showed promising results for ac- The observed behaviour shows that in general commodating any frequency-dependence us- the ampliﬁer measured ﬁts into the broad- ing serial block models. The measurement band category. The frequency dependence of work was then expanded to enable the gen- its distortion behaviour is not accommodated eration of diﬀerent pairs of input tones from by the simple two-block model proposed ear- six chosen discrete frequencies. Six pairs of lier. Consequently, a new model which tones were chosen to further investigate any could accommodate the inherent frequency- frequency dependence by generating distor- dependence of the distortion is necessary. tion products which fell at the same or similar output frequencies when generated by diﬀer- ent pairs of tones, and to generate a range of output product frequencies by keeping one 4 Frequency-dependent dis- input tone frequency constant while varying tortion cascode model the other. The results of these later measurements of Previous examination of the interaction be- second-order and third-order intermodula- tween devices and the circuit in which they tion products partly conﬁrm Hamilton and operate has resulted in FET models that Stoneback’s results. An ampliﬁer set with no speciﬁcally address frequency-dependent dis- tortion . It is shown that commonly-used circuit layouts – common gate, current mir- ror, cascode and common source – produce i diﬀerent frequency-dependent distortion be- haviour. vout -vout Of the circuits reported in , the cascode circuit (Figure 1a) displayed narrowband be- haviour most like that observed in the mea- + surements taken to that time and was cho- vin sen for further investigation of its broadband behaviour. An ideal current-source version of the model is shown in Figure 1b. Note (a) FET representation that even though both current sources are modeled with only second-order nonlinearity (in the form of v 2 ), the circuit produces all i = -gmvout + gm'vout2 odd- and even-order output distortion prod- ucts. This is due to the eﬀect of the capacitor, vout which inﬂuences the dependence of the upper current source on vout . vin + i = gmvin + gm'vin2 Simulation was performed in SPICE, inject- + ing two tones that were swept over several decades to give an overall indication of the behaviour of the cascode circuit. A ﬁxed 10% separation between the two tones, rather (b) Ideal current source than the more commonly used ﬁxed fre- representation quency diﬀerence, was maintained across the sweep range. SPICE performs a time do- Figure 1: Cascode circuit models main simulation and calculates output pri- mary tone and distortion product levels us- ing a high-dynamic range spectrum (FFT) In Figure 2, results are shown for two ideal function. The output frequency resolution sources and two sources with gm mismatched must be carefully chosen to ensure the accu- by 1% to illustrate the non-ideal case. In rate recording of individual frequency com- the primary tones, the eﬀect of the output ponents without interference from window- capacitor forming a low-pass ﬁlter with the ing functions or adjacent frequency compo- transconductance of the upper FET (ideal nents. To accommodate both increasing fre- current source in this case) can be seen. As quency and a ﬁxed frequency resolution in frequency increases, the distortion products the time-domain SPICE simulation, the num- increase in level before rolling oﬀ as more ber of simulation points and therefore the current ﬂows through the capacitor. For simulation time would need to be increased as matched sources (Figure 2a), distortion lev- the input frequencies increase. Maintaining a els drop consistently with decreasing frequen- relative separation between tones enabled the cies, whereas distortion levels in Figure 2b scaling of the required frequency resolution level out at lower frequencies due to the gm and therefore kept simulation times reason- mismatch. able. The broadband distortion behaviour of the -20 -20 -30 -40 -40 -60 -50 -60 -80 Level (dB) Level (dB) -70 -80 -100 -90 -120 -100 Carrier 2ndO sum 2ndO diff -110 Carrier -140 3rdO sum 2ndO sum -120 2ndO diff 3rdO diff 3rdO sum -160 -130 3rdO diff 2 3 45 2 3 45 2 3 45 2 3 45 2 3 45 2 3 45 2 3 45 2 3 45 2 3 45 2 3 45 10 -1.0 10 0.0 10 1.0 10 2.0 10 3.0 10 4.0 10 -1.0 10 0.0 10 1.0 10 2.0 10 3.0 10 4.0 Carrier frequency (MHz) Carrier frequency (MHz) (a) Matched sources (b) Sources mismatched by 1% Figure 2: Behaviour of the cascode ideal current source model from 10 kHz to 10 GHz to two tones with ∆f = 10 % model has been investigated across the down- complex. Third-order sum products dis- stream cable ampliﬁer range, using SPICE play some level constancy with product fre- simulations. Figures 3–6 show the output quency (although not as closely as second- levels of distortion for second-order products order products), and increase in level before A + B and A − B, and third-order products rolling oﬀ with increasing carrier level, espe- 2A+B and 2A−B, respectively. (2B +A and cially along the axis of the “squared” carrier 2B − A charts are mirror-images of 2A + B (ie carrier A for the product 2A + B). The and 2A − B, respectively). third-order diﬀerence plot shows rapid roll-oﬀ with decreasing carrier frequency, especially The eﬀect of the capacitor on the third-order at low product frequencies, and shows similar distortion mechanism can be seen by com- behaviour to the third-order sum products as paring Figures 3 and 5 with Figures 4 and carrier frequency increases. It is likely that 6, respectively (especially for the diﬀerence the “saddle” feature in the third-order diﬀer- products (Figures 5 and 6)). ence plot is a result of poles and zeros in the output plane. Both second-order products roughly increase in level with increasing carrier frequencies, To accommodate the ampliﬁer’s “wideband” and display diﬀerent behaviour with respect frequency response, low-pass and high-pass to product frequencies – sum products are ﬁlter characteristics must also be modeled, close to constant in level with respect to and perhaps included separately from the product frequencies, whereas diﬀerence prod- frequency-dependent distortion model. An ucts vary with product frequency, especially example of a high-pass ﬁlter is discussed here for low product frequencies. to illustrate the eﬀect on the cascode model’s behaviour. Behaviour of third-order products is more A simple RC single-pole high-pass ﬁlter with 5 Conclusion -3dB frequency set at 78MHz, about the mea- sured ampliﬁer’s lower cutoﬀ frequency, was placed either before or after the cascode cir- Characterization of the distortion behaviour cuit and simulated in SPICE. The cascode of a broadband cable ampliﬁer has revealed ampliﬁer model’s bandwidth before the ad- that the distortion mechanism is dependent dition of the high-pass ﬁlter is as shown in upon the frequency of the input carriers. Figure 2a. The results of those simulations This frequency dependence has a signiﬁcant for the third-order products only are shown eﬀect that is independent of the frequency in Figures 7–10. response of the ampliﬁer. A model which demonstrates a frequency-dependent distor- Comparison of Figures 7 and 8 with the un- tion mechanism has been presented. ﬁltered result of Figure 4 shows that a ﬁlter at the input has a greater eﬀect on the out- put product levels than does the same ﬁlter 6 Acknowledgements placed at the output. This is because the output product level results from the scaled multiplication of the input tones generating The authors acknowledge the support of the the product – for example, kA2 B for prod- Australian Research Council and wish to ucts 2A + B and 2A − B. The input ﬁlter thank Scientiﬁc Atlanta for the loan of a ca- aﬀects the levels of the generating tones for ble ampliﬁer and Professor David Skellern, each product, whereas the output ﬁlter af- Macquarie University, for his suggestion of fects the output product levels more directly. the term “bandpass.” The direct aﬀect of the output ﬁlter on prod- uct levels versus the indirect aﬀect of the in- References put ﬁlter is even clearer when diﬀerence prod- ucts (Figures 6, 9 and 10) are compared. The  Steel, J.G., Parker, A.E. and Skellern, form of the output 2A − B plane for the in- D.J., “Characterization of Cable Am- put ﬁlter model is essentially the same as pliﬁers for Broadband Network Applica- that for the unﬁltered version, except that tions,” 49th ARFTG Conference Digest, products with one or both input tones in the The Brown Palace Hotel, Denver, 13 June low frequency range are attenuated by the in- 1997, IEEE, pp. 39–45. put ﬁlter, leading to a “folding over” of the output plane. In the output ﬁlter case, all  Hamilton, J. and Stoneback D., “The Ef- low-frequency products are heavily attenu- fect of Digital Carriers on Analog CATV ated and the output plane “saddle” is slightly Distribution Systems,”1993 NCTA Tech- ﬂattened due to the combined eﬀect of the nical Papers, National Cable Television high-pass ﬁlter and the inherent cascode re- Association, pp. 100–111. sponse.  Webster, D.R., Haigh D.G., Passiopou- The simulation results show some features los G. and Parker, A.E., “Distortion necessary to model the distortion frequency in short channel FET circuits,” in Low dependence observed in the measured ampli- Power HF Microelectronics, A Uniﬁed ﬁer. Work is continuing to determine the ap- Approach, G.A.S. Machado, Ed., Ch. 24, propriate conﬁguration to better ﬁt the mea- pp. 929–958, IEE, London, Jan. 1996. surements. 900 800 16 00 700 Carrier B frequency (MHz) 14 00 600 500 400 12 00 -52 300 -5 -5 4 60 10 200 -5 6 -5 0 -53 00 7 5 40 80 -5 0 0 8 200 300 400 500 600 700 800 900 Carrier A frequency (MHz) Figure 3: Behaviour of cascode ideal current source model second-order sum (A + B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate sum product frequency. 900 26 00 800 -87 700 -862 Carrier B frequency (MHz) 40 600 0 -85 500 22 00 400 -84 -83 -82 300 -81 80 12 60 14 10 20 0 00 0 00 00 -8 00 200 0 -82 16 -81 18 00 00 200 300 400 500 600 700 800 900 Carrier A frequency (MHz) Figure 4: Behaviour of cascode ideal current source model third-order sum (2A + B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate sum product frequency. 900 0 70 800 0 0 60 10 0 50 700 0 20 Carrier B frequency (MHz) 0 600 40 0 30 500 0 30 0 40 400 200 4 0 50 -6 -66 0 300 10 0 -70 60 200 2 -6 0 8 6 -6 -5 -5 200 300 400 500 600 700 800 900 Carrier A frequency (MHz) Figure 5: Behaviour of cascode ideal current source model second-order diﬀerence (A − B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate diﬀerence product frequency. 900 0 60 -8 800 2 00 10 0 40 700 Carrier B frequency (MHz) 600 00 -83 12 0 20 500 400 00 14 300 -88 -90 3 200 -8 -95 00 -0 7 -86 -85 8 -82 0 16 -84 20 0 0 40 60 80 200 300 400 500 600 700 800 900 Carrier A frequency (MHz) Figure 6: Behaviour of cascode ideal current source model third-order diﬀerence (2A − B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate diﬀerence product frequency. 900 26 00 800 -87 700 Carrier B frequency (MHz) -86 24 0 600 0 22 500 - 85 00 -80 400 -83 -84 300 -84 -85 80 12 60 14 10 20 0 00 0 00 00 00 200 -8 -86 3 16 18 -81 -82 00 00 200 300 400 500 600 700 800 900 Carrier A frequency (MHz) Figure 7: Behaviour of cascode ideal current source model with input high-pass ﬁlter third-order sum (2A + B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate sum product frequency. 900 26 -87 00 800 -8 700 6 Carrier B frequency (MHz) 24 00 600 -85 2 500 20 0 400 -84 -81 300 -82 -83 80 12 60 14 10 20 0 00 0 00 00 00 200 -82 16 18 00 00 200 300 400 500 600 700 800 900 Carrier A frequency (MHz) Figure 8: Behaviour of cascode ideal current source model with output high-pass ﬁlter third- order sum (2A + B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate sum product frequency. 900 -82 800 -8 2 00 10 0 40 700 3 -8 Carrier B frequency (MHz) 600 00 12 0 500 20 400 00 14 -86 -83 300 -95 -85 -84 -90 -84 -100 -88 -87 200 0 0 20 0 0 40 60 80 200 300 400 500 600 700 800 900 Carrier A frequency (MHz) Figure 9: Behaviour of cascode ideal current source model with input high-pass ﬁlter third-order diﬀerence (2A − B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate diﬀerence product frequency. 900 0 -95 60 -82 800 3 00 -8 10 4 0 -8 40 700 -86-85 -87 Carrier B frequency (MHz) 600 00 12 0 -84 20 500 8 -8 -95 400 -90 00 14 -84 300 -88 -86 -85 -90 200 00 3 20 7 0-8 0 -8 0 16 0 40 60 80 200 300 400 500 600 700 800 900 Carrier A frequency (MHz) Figure 10: Behaviour of cascode ideal current source model with output high-pass ﬁlter third- order diﬀerence (2A − B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate diﬀerence product frequency.