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Frequency-Dependent Distortion Mechanism in a Broadband Amplifier
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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 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 – [1], 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 [2] 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 [1]) 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 [3]. 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 [3], 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
[1] 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 [2] 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.
[3] 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.
