CPU time (h:min:sec)—simulation of the circuit depicted in Figure 3.
This chapter is devoted to the discussion of a hybrid frequency-time CAD tool especially designed for the efficient numerical simulation of nonlinear electronic radio frequency circuits operating in an aperiodic slow time scale and a periodic fast time scale. Circuits driven by envelope-modulated signals, in which the baseband signal (the information) is aperiodic and has a spectral content of much lower frequency than the periodic carrier, are typical examples of practical interest involving such time evolution rates. The discussed method is tailored to take advantage of the circuits and signals heterogeneity and so will benefit from the time-domain latency of some state variables in the circuits. Because the aperiodic slowly varying state variables are treated only in time domain, the proposed method can be seen as a hybrid scheme combining multitime envelope transient harmonic balance based on a multivariate formulation, with a purely time-step integration scheme.
- partial differential equations
- numerical simulation
- radio frequency circuits
- time-frequency analysis
In the last two decades, radio frequency (RF) and microwave system design has been found as a significant part of the electronic semiconductor industry’s portfolio. Over the years, the necessity of continuously providing new wireless systems’ functionalities and higher transmission rates, as also the need to improve transmitters’ efficiency, has been gradually reshaping wireless architectures. Heterogeneous circuits combining baseband blocks, digital blocks, and RF blocks, in the same substrate, are commonly found today. Hence, RF and microwave circuit simulation has been conducted to an increasingly challenging scenario of heterogeneous broadband and strongly nonlinear wireless communication circuits, presenting a wide variety of slowly varying and fast changing state variables (node voltages and branch currents). Thus, RF and microwave design has been an important booster for numerical simulation and device modeling development.
In general, waveforms processed by wireless communication systems can be expressed by a high-frequency RF carrier modulated by some kind of slowly varying baseband aperiodic signal (the information signal). Therefore, the evaluation of any relevant information time window requires the computation of thousands or millions of time instants of the composite modulated signal, turning any conventional numerical time-step integration of the circuits’ systems of differential algebraic equations highly inefficient. However, if the waveforms do not require too many harmonic components for a convenient frequency-domain representation, this category of circuits can be efficiently simulated with hybrid time-frequency techniques. Handling the response to the slowly varying baseband information signal in the conventional time step by time step basis, but representing the reaction to the periodic RF carrier as a small set of Fourier components (a harmonic balance algorithm for computing the steady-state response to the carrier), hybrid time-frequency techniques are playing an important role in RF and microwave circuit simulation.
Beyond overcoming the signals’ time-scale disparity, the partitioned time-frequency technique discussed in Section 3.2 is also able to efficiently simulate highly heterogeneous RF networks, by splitting the circuits into different subsets (blocks) and computing their state variables with distinct numerical schemes.
2. Theoretical background material
2.1. Mathematical model of an electronic circuit
Dynamic behavior of an electronic circuit can be modeled by a system of differential algebraic equations (DAE) involving electric voltages, currents and charges, and magnetic fluxes. The DAE system can, in general, be formulated as
This system of (1) can be constructed from a circuit description using, for example, nodal analysis, which involves applying Kirchhoff currents’ law to each node in the circuit, and applying the constitutive or branch equations to each circuit element. Hence, it represents the general mathematical formulation of lumped problems. However, as reviewed in , this DAE circuit model formulation can also include linear distributed elements. For that, the distributed devices are substituted by their lumped-element equivalent circuit models, or are replaced, as whole sub-circuits, by reduced order models derived from their frequency-domain characteristics. It must be noted that the substitution of distributed devices by lumped-equivalent models is especially reasonable when the size of the circuit elements is small in comparison to the wavelengths, as is the case of most emerging RF technologies integrating digital high-speed CMOS baseband processing and RFCMOS hardware in the same substrate.
2.2. Transient simulation
Obtaining the solution of (1) over a specified time interval with a specific initial condition
2.3. Steady-state simulation
Although SPICE-like computer programs (which were initially conceived to compute the transient response of electronic circuits) are still widely used nowadays, RF and microwave designers’ interest normally resides on the steady-state response. The reason for that is some properties of the circuits are better described, or simply only defined, in steady-state (e.g., harmonic or intermodulation distortion, noise, power, gain, impedance, etc.). Initial value solvers, as linear multistep methods, or Runge-Kutta methods, which were tailored for finding the circuit’s transient response, are not adequate for computing the steady-state because they have to pass through the lengthy process of integrating all transients, and expecting them to vanish.
Computing the periodic steady-state response of an electronic circuit can be formulated as finding out a starting condition (left boundary),
In order to numerically solve (2), a solution that simultaneously satisfies the differential system and the two-point periodic boundary condition has to be computed. A particular technique has been found especially useful for RF circuit simulation: the
2.4. Harmonic balance
Harmonic balance (HB) [6–8] handles the circuit, its excitation and its state variables in the frequency-domain. Because of that, it benefits from allowing the direct inclusion of distributed devices (like dispersive transmission lines), or other circuit elements described by frequency-domain measurement data. Frequency-domain methods differ from time-domain steady-state techniques in the way that, instead of representing waveforms as a collection of time samples, they represent those using coefficients of sinusoids in trigonometric series. As a consequence, under moderate nonlinearities, the steady-state solution is typically achieved much more easily in the frequency domain than in the time domain.
In periodic steady-state, any stimulus
The system of (4) can be rewritten as
2.5. Multivariate formulation
The multivariate formulation is a powerful strategy that emerged in the late 1990s, playing an important role in RF circuit simulation today. It was first introduced by Brachtendorf et al.  as a sophisticated derivation of quasi-periodic harmonic balance, followed by Roychowdhury , who demonstrated that the multivariate formulation can be an efficient strategy to analyze circuits running on distinct time scales. The multivariate formulation uses multiple time variables (artificial time scales) to describe the multirate behavior of the circuits. Thus, it is suitable to describe typical multirate regimes of operation present in RF and microwave systems, as is the case of circuits handling amplitude and/or phase-modulated signals, quasiperiodic signals, or any other kind of multirate signals containing a periodic component.
The main achievements of the multivariate formulation are due to the fact that multirate signals can be represented much more efficiently if they are defined as functions of two or more time variables (artificial time scales), i.e., if they are defined as multivariate functions [11–16]. Therefore, as we see in Section 2.5.2, circuits will be no longer described by ordinary differential algebraic equations in the one-dimensional time
2.5.1. Multivariate representations
The multivariate (multidimensional) strategy is easily illustrated by applying it to a bi-dimensional problem (two distinct time scales). So, let us consider, for example, an amplitude-modulated RF carrier of the form
Consider now the following bidimensional definition for
The univariate form,
2.5.2. Multirate partial differential algebraic equations’ systems
Let us consider a general nonlinear RF circuit described by the differential algebraic equations’ system of (1), and let us suppose that this circuit is driven by the envelope-modulated signal of (12). Considering the above stated, we are able to reformulate the excitation
The mathematical relation between (1) and (15) establishes that if and satisfy (15), then the univariate forms and satisfy (1) . Therefore, univariate solutions of (1) are available on diagonal lines along the bivariate solutions of (15), i.e.,
2.5.3. Initial and boundary conditions for envelope-modulated regimes
Dynamical behavior of RF circuits driven by stimuli of the form of (12) can be described by the MPDAE system of (15) together with a set of initial and periodic boundary conditions. In fact, bivariate forms of the circuits’ state variables can be achieved by computing the solution of the following initial periodic-boundary value problem
The reason why bivariate envelope-modulated solutions do not need to be evaluated on the entire domain (which would be computationally very expensive and would turn the multivariate strategy useless), and are restricted to the rectangle , is because the solutions repeat along the
3. Hybrid time-frequency techniques for computing the solution of MPDAEs
In this section, we will finally discuss the hybrid time-frequency numerical techniques that can be used to evaluate the solution of MPDAEs describing the operation of nonlinear electronic radio frequency circuits running in an aperiodic slow time scale and a periodic fast time scale. Section 3.1 addresses an efficient technique often referred to as
3.1. Multitime envelope transient harmonic balance
Let us consider the initial-boundary value problem of (17) and let us define a semi-discretization of the rectangular domain in the
The HB system of (20) can be rewritten as
The system of (25) requires the computation of the Jacobian matrix , i.e., the derivative of the vector , with respect to the vector ,
This matrix has a block structure, consisting of
3.2. Partitioned time-frequency technique
Although multitime ETHB can take advantage of the signals’ time rate disparity, it does not take into account the circuit’s heterogeneities, i.e., it uses the same numerical algorithm to compute all the circuit’s state variables. Thus, if the circuit evidences some heterogeneity (e.g., modern wireless architectures combining RF, baseband analog circuitry, and digital components in the same circuit), this tool cannot benefit from such a feature. This lack of ability to perform some distinction between nodes or blocks within the circuit had already been identified by Rizzoli et al.  and is the main limitation of multitime ETHB. To cope with this deficiency, the partitioned time-frequency technique separates the circuit’s state variables (node voltages and branch currents) into fast (
With the purpose of providing an elucidatory explanation of the partitioned time-frequency technique, let us consider a typical wireless system, composed of RF and baseband blocks. In such a case, the state variables in the RF block can be described as fast carrier envelope modulated waveforms defined as
In the following we provide a brief theoretical description of the partitioned time-frequency technique fundamentals. For that, let us now consider the bivariate forms of
By considering this, we can easily deduce that the size of the vector defined by (21) will be significantly decreased, as well as the total number of equations in the HB system of (23). Furthermore, another crucial aspect is that we are no longer forced to carry out the conversion between time and frequency domains for the latent state variables expressed in the form of (35), as well as for the components of corresponding to latent blocks of the circuit. Given that the
Considerable Jacobian matrix size reductions will also be achieved with this technique. Indeed, by considering the latency of state variables in some parts of the circuit, some blocks of the Jacobian matrix (26) are simply reduced to 1×1 scalar elements. These scalar elements contain the dc sensitivity of to the latent components of .
With the state variable and the error function vector size reductions, as also the resulting Jacobian matrix size reduction, it is possible to avoid dealing with large linear systems in the iterations of (25). Thus, a less computationally expensive Newton-Raphson iterative solver is required to solve (23). In conclusion, partitioning the circuit into active and latent sub-circuits (blocks) can lead us to significant computation and memory savings when computing the solution.
4. Efficiency of the partitioned time-frequency technique
The effectiveness of the multitime ETHB technique is nowadays widely recognized by the RF and microwave community. The efficiency of the partitioned time-frequency simulation technique described in the previous section was also already established, as a consequence of the considerable reductions in the computational effort required to obtain the numerical solution of several RF circuits with distinct topologies and levels of complexity . Even so, a brief comparison between this method, the previous state-of-the-art multitime ETHB and a conventional univariate time-step integration scheme (SPICE-like simulation), is included in this section. This will help the reader to get a perception of the potential of the partitioned hybrid technique. For that, we considered the RF mixer (frequency translation device) depicted in Figure 3 as the illustrative application example. The circuit was simulated in MATLAB with three different techniques: (i) the partitioned time-frequency simulation technique, (ii) the multitime ETHB, and (iii) the Gear-2 multistep method  (a time-step integrator commonly used by SPICE-like commercial simulators).
Numerical computation times for simulations in the [0, 1.0
By comparing the CPU times obtained with the methods, we can attest the superiority of the partitioned time-frequency method. Indeed, speedups of more than two times were obtained with this method in comparison to multitime ETHB. We can also attest the inefficiency of univariate time-step integration when dealing with RF problems. Finally, it must be noted that the efficiency gain of the partitioned time-frequency technique was achieved without compromising accuracy. Indeed, the maximum discrepancy between solutions computed with this technique and multitime ETHB was in the order of 10−7 for all the state variables of the circuit.
|[0, 1.0 μs]||00:00:04.9||00:00:11.3||00:19:21|
|[0, 10.0 μs]||00:00:39.6||00:01:35.1||02:47:33|
In this chapter, we have presented a partitioned time-frequency numerical technique especially designed for the efficient simulation of RF circuits operating in a periodic fast time scale and an aperiodic slow time scale. This technique can be viewed as a wise combination of multitime ETHB based on a multivariate formulation, with a conventional univariate time-step integration scheme. With this technique fast changing (active) state variables are computed in a bivariate mixed time-frequency domain, whereas slowly varying (latent) state variables are evaluated in the natural one-dimensional time domain. By partitioning the circuits into active and latent parts and exploiting the fact there is no obligation to perform conversion between time and frequency for latent blocks, considerable reductions in the computational effort can be achieved without compromising the accuracy of the results. Although the speedups obtained with the simulation of the illustrative application example presented in Section 4 are already notable, it must be noted that higher efficiency gains should be expected when simulating RF networks containing a number of latent blocks larger than the active ones.
This work is funded by National Funds through FCT - Fundação para a Ciência e a Tecnologia, under the project UID/EEA/50008/2013.
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