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Computation of X-Parameters Using Multipoint Moment Expansion
Abstract
Recently X-parameters
1
have been proposed as an extension to the concept of S-parameters in order to create a macromodel that captures nonlinear behavior. In this paper, we develop a relation between the X-parameters and circuit moments. Higher order multidimensional moments are then used to efficiently compute the X-parameters as a piecewise polynomial approximation with respect to input power. The proposed approach is shown to be accurate and CPU efficient compared to the brute force methods.
... This includes time domain approaches such as Latency Insertion Method [2]. Another type of simulation method is the use of Harmonic Balance moments to compute the X-parameters for circuits excited with single or multiple input frequencies [3][4][5]. This approach allows for the computation of a complete set of X-parameters for the given system. ...
... This paper presents a general adjoint based method for computing the required subset of X-parameters. The proposed approach is based on the fact that the Xparameters were reformulated in [3,4] as a sensitivity with respect to a parameter β. The proposed method is shown to be as accurate while considerably faster for the computation of some subsets of X-parameters. ...
... But for the case of a mixer circuit, there are two large signals present, one of them is applied at the Local Oscillator i.e. port 1, and the other large signal is at the Radio Frequency input port i.e. port 2. Therefore, the set of input waves A 1,1 and A 2,1 are chosen as the linearization points, where A 1,1 and A 2,1 are the input waves for the fundamental tone at the local oscillator (port 1) and RF input (port 2) respectively. The output waves of the linearized system are described by the following equation [3], ...
... Consequently, we present in this paper a method of calculating the first derivative of the X-parameters with respect to the value of a linear element in the circuit. This is derived from the Taylor series expansion of the HB and X-parameter moments, similar to [4][5][6][7]. This results in a closed form value for the linear sensitivity of the X-parameters with regards to changes in a linear circuit element's magnitude. ...
... B dc ∈ R N contains the contributions of all dc sources and B ∈ R N contains the Fourier coefficients of all the smallsignal sources including any additional small signal at the fundamental tone. The relationship between the Fourier coefficients and the X-parameters, as shown in [7], is ...
... The following subsections present a methodology to derive a closed form expression of the first derivative of the X-parameters with respect to λ for sensitivity analysis. The methodology uses the moments of the Taylor series expansions, as in [4][5][6][7] to derive the sensitivity. ...
... However, a nonlinear steady-state solution was still required at each operating point, and the computation of the X-parameters themselves still required multiple solutions of large and relatively dense linear systems of equations, a significant computational bottleneck in the presence of multitone RF signals. In [5], a new approach for the computation of the X-parameters based on the multi-dimensional high-order moments, as well as a multipoint expansion using a binary search scheme was proposed. The proposed method significantly reduced the number of HB solutions required. ...
... The proposed method significantly reduced the number of HB solutions required. However, the method in [5] is limited to circuits and packages with single-tone inputs. In RF simulations, circuits with multi-tone signals such as mixers are quite common, and therefore the ability to efficiently compute multi-tone X-parameters becomes necessary. ...
... Circuits with multi-tone inputs present significant computational challenges due to the presence of a significantly increased number of harmonic and intermodulation frequencies in the system formulation. In this paper, we will show how 1 X-parameters is a registered trademark of Agilent Technologies the method proposed in [5] can be extended to cover the computation of multi-tone X-parameters and highlight the required modifications to the computation algorithm of both the multi-dimensional moments using multi-point expansions in addition to the binary search algorithm. We will show that the method maintains its CPU cost advantage over the traditional brute force approach. ...
The efficient generation of X-parameter macromodels for nonlinear RF circuit blocks that capture the essential nonlinear behavior of the system has become increasingly useful and even necessary. Recent work on X-parameter computations have provided clear relations between the X-parameters and circuit moments. In this paper, we show how higher order multi-dimensional moments evaluated using multi-point expansions can be used to efficiently compute the X-parameters for two-tone inputs, which are commonly found in general RF circuits.
The polyharmonic distortion parameters of a nonlinear structure or circuit block, often referred to as X-parameters,* relate the different harmonics of the incident waves to those of the reflected waves in the frequency domain. Recently, a closed form derivative-based approach for calculating the X-parameters sensitivity with respect to linear circuit elements has been developed for a given input amplitude level. In this manuscript, we build on those results and present an efficient and accurate moments based method for the sensitivity analysis with respect to both linear and nonlinear circuit parameters, parameterized with respect to the input amplitude. The proposed method allows for the accurate and efficient computation of X-parameter sensitivity terms without the need for CPU expensive sweeps or brute force perturbation.
This paper presents an extended nonlinear black-box behavioral model in the frequency domain, automated experimental identification of the model from large-signal vector network measurements, and extensive experimental validation of its application to real microwave ICs. The model is a broad-band extension of the multi-harmonic "linearized scattering function" theory. The characterization is based on large-signal vector network measurements where harmonic perturbations are applied, in phase and in quadrature, to a component excited at the input by a large amplitude tone. The experiment design and model generation are simple and highly automated. The derived model is shown to be valid for both small and large amplitude drive signals, correctly predict even and odd harmonics, and simulate, accurately, load-pull behavior far from 50 ohms. The model is implemented in Agilent ADS.
For more than a quarter of a century, microwave engineers have had the benefit of a foundation of mutually interacting components of measurement, modeling, and simulation to design and test linear components and systems. S-parameters are perhaps the most successful behavioral models ever. They have the powerful property that the S-parameters of individual components are sufficient to determine the S-parameters of any combination of those components. S-parameters of a component are sufficient to predict its response to any signal, provided only that the signal is of sufficiently small amplitude. We have presented the PHD modeling approach. It is a black-box frequency-domain model that provides a foundation for measurement, modeling, and simulation of driven nonlinear systems. The PHD model is very accurate for a wide variety of nonlinear characteristics, including compression, AM-PM, harmonics, load-pull, and time-domain waveforms. The PHD model faithfully represents driven nonlinear systems with mismatches at both the fundamental and harmonics. This enables the accurate simulation of distortion through cascaded chains of nonlinear components, thus providing key new design verification capabilities for RF and microwave modules and subsystems
We propose a new method for the steady state analysis of
periodically excited nonlinear microwave circuits. It is a modified and
more efficient form of Newton-Raphson iteration based harmonic balance
(HB) technique. It solves the convergence problems of the HB technique
at high drive levels. The proposed method makes use of the parametric
dependence of the circuit responses on the excitation level. It first
computes the derivatives of the complex amplitudes of the harmonics with
respect to the excitation level efficiently and then finds the Pade
approximants for the amplitudes of the harmonics using these derivatives
The nodal method has been widely used for formulating circuit equations in computer-aided network analysis and design programs. However, several limitations exist in this method including the inability to process voltage sources and current-dependent circuit elements in a simple and efficient manner. A modified nodal analysis (MNA) method is proposed here which retains the simplicity and other advantages of nodal analysis while removing its limitations. A simple and effective pivoting scheme is also given. Numerical examples are used to compare the MNA method with the tableau method. Favorable results are observed for the MNA method in terms of the dimension, number of nonzeros, and fill-ins for comparable circuit matrices.
We present an optimal experiment design methodology and a superior and fully automated model generation procedure for identifying a class of broad-band multiharmonic behavioral models in the frequency domain. The approach reduces the number of nonlinear measurements needed, minimizes the time to generate the data from simulations, reduces the time to extract the model functions from data, and when used for simulation-based models, takes maximum advantage of specialized simulation algorithms. The models have been subject to extensive validation in applications to real microwave integrated circuits. The derived model is valid for both small and large amplitude drive signals, correctly predicts even and odd harmonics through cascaded chains of functional blocks, simulates accurately load-pull behavior away from 50 Ω, and predicts adjacent channel power ratio and constellation diagrams in remarkably close agreement to the circuit model from which the behavioral model was derived. The model and excitation design templates for generating them from simulations are implemented in Agilent Technologies' Advanced Design System.
As the complexity of radio frequency circuits increases, the generation of macromodels for circuit blocks that can be used in system simulations has become increasingly useful and even necessary. Recently X-parameters1 have been proposed as an extension to the concept of S-Parameters in order to create a macromodel that captures some of the key nonlinearities in the circuit. In this paper, an efficient semi-analytical approach for the computation of X-parameters is presented. More specifically, a clear relation between the X-parameters and the circuit moments is developed, which in turn facilitates the application of moments to further improve the efficiency of X-parameter generation.
The main idea of black-box macromodeling is to approximate the dynamic behavior of complex systems in terms of lowcomplexity models or equivalent circuits. Such compact models can be derived through robust numerical algorithms, such as the Vector Fitting scheme, starting from frequency-or time-domain responses of the system, and without any specific knowledge of its internal structure. The excellent accuracy that can be achieved, combined with the reduced size of the models, has led to a widespread adoption of this approach in several electrical and electronic applications, allowing designers to perform numerical simulations at the system level with high efficiency. This paper reviews the basics of black-box macromodeling and illustrates several application scenarios that are relevant for the EMC community, including Signal and Power Integrity, lossy transmission line modeling, electromagnetic full-wave simulation, network equivalencing and transformer modeling.
Loewner Matrix interpolation based macromodeling methods can generate an accurate macromodel for large and complex structures. However, there is less flexibility in the choice of the macromodel order due to the need to ensure passivity. In this paper, an algorithm is proposed to obtain a reduced order model based on Loewner Matrix interpolation while still maintaining the passivity of the macromodel. This is achieved by first relaxing the passivity requirement during the order selection process, then a new passivity enforcement scheme with significantly improved speed and convergence properties is applied. The proposed approach results in a significantly reduced macromodel size while maintaining accuracy and passivity conditions. The
efficiency of the new approach is shown in the examples.
Volterra series representation is a powerful mathematical model for nonlinear circuits. However, the difficulties in determining higher-order Volterra kernels limited its broader applications. In this work, a systematic approach that enables a convenient extraction of Volterra kernels from X-parameters is presented. A concise and general representation of the output response due to arbitrary number of input tones is given. The relationship between Volterra kernels and X-parameters is explicitly formulated. An efficient frequency sweep scheme and an output frequency indexing scheme are provide. The least square linear regression method is employed to separate different orders of Volterra kernels at the same frequency, which leads to the obtained Volterra kernels complete. The proposed Volterra series representation based on X-parameters is further validated for time domain verification. The proposed method is systematic and general-purpose. It paves the way for time domain simulation with X-parameters and constitutes a powerful supplement to existing blackbox macro-modeling methods for nonlinear circuits.
Sensitivity analysis is a critically important aspect in the simulation of integrated circuits. It allows the designer to perform vital tasks such as circuit optimization and design space exploration. Recently, an efficient moments-based approach was presented for computing key distortion figures of merit in radio frequency circuits from the harmonic balance equations, without the need to perform a harmonic balance simulation. In this paper, we show how the moments method can also be used to perform an efficient and accurate sensitivity analysis through the computation of the expansions of the moment vectors with respect to a circuit parameter, i.e., the multidimensional moments. We show that the sensitivities of key distortion figures of merit, including the 1 dB compression point and different intercept points, at all outputs in the circuit can be obtained with minimal additional computation cost to that of the original algorithm, thereby providing insight into the circuit performance.
X-parameters have been shown to have a wide array of applications in the modeling of nonlinear devices and systems. This work analyzes the large-signal portion of an X-parameter model of a buffer to determine the extent of its nonlinearity and use for modeling input/output buffers.
Nonlinear distortion analysis is one of the most computationally challenging aspects in the simulation of radio frequency circuits. Recently, an efficient moments-based approach to determine the third-order intercept point from the general harmonic balance equations, without the need to perform a harmonic balance simulation, was presented. In this article, the efficient computation of another important performance figure of merit, that of the 1-dB compression point, using moments analysis is presented. In addition, the computations of the second- and fifth-order intercept points are also presented. As a result, moments analysis becomes a comprehensive approach for quantifying nonlinear distortion using several key distortion figures of merit with a significant speed-up over traditional harmonic balance methods. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2014.
X parameters have been shown to have a wide array of applications in the modeling of nonlinear devices and systems. In this work, the polyharmonic distortion (PHD) model and the latency insertion method (LIM) are combined to generate X parameters describing the nonlinear relationship between power waves. This technique leverages the speed and convergence advantages of the LIM simulation method to generate frequency-domain models. Results are compared with those of other methods.
Accurate computer simulation has helped reduce the cost and time required to complete many new integrated circuit designs. This is because a design can be corrected and its performance tuned faster and more economically using computer simulation than by repeated fabrication and testing of prototypes. However, to tune the performance of many analog and microwave integrated circuit designs, prototyping is still used because the commonly available computer simulation programs cannot simulate efficiently the test scenarios of interest. The main difficulty simulating general analog circuit designs is computing steady-state quantities, such as harmonic distortion, for circuits with a widely spread response spectrum. And in the specific case of microwave designs, this problem is compounded because microwave circuits typically include linear time-invariant distributed devices such as dispersive transmission lines.
Obtaining the value of the third order intercept point using traditional simulation techniques typically requires a nonlinear steady state analysis with multitone inputs, which is very computationally expensive. In this paper, a new method is presented for the computation of the third order intercept point. Using the proposed approach, the necessary Volterra kernels are computed directly from the harmonic balance equations. The only computation cost is that of solving a set of sparse linear equations. Furthermore, only one input tone is required in this case, which greatly reduces the size of the equations and thus the computation cost.
Linearity is one of the main design requirements for RF circuits, and one of the main figures of merit used to measure linearity is the third-order intercept point (IP3). However, obtaining the IP3 using traditional harmonic balance simulation requires multitone inputs, which significantly increases the computational costs. On the other hand, analytical approaches based on obtaining closed-form expressions for Volterra functional series are too complex to be fully automated for arbitrary circuit topologies. In this paper, a method for the numerical evaluation of the Volterra kernels required for obtaining the IP3 is proposed. The proposed approach is applied directly to the harmonic balance equations and is thus independent of circuit topology and type of nonlinearity. Furthermore, it is shown that the computation cost of this approach is significantly less than a harmonic balance simulation.
Computing the steady-state response of large nonlinear circuits is
becoming a key simulation requirement due to the rapid market growth of
RF silicon integrated circuits. In this paper, we describe a nonlinear
circuit reduction algorithm for finding the steady-state response. The
proposed algorithm uses a congruent transformation-based technique to
reduce the harmonic-balance equations into a much smaller set of
equations. The main feature of the reduced circuit is that it shares
with the original one a certain number of the derivatives with respect
to the RF input power, steady-state analysis is then carried out on the
reduced circuit instead of the original circuit