Abstract — RF circuits exhibit several distinguishing characteristics that make them difficult to simulate

Gaussian-elimination based shooting-Newton methods, a commonly used approach for computing steady-state solutions, grow in computational complexity like N³ where N is the number of circuit equations. Just using iterative methods to solve the shooting-Newton equations results in an algorithm which is still order N because of the cost of calculating the dense sensitivity matrix. Below, a matrix-free Krylov-subspace approach is presented, and the method is shown to reduce shooting-Newton computational complexity to that of ordinary transient analysis. Results from several examples are given to demonstrate that the matrix-free approach is more than ten times faster than using iterative methods alone for circuits with as few as 400 equations.

Abstract — The principles employed in the development of modern RF simulators are introduced and the various techniques currently in use, or expected to be in use in the next few years, are surveyed. Frequencyand time-domain techniques are presented and contrasted, as are steady-state and envelope techniques and large- and small-signal techniques. I. RF CIRCUITS The increasing demand for low-cost mobile communication systems has greatly expanded the need for simulation algorithms that are both efficient and accurate when applied to RF communication circuits. RF circuits have several unique characteristics that are barriers to the application of traditional circuit simulation techniques. Over the last decade, researchers have developed many special purpose algorithms that overcome these barriers to provide practical simulation for RF circuits, often by exploiting the very characteristic that represented the barrier to traditional methods. Despite dramatic progress, the average design cycle of an RFIC is still twice the length of that for other ICs found in a communication system, such as a cellular phone. This represents a significant practical barrier to integration of the RF and baseband sections of a transceiver onto a single chip. Clearly, more progress is necessary. This paper is a overview of RF simulation methods that seeks to provide an understanding of how the various methods address the RF simulation problem, and how they relate to each other. It begins by describing the unique characteristics of RF circuits. The basic solution methods of transient analysis, harmonic balance, and shooting methods are presented and contrasted. Small-signal analysis versions of both harmonic balance and shooting methods are covered. Composite methods are next. These methods apply the base

RF integrated circuit designers make extensive use of simulation tools which perform nonlinear periodic steady-state analysis and its extensions. However, the computational costs of these simulation tools have restricted users from examining the detailed behavior of complete RF subsystems. Recent algorithmic de- velopments, based on matrix-implicit iterative methods, is rapidly changing this situation and providing new faster tools which can easily analyze circuits with hundreds of devices. In this paper we present these new methods by describing how they can be used to accelerate finite-difference, shooting-Newton, and harmonic- balance based algorithms for periodic steady-state analysis.

Widely-separated time scales appear in many electronic circuits, making traditional analysis di#cult or impossible if the circuits are highly nonlinear. In this paper, an analytical formulation and numerical methods are presented for treating strongly nonlinear multi-rate circuits e#ectively. Multivariate functions in the time domain are used to capture widely separated rates e#ciently, and a special partial differential equation #the MPDE# is shown to relate the multivariate forms of a circuit's signals. Time-domain and mixed frequency-time simulation algorithms are presented for solving the MPDE. The new methods can analyze circuits that are both large and strongly nonlinear. Compared to traditional techniques, speedups of more than two orders of magnitude, as well as improved accuracy, are obtained. 1 Introduction Consider a fast #e.g., 1Ghz# pulse train mixed with a slow #1Khz# sinusoid, or the same pulse train but with a slowlyvarying duty cycle. These are instances of multi-rat...

Data structures such as *BMDs, HDDs, and K*BMDs provide compact representations for functions which map Boolean vectors into integer values, but not floating point values. In this paper, we propose a new data structure, called Multiplicative Power Hybrid Decision Diagrams (*PHDDs), to provide a compact representation for functions that map Boolean vectors into integer or floating point values. The size of the graph to represent the IEEE floating point encoding is linear with the word size. The complexity of floating point multiplication grows linearly with the word size. The complexity of floating point addition grows exponentially with the size of the exponent part, but linearly with the size of the mantissa part. We applied *PHDDs to verify integer multipliers and floating point multipliers before the rounding stage, based on a hierarchical verification approach. For integer multipliers, our results are at least 6 times faster than *BMDs. Previous attempts at verifying floating point multipliers required manual intervention. We verified floating point multipliers before the rounding stage automatically.

The design of the radio frequency (RF) section in a communication integrated circuit (IC) is a challenging problem. Although several computer-aided analysis tools are available for RFIC design, they are not effectively used, because there is a lack of understanding about their features and limitations. These tools provide fast simulation of RFIC's. However, no single tool delivers a complete solution for RFIC design. This paper describes the shortcomings of conventional SPICE-like simulators and the analyses required for RF applications with an emphasis on accurate and efficient simulation of distortion and noise. Various analysis methods, such as harmonic balance, shooting method, mixed frequency-time methods, and envelope methods, that are currently available for RFIC simulation are presented. Commercial simulators are compared in terms of their functionalities and limitations. The key algorithmic features and the simulator-specific terminology are described. Index Terms---Circuit simulation, cyclostationary noise, distortion, envelope method, frequency-domain methods, harmonic distortion, intermodulation, linear time-varying analysis, mixed frequency-time methods, mixer noise, noise, periodic steady-state, phase noise, quasiperiodic steady-state, RFIC simulation, SPICE harmonic balance, shooting method, time-domain methods. I.

Simulating the transient behavior of switching power converter circuits is computationally expensive because these circuits are docked at a frequency whose period is orders of magnitude smaller than the time interval of interest to the designer. It is possible to reduce the simulation time without compromising much accuracy by exploiting the proper D that the behavior of switching converters in a given high-frequency clock ccle is similar, but not identical, to the behavior in the preceding and following cycles. In particular, the envelope of the high-frequency clock can be followed by accurately computing the circuit behavior over occasional cycles. In this paper the implementation of an envelope-following method that is particularly efficient for open-loop switching power converters with fixed clock frequencies is described, and results demonstrating the methnd's effectiveness are presented.

Matrix-implicit Krylov-subspace methods have made it possible to efficiently compute the periodic steady-state of large circuits using either the time-domain shooting-Newton method or the frequencydomain harmonic balance method. However, the harmonic balance methods are not so efficient at computing steady-state solutions with rapid transitions, and the low-order integration methods typically used with shooting-Newton methods are not so efficient when high accuracy is required. In this paper we describe a Time-Mapped Harmonic Balance method (TMHB), a fast Krylovsubspace spectral method that overcomes the inefficiency of standard harmonic balance in the case of rapid transitions. TMHB features a non-uniform grid to resolve the sharp features in the signals. Results on several examples demonstrate that the TMHB method achieves several orders of magnitude improvement in accuracy compared to the standard harmonic balance method. The TMHB method is also several times faster than the standard harmonic balance method in reaching identical solution accuracy.

Interdependent characterization of latch setup/hold times is a core component of techniques for pessimism reduction via Setup/Hold Interdependence Aware Static Timing Analysis (SHIA-STA) [1], [2]. We present an efficient and novel method for such characterization, by formulating the interdependent setup-hold time problem as an underdetermined nonlinear equation h(τs,τh) =0, which we derive from the latch’s state-transition function. We solve this equation numerically using a Moore-Penrose Newton method. Further, we use null-space information from the Newton’s Jacobian matrix to efficiently find constant-clock-to-Q contours (in the setup/hold time plane), via an Euler-Newton curve tracing procedure. We validate the method on TSPC and C 2 MOS registers, obtaining speedups of more than 20 × over prior approaches while achieving superior accuracy. This speedup increases linearly with the precision with which curve tracing is desired. In view of the importance and large computational expense of latch characterization in industry today, the new technique represents a significant enabling technology for dramatically speeding up industrial timing closure flows.