Pole-zero Identification Research Articles

Characterization Technique to Reveal Critical Resonances in Nonlinear RF Circuits

The presence of critical resonances can significantly impact the stable and robust operation of microwave amplifiers, leading to spurious oscillations in the worst case. In this work, a technique based on noise measurements performed with a spectrum analyzer is presented to experimentally characterize these resonances. Broadband white noise is injected into the input of the circuit, so the critical resonances are clearly revealed above the noise floor of the measuring instrument, without the need of exigent settings and far from the bifurcation point. In addition to detecting potentially risky resonances, the actual position of the critical poles on the pole-zero map can be estimated from these noise measurements by applying pole-zero identification techniques. The noise injection technique has been tested in two nonlinear prototypes in microstrip technology: an analog frequency divider and an <inline-formula> <tex-math notation="LaTeX">${L}$ </tex-math></inline-formula>-band amplifier.

Two-Level Stability Analysis of Complex Circuits

A new methodology is proposed for the small- and large-signal stability analysis of complex microwave systems, containing multiple active blocks. It is based on a calculation of the system characteristic determinant that ensures that this determinant does not exhibit any poles on the right-hand side (RHS) of the complex plane. This is achieved by partitioning the structure into simpler blocks that must be stable under either open-circuit (OC) or short-circuit (SC) terminations. Thus, the system stability is evaluated using a two-level procedure. The first level is the use of pole-zero identification to define the OC- or SC-stable blocks, which, due to the limited block size, can be applied reliably. In large-signal operation, the OC- or SC-stable blocks are described in terms of their outer-tier conversion matrices. The second level is the calculation and analysis of the characteristic determinant of the complete system at the ports defined in the partition. The roots of the characteristic determinant define the stability properties. The Nyquist criterion can be applied since, by construction, the determinant cannot exhibit any poles in the RHS. In addition, one can use pole-zero identification to obtain the values of the determinant zeroes. Because the determinant is calculated at a limited number of ports, the analysis complexity is greatly reduced.

Odd-Mode Instability Analysis of f T-Doubler Hybrid Power Amplifiers Based on GaN-HEMT

This brief is aimed to investigate and resolve the odd-mode instability of a hybrid high-frequency power amplifier (PA) based on the f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> T</sub> -doubler technique. A description of odd-mode stability based on pole-zero identification is performed to provide insight into the causes of the odd-mode oscillation. The unwanted right half poles (RHPs) at frequencies less than the unity gain frequency, resulting from the circuit configuration, corrupt the PA operation. Reducing the length of the interconnects and tuning the values of the circuit elements is a conventional approach to get rid of the RHPs. However, in practice, in hybrid technology, there is poor control over interconnects. Besides, changing the circuit elements to prevent the odd-mode instability results in performance degradations such as a reduction in power gain, output power, power added efficiency (PAE), and unity current gain frequency ( f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> T</sub> ). A proposed f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> T</sub> -doubler PA using Wilkinson combiner with a calculated odd-mode resistor is exploited to overcome the odd-mode oscillation and relax the trade-off between odd-mode stability and performance of the PA. The measurement of the modified f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> T</sub> -doubler GaN-HEMT PA with Wilkinson combiner demonstrates a broadband PA with a stable behavior.

Pole-Zero Identification: Unveiling the Critical Dynamics of Microwave Circuits Beyond Stability Analysis

The term pole-zero identification refers to obtaining the poles and zeros of a linear (or linearized) system described by its frequency response. This is usually done using optimization techniques (such as least squares, maximum likelihood estimation, or vector fitting) that fit a given frequency response of the linear system to a transfer function defined as the ratio of two polynomials [1], [2]. This kind of linear system identification in the frequency domain has numerous applications in a wide variety of engineering fields, such as mechanical systems, power systems, and electromagnetic compatibility. In the microwave domain, rational approximation is increasingly used to obtain black-box models of complex passive structures for model order reduction and efficient transient simulation. An extensive bibliography on the matter can be found in [3]-[6]. In this article, we focus on a different application of pole-zero identification. We review the different ways in which pole-zero identification can be applied to nonlinear circuit design, for power-amplifier stability analysis, and more. We provide a comprehensive view of recent approaches through illustrative application examples. Other uses for rational-approximation techniques are beyond the scope of this article.

Stability and Bifurcation Analysis of Multi-Element Non-Foster Networks

A stability and bifurcation analysis of multi-element non-Foster networks is presented, illustrated through its application to non-Foster transmission lines. These are obtained by periodically loading a passive transmission line with negative capacitors, implemented with negative-impedance converters (NICs). The methodology takes advantage of the possibility to perform a stability analysis per subintervals of the perturbation frequency. This will allow an independent analytical study of the low-frequency instability, from which simple mathematical criteria will be derived to prevent bias-network instabilities at the design stage. Then, a general numerical method, based on a combination of the Nyquist criterion with a pole-zero identification of the individual NIC, will be presented, which will enable the detection of both low- and high-frequency instabilities. A bifurcation analysis of the multi-element non-Foster structure will also be carried out, deriving the bifurcation condition from a matrix-form formulation of the multi-element structure. The judicious choice of the observation ports will enable a direct calculation of all the coexisting bifurcation loci, with no need for continuation procedures. These bifurcation loci will provide useful insight into the global-stability properties of the whole NIC-loaded structure.

Stability Analysis of Digital Microwave Power Amplifiers

This paper presents a stability analysis of microwave power amplifiers (PAs) driven by binary pulse trains, as in the case of class-S PAs. First, using a simplified digital PA test bench in class-D configuration, different qualitative behaviors are obtained when varying the pulsewidth, including subharmonic and incommensurable oscillations. The mechanisms affecting the stability properties are studied with a harmonic balance-based formulation, by means of pole-zero identification and bifurcation detection. A sufficiently high number of harmonic components must be considered, together with the Krylov decomposition for an efficient computation of the inverse of the Jacobian matrix. It is demonstrated that, when varying the pulsewidth, the distinct pairs of complex-conjugate poles may shift to the right-hand side of the complex plane and, therefore, lead to different kinds of unstable behavior. This phenomenon is related to the dependence of the critical poles on the average value of the input signal. Boundaries of the various types of unstable behavior are traced in the plane defined by the pulse repetition rate and pulsewidth, using bifurcation detection techniques. All the predicted phenomena have been confirmed experimentally. In a second step, the algorithms derived from the simple class-D circuit are transferred to study the stability of a more complex tri-band class-S amplifier. It has been analyzed versus the input bitrate, obtaining a fully stable behavior that has been validated experimentally.

Prediction of odd-mode instabilities under output mismatch effects

A methodology is presented to predict odd-mode instability in power amplifiers under output mismatch effects, as in the case of amplifiers connected to an antenna. This kind of instability is often observed in multi-device configurations, due to their symmetry properties. Unlike the single-ended situation, there is a cancellation of odd multiples of the oscillation frequency at the circuit output, so there is no impact of the load-impedance values at the sideband frequencies. The odd-mode instability only depends on the impedance terminations at the fundamental frequency and its harmonic terms, and can only be detected within the circuit unstable loop, instead of the antenna-connection terminals. The possible unstable modes are related with the eigenvectors of an outer tier conversion matrix accounting for the symmetry properties of the circuit topology. Under sufficient low-pass filtering of the amplifier output network, the analysis parameters can be limited to the magnitude and phase of the reflection coefficient at the fundamental frequency. This analysis involves a computationally efficient graphical technique to detect potential instabilities and a bifurcation-detection method to determine the instability boundaries in the Smith chart. The two main types of instability from periodic regime are considered, respectively associated with incommensurable and subharmonic oscillations. Results have been validated through pole-zero identification and experimental measurements.

Hysteresis and Oscillation in High-Efficiency Power Amplifiers

Hysteresis in power amplifiers (PAs) is investigated in detail with the aid of an efficient analysis method, compatible with commercial harmonic balance. Suppressing the input source and using, instead, an outer-tier auxiliary generator, together with the Norton equivalent of the input network, analysis difficulties associated with turning points are avoided. The turning-point locus in the plane defined by any two relevant analysis parameters is obtained in a straightforward manner using a geometrical condition. The hysteresis phenomenon is demonstrated to be due to a nonlinear resonance of the device input capacitance under near optimum matching conditions. When increasing the drain bias voltage, some points of the locus degenerate into a large-signal oscillation that cannot be detected with a stability analysis of the dc solution. In driven conditions, the oscillation will be extinguished either through synchronization or inverse Hopf bifurcations in the upper section of the multivalued curves. For an efficient stability analysis, the outer-tier method will be applied in combination with pole-zero identification and Hopf-bifurcation detection. Departing from the detected oscillation, a slight variation of the input network will be carried out so as to obtain a high-efficiency oscillator able to start up from the noise level. All the tests have been carried out in a Class-E GaN PA with measured 86.8% power-added efficiency and 12.4-W output power at 0.9 GHz.

Generalized Stability Criteria for Power Amplifiers Under Mismatch Effects

Potential instability of power amplifiers (PAs) under mismatch effects is analyzed, with emphasis on the ease and generality of application of the stability criteria. The methodology is based on the evaluation of a large-signal version of the $\mu$ factor, considering mismatch effects in the fundamental frequency and three relevant sidebands: the baseband, the lower sideband and the upper sideband. This requires an outer-tier scattering-type conversion matrix of order 3 $\times$ 3 to be obtained, with the rest of sideband equations acting as an inner tier. It is taken into account that the circuit behaves nonlinearly with respect to the termination at the fundamental frequency. The consideration of three sidebands will enable the prediction of the two major forms of large-signal instability: incommensurable oscillations and frequency divisions by two. The analysis is preceded by an evaluation of the circuit own stability properties (proviso) under open and short circuit terminations at the sidebands, for all possible values of the termination at the fundamental frequency. Three different $\mu$ factors can be defined between any two ports of the scattering matrix. The analysis of the relationships between these factors and their continuity properties will allow the derivation of a single number able to characterize the PA potential instability for each fundamental-frequency termination. Results have been exhaustively validated with independent circuit-level simulations based on pole-zero identification and with measurements, using a variable output load and loading the PA with an antenna.

Stability Analysis of Power Amplifiers Under Output Mismatch Effects

Novel criteria are presented for small- and large-signal stability analysis of microwave circuits under output mismatch effects. They are based on concepts from bifurcation theory, which enable the derivation of general equations for the stability boundaries that depend on the termination load. The boundary calculation does not require any optimization of the circuit parameters. Instead, it is based on the direct calculation of an admittance function obtained through circuit linearization about either the dc solution, in the small-signal stability analysis, or the periodic solution, in the large-signal one, with low computational cost. As will be demonstrated both with homotopy and bifurcation concepts, the Hopf locus obtained in this way delimits a stable region and a potentially unstable one. In a large-signal stability analysis, the sidebands that influence the stability properties are identified, which depend on the particular amplifier topology and its filtering effects. The stability boundaries are then calculated with the conversion matrix approach for all load impedance values at the relevant harmonic and sideband frequencies. For the derivation of absolute stability conditions, a new scattering type matrix is proposed describing the circuit response from the output reference plane with respect to the two sidebands about the input carrier. The new methodologies and formulations have been validated by independent simulations using pole-zero identification, and by measurements.

Stability and Phase-Noise Analysis of Pulsed Injection-Locked Oscillators

An investigation of the stability and phase noise of oscillators driven with pulsed signals is presented. The aim is to generalize the theory of sinusoidal injection-locked oscillators to the case of oscillators driven with multiharmonic sources. As will be shown, under the pulse injection, a multiresonance phenomenon occurs in the circuit, leading to broad ultra-subharmonic synchronization bands. The stability phenomena delimiting these bands are analyzed through the detection of the oscillation-burst maxima, in the time domain, and the results are validated with pole-zero identification. The rich nonlinear dynamics is investigated, with detailed explanations of the various bifurcations detected. The phase-noise response of the pulse injection-locked oscillator is analyzed and related to the stability properties of the synchronized steady state. The investigation is illustrated through application to a pulse injection oscillator at 5.7 GHz.

Experimental Characterization of Stability Margins in Microwave Amplifiers

This paper proposes a method for the experimental estimation of the stability margins in microwave amplifiers. The approach is based on measuring a closed-loop frequency response representing the linearization of the circuit about a steady-state solution. Critical poles of the amplifier are then obtained by applying conventional pole-zero identification techniques to the measured frequency response. As circuit parameters are modified, the evolution of these critical poles on the complex plane provides a practical way to assess the robustness of the design regarding its stability. Two types of common instabilities in microwave amplifiers are studied: low-frequency bias oscillations and parametric oscillations. For the low-frequency oscillations, the approach proposes the inclusion of an observation RF port into the amplifier bias path to experimentally obtain the critical poles of the circuit from a reflection coefficient measurement. Pole-placement techniques are then applied to increase the stability margin of detected critical resonances. For the parametric oscillations, pole-zero identification is applied to a frequency response obtained from a mixer-like characterization equivalent to the measurement of a “hot” reflection coefficient. The methodology is applied to two amplifier prototypes: an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L$</tex></formula> -band field-effect transistor amplifier and a dual-mode WiFi-WIMAX amplifier that exhibit different kinds of unstable behavior.

Systematic Approach to the Stabilization of Multitransistor Circuits

This paper proposes a systematic approach for the elimination of spurious oscillations in circuits with multiple active elements. The procedure is based on detecting the sensitive parts (nodes or branches) of the complex circuit at which we can have a strong control of the dynamics responsible for the instability. To do so, stability analyses based on pole-zero identification are performed at multiple observation ports. Once sensitive locations are detected, stabilization is achieved by adding series or shunt stabilization networks at the suitable node or branch. Standard techniques from linear control theory (pole placement strategies) are used to automatically calculate the values of the stabilization elements ensuring circuit stability. Here, the methodology is applied to the stabilization of a two-stage <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ku</i> -band high-power amplifier for active antenna space applications.

Parametric oscillations in distributed power amplifiers

Parametric oscillations in high-efficiency distributed power amplifiers are analysed. Large-signal stability verification based on pole-zero identification is proposed in order to understand the origin of the instability and to choose the optimal value and place of the stabilisation resistor that ensures circuit stability while minimising degradation of circuit performance. The work is illustrated through the stabilisation of a three-stage high-efficiency LDMOS distributed power amplifier.

Complete Stability Analysis of Multifunction MMIC Circuits

This paper describes a systematic methodology for complete stability analysis of nonlinear microwave multifunction circuits. The proposed strategy has two different stages: the stability analysis of a nominal steady-state solution and the use of continuation techniques to efficiently determine the unstable operation ranges. The stability analysis is demanding due to the multiple loops contained in the large multifunction circuit. The first step is to check the possible fulfillment of the oscillation startup conditions at different circuit nodes followed by pole-zero identification. Given the complexity of the circuit topology, a systematic technique is necessary for the selection of the observation nodes. This has been applied at both the lumped-element schematic and the layout levels. These stability analyses have been carried out at small-signal (linear) and large-signal (nonlinear) since the multifunction circuit includes a nonlinear mixer. In the case of instability, the origin of the oscillation and its characteristics are analyzed versus the critical circuit parameters through the application of continuation techniques to the steady-state oscillatory solution. Moreover, sensitivity yield analysis and variations of environmental conditions combined with the stability techniques have also been taken into account and integrated into the design cycle. The proposed systematic approach has been successfully applied to determine and correct an oscillation of a multifunction monolithic-microwave integrated-circuit converter. It has also been proven in other multifunction circuits in the same way.

Nonlinear Design Technique for High-Power Switching-Mode Oscillators

A simple nonlinear technique for the design of high-efficiency and high-power switching-mode oscillators is presented. It combines existing quasi-nonlinear methods and the use of an auxiliary generator (AG) in harmonic balance. The AG enables the oscillator optimization to achieve high output power and dc-to-RF conversion efficiency without affecting the oscillation frequency. It also imposes a sufficient drive on the transistor to enable the switching-mode operation with high efficiency. Using this AG, constant-power and constant-efficiency contour plots are traced in order to determine the optimum element values. The oscillation startup condition and the steady-state stability are analyzed with the pole-zero identification technique. The influence of the gate bias on the output power, efficiency, and stability is also investigated. A class-E oscillator is demonstrated using the proposed technique. The oscillator exhibits 75 W with 67% efficiency at 410 MHz

Efficient scheme of pole-zero system identification based on multilayer neural network

A novel scheme to identify adaptive pole-zero filters, based on the multilayer neural network approach, is proposed using the conventional back propagation algorithm. The performance of the neural network is studied in a system identification application and simulation results are presented showing identical performance but computational superiority over a frequency domain adaptive pole-zero filter reported in the literature.

A recursive algorithm for linear system identification

This paper deals with the pole-zero identification of a linear system from a measured input-output record. It is shown that the minimization of a modified version of the squared Kalman equation error can be implemented by an order recursive algorithm in the time domain. The algorithm is based on the Gram-Schmidt orthogonalization of intertwined Krylov sequences involving a skew self-adjoint linear operator.

Speech analysis by homomorphonic prediction

Linear prediction (LPC) is a generally accepted method for obtaining all-pole speech representations. However, in many situations (e.g., nasalization studies) spectral zeros are important and a more general modeling procedure is required. Unfortunately the need for pitch synchronization has limited the success of available techniques. This talk deals with a new approach to pole-zero identification, based on homomorphic prediction, which avoids the synchronization problem. Homomorphonic prediction is a three step process. Homomorphic deconvolution is used to obtain a minimum-phase estimate of the vocal tract impulse response. Such a signal by definition is properly time synchronized. Linear prediction is applied to this waveform to identify its poles. Next, the LPC “residual” (error signal) is computed by inverse filtering. This signal contains the information about the zeros. Its z transform is approximated by a polynomial either through a weighted least-squares procedure (as in Shanks' method) or by spectral inversion followed by a second evaluation on real speech data will be presented. [Supported in part by ARPA, NSF, and the Fannie and John Hertz Fonudation.]