* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Formulations and a Computer-Aided Test Method for the
Alternating current wikipedia , lookup
Sound reinforcement system wikipedia , lookup
Mains electricity wikipedia , lookup
Scattering parameters wikipedia , lookup
Transmission line loudspeaker wikipedia , lookup
Spectral density wikipedia , lookup
Control theory wikipedia , lookup
Immunity-aware programming wikipedia , lookup
Pulse-width modulation wikipedia , lookup
Resistive opto-isolator wikipedia , lookup
Dynamic range compression wikipedia , lookup
Audio power wikipedia , lookup
Electronic engineering wikipedia , lookup
Switched-mode power supply wikipedia , lookup
Signal-flow graph wikipedia , lookup
Analog-to-digital converter wikipedia , lookup
Public address system wikipedia , lookup
Rectiverter wikipedia , lookup
Control system wikipedia , lookup
Opto-isolator wikipedia , lookup
Negative feedback wikipedia , lookup
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- c 2012 IEEE 0278-0070/$31.00 1882 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 1883 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. 1884 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 1885 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 1886 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 2012 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 1888 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 2012 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 1889 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 1890 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 2012 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.