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The Value Of RF Harmonic Balance Analyses For Analog Verification

Frequency domain periodic large and small signal analyses.

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By Pradeep Thiagarajan and Scott Guyton

The world we live in is intricately connected by electronic systems that are expected to function flawlessly to satisfy consumer needs. Functionality violations beyond certain tolerance levels are frowned upon and negatively impact the quality level of products. These systems are required to function accurately, in tandem with other interdependent systems. ICs are at the core of these systems where analog and digital signals need to co-exist to accomplish desired functionality. Furthermore, the demand for enhanced connectivity with more and faster access to data continues to drive technology towards higher data rates and increased functional capabilities. Designers are under tight deadlines to create competitive circuits that meet stringent specifications covering performance, power, noise, and reliability. There is a constant push to minimize silicon re-spins and that requires highly accurate verification. Specialized analyses beyond typical methods need to be considered for analog circuits.

While advanced process technologies provide the benefits of lower power and higher performance, designers must get more innovative to address growing design complexity due to a variety of factors. The prominent influencers are higher circuit density with noise vulnerability, decreasing supply voltage levels, increasing device and interconnect resistance and capacitance parasitics. To properly verify these designs before silicon fabrication, specialized radio frequency (RF) analyses are required to accurately predict silicon behavior for which a high-performance RF simulation engine is needed. Both single and multi-tone Harmonic Balance analyses are essential in addressing these challenges for linear and moderately nonlinear periodic circuits (such as LNA, PA, Mixer, Rx, Tx, etc.) along with some of the most common building blocks including VCOs (LC-tank, ring) and oscillators (xtal, RTC, etc.).

Why do we need periodic analysis?

Why not just run the traditional AC, DC, transient, and transient noise analysis? Why require an RF engine with periodic analysis? To answer that, we need to understand what RF is first. RF applies to frequencies between 3 kHz to 300 GHz. A subset from 30 GHz to 300 GHz is millimeter wave. These invisible waves are all around us as we immerse ourselves with the latest technological advancements in the mobile communication and networking world. The analysis for any system that processes these frequencies needs to account for signal settling time and noise effects from within and around the system that can interfere with other signals in the system.

Some designers may be familiar with transient noise analysis (TN) where the use-model is straightforward. It produces noisy transient waveforms generated from random noise in circuit elements including resistors and transistors. The waveforms can often be postprocessed to get noise-related metrics. TN analysis is an effective method for non-periodic circuits like PLLs and ADCs since periodic analysis may not be viable or may be very difficult to converge. However, for periodic driven and autonomous circuits, RF analyses is a viable alternative. The disadvantages of running TN includes much longer simulation times, careful attention required for postprocessing, and no identification of top noise contributors.

With the proliferation of mobile and wireless applications, device noise can be a limiting factor in meeting the increasingly stringent target specifications for RF blocks. Traditional SPICE simulators cannot be used to predict this type of periodic noise since a long transient is required to allow the circuit to settle. It also entails several simulations and postprocessing steps that can be cumbersome and prone to user error. Traditional SPICE also cannot accurately determine the noise in RF circuits such as mixers, LNAs, frequency dividers, oscillators, or PLLs. Noise calculations in SPICE are based on small-signal linearized analysis of the circuit at its DC operating point, and this linearization affects proper frequency translation of noise due to circuit non-linearities.

What is periodic analysis? RF Shooting Newton versus Harmonic Balance

In our experience, we have found that many designers find RF analysis confusing and even intimidating as it is not as straightforward as traditional analysis. Let’s delve more into it.

Today, the two primary RF analysis methods are time-domain engine (Shooting Newton or SN) and frequency-domain engine (Harmonic Balance or HB) to verify periodic circuits for linearity, noise, and gain. Both methods have their limitations and advantages. It is imperative to apply the most appropriate method, analyses, and options to maximize throughput for the required accuracy to garner maximum coverage and confidence before tapeout.

Periodic steady-state analysis can be thought of as an extension of SPICE operating point analysis. In SPICE, you apply DC signals to the circuit and the simulator computes the steady-state solution. That solution is the DC operating point at which you perform subsequent small-signal analyses. In a periodic steady-state large signal analysis, you drive the circuit with one or more periodic sources. The steady-state response is the response that results after any transient effects have dissipated. The large signal solution is the starting point for small-signal analyses, including periodic AC, periodic transfer function, periodic noise, periodic stability, and periodic scattering parameter analyses.

Designers refer periodic steady state analysis in time domain as “PSS” and corresponding frequency domain notation as “HB.” Figure 1 below showcases the iteration methodology difference between SN and HB. After calculating the DC operating point and running for some initial transient stabilization time (tstab), we assume tstab is close to periodic steady state. In SN, we start iterating in the time domain to solve for periodic steady-state solution. HB, on the other hand, iterates in frequency domain to solve for a converged steady-state solution using Kirchoff’s Current Law (KCL) criterion. While the time domain view using PSS can show the net voltage waveform, the frequency domain view using HB can show output power at specific loading with various harmonics.

Fig. 1: Shooting Newton versus Harmonic Balance computation windows.

Typical applications and measurements for RF analyses

RF analyses are well suited for periodic driven and autonomous circuits. Driven circuits are driven by periodic time-varying independent sources and have a periodic solution. The circuit’s steady-state response is also periodic with the fundamental period. Mixers, dividers, low noise amplifiers, transmit and receive chains, and power amplifiers are common examples of driven circuits. In contrast, autonomous circuits contain non-time-varying independent sources. An autonomous circuit oscillates due to positive feedback, which results in a steady-state oscillation. Periodic autonomous circuits produce periodic time-varying outputs from non-time-varying sources. The steady-state waveforms for the circuit are all periodic with the identical fundamental period. Ring oscillators, LC-tank oscillators, and crystal oscillators are periodic autonomous circuits.

Typical measurements from RF analyses focus on voltage gain (in dB), power (in dBm) and linearity effects (such as P1dB 1dB compression point and IP3 third order intercept point). Noise effects also can be accurately measured whereby Noise factor and Noise figure metrics are used. Time domain jitter also can be seen via frequency domain for long-term and short-term phase variation, where phase is inversely proportional to the wave’s frequency. Phase noise in dBc/Hz is another powerful metric that can then be postprocessed as relevant to the application to see the RMS jitter that is used extensively to build jitter budgets for systems.

Harmonic Balance is very well-suited for designs that have many reactive components as well as circuits in which time constants are large compared to the period of the simulation frequency, such as dispersive transmission lines. Many linear models are best represented in the frequency domain at high frequencies. Usage examples include determining the spectral content of voltages or currents, calculating IP3, total harmonic distortion (THD), and intermodulation distortion components, performing non-linear noise analysis, and load-pull analyses for amplifiers, etc.

Shooting Newton is very well-suited for most periodic driven circuits, especially non-linear circuits with sharp transitions such as switched-cap filter, divider and phase-frequency detector. Oscillators, such as ring and crystal, can also be target circuits. SN accommodates a high number of harmonics. However, runtime and memory usage increase with time points or number of tones. SN is usually slower than HB for linear or moderately non-linear circuits.

How does RF Harmonic Balance work?

HB is a frequency-domain analysis technique for simulating distortion in nonlinear electrical circuits and systems. In essence, it calculates the steady-state response of non-linear differential equations. It starts with KCL in the frequency domain and a chosen number of harmonics. A sinusoidal signal applied to a non-linear component will generate harmonics of the fundamental frequency. The assumption is that the solution can be represented by a linear combination of sinusoids after which voltage and current sinusoids are balanced to satisfy Kirchhoff’s law.

HB analysis is most applicable for circuits that exhibit a linear to moderately nonlinear behavior, require high numerical accuracy and dynamic range, and use frequency-domain models such as tabulated S-parameters. It is usually the method of choice for simulating circuits that are most naturally handled in the frequency domain such as analog RF and microwave designs that are excited with sinusoidal signals.

To perform an HB simulation, you only need to specify one or more fundamental frequencies and the order for each fundamental frequency. Within the context of high-frequency circuit simulation, HB offers several benefits over conventional time-domain transient analysis, wherein it obtains frequency-domain voltages and currents, directly calculating the steady-state spectral content of voltages or currents in the circuit. The frequency integration required for transient analysis is prohibitive in many practical cases.

Analog FastSPICE platform

Siemens EDA’s Analog FastSPICE platform (AFS) is a simulation technology that provides superior nm-accurate circuit simulation, mixed-signal simulation, and full-spectrum device noise analysis. AFS is foundry certified by the world’s leading foundries delivering SPICE accuracy. It supports industry-standard netlist syntax and is seamlessly integrated into industry EDA design environments. The RF engine in AFS supports Shooting Newton and Harmonic Balance methods with recent innovations. For silicon-accurate characterization, the AFS platform includes a comprehensive full-spectrum device noise analysis and integrates with Solido Variation Designer to deliver full variation-aware design coverage in orders-of-magnitude fewer simulations, but with the accuracy of brute force techniques.

The recently announced Analog FastSPICE eXTreme (AFS XT) technology further enhances performance for large post-layout netlists. There is no additional cost to existing and new customers. AFS XT can handle over 300 million element transient capacity and delivers mixed-signal simulation with Symphony Mixed-Signal platform.

RF Harmonic Balance offerings

The AFS Harmonic Balance suite comprises of large signal HB analysis and small signal AC, transfer function, noise, stability, and scattering parameter analyses. While single tone is supported for driven circuits and autonomous circuits, multi-tone is supported for driven circuits. It is most applicable for circuits with high dynamic range, for linear and moderately non-linear circuits and for circuits with distributed elements.

AFS HB solvers are further optimized for performance and convergence via automated intelligence algorithms. AFS HB utilizes multi-threaded capability for an additional performance speedup versus single core operations. The AFS multi-core parallel feature (MCP) speeds up these simulations by 3.5x to 7x by efficiently using multi-core shared memory machines for Monte Carlo, sweeps, and corner simulations.

Table 1 below summarizes the AFS RF Harmonic Balance offerings.


Table 1: AFS Harmonic Balance analyses.

Large signal HB analysis determines periodic time varying operating points for single tone, and quasi-periodic time varying operating points for multi tone. All signals must be co-periodic with the fundamental frequency. After a periodic steady-state solution is found, the listed small signal analyses can be run.

Small signal summary

  • HBAC is useful for measuring intermodulation distortion, conversion gain and RF-to-LO isolation etc., (for example, LNAs, mixers). Transfer function is computed from one input to multiple outputs and includes frequency conversion effects.
  • HBXF is useful to determine image and sideband rejection and power supply rejection. HBXF is the inverse of HBAC and determines the transfer functions from multiple input to one output.
  • HBSTB is useful to determine stability in oscillators and switched capacitor circuits. It can be used to measure loop gain when a large periodic signal is present that may cause non-linearity.
  • HBSP analysis is useful to determine the power gain of the system. It computes the periodic scattering matrix of the circuit at the specified ports and harmonics.
  • HB NOISE is useful to measure phase noise in oscillators, noise figure in LNAs, and noise of mixers. HBNOISE, unlike conventional noise analysis, computes frequency conversion effects and noise folding. It accounts for thermal noise, shot noise, and flicker noise. It essentially includes the effect of a periodically time-varying bias point on the device noise where the result is the sum of the noise contributions from both the up-converted and down-converted frequency bands.

Capacity, performance and memory for RF HB analyses

Achieving proper convergence in HB can be a challenge for complex designs. Relaxation of convergence parameters is a logical approach, but it takes a toll on accuracy. Furthermore, verification complexity increases with increasing harmonics. Extracted netlists further introduce much more convergence complexity compared to schematics. Capacity refers to the number of elements contained in the netlist including active and passive devices, parasitic elements, etc. As shown in figure 2, AFS RF analyses can easily handle designs with high element count. The ability to handle large netlists is crucial for RF analyses, especially as we move towards advanced process nodes.


Fig. 2: AFS RF HB capacity examples.

Ultimately, faster simulations without accuracy loss are needed to meet aggressive design deadlines. Figure 3 shows examples of runtime and memory utilization benefits of AFS XT from AFS for a wide range of circuits.


Fig. 3: Comparisons of AFS versus AFS XT (value greater than 1.0 signifies speed up with AFS XT and less memory utilization with AFS XT).

Summary

For periodic driven and autonomous circuits, choose your analysis engine wisely with the capabilities and benefits that AFS RF engines provide with both Shooting Newton and Harmonic Balance. Siemens EDA’s Analog FastSPICE platform offers a Harmonic Balance engine for both single-tone and multi-tone analysis and provides boosted performance with superior convergence without compromising accuracy. The Harmonic Balance suite comprises of large signal HB analysis and a variety of small signal analyses. Take further advantage of AFS RF HB’s multi-threaded capability for additional speedup for designs needing extensive sweeps, corners, and statistical analysis.

References

  1. https://eda.sw.siemens.com/en-US/ic/analog-fastspice/
  2. https://semiengineering.com/radio-frequency-technology-is-found-everywhere-in-daily-life/

Scott Guyton is director, solutions architecture at Siemens EDA.



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