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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 2012
1881
Formulations and a Computer-Aided Test Method
for the Estimation of IMD Levels in an Envelope
Feedback RFIC Power Amplifier
Nicolas G. Constantin, Member, IEEE, Kai H. Kwok, Senior Member, IEEE, Hongxiao Shao, Member, IEEE,
Cristian Cismaru, Senior Member, IEEE, and Peter J. Zampardi, Senior Member, IEEE
Abstract—This paper presents new formulations, together with
an efficient computer-aided test approach intended for radio
frequency integrated circuit power amplifiers (PAs), allowing
the estimation of linearity requirements for the circuit blocks
typically found in the error signal path of an envelope feedback
amplifier. The formulations are based on a three-tone excitation,
allowing analysis of intermodulation distortion (IMD) within the
feedback system using parameterized peak-to-average envelope
voltage. They are also based on a fifth-degree representation, and
may be extended to higher degrees of nonlinearities in the RF PA
block, enabling IMD analysis of envelope feedback amplifiers at
low power. The approach proposed in this paper circumvents the
difficulty of measuring error signals during closed-loop operation
for troubleshooting purposes. This approach is also very useful
for computer-aided test setups intended for development work
independent of the often idealized circuit simulation environment.
Index Terms—Amplifier, envelope feedback, intermodulation
distortion.
I. Introduction
E
FFICIENT characterization and test methods used in the
design and/or production phases of RF integrated circuits
(RFIC) and highly integrated subsystems for wireless communication are of paramount importance because of their impact
on development time and costs and, ultimately, profit margins.
This drives the study of various methodologies combining
different techniques for this purpose, such as using statistical
data based calculations or signal processing in conjunction
with analytical formulations to minimize test time, test setup requirements and manipulation during the characterization
and test of RFICs, which are applicable at the design stage
and at the production stage [1], [2]; statistical performance
models of the circuit under test and test measurements using
limited circuit simulation data in order to anticipate and refine
test generation programs early in the design [3]; and efficient equivalent-circuit modeling of advanced RFIC modules,
Manuscript received October 4, 2011; revised March 16, 2012; accepted
June 11, 2012. Date of current version November 21, 2012. This paper was
recommended by Associate Editor A. Ivanov.
N. G. Constantin is with the École de Technologie Supérieure, Montreal,
QC H3C 1K3, Canada (e-mail: [email protected]).
K. H. Kwok, H. Shao, C. Cismaru, and P. J. Zampardi are with
Skyworks Solutions, Inc., Newbury Park, CA 91320 USA (e-mail:
[email protected]; [email protected]; cristian.
[email protected]; [email protected]).
Digital Object Identifier 10.1109/TCAD.2012.2207954
which are part of hybrid integrated computer-aided design
(CAD) platforms using evolutionary simulation algorithms and
analytical-modeling equations [4].
The method we present in this paper addresses the need
for developing efficient RFIC testing and characterization
approaches for increasingly complex RFIC power amplifier
(PA) modules, specifically during the development phases,
where the impact is, this time, more on accelerating the design
convergence and reducing the time to market.
The gated envelope feedback (GEF) PA architecture [5],
[6] was demonstrated as an attractive method for automatically switching RF transistor arrays on and off in RFIC
PAs as a function of power, reducing current and, hence,
improving power efficiency. Its block diagram is shown in
Fig. 1. It is comprised mainly of the two circuit sections
shown within dotted lines: 1) a variable gain RF amplifier
chain that amplifies the envelope varying RF signal with an
electronically adjustable gain, as part of a feedback process;
and 2) a feedback and gating circuit that performs automatic
switching of these RF transistor arrays at predetermined average input RF power thresholds while maintaining gain regulation during the RF transistor switching, through envelope
feedback.
From this basic description, it is obvious that the design
of such a complex PA system should benefit from extensive
circuit simulation using accurate component models, as a
function of the actual applications modulated input excitation,
in order to ensure meeting linearity requirements.
On the other hand, estimation of multitone intermodulation
distortion (IMD) in such a system independently of the circuit simulation environment, using design equations that are
implemented in a computerized experimental test set-up, has
some key advantages: 1) facilitating IMD analyses that are
function exclusively of data obtained from the actual circuit
implementation; 2) allowing rapid design convergence during
experimental development; and 3) circumventing the often
difficult and time consuming task of correlating the initial
simulation phase results with measurement results (where
the conditions may not exactly match), at the time when
the RFIC PA designer needs to make the next iteration in
the design convergence process. The difficulty is exacerbated
when trying to probe very low amplitude (a few millivolts typically) envelope feedback error signals in closed-
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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 2012
using the proposed equations can be performed exclusively
with test setups and data independent of the circuit simulation
environment.
Section II gives a description of the GEF PA operation
and briefly summarizes envelope detector linearity measurements from [5], emphasizing the need for design equations
that facilitate IMD characterization and testing of envelope
feedback systems. Section III describes the derivation of the
new formulations and the proposed method. In Section IV, the
equations and the proposed method are used to estimate the
IMD requirements of the envelope detector with specific input
excitation conditions, and to validate its design in the context
of the implementation presented in [5].
II. GEF PA and Detector Linearity
A. GEF Amplifier Operation
Fig. 1.
GEF RFIC PA architecture (from [5]).
loop operation, since noise induced in the RF envelope
detectors or in the error comparator through the probing
interface drastically alters the actual operation conditions
compared to the simulated conditions, preventing conclusive
comparisons.
Thorough analytical methods and exact formulations for
distortion analysis based on the accurate modeling of the
nonlinearities at the semiconductor device level or at the level
of a stand-alone PA component are available [7], [8].
However, there has been limited work on multitone IMD
formulations allowing estimation of specific IMD levels at any
node within the more complex architecture of an envelope
feedback PA system, as a function of the system level and
circuit block parameters. The expressions in [9, eq. (2)–(7)]
are limited only to a symbolic representation of the squaring
and cubing of the input envelope information, but not function
of the resulting output two-tone signal and IMD product levels
and their distinct frequencies. Also, those expressions only
represent the condition that exists in the amplifier system
before the feedback correction process, and do not express
a steady-state operation of the feedback.
In our work, equations that are easily implemented in a
computerized test set-up are derived for computing small
envelope signal IMD levels at any node in a typical envelope
feedback PA operating in a closed loop. Also, we propose a
method based on these equations allowing the estimation of
IMD and gain performance requirements for the RF envelope
detectors and other building blocks in the error signal path,
as a function of: 1) the input excitation variable peak-toaverage envelope voltage ratio; 2) the AM-AM response of
the RF amplifier block alone (in easily characterized openloop conditions); 3) the small signal feedback loop gain,
also easily measured in open loop; and 4) the overall IMD
levels that are desired in the closed-loop operation of the
amplifier system. A numerical example of IMD analysis
applied to the envelope detectors used in the RFIC PA design
implemented in [5] confirms the utility of this method. The
example is based on simulation data, and the same method
Fig. 1 illustrates the partitioning of the GEF PA full on-chip
circuits on two GaAs HBT ICs.
The role of the variable gain RF amplifier section is to
amplify the envelope varying RF input signal with a total gain
that is adjusted by the amplified feedback error signal (CTRL).
It is comprised of: 1) a variable gain stage (var. gain block) and
adjustable bias circuits to control the overall PA gain through
feedback; 2) RF transistor arrays in the intermediate and power
stages (IS and PS) that are fragmented so that half of each
array may be turned on or off independently, function of the
average input RF power; and 3) interstage impedance matching
networks (Zm). The output matching network (VAR Zm) is
implemented off chip to minimize loss and allow tuning.
The feedback and gating circuit section is enabled (or
disabled with almost no current consumption) via the GATE
signal. Its role is to detect the input and output envelope power,
to trigger automatic RF transistor switching as a function
of the average input RF power, and to regulate the PA’s
overall RF power gain upon automatic switching through envelope feedback. It is comprised of the following subsections:
1) envelope detectors (Di and Do ) for the comparison of the
instantaneous input and output envelope signals, as part of the
feedback process, and to provide a measure (V cnd signal) of
the average input RF power (P in ) to the switching circuitry;
2) a CONDITIONER block to minimize voltage offsets between the detector outputs, by providing equal impedance
loading and isolation from the switching circuitry; 3) a
comparator C to generate the envelope error signal; 4) an
analog error amplifier A, with adjustable biasing to improve
gain flatness over a large power range, and with an off-chip
phase compensator (not shown) for stability; 5) hysteresis
comparators (IHC and PHC) with outputs that toggle between
logic high and logic low depending on predetermined levels
of Vcnd , for the RF transistor switching; and 6) switches that
disable the feedback and gating circuit section when GATE is
at logic OFF.
The envelope feedback signal path is highlighted with
thicker lines in Fig. 1. The forward path includes Di , C, A,
and the variable gain RF amplifier chain. The reverse path
includes an RF attenuator (RF ATT.) followed by Do .
CONSTANTIN et al.: FORMULATIONS AND A COMPUTER-AIDED TEST METHOD FOR THE ESTIMATION OF IMD LEVELS
Fig. 2. Schematic of the input (Q1) and the output (Q2) envelope detectors
(Di, Do in Fig. 1). The envelope comparator is shown as the load (from [5]).
The GEF PA operation is essentially as follows. At high P in ,
GATE at logic OFF disables the envelope feedback circuitry,
and IS and PS are fully operational. Below some P in threshold,
GATE is activated at logic ON to enable envelope feedback and
to reconfigure VAR Zm for low power impedance matching.
As P in further decreases, IHC and PHC automatically turn
OFF half of PS and half of IS at different thresholds, to reduce
current consumption at low power. The gain variation from
turning on or off these transistors is cancelled through the
envelope feedback, thus maintaining a constant gain.
B. Detector Circuitry and Amplifier Linearity Performances
Fig. 2 shows the schematic of the RF envelope detectors (Di
and Do in Fig. 1 and built with Q1 and Q2, respectively, in
Fig. 2) that were implemented and described in [5], followed
by the envelope comparator circuit (shown as the output load
only).
The RF to analog conversions by Q1 and Q2, which constitute envelope detection by each transistor, rely on the wellknown dependence of collector average current and voltage on
the RF input level, typical of transistors operated in class-AB
[8]. The detected envelope voltage signals at the collectors of
Q1 and Q2 are filtered by R11, C3, L3 and R12, C4, L4,
and then applied to the envelope error comparator circuit. To
improve the detectors’ dynamic range, self-triggered nonlinear
compensation networks (built with resistors and diodes in the
collector circuits of Q1 and Q2) are activated when the average
input RF power (P in ) applied to the RF INPUT terminal in
Fig. 1 reaches a threshold power level of ∼ −12 dBm [5]. Part
of these nonlinear effects is inevitably reflected at the output
of the PA and cannot be canceled by feedback [5].
The gain and linearity performance of the RFIC PA illustrated in Fig. 1 was measured using a CDMA2000-1X
modulated excitation at 1.88 GHz. The results shown in Fig. 3
indicate that the standard minimum adjacent channel power
rejection (ACPR) specification for CDMA2000-1X (−42 dBc
in a 30 kHz bandwidth and at a 885 kHz offset [10]) can be met
with this GEF implementation, despite the strongly nonlinear
conversion gains in the two branches of the envelope detector
circuitry (Fig. 2).
Better design margins than in Fig. 3 are usually desired,
typically ACPR values of ∼−50 dBc or better at 885 kHz
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Fig. 3. Measured PA gain and ACPR at 885 kHz offset, with a CDMA20001X (RVS− RC1− FCH− 9.6 kb/s) input excitation (from [5]).
offset [11]. A particularly interesting case for linearity analysis
is the −12 dBm input power threshold, since this is the
triggering point for the nonlinear compensation network in
Fig. 2, and it was shown to coincide with perturbations in
the closed-loop PA gain [5]. The 16 dB PA gain at this level
[5] yields an output power (P out ) of ∼4 dBm with a −51 dBc
ACPR performance, as shown in Fig. 3. However, improvements in the linearity performance at the −12 dBm threshold
improve the linearity at higher levels because of the gradually
changing ACPR as P out increases (Fig. 3). Hence, evaluating
the detectors’ contribution to multitone IMD performance
degradation in the amplifier system at P in = −12 dBm is a
useful example for our proposed IMD test method.
III. Multitone Linearity Analysis
A. Test Setups Applicable to Memoryless Behavioral Models
As detailed in [14], two types of memory effects are considered in PA design: electrical and thermal. Electrical memory effects are caused by frequency-dependent impedances
due to energy-storage elements (capacitors and inductors),
which introduce phase shifts in the electric signals (and the
associated time dependence in the PA response). Thermal
memory effects are caused by time-dependent variations in
junction temperatures of semiconductor devices, function of
the intensity of the modulated electric signals being amplified,
thereby introducing time constants in the electrical transfer
functions of the devices. In the context of multitone IMD
analysis [7], [8], [14], the consequence of memory effects is
that each discrete IMD product in the frequency spectrum is
a result of a vector sum of phasor quantities originating from
the PA nonlinear processing of the multitone input signal.
However, these effects have minimal impact on IMD performances when the memory time constants are much smaller
than the period of a low-frequency envelope signal (e.g.,
1 kHz). Regarding electrical memory, the PA circuitry may be
adjusted so that the time constants in the bias circuits and the
delays associated with the phase shifts in the RF signal exhibit
much lower values than the 1 ms envelope period. Regarding
thermal memory, the two ICs of Fig. 1 may be implemented
on separate dies, including on-chip probe access and providing
high thermal isolation between each die and their test fixtures.
Then, the thermal time constants are very short as well.
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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 2012
We have experimentally measured the thermal time constant
for InGaP/GaAs HBTs. The transient current and voltage on
the base and collector terminals were measured with a time
interval of as low as 20 ns using a Keithley 4200 with the
4225 ultrafast I–V pulsed measurement unit. Step voltages
of various values were applied to the base of the device,
synchronized with a separate step voltage in the collector. This
test is similar to previous experimental work [15], [16]. For a
56 μm2 emitter area device on a die that was attached with
comparable heat sinking conditions as in typical PA modules,
the collector current stabilizes after approximately 200 ns and
begins increasing in an asymptotical fashion afterward to reach
a stable value after a few microseconds. The asymptotical
curve represents the self-heating transient under base voltage
drive. Fitting this portion of the curve allows us to extract a
thermal time constant of 0.8 μs for this device.
Assuming a linear scaling, the time constant for a 48 × 56
μm2 (∼2700 μm2 ) emitter HBT (one half-array at PS
in Fig. 1) would be in the order of ∼48 times longer,
corresponding to a thermal-filtering cutoff frequency of
∼4.2 kHz [i.e., 1/(2π·38.4 μs)]. However, the thermal cutoff
frequency does not decrease linearly with the emitter area [17],
because the rate of increase of the thermal capacitance with
emitter area slows down as the device spacing becomes smaller
than the die thickness and/or the device gets closer to the die
boundary, suggesting a cut-off frequency above 4.2 kHz for
48×56 μm2 devices with typical RFIC PA die attach. Then,
with a test set-up using high thermal isolation between each die
and its respective test fixture, the thermal cut-off frequency of a
48×56 μm2 array will be well above 4.2 kHz, with negligible
related amplitude and phase influence (negligible memory
effects) in the dynamics of a 1 kHz AM modulation. This is
consistent also with the simulated data in [14, Fig. 3.11(b)],
for a much larger silicon device (160 000 μm2 emitter area)
without any heat sinking. It shows a thermal-filtering corner
frequency of ∼3 kHz, implying that corner frequencies well
above 10 kHz may be reached with a 2700 μm2 emitter
transistor on a die with high thermal isolation from the test
fixture.
The above electrical and thermal test conditions are applicable for IMD estimation with a memoryless model.
B. Behavioral Model
The proposed formulations are based on a memoryless
behavioral model. They are also based on a three-tone input excitation equivalent to an AM signal because of the
following advantages: 1) the intuitiveness that a multitone
test provides and the associated simplicity of the equations,
which is particularly useful for complex PA architectures;
2) the suitability of AM (three-tone) for small and variable
envelope peak-to-average ratios to facilitate analyses that are
localized in the power domain (e.g., across the −12 dBm
threshold of interest discussed earlier); and 3) the convenience
of symmetrical AM sideband levels on the spectrum analyzer
display, since it provides an easily observable reference when
experimentally verifying the zero memory behavior and AMAM only response (note that memory and AM-PM introduce
asymmetry [12], [14]).
Fig. 4.
Envelope feedback block diagram for multitone analysis.
The behavioral model is shown in Fig. 4, and represents
the weakly nonlinear processing of the envelope information
within the GEF amplifier architecture of Fig. 1 (excluding all
switched control circuitry).
The variable gain function of the RF amplifier chain is
approximated by a gain control element (explained in SectionIII-D) followed by an amplifier block that is defined as a
power series (G in Fig. 4), assuming an ideal output bandpass filter centered at the carrier frequency ωc . Typically, Vout
corresponds to the voltage node between the output of the RF
amplifier chain and the output matching network (not shown).
Hence, the frequency selective loading effect of this matching
network provides the band-pass filtering function.
The attenuation in the feedback path (RF ATT. in Fig. 1)
is represented by H. The RF-to-analog conversion gain of
the detectors is represented by fD in Fig. 4 (this is further
explained in Section-III-C), and the gains of the comparator
and the error amplifier are represented by C and Av .
The electrical condition associated with the memoryless
approximation is that the highest order mixing product considered in the envelope signal path (i.e., 4ωx in the frequency
spectrum of Ve in Fig. 4) is small enough compared to any
cutoff frequency in the envelope frequency response so that
the phase shifts in the feedback loop may be neglected.
C. IMD Test Method
The proposed method relies on equations for computeraided test and characterization of the actual PA implementation
to estimate the linearity requirements of circuit blocks Di ,
Do , C, and A (Fig. 1) in open-loop conditions, through a
comparative study based on computed benchmarking values
and measured values. It consists of the following.
1) Measuring the multitone, small envelope amplitude IMD
levels at the output of each of these circuit blocks
separately in open-loop conditions and with a specific
multitone excitation.
CONSTANTIN et al.: FORMULATIONS AND A COMPUTER-AIDED TEST METHOD FOR THE ESTIMATION OF IMD LEVELS
2) Making the following idealized a priori assumptions,
only for the purpose of computing the benchmarking
values:
a) to consider that both envelope detectors (Di and
Do in Fig. 1) perform an ideal detection of the
instantaneous RF envelope amplitude of their respective AM modulated input RF signal (hence
with the same constant RF-to-analog conversion
gain fD for both detectors);
b) that the gains of blocks C and A (Fig. 1) are linear;
c) accordingly, to consider that the multitone error
signal (represented by Ve0 to Ve4X in Fig. 4) and
their amplified levels along the feedback error
signal path are representative of the distortion in
the variable gain RF amplifier chain (Fig. 4) only.
3) Computing the best (lowest) achievable closed-loop
IMD levels (which will serve as the benchmarking
values) along this error signal path with the help of the
design equations, given the above idealized assumptions
and using the same multitone excitation conditions as
during the measurements.
4) Verifying the sufficiency (or insufficiency) of the measured IMD performances of Di , Do , C and A based
on a comparison with the benchmarking values, using
the following criterion: IMD performance compliance
with respect to the calculated benchmarking figures
ensures that the circuit blocks in the error signal path
do not degrade the overall output IMD performances.
The design margin considered depends on the safeguard
required to guarantee performance compliance after the
idealized assumptions. Alternatively, deviation from the
ideal performance by a known error margin still allows
setting design goals for performance improvement.
In determining the benchmarking IMD levels, the idealized assumptions (step 2) implies that fD , C and Av can
be regrouped as cascaded linear gain blocks following an
ideal comparator, as shown in Fig. 4, with a combined low
frequency gain
E = fD · C · Av .
(1)
Also, the IMD benchmarking levels at the output of any of
these three circuit blocks are modeled as IMD product levels
referred to the output of the ideal comparator, and then scaled
up by the low frequency gain factors of the circuit blocks in
between.
D. Approximation for the Gain Control and Justification
The justifications for the approximation are: 1) it allows a
drastic simplification of the formulations; and 2) it is a reasonable approximation in the context of estimating benchmarking
figures for a comparative study with conservative safeguards.
Fig. 5(a) represents the variable gain RF amplifier chain IC
shown in Fig. 1. Fig. 5(b) is the approximate model used in
Fig. 4, including the envelope amplitude Vae− 1 of the input
RF signal Va− 1 (after attenuator F in Fig. 4), an envelope
amplitude variable Vae that stems from the approximation, the
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Fig. 5. (a) Variable gain amplifier with a gain control signal Vp .
(b) Approximate model, assuming only small variations in Vae and Vp .
Fig. 6.
AM-AM curves illustrating the limitations of the model in Fig. 5(b).
envelope amplitude Voe of the RF output Vout , and the gain
control signal VP .
To explain the approximate model in Fig. 5(b), we refer
to Fig. 6. The bottom curve represents the PA’s AM-AM
response, characterized as a power series [G in Fig. 5(b)] for
a given quiescent condition on VP .
Consider a very small change in VP , which translates into a
small variation in Voe , thus reaching Voe− 1 , for any given constant amplitude Vae− 1 within a narrow range (note that in Fig. 6
the Vae range is large, and a large gain adjustment is shown
intentionally to represent a significant change in the AMAM profile for the purpose of the analysis in Section-III-E).
Given the power series (G in Fig. 5), which allows close
approximation of the AM-AM relation between Vae and Voe in
the vicinity of Vae− 1 (or Vae− 2 ) after the small variation in VP ,
the change in Voe is referred (based on this approximation) to
the input with a small change in Vae , defined by an incremental
quantity Vae added to the initial amplitude Vae− 1 (or Vae− 2 ).
Hence, the model approximates the small gain adjustment with
the quantity Vae− 1 + Vae1 (or Vae− 2 + Vae2 ) applied to the
power series G. It is assumed that over a narrow Vae range
(Fig. 6), a single Vae value can approximate Vae1 and Vae2
for the same variation VP considered, which allows equating
Vae = P·VP , P being a constant proportionality factor, and
therefore approximating Voe as the incremental quantity that
results from (Vae + P·VP ) applied to the power series G.
Simulation was performed on the design in [5] in an open
loop, with VP = 50 mV, Vae− 1 = 68 mV (Fig. 6), and
Vae− 2 = 85 mV (a 2 dB ratio). The differences (in %) between
the Vae1 and Vae2 required to get the same Voe1 and Voe2
changes, but on the AM-AM characterized curve, are 7% when
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the quiescent value VP− DC = 1.35 V, 4% for VP− DC = 2.05 V,
and 16% for VP− DC = 2.65 V. The 1.35 to 2.65 V is the
full usable range of CTRL in Fig. 1 for this design, and
VP = 50 mV exceeds the VP voltage swing required during
feedback operation when Vae varies by 2 dB. Moreover, the
largest difference (16% at 2.65 V) is only in the extreme case
where the RF gain is forced to its minimum, resulting in
the biggest AM-AM profile change, hence the largest Vae
to represent Voe . A normal feedback operation does not
reach this extreme condition. Therefore, these relatively small
differences between Vae1 and Vae2 show that, for such VP
amplitudes and Vae ranges, the approximation is acceptable.
E. Limitations From the Approximation of the Gain Control
The limitations from this approximate model are analyzed
next. As the gain keeps increasing with VP , the Voe profile
(associated with the same constant Vae range in Fig. 6) deviates
too much from that of the AM-AM curve used to define G,
and thus relating the change in Voe to the quantity (Vae + Vae )
becomes less accurate. Therefore, the model in Fig. 5(b) is
applicable only for: 1) small excursions in Vae ; 2) operating
conditions where the d.c. component of the control signal VP
is very close to known quiescent values of VP used for the
power series characterization; and 3) small variations in VP .
In the design equations, P will be considered as a linear
function of VP , for a constant amplitude Vie representing
the average envelope amplitude of the RF signal Vi in Fig. 4,
at the power of interest. This approximation is analyzed next.
Simulation performed on the RF amplifier chain alone
shows that, for a given constant envelope amplitude Vieo , a very
small variation in VP yields an almost proportional variation
in the envelope signal gain G(VP ) defined as Voe (VP )/Vieo and
which is equal to GVp0 in the static condition where VP = 0,
Vie = Vieo . From this observation, the following approximation
can be made assuming small variations in VP :
Voe (Vp , Vie ) ∼
(2)
= G(Vp ) · Vie Vi e = Vi e o
G(Vp ) = m · Vp + GVp0
(3)
with m being the constant that reflects the VP to G(VP )
proportionality approximation. Substituting (3) into (2) yields
Voe (Vp , Vie ) = m · Vp · Vie + GVp0 · Vie Vi e =V i eo
(4)
and assuming F in Fig. 4 may be set to 1 for the purpose of
evaluating P, the following limit considerations and derivatives
apply and are useful (given that the signal Va in Fig. 4 does
not exist physically in the circuit) for the evaluation of the
parameter P through simulation or measurements on the RF
amplifier chain:
lim Vp → 0 : Vae (Vp , Vie ) F = 1
Vi e = Vi e o
Voe (Vp , Vie ) =
(5)
F =1
GVp0
Vi e = Vi e o
and thus
∂Vae (Vp , Vie ) P=
F = 1
∂Vp
Vi e = Vi e o
1
=
· m · Vieo G
Vp0
1
∂Voe (Vp , Vie ) =
·
F = 1
GVp0
∂Vp
Vi e = V i e o
F= 1
(6)
The dependence of P on Vieo in (6) indicates that despite
the limit consideration VP → 0 for the estimation of P, the
representation of the effects of a small gain increment in the
amplifier chain with the small incremental quantity Vae =
P·VP and with P constant will yield an increasing error as the
amplitude of Vie departs from Vieo as a result, for example, of
the modulation of the input signal. Nonetheless, because of the
drastic simplification it brings to the formulations, a constant
gain control parameter P will be assumed, and the uncertainty
due to the dependency of P on Vie will be mitigated through
the limitation of the peak-to-average ratio of the envelope
signal Vie . Thus, P will be approximated with
1
Voe ∼
P =
·
.
(7)
GVp0 Vp Vi e = Vi e o = constant, F = 1, Vp → 0
A numerical example based on the RF amplifier design described in [5], presented in a later section, shows that the above
approximations yield acceptable errors in the calculation of the
IMD benchmarking levels, in the context of estimating (with
conservative safeguards) the IMD performance requirements
for the circuit blocks in the error signal path.
F. Degree of Nonlinearity for the PA Block
A memoryless and weakly nonlinear system may be analyzed with power series [8], [9], [11]–[13] to relate an input
signal V (t) to the output signal Vout (t), that is
Vout (t) = a1 V (t) + a2 V 2 (t) + a3 V 3 (t)+a4 V 4 (t) + a5 V 5 (t) + ...
(8)
with an being scalar coefficients. The analysis of the distortion
introduced by the dynamic biasing of an RF amplifier through
envelope feedback operation may require a representation
of the RF amplifier chain with a high degree polynomial
function. The proposed formulations will be limited to a
fifth degree power series representation of the RF PA block
(G in Fig. 4), and the frequency spectrum of the benchmarking multitone signals limited to (ωc ± 4ωx ) as depicted
in Fig. 4, for conciseness in the formulations presented in
this paper. Note, however, that the proposed method may be
extended with equations using even higher degree polynomials and a broader multitone representation for enhanced
accuracy.
G. Derivation of the Equations
An expression of the form Vae (t)·cos(ωc t) is used to represent Va in Fig. 4, with Vae (t) describing the multitone envelope
information, and ωc the carrier frequency. Assuming an ideal
output band-pass filter at ωc to pass only the information
CONSTANTIN et al.: FORMULATIONS AND A COMPUTER-AIDED TEST METHOD FOR THE ESTIMATION OF IMD LEVELS
centered at ωc , it may be shown by expanding (8) with a fifth
degree polynomial that Vout (t) is in the form
3
5
3
5
Vout (t) = a1 Vae (t) +
a3 Vae
a5 Vae
(t) +
(t) cos ωc t.
4
8
(9)
The three-tone input signal (see the frequency spectrum of
Vi in Fig. 4) may be formulated as
Vi (t) = (J + 2K cos(ωx t)) cos(ωc t)
(10)
and thus the input envelope information may be expressed as
Vie (t) = J + 2K cos(ωx t).
(11)
In the same way, the band-limited output IMD benchmarking levels depicted in Fig. 4 may be formulated as
Voe (t) = L + 2M cos(ωx t) + 2N cos(2ωx t)
+2Q cos(3ωx t) + 2R cos(4ωx t).
(12)
The idealized assumptions made for the circuit blocks in
the error signal path (Section III-C) and the approximation for
the gain control allow expressing the instantaneous envelope
signal Vae (t) as well as the envelope error information Vee (t)
at the output of the ideal comparator (Fig. 4) in terms of
linear functions of Vie (t) and of Voe (t), using the low frequency
envelope signal gains H, E, P, and F, that is
Vae (t) = FVie (t) + EPVee (t)
(13)
Vee (t) = Vie (t) − HVoe (t)
(14)
and
with E defined by (1). Hence, (13) may be expanded to
Vae (t) = FVie (t) + EPVie (t) − EPHVoe (t).
(15)
Substituting (11), (12), and (14) into (13) yields the following general expression, which defines the envelope information
applied to the input of the PA block:
Vae (t) = U + V cos(ωx t) + W cos(2ωx t)
+X cos(3ωx t) + Y cos(4ωx t)
(16)
U = FJ + EP(J − HL)
(17)
V = 2(FK + EPK − EPHM)
(18)
W = −2EPHN
(19)
X = −2EPHQ
(20)
Y = −2EPHR.
(21)
with
The benchmarking multitone envelope information Voe (t)
was defined with (12), and the nonlinear processing of Vae (t)
by the PA block was formulated with (9). Accordingly, solving
for the closed loop, band limited inter-modulation products
present at the input and the output of the amplifier system
of Fig. 4 (and thereafter at any node in the system) under
1887
the assumptions made above requires equating the envelope
expressions from the two solutions, that is
L + 2M cos(ωx t) + 2N cos(2ωx t)+ 2Q cos(3ω
xt)
3
5
(t) + 58 a5 Vae
(t)
+2R cos(4ωx t) =a1 Vae (t) + 43 a3 Vae
(22)
with Vae (t) defined in (16)–(21).
After expansion of the right-hand side terms in (22), solving
for only the constant term and the cos(ωx ), cos(2ωx ), cos(3ωx ),
and cos(4ωx ) terms allows determining the frequency components L, M, N, Q and R in the amplifier system’s output
envelope spectrum, as a function of the multitone excitation
parameters J, K, and the system parameters a1 , a3 , a5 , F, E,
P, and H. Alternatively, the excitation levels J and K and
the feedback parameters that would be required in order to
meet some desired closed loop IMD goals at the output of
the amplifier may be determined. Either case requires solving
a system of five nonlinear equations and a certain number
of unknowns among the above 14 parameters. The resulting
system of equations is
Voe(L) − L = 0
Voe(M) − 2M = 0
Voe(N) − 2N = 0
Voe(Q) − 2Q = 0
Voe(R) − 2R = 0
(23)
with the five terms Voe (L), Voe (M), Voe (N), Voe (Q), and
Voe (R) used to calculate [with (23)] the envelope amplitudes
associated with the center frequency ωc and with the ωx , 2ωx ,
3ωx and 4ωx related mixing products, respectively, and which
are depicted at the output of the amplifier system of Fig. 4.
The general forms of these terms as functions of U, V, W, X,
and Y [which are defined with (17)–(21)]) are provided in the
Appendix. The correctness of these terms as solutions to the
expansion of the right-hand side polynomial of (22) may be
verified by subtracting the quantity
Voe(L) + Voe(M) cos(ωx t) + Voe(N) cos(2ωx t)
+Voe(Q) cos(3ωx t) + Voe(R) cos(4ωx t)
from this same polynomial [with Vae (t) expanded through (16)]
and observing that the result does not contain any term of
frequency inferior to 5ωx .
With the system of equations (23) solved, and thus with
J, K, L, M, N, Q, and R known as solutions to this system
of equations or any of them known as input variables, then
the general expression for the benchmarking IMD multitone
information at the output of the amplifier may be readily
obtained using (12).
From the substitution of (11) and (12) into (14), and using
the solutions to (23), the general expression representing the
benchmarking IMD multitone information at the output of the
ideal comparator of Fig. 4 may then be formulated as
Vee (t) = (J − HL) + 2(K − HM) cos(ωx t)
−2HN cos(2ωx t) − 2HQ cos(3ωx t)
−2HR cos(4ωx t)
(24)
and from (13), the general expression for the envelope signal
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at the input of the RF PA block of Fig. 4 may be formulated as
Vae (t) = J(F + EP) − EPHL
+(2K(F + EP) − 2EPHM) cos(ωx t)
−2EPHN cos(2ωx t) − 2EPHQ cos(3ωx t)
−2EPHR cos(4ωx t).
(25)
IV. Localized Linearity Analysis and Validation
of the RF Envelope Detectors’ Performances
A. Behavioral Parameters for Localized IMD Analysis
In this section, the IMD performance associated with the
envelope detector circuitry of Fig. 2 in a specific excitation
condition is analyzed and validated with the help of the proposed equations, in accordance with the test method described
in Section III-C. The small envelope signal feedback loop
parameters were extracted through simulation on the different
circuit blocks of the RFIC PA design presented in [5] in openloop conditions. The analysis is localized at the power level
of interest discussed in Section II, i.e., using a small input envelope power sweep in the vicinity of the −12 dBm threshold.
Note that all the behavioral model parameters that are
derived below could also be obtained exclusively through
open-loop measurements during an experimental design phase.
B. Small Envelope Signal Loop Parameters of the RFIC PA
The simulated detector conversion gain fD (Fig. 4) is equal
to 2.466 (7.84 dB). The comparator and error amplifier voltage
gains (C and Av ) are equal to 6.187 and 61.31, respectively.
Hence, the gain E defined by (1) and used in (17)–(21) is 935.
Simulation performed on the RF amplifier chain with a
continuous wave (CW) excitation at 1.88 GHz, a constant input
power of −12dBm, and with small variations of the gain
control voltage (Vp in Fig. 4) allows determining a gain control
parameter P equal to 0.045 [using (7)]. The gain H is 0.124,
and since no attenuation or amplification precedes the input
of the variable RF amplifier chain [5], F is set to 1.
C. Power Series Representation of the PA Block
Fig. 7 shows the simulated amplitude and phase responses
of the RF amplifier chain reported in [5] at low power, in the
hardware state where half the RF transistor array in both IS
and PS in Fig. 1 have been shut off.
The impedance matching (Zm, VAR Zm in Fig. 1) is the
same as designed in [5] and kept constant. It is well known
that impedance matching affects IMD performances [12], [14].
However, our equations do not use impedance matching as a
parameter, and our test method is applicable with any constant
Zm and VAR Zm. Hence, the study of IMD versus impedance
matching is not within the scope of this paper.
The output signal is the steady-state harmonic balance
solution for every amplitude step of the CW excitation. The
band-pass output matching network attenuates the harmonics
by 12 dB at 2ωc , 20 dB at 3ωc , and even more at 4ωc and
above. Thus, it will be assumed that Fig. 7 corresponds to the
1.88 GHz centered band-limited static AM-AM and AM-PM
responses associated with Va and Vout in Fig. 4.
Fig. 7. Amplifier block’s simulated band-limited input to output gain and
phase relationship, and a polynomial curve fitting based on a1 , a3 , and a5 .
TABLE I
Input Variables to the System of Equations (23) for the
Calculation of the Output IMD Benchmarking Levels
L, M, N, Q, and R Depicted in Fig. 4
J (mV)
68
K (mV)
8.5
F
1
P
E
0.045 935
H
a1
0.124 6.8436
a3
518
a5
−41899
Using a mathematical curve fitting tool, the fifth degree
power series was extracted to describe with sufficient accuracy the corresponding memoryless input-to-output voltage
relationship, i.e., with the scalar polynomial coefficients a1 =
6.8436, a3 = (3/4)·a3 = 388.5, and a5 = (5/8)·a5 = −26187,
and which allow defining the Vae (t) to Vout (t) relationship as
in (9). Coordinates that are solutions to the polynomial curve
fitting with a1 , a3 , and a5 are plotted in Fig. 7 to demonstrate
a satisfactory match to the simulated band-limited response.
The coefficients a1 , a3 , and a5 are used for solving the system
of equations (23).
Simulation allows determining that a 1.88 GHz continuous
wave input power of −12 dBm translates into an input signal
amplitude of 68 mV at the junction represented by Vi in Fig. 4.
For a localized linearity analysis in the power domain, the
excitation considered is limited to a 2 dB maximum peak-toaverage envelope voltage ratio (i.e., an average-to-minimum
ratio of ∼ 2.5 dB). Accordingly, J and K in (11) may be set
to 68 mV and 8.5 mV, respectively. Table I summarizes the
above defined input variables to the system of equations (23).
D. Solutions to the Equations and AM-PM Estimation
Solving the system of equations (23) with a mathematical
software (e.g., MATLAB) using the Voe (L ∼ R) terms given
in the Appendix and the input variables listed in Table I yields
the solutions for L, M, N, Q and R (Fig. 4). These solutions
allow calculating the IMD benchmarking levels at the output
of the ideal comparator using (24). They are summarized in
Table II.
CONSTANTIN et al.: FORMULATIONS AND A COMPUTER-AIDED TEST METHOD FOR THE ESTIMATION OF IMD LEVELS
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TABLE II
Calculated IMD Benchmarking Levels L, M, N, Q, R, Ve0 , Vex ,
Ve2x , Ve3x , and Ve4x (Depicted in Fig. 4), and Function of the
Variables of Table I
Vout Tones
Amplitude (mV)
Amplitude (dBmV)
(dBc) rel. to L
Ve Tones
Amplitude (mV)
Amplitude (dBmV)
(dBc) rel. to Vex
L
584.4
54.8
0
Ve0
−1.45e-3
−56.8
−31.2
M
N
68.8
−5.5e-3
36.8
−45.1
−18
−99.9
Vex
Ve2x
−0.053 1.37e-3
−25.6
−57.2
0
−31.6
Q
−6.9e-3
−43.2
−98
Ve3x
1.71e-3
−55.3
−29.7
R
−691e-6
−63.2
−118
Ve4x
171e-6
−75.3
−49.7
Note that Ve0 multiplied by E (Table I) represents a d.c.
offset that is generated through feedback as a gain compensation mechanism, and added to the gain control signal Vp
in Fig. 4. It was also discussed in Section III-C that the
approximation for the gain control element in Fig. 4 is based
on the assumption of small deviations in Vp from the d.c.
condition used for the AM-AM characterization. The Ve0 ·E
d.c. offset value may be used to determine a new quiescent
condition in the gain control signal for the extraction of a
new AM-AM response of the RF amplifier chain (Fig. 7) and
the associated polynomial coefficients. Therefore, a computeraided and automated iterative process that consists of solving
(23), determining new quiescent conditions for Vp based on
Ve0 ·E, extracting new polynomial coefficients, and solving (23)
again, until Ve0 is minimized, will ensure a minimum of error
in the IMD benchmarking levels. The calculated d.c. offset due
to Ve0 from Table II (i.e., −1.45 μV times 935 = −1.36 mV)
is very small compared to the 50 mV Vp used in the model
discussion in Section III-D. Therefore, the approximation for
the gain control element in Fig. 4 is applicable in this example.
Though it is assumed that the 1.6° (0.0279 rad.) peak
phase deviation shown in Fig. 7 does not affect the envelope
response, the PM side-bands at ωc ± 2ωx will be used
to represent an equivalent IMD benchmarking level that is
associated with N in Fig. 4 and scaled down by the factor
2·H [as per (24)], and thus referred to the output of the ideal
comparator. Using the general PM representation [18]
n=
∞
f (t) = A
Jn (β) cos(ωc + nωx )t
(26)
n= −∞
these side-band levels relative to L in Table II may be very
closely approximated [18] by setting n = 2 and β = 0.0279 in
β2
β2
Jn (β) = J−n (β) =
1−
(27)
8
12
which yields −80.2 dBc, i.e., −25.4 dBmV at Vout , and
−37.5 dBmV when referred to the output of the comparator.
E. Validation of the Solutions
The values reported in Table II, and which are solutions to
(23), may be verified as being the exact IMD solutions to the
amplifier system model shown in Fig. 4 (given the parameters
of Table I) by simulating its envelope response with the setup
shown in Fig. 8, using an excitation Vie defined by (11) with
Fig. 8.
Block diagram for the validation of the solutions in Table II.
Fig. 9. Simulation setup for the evaluation of the accuracy of the solutions
calculated with (23) and reported in Table II, using an RF excitation and the
complete RF amplifier chain used for the extraction of the curves in Fig. 7.
J = 68 mV, K = 8.5 mV and an arbitrary ωx value, and limiting
the harmonic content to 4ωx and below.
The simulation setup of Fig. 9 may also be used to estimate
the accuracy of the solutions of Table II, this time with respect
to the RF response of the RF amplifier chain. The variable
gain RF amplifier chain and the output matching circuit are
the same ones presented in [5]. The simulation was performed
with the same hardware state (defined by the logic states of
ISS and PSS in Fig. 1), the same quiescent level for the gain
control signal CTRL (Vp in Fig. 4), and the same bias and
load conditions used for the extraction of the AM and PM
responses shown in Fig. 7. A 68 mV, 1.88 GHz input carrier,
and an 8.5 mV, 1 kHz AM modulating signal were used.
LOAD1 and LOAD2 are used to maintain the same load
conditions associated with Fig. 7. Ideal AM demodulators are
used to extract the input and feedback envelope information.
An ideal shunt band-pass filter centered at ωc = 1.88 GHz
may be connected or disconnected from the output of the RF
amplifier chain. When connected, it does not affect the output
load or phase shift in the output signal Vout at ωc , but all
harmonic signal voltages on the Vout node are zeroed.
The simulation results from the setup of Fig. 9 are reported
in Table III. In the case with no filter there is a difference of
∼ 5.7 dB between the simulated N and Ve2x components and
the calculated N and Ve2x solutions from Table II. This difference may be attributed mainly to the IMD mixing products
falling on ωc ± 2ωx and that stem from the nonideal filtering
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TABLE III
Simulated Results From the Setup of Fig. 9
Tones
(dBmV)
No filter
(dBmV)
With filter
L
M
N
Ve2x
Peak Phase
Deviation
54.8 36.7 −50.8 −62.9 2.0°
PM Sideband
(dBmV) at
ωc ± 2ωx
−21.5
54.9 36.7 −44.2 −56.3 1.2°
−30.3
of the out-of-band tones that are centered across the harmonic
frequencies and reinserted in the feedback loop.
With the filter connected, the difference between the calculated and simulated results is reduced to 0.9 dB. This is an
expected result, since the ideal shunt band-pass filter limits the
voltage at the Vout node to only the envelope information that
is centered at ωc = 1.88 GHz (i.e., with no harmonics), which is
a condition that conforms with the band-limitation assumption
made when solving (23). Hence, the computation of the
IMD benchmarking levels with the proposed equations yields
better precision when analyzing systems where the out-of-band
frequency components at the Vout node are highly attenuated.
The residual 0.9 dB difference may be attributed to the approximation that is inherent to the modeling of the PA block
with a power series, and the approximation made in modeling
the gain control element. Therefore, it also indicates that the
approximation made with (7) yields an error that is acceptable.
In both cases, the simulated phase deviation due to AMPM conversion is slightly different from the 1.6° in Fig. 7
[and which corresponds to the −25.4 dBmV calculated with
(27)]. This is justified by the fact that the phase curve shown
in Fig. 7 relates only to variations in the input signal amplitude
under fixed bias conditions, whereas the AM-PM conversion
in the setup of Fig. 9 accounts also for the feedback dynamics
through to the gain control and biasing of the RF amplifier.
F. Comparison With the Detectors’ Linearity Performances
In conformity with the test method discussed in Section III,
we have performed a simulation on the two detector branches
of Fig. 2 as stand-alone circuit blocks in open-loop conditions,
in order to evaluate the IMD levels that would be added in the
differential signal between their outputs (Vdeti and Vdeto ) at the
frequency 2ωx . The simulation was carried out with different
values of a parameter α, which is used as a scaling factor
applied to all resistor values in the nonlinear compensation
network of the collector circuitry in the input envelope detector
(Fig. 2), as a simple means to emulate some level of hardware mismatch between the two detector branches. For this
simulation, the same input multitone excitation that was used
for solving (23) (i.e., from Table I) was applied to the input
detector, with the frequency spacing ωx set to 1 kHz. Though
the solutions from (23) (Table II) would allow including the
(ωc ± 3ωx ) and (ωc ± 4ωx ) frequency components in the
excitation used for the output envelope detector, the simulation
set-up was simplified by using ωc and (ωc ± ωx ) only, given
the very low levels of Q and R (Table II ) in this particular
case. The amplitudes used were the values L = 584.4 mV and
Fig. 10. IMD benchmarking levels that were calculated with the design
equations (horizontal lines), and a curve showing the simulated 2ωx IMD
level that would be added by the envelope detectors in the error signal path,
as a function of the hardware mismatch factor α.
M = 68.8 mV, multiplied by 0.124 in order to account for the
attenuation H in Fig. 4.
Note also that with these excitation conditions, if the envelope detectors were perfectly linear, there would be no 2ωx
IMD product at their outputs.
The curve shown in Fig. 10 describes the simulated openloop 2ωx IMD level that would be added by the detectors
(but scaled down by their 7.84 dB conversion gain to allow a
comparison when referred to the output of the ideal comparator
of Fig. 4) with the above excitation conditions, and as a
function of α. The −57.2 and −37.5 dBmV benchmarking
values calculated with (23) (i.e., from Table II) and (27),
respectively, are also represented by straight lines.
The −57.2 dBmV benchmarking value corresponds to (ωc ±
2ωx ) AM-AM output mixing products that are as low as
−99.9 dBc (Table II), and thus allows concluding that the
AM-AM linearity performance that would be achievable at
the −12 dBm threshold, given the feedback loop parameters
and the nonlinearities in the RF amplifier chain only, is likely
to be well within the requirements to meet adequate overall
linearity performances. This is a direct implication of the
fact that −12 dBm lies well within the effective feedback
operation range [5], where AM-AM linearization capability
is maximized.
However, the detectors’ 2ωx IMD curve allows concluding
that even when the detectors are perfectly matched (α = 1),
the nonlinearity they introduce in the vicinity of the −12 dBm
threshold for the compensation mechanism would significantly
impact the AM-AM linearity performances of the amplifier,
since the 2ωx IMD product level that they would generate
(−47 dBmV), when referred to the differential output of the
ideal comparator of Fig. 4, is at a significantly higher level than
the calculated closed loop and ideal case benchmarking level
(−57.2 dBmV). In addition, this curve provides a valuable
insight into how this linearity degradation is worsened with
an increasing hardware mismatch.
The detectors’ optimum linearity performance is at an α
value slightly higher than 1. This can be explained by a
distortion cancellation effect. Slightly different (i.e., α =1)
CONSTANTIN et al.: FORMULATIONS AND A COMPUTER-AIDED TEST METHOD FOR THE ESTIMATION OF IMD LEVELS
nonlinear processing of the two different sets of multitone excitation signals described above may result in output common
mode 2ωx IMD products that approach each other in amplitude
and phase for a particular α condition, and producing a
differential quantity that tends to vanish.
The 19.7 dB difference between the predominant
−37.5 dBmV benchmarking level (AM-PM) and the
theoretically best achievable AM-AM IMD performance
(−57.2 dBmV) gives a valuable indication of the extent to
which the AM-PM conversion through the RF amplifier chain
alone dictates the PA linearity performance.
Despite the linearity degradation that is caused by the
detectors, the IMD levels being considered here tend to
validate, in fact, the detector design as being suitable for this
application and in these particular operating conditions, given
the very low benchmarking values (i.e., −57.2 dBmV). This
is also in agreement with the ACPR specification compliant
experimental result obtained at this power level, as shown in
Fig. 3 (i.e., −51 dBc at Pout = 4 dBm).
1891
+ 24UV2 X2 + 24UV2 Y2 +24UVW2 X+24U2 W2 Y + 24UX2 Y2
+ 24UW2 X2 + 24UW2 Y2 + 8UV3 X + 12V2 W2 Y + 6W2 Y3
+ 2W3 X2 + 48U2 VWX)
Voe(M) = a1 V
+ (9/16)a3 (V3 + 4UVW + 4UWX + 2VWY + 2WXY + 4UXY
+ 4VU2 +W2 X+V2 X+2VW2 +2VX2 +2VY2 ) + (25/128)a5 (2V5
+ 24UWX3 +48UWXY2 +48V2 WXY+48UVWX2 + 24UW3 X
+ 48VWX2 Y + 24U2 V3 + 24UX3 Y + 16VW3 Y + 18W2 XY2
+ 12WX3 Y + 5V4 X + 24UVX2 Y + 32U3 WX + 72UV2 WX
+ 24U2 W2 X + 48U2 WXY + 16V3 WY + 24VW2 Y2 + 16U4 V
+ 32U3 VW+48UW2 XY+48U2 VX2 +72UV2 XY + 24U2 V2 X
+ 48U2 VWY + 16W3 XY + 6V2 X3 + 32U3 XY + 8UV3 Y
+ 48U2 VW2 + 6VY4 + 32UV3 W + 48UVWY2 + 48U2 VY2
+ 12V2 XY2 + 12VWY3 + 30VW2 X2 + 30V2 W2 X + 24UXY3
+ 24UVW3 + 30VX2 Y2 + 4W4 X + 12V3 Y2 + 12V3 X2
V. Conclusion
The proposed formulations and computer-aided IMD
test method facilitated investigations of the performance
requirements for the circuit blocks in the error signal path of
an envelope feedback RFIC PA, as well as their performance
validation. They allowed estimation of IMD levels throughout
the PA structure during closed-loop feedback operation,
based on open-loop characterization data (simulated or
measured). As demonstrated by the application example
provided, similar analyses performed at carefully chosen
input power levels and peak-to-average envelope voltage
ratios provided valuable insights into the system level design.
The use of the proposed equations and computer-aided test
method during experimental work were particularly helpful to
circumvent the difficulties encountered when experimentally
probing an envelope feedback RFIC PA during closed-loop
operation. The proposed approach for estimating closed-loop
performances based on open-loop measurements may be
extended to envelope feedback based automatic gain control
of PAs or transmitters. The equations would then have to be
reformulated with complex polynomials to include memory
effects, with the appropriate envelope feedback bandwidth.
APPENDIX
Voe(L) = a1 U + (3/4)a3 U3
+ (3/16)a3 (6VXY + 6VWX + 3W2 Y+6UW2 + 6UX2 + 6UY2
+ 6UV2 + 3V2 W) + (5/8)a5 U5 + (25/128)a5 (V4 Y + 6UV4
+ 12V2 WX2 + 48U2 VXY + 12V3 XY + 24VW2 XY + 16U3 W2
+ 16U3 Y2 + 16U3 X2 + 6UX4 + 6UW4 + 6UY4 + 12VWX3
+ 12W2 X2 Y + 4W4 Y + 12VW3 X + 24UV2 WY + 6WX2 Y2
+ 12V3 WX + 4V4 W + 12VX3 Y
+ 14V3 W2 + 6VW4 + 48UVW2 Y + 6W2 X3 + 12WXY3
+ 6VX4 + 2X3 Y2 )
Voe(N) = a1 W
+ (9/16)a3 (W3 + 4U2 W + 4UVX + 4UWY + 2VXY + 2VWX
+ 2UV2 + 2V2 W + 2WX2 + 2WY2 + X2 Y + V2 Y)
+ (25/128)a5 (2W5 + 30V2 WX2 + 48VW2 XY + 24V2 WY2
+ 24UV3 X + 12V2 W3 + 16U3 V2 + 8UV4 + 6WX4 + 24U2 W3
+ 12VX3 Y + 24UV2 X2 + 24UV2 Y2 +36VWXY2 + 48UVXY2
+ 20V3 WX + 16VW3 X + 6X2 Y3 + 72UVW2 X + 24WX2 Y2
+ 4V4 Y + 48U2 VWX + 32U3 VX + 48U2 WX2 + 24UVX3
+ 36UV2 W2 +48U2 WY2 +24V2 X2 Y+24U2 X2 Y + 48UV2 WY
+ 24U2 V2 Y + 16U4 W + 12UW2 X2 + 32U3 WY + 6WY4
+ 7V4 W + 24UWY3 + 96UVWXY + 12UX2 Y2 + 12VXY3
+ 4X4 Y + 12W3 X2 + 14W3 Y2 + 48U2 V2 W + 48U2 VXY
+ 48UWX2 Y + 12VWX3 + 18W2 X2 Y + 24V2 W2 Y + 16V3 XY
+32UW3 Y + 6V2 Y3 )
Voe(Q) = a1 X
+ (3/16)a3 (V3 +12U2 X+6VWY + 12UVW+12UVY + 6WXY
+ 6V2 X + 3VW2 + 6W2 X + 6XY2 + 3X3 ) + (25/128)a5 (V5
+ 12V2 X3 + 24U2 X3 + 48UVW2 Y + 48U2 V2 X + 48UW2 XY
+ 36VWX2 Y+8U2 V3 +2X5 +30V2 XY2 + 8UW3 X + 48U2 XY2
+ 18VW2 Y2 +18VW2 X2 +12W3 XY+4VW4 +6V4 X+72UVX2 Y
+ 48V2 WXY+48U2 WXY+16V3 WY+72UVWX2 +32U3 VW
+ 24U2 VW2 + 24UVW3 + 24UV2 XY + 16WX3 Y + 30V2 W2 X
+ 12VXY3 + 24UV2 W2 + 6V2 X2 Y + 24U2 V2 W + 12V2 WY2
+ 48UV2 WX+32U3 VY+6XY4 +16U4 X+48U2 VWY+24UVY3
+ 12W2 X3 + 48UVWY2 + 6VX2 Y2 + 48U2 W2 X + 16VW3 Y
+ 12VWY3 + 24UV3 W + 24W2 XY2 + 12X3 Y2 + 24UWXY2
+ 24VWXY2 + 24UWX2 Y + 16U3 V2 + 6V2 W3 + 48UVWXY
+ 6W4 X + 4V3 Y2 +6V3 X2 + 10V3 W2 + 24UV3 Y + 12WXY3 )
1892
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 2012
Voe(R) = a1 Y
+ (9/16)a3 (Y3 + 4U2 Y + 4UVX + 2VWX + 2X2 Y + 2V2 Y
+ V2 W+WX2 +2UW2 +2W2 Y)+(25/128)a5 (2Y5 + 48U2 V2 Y
[17] S. Russo, “Measurement and simulation of electrothermal effects
in solid-state devices for RF applications,” Ph.D. dissertation, Dept.
Biomed., Electron. Telecommun. Eng., Univ. degli studi di Napoli
Federico II, Napoli, Italy, 2010, p. 125.
[18] F. G. Stremler, Introduction to Communication Systems, 3rd ed. Reading,
MA: Addison-Wesley, 1992, pp. 306–314, 735, 736.
+ 2UV4 + 16U3 W2 + 6X4 Y + 24U2 WX2 + 36VWXY2
+ 24UWX2 Y + 36VW2 XY + 96UVWXY + 4WX4 + 24UV3 X
+ 16VW3 X + 24U2 V2 W + 16V3 WX + 24UW2 X2 + 12V2 Y3
+ 12W2 Y3 + 48U2 VWX + 18WX2 Y2 + 32U3 VX + 24UVX3
+ 8V3 XY + 24UV2 W2 + 24W2 X2 Y + 18V2 WY2 + 48UV2 WY
+ 48U2 W2 Y + 4VX3 Y + 16U4 Y + 30V2 X2 Y + 24V2 W2 Y
+ 48UVW2 X + 7W4 Y + 8UW4 + 24V2 WX2 + 72UVXY2
+ 48U2 X2 Y + 36UW2 Y2 + 24U2 Y3 + 8V2 W3 + 12UV2 X2
+ 12X2 Y3 + 4V4 W + 6V4 Y + 6W3 X2 + 12VWX3 )
References
[1] E. Acar and S. Ozev, “Low-cost characterization and calibration of RF
integrated circuits through I–Q data analysis,” IEEE Trans. Comput.Aided Des. Integr. Circuits Syst., vol. 28, no. 7, pp. 993–1005, Jul.
2009.
[2] E. S. Erdogan and S. Ozev, “A multisite test solution for quadrature
modulation RF transceivers,” IEEE Trans. Comput.-Aided Des. Integr.
Circuits Syst., vol. 30, no. 9, pp. 1421–1425, Sep. 2011.
[3] H.-G. Stratigopoulos, S. Mir, and A. Bounceur, “Evaluation of analog/RF test measurements at the design stage,” IEEE Trans. Comput.Aided Des. Integr. Circuits Syst., vol. 28, no. 4, pp. 582–590,
Apr. 2009.
[4] H.-C. Tseng, “A hybrid evolutionary modeling/optimization technique
for collector-up/down HBTs in RFIC and OEIC modules,” IEEE Trans.
Adv. Packag., vol. 30, no. 4, pp. 823–829, Nov. 2007.
[5] N. G. Constantin, P. J. Zampardi, and M. N. El-Gamal, “Automatic
hardware reconfiguration for current reduction at low power in RFIC
PAs,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 6, pp. 1560–1570,
Jun. 2011.
[6] N. G. Constantin, P. J. Zampardi, and M. N. El-Gamal, “A gated
envelope feedback technique for automatic hardware conditioning of
RFIC PAs at low power levels,” in Proc. IEEE MTT-S Int. Microw.
Symp. Dig., Jun. 2007, pp. 139–142.
[7] P. Wambacq and W. Sansen, Distortion Analysis of Analog Integrated
Circuits. Dordrecht, The Netherlands: Kluwer Academic, 1998, chs. 3,
4, 6, 7, pp. 458–469.
[8] S. A. Maas, Nonlinear Microwave Circuits. Norwood, MA: Artech
House, 1988, chs. 2, 4.
[9] H. Park, D. Baek, K. Jeon, and S. Hong, “A predistortion linearizer using
envelope-feedback technique with simplified carrier cancelation scheme
for class-A and class-AB power amplifiers,” IEEE Trans. Microw. Theory
Tech., vol. 48, no. 6, pp. 898–904, Jun. 2000.
[10] Recommended Minimum Performance Standards for cdma2000 Spread
Spectrum Mobile Stations, release C, ver. 2.0, Feb. 2006 [Online].
Available: http://www.3gpp2.org
[11] Q. Gu, RF System Design of Transceivers for Wireless Communications.
Berlin, Germany: Springer Science, 2005, pp. 29–37, 449–455.
[12] S. C. Cripps, RF Power Amplifiers for Wireless Communications. Norwood, MA: Artech House, 1999, pp. 181–197.
[13] J. Rogers and C. Plett, Radio Frequency Integrated Circuit Design.
Norwood, MA: Artech House, 2003, pp. 23–35.
[14] J. Vuolevi and T. Rahkonen, Distortion in RF Power Amplifiers. Boston,
MA: Artech House, 2003.
[15] T. C. Kleckner, “Self-heating and isothermal characterization of heterojunction bipolar transistors,” M.A.Sc. thesis, Dept. Electric. Comput.
Eng., Univ. British Columbia, Vancouver, BC, Canada, 1998.
[16] M. Busani, R. Menozzi, M. Borgarino, and F. Fantini, “Dynamic thermal
characterization and modeling of packaged AlGaAs/GaAs HBTs,” IEEE
Trans. Comp. Packg. Tech., vol. 23, no. 2, pp. 352–359, Jun. 2000.
Nicolas G. Constantin (S’04–M’09) received the
B.Eng. degree from the École de Technologie
Supérieure (ÉTS), University of Quebec, Montreal,
QC, Canada, in 1989, the M.A.Sc. degree from
the École Polytechnique de Montréal, Montreal, in
1994, and the Ph.D. degree from McGill University,
Montreal, in 2009, all in electrical engineering.
He is currently an Assistant Professor with ÉTS.
He was engaged in the design of microwave
transceivers for point-to-point radio links from 1989
to 1992. He was an RF Design Engineer with NEC,
Victoria, Australia, from 1996 to 1998, working on the development of RF and
microwave transceivers for mobile telephony. He was a Design Engineer with
Skyworks Solutions, Inc., Newbury Park, CA, from 1998 to 2002, where he
developed GaAs HBT RF integrated circuit (RFIC) power amplifiers (PAs) for
wireless communications. With Skyworks Solutions, Inc., he was also actively
involved in research on smart biasing and efficiency improvement techniques
for RFIC PAs, and holds three patents on the subject. His current research
interests include design and test of RFICs for wireless communications,
including PAs and front-end modules.
Dr. Constantin was a recipient of the Doctoral Scholarship from the Natural
Science and Engineering Research Council of Canada, and the Doctoral
Scholarship from ÉTS.
Kai H. Kwok (S’88–M’01–SM’07) received the
B.A.Sc. degree in computer engineering, and the
M.A.Sc. and Ph.D. degrees in electrical engineering
from the University of Waterloo, Waterloo, ON,
Canada, in 1993, 1995, and 2001, respectively. His
Masters thesis was focused on the boron diffusion
characteristics in germanium-implanted silicon, and
his Ph.D. dissertation was focused on the fabrication,
analysis, and optimization of silicon-germanium heterojunction bipolar transistors (HBTs).
He joined GaAs Technology Platform of Conexant
Systems (now Skyworks Solutions, Inc.), Newbury Park, CA, in 2001 as a
Process Development Engineer responsible for physical device simulations
for improving the performance of the InGaP HBTs. He has been an Electrical Staff Engineer with the Device Design, Modeling, and Technology
Characterization Group, Skyworks Solutions, Inc., since 2003, where he is
responsible for the passive device modeling for the III–V HBT technologies
used in wireless applications.
Hongxiao Shao (M’00) received the Ph.D. degree in
condensed matter physics from Rutgers University,
Piscataway, NJ, in 1992.
Since 2000, he has been with the Advanced Development Group, Conexant/Mindspeed/Skyworks Solutions, Inc., Newbury Park, CA, responsible for
design kit and design environment development for
semiconductor process technologies and package
technologies. He joined Rice University, Houston,
TX, as a Post-Doctoral Researcher in 1992, where
he focused on the studies of dynamic effects of impurities at surfaces, and quantum dot systems. In 1993, he joined the Quantum
Electronic Structure Institute, University of California, Santa Barbara, as a
Research Fellow, continuing his studies on quantum electronics and research
on stochastic processes of surface morphological phenomena. In 1995, he
joined the Research and Development Staff, EDS, Orange County, CA, and
then HP/Agilent, Westlake Village, CA, working on computer-aided design
and engineering solutions for electronic systems. His current research interests
include simulation technologies in time and frequency domains for electronic
systems, and the system level front to back design methodology.
CONSTANTIN et al.: FORMULATIONS AND A COMPUTER-AIDED TEST METHOD FOR THE ESTIMATION OF IMD LEVELS
Cristian Cismaru (M’96–SM’05) received the M.S.
degree in control systems from “Gheorghe Asachi”
Technical University, Iasi, Romania, in 1993, and
the M.S. and Ph.D. degrees in electrical engineering
from the University of Wisconsin, Madison, in 1997
and 1999, respectively, where he studied plasma
processing-induced damage to semiconductor devices.
Since 2001, he has been a Principal Engineer with
the Device Design and Technology Characterization
Group, Skyworks Solutions, Inc., Newbury Park,
CA, where he is responsible for the development, test, and characterization
of III–V and silicon-germanium device technologies used in power amplifier
applications. He joined Conexant Systems (now Skyworks Solutions, Inc.),
Irvine, CA, in 1999 as a Process Development Engineer responsible for
design, layout, physical verification, electrical simulation, and test of the
process control modules used in submicron technologies. His research yielded
a method for reducing the charging damage to submicron MOS devices by
employing the vacuum ultraviolet radiation created by processing plasmas,
which later became the subject of a U.S patent.
1893
Peter J. Zampardi (S’93–M’96–SM’02) received
the B.E. degree in engineering physics from the
Stevens Institute of Technology, Hoboken, NJ, in
1986, the M.S. degree in applied physics from
the California Institute of Technology (Caltech),
Pasadena, in 1988, and the Ph.D. degree from the
University of California, Los Angeles, in 1997.
At Caltech, he studied MBE-grown GaAs/AlGaAs
structures and investigated tellurium clustering in
ZnSe:Te for use in visible light emitters.
He joined the Ring Laser Gyroscope Test Laboratory, Rockwell, in 1988, where he was responsible for development and testing
of ring laser gyros for use in inertial measurement units. He joined the Optics
Technology Department, Rockwell Science Center, Thousand Oaks, CA, in
1990, where he developed processes and procedures for the characterization
and fabrication of IR etalon filters. In 1991, he joined the High Speed
Circuits Department where he performed device and circuit development,
characterization, and modeling of GaAs, InP, SiGe HBTs, MESFET, HEMT,
BiFET, and RTD technologies. In 1999, he led the technical development
of SiGe RF models for the Analog and Mixed Signal Foundry Business,
IBM, Burlington, VT. Since 2000, he has been with Conexant (now Skyworks
Solutions, Inc.), Newbury Park, CA, where he is the Technical Director of the
Device Design, Modeling, and Characterization Group. The groups’ interests
are technologies, characterization, modeling, and circuit design for wireless
applications. He has authored and co-authored over 170 papers related to
circuits and devices, and two book chapters. He holds six U.S. patents.
Dr. Zampardi actively participates in several IEEE technical conference
committees.