Frequency-Dependent Distortion Mechanism in a Broadband Amplifier Jodi Steel, Anthony Parker Electronics Department, Macquarie University, Australia jodis, tonyp@ieee.org March 25, 1999 Abstract Investigation has been undertaken to assess the feasibility of black-box models for broadband cable amplifiers. Frequencydependence in the distortion behaviour of these amplifiers is not accounted for in earlier-proposed two-block models. A new model with inherent frequency dependence in its distortion behaviour is presented. 1 Introduction Recognizing the need for a new approach to the design of cable networks with the advent of digital services, work began in 1994 to study the characteristics of digital networks and the competitive environment in which they would operate. With both high costs and high risks involved in establishing new networks or upgrading existing networks to carry broadband digital services, accurate simulation of the networks during design can assist in technical risk identification and management. To facilitate the design of digital networks using simulation, models of cable network elements cables, amplifiers, passive devices are required. The link-budget approach used to design analog networks does not consider the use of equalization and error correction schemes to reduce the impacts of system impairments and is therefore likely to result in inefficient digital networks. While simulation models that require the inclusion of a measured system frequency response are a very useful tool in the understanding of digital system behaviour, they do not readily enable the design or upgrading of networks. The aim of the work has been to develop broadband element models, which can be determined from black box measurements, for use in simulation for design of broadband cable networks. The complexity of the design and behaviour of elements such as cable amplifiers and their commercial sensitivity renders the simulation of such elements by circuit-level models impractical. To be most useful to network designers, a model should be able to be fitted using a small number of simple measurements of a network element, without the need for highly-specialized and expensive equipment. Particular focus has been placed on the characterization of distortion behaviour of broadband cable amplifiers for low signal levels; that is, below levels where clipping or effects of order greater than three are significant. Early work on this project was reported in [1], which described a block-model approach to the characterization and modeling of cable
amplifiers and other network elements. Subsequent work has revealed greater complexity in the distortion behaviour of a representative broadband cable amplifier than can be readily accommodated in the block model. This paper presents the results of the characterization work and proposes a new model to represent the observed behaviour. 2 Distortion classification The importance of distortion behaviour in a given network depends on the relative bandwidth of that network, which determines the type and placement of in-band distortion products. In narrowband networks, only odd-order difference products fall in-band; only thirdorder difference products are normally considered significant. Accordingly, only the distortion products associated with adjacent tones will affect a given carrier of interest. Since the variations in network or amplifier frequency responses are typically small or even negligible over a narrow frequency range, little or no variation is present in the narrowband distortion levels. In other words, the distortion products do not display frequency dependence. In other networks with broader frequency ranges, the levels of distortion produced by a given pair of input tones are more likely to display frequency dependence, according to the frequencies of the input tones, the frequencies of the distortion products, or both. The distortion frequency dependence may be of two types. In the first, the distortion mechanism is able to be separated from the frequency dependence, leaving a frequencyindependent nonlinearity and a linear frequency response. This is the basis for amplifier block models, which use a combination of filter and nonlinear blocks (usually only two or three blocks in total) in series. In the second case, the distortion and frequency-dependence are inseparable. For clarity, we have termed these cases wideband and broadband, respectively. Table 1 summarizes the classification of distortion behaviour from a modeling perspective. As discussed above, narrowband networks are a simple modeling case where both the frequency response and any frequency dependence in the distortion behaviour can be ignored. For completeness, a fourth category, which we have named bandpass, has been included for cases where an amplifier may have a frequency response that is essentially flat over the band of interest and yet still displays frequency-dependence in its distortion behaviour. 3 Characterization results Hamilton and Stoneback [2] performed distortion characterization on several amplifiers using 77 or more tones. Assuming that thirdorder intermodulation beats would be insignificant with respect to triple beats, they calculated a measure they termed Individual Triple Beat (ITB) by dividing the measured CTB (composite triple beat) by the expected number of triple beats for a particular frequency. They proposed that the level of a triple beat depended on the frequency at which the beat fell and not on the frequency of the generating tones. Their results showed a strong frequency-dependence in the ITB levels and indicated that the triple beats did indeed depend only on the frequency at which the beat fell. Measurements of discrete distortion products were taken for this work using a standard two-tone test method, even though thirdorder triple-beat products (ie of type A±B ± C) are more numerous than third-order intermodulation 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 include filter Broadband NO NO Simple serial block model not possible, new model under investigation Bandpass YES NO Tunable narrow-band version of broadband model Table 1: Classification of distortion behaviour The two-tone test method is consistent with the aim to fit models to simple measurements without the need for expensive and specialist equipment. For measurements using more than two input tones, the equipment and setup complexity increases significantly unless specialist equipment such as a multiple-tone matrix generator is used. Both second- and third-order distortion products can be characterized using two-tone measurements and triple-beat product behaviour is expected to be predictable from those results. Early characterization work (reported in [1]) using only one pair of input tones and for three different amplifier tilt levels (including no tilt) showed promising results for accommodating any frequency-dependence using serial block models. The measurement work was then expanded to enable the generation of different pairs of input tones from six chosen discrete frequencies. Six pairs of tones were chosen to further investigate any frequency dependence by generating distortion products which fell at the same or similar output frequencies when generated by different pairs of tones, and to generate a range of output product frequencies by keeping one input tone frequency constant while varying the other. The results of these later measurements of second-order and third-order intermodulation products partly confirm Hamilton and Stoneback s results. An amplifier set with no gain tilt was used to ensure any frequencydependence observed was inherent to the amplifier. For a given pair of input tones and for products of the same order, distortion product output levels did increase as the frequency of the output products increased, except for those products which fell outside the amplifier s passband and were therefore attenuated. However, when results from the six different pairs of input tones were combined, the distortion products showed only a general trend of increasing level with increasing output frequency there is also some dependence on the frequency of the generating tones. The observed behaviour shows that in general the amplifier measured fits into the broadband category. The frequency dependence of its distortion behaviour is not accommodated by the simple two-block model proposed earlier. Consequently, a new model which could accommodate the inherent frequencydependence of the distortion is necessary. 4 Frequency-dependent distortion cascode model Previous examination of the interaction between devices and the circuit in which they operate has resulted in FET models that specifically address frequency-dependent dis-
tortion [3]. It is shown that commonly-used circuit layouts common gate, current mirror, cascode and common source produce different frequency-dependent distortion behaviour. Of the circuits reported in [3], the cascode circuit (Figure 1a) displayed narrowband behaviour most like that observed in the measurements taken to that time and was chosen for further investigation of its broadband behaviour. An ideal current-source version of the model is shown in Figure 1b. Note that even though both current sources are modeled with only second-order nonlinearity (in the form of v 2 ), the circuit produces all odd- and even-order output distortion products. This is due to the effect of the capacitor, which influences the dependence of the upper current source on v out. Simulation was performed in SPICE, injecting 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 than the more commonly used fixed frequency difference, was maintained across the sweep range. SPICE performs a time domain simulation and calculates output primary tone and distortion product levels using a high-dynamic range spectrum (FFT) function. The output frequency resolution must be carefully chosen to ensure the accurate recording of individual frequency components without interference from windowing functions or adjacent frequency components. To accommodate both increasing frequency and a fixed frequency resolution in the time-domain SPICE simulation, the number of simulation points and therefore the simulation time would need to be increased as the input frequencies increase. Maintaining a relative separation between tones enabled the scaling of the required frequency resolution and therefore kept simulation times reasonable. + + + v in -v out v out (a) FET representation v in v out i i = -g m v out + g m'v out 2 i = g m v in + g m'v in 2 (b) Ideal current source representation Figure 1: Cascode circuit models In Figure 2, results are shown for two ideal sources and two sources with g m mismatched by 1% to illustrate the non-ideal case. In the primary tones, the effect of the output capacitor forming a low-pass filter with the transconductance of the upper FET (ideal current source in this case) can be seen. As frequency increases, the distortion products increase in level before rolling off as more current flows through the capacitor. For matched sources (Figure 2a), distortion levels drop consistently with decreasing frequencies, whereas distortion levels in Figure 2b level out at lower frequencies due to the g m mismatch. The broadband distortion behaviour of the
-20-20 -40-30 -40-60 -50-60 Level (db) -80-100 Level (db) -70-80 -90-120 -140-160 Carrier 2ndO sum 2ndO diff 3rdO sum 3rdO diff -100-110 -120-130 Carrier 2ndO sum 2ndO diff 3rdO sum 3rdO diff 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 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 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 downstream cable amplifier range, using SPICE simulations. Figures 3 6 show the output levels of distortion for second-order products A + B and A B, and third-order products 2A+B and 2A B, respectively. (2B +A and 2B A charts are mirror-images of 2A + B and 2A B, respectively). The effect of the capacitor on the third-order distortion mechanism can be seen by comparing Figures 3 and 5 with Figures 4 and 6, respectively (especially for the difference products (Figures 5 and 6)). Both second-order products roughly increase in level with increasing carrier frequencies, and display different behaviour with respect to product frequencies sum products are close to constant in level with respect to product frequencies, whereas difference products vary with product frequency, especially for low product frequencies. Behaviour of third-order products is more complex. Third-order sum products display some level constancy with product frequency (although not as closely as secondorder products), and increase in level before rolling off with increasing carrier level, especially along the axis of the squared carrier (ie carrier A for the product 2A + B). The third-order difference plot shows rapid roll-off with decreasing carrier frequency, especially at low product frequencies, and shows similar behaviour to the third-order sum products as carrier frequency increases. It is likely that the saddle feature in the third-order difference plot is a result of poles and zeros in the output plane. To accommodate the amplifier s wideband frequency response, low-pass and high-pass filter characteristics must also be modeled, and perhaps included separately from the frequency-dependent distortion model. An example of a high-pass filter is discussed here to illustrate the effect on the cascode model s behaviour.
A simple RC single-pole high-pass filter with -3dB frequency set at 78MHz, about the measured amplifier s lower cutoff frequency, was placed either before or after the cascode circuit and simulated in SPICE. The cascode amplifier model s bandwidth before the addition of the high-pass filter is as shown in Figure 2a. The results of those simulations for the third-order products only are shown in Figures 7 10. Comparison of Figures 7 and 8 with the unfiltered result of Figure 4 shows that a filter at the input has a greater effect on the output product levels than does the same filter placed at the output. This is because the output product level results from the scaled multiplication of the input tones generating the product for example, ka 2 B for products 2A + B and 2A B. The input filter affects the levels of the generating tones for each product, whereas the output filter affects the output product levels more directly. The direct affect of the output filter on product levels versus the indirect affect of the input filter is even clearer when difference products (Figures 6, 9 and 10) are compared. The form of the output 2A B plane for the input filter model is essentially the same as that for the unfiltered version, except that products with one or both input tones in the low frequency range are attenuated by the input filter, leading to a folding over of the output plane. In the output filter case, all low-frequency products are heavily attenuated and the output plane saddle is slightly flattened due to the combined effect of the high-pass filter and the inherent cascode response. The simulation results show some features necessary to model the distortion frequency dependence observed in the measured amplifier. Work is continuing to determine the appropriate configuration to better fit the measurements. 5 Conclusion Characterization of the distortion behaviour of a broadband cable amplifier has revealed that the distortion mechanism is dependent upon the frequency of the input carriers. This frequency dependence has a significant effect that is independent of the frequency response of the amplifier. A model which demonstrates a frequency-dependent distortion mechanism has been presented. 6 Acknowledgements The authors acknowledge the support of the Australian Research Council and wish to thank Scientific Atlanta for the loan of a cable amplifier and Professor David Skellern, Macquarie University, for his suggestion of the term bandpass. References [1] Steel, J.G., Parker, A.E. and Skellern, D.J., Characterization of Cable Amplifiers for Broadband Network Applications, 49th ARFTG Conference Digest, The Brown Palace Hotel, Denver, 13 June 1997, IEEE, pp. 39 45. [2] Hamilton, J. and Stoneback D., The Effect of Digital Carriers on Analog CATV Distribution Systems, 1993 NCTA Technical Papers, National Cable Television Association, pp. 100 111. [3] Webster, D.R., Haigh D.G., Passiopoulos G. and Parker, A.E., Distortion in short channel FET circuits, in Low Power HF Microelectronics, A Unified Approach, G.A.S. Machado, Ed., Ch. 24, pp. 929 958, IEE, London, Jan. 1996.
900 1-52 1 1-56 -55-54 -53-57 -58 900 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 2 2 2 1-81 -81-80 1 0 1 1 900 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.
900 100-66 -64 100-70 -62-60 -58-56 900 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 1 1-88 -95-90 1 900 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 2-80 2 2 1 1 0-81 1 1 900 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 2 2 2-81 1 1 0 1 1 900 Figure 8: Behaviour of cascode ideal current source model with output high-pass filter thirdorder sum (2A + B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate sum product frequency.
900 1 1-95 -90-100 -88 900 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. 900-95 -90-88 -95 1 1-88 -90 1 900 Figure 10: Behaviour of cascode ideal current source model with output high-pass filter thirdorder difference (2A B) product for input tones from 105 MHz to 905 MHz. Diagonal lines indicate difference product frequency.