Techniques for Time Domain Transient Analysis

Cadence System Analysis

Key Takeaways

All physical systems exhibit some transient behavior in their dynamics, including electronics.
Transient analysis techniques help you understand transitions between different electrical states.
When examining measurements or simulation data, some basic transient analysis techniques can be used to understand transitions between electrical states.

Time domain transient analysis techniques operate in the time domain and provide a complete view of signal transitions between different states

When you turn on or off an LED, there is a slow transition as the light brightens and fades. Transient behavior of this type is very simple, but it is a fundamental reaction as an electronic system changes states. Signal transitions in the time domain, and their relationship to important system parameters, can be best understood using transient analysis.

Designers that are new to working with simulation data may think that time domain transient analysis is a simple matter of looking at waveforms on a graph. The reality is that there are a range of mathematical techniques and some commercial software applications used in transient analysis.

With some creative approaches and powerful design tools, you can gain a complete understanding of the transient behavior in your electronics. You often don’t need to complete a large set of calculations by hand, you can gain important insights directly from looking at time domain responses and some simple mathematical steps.

Getting Started With Time Domain Transient Analysis

Transient analysis involves a set of techniques used to analyze simulation data or experimental results in the time domain, specifically when the system under study is transitioning between two states. If this sounds vague, consider an RC circuit that has been charged up to 5 V; if the source voltage suddenly changes to 7 V, there will be a transition in the time domain between these two states. Some of the principle methods in time domain transient analysis include:

Stability analysis: This is a generalization of Laplace domain analysis, but it can be applied to coupled nonlinear systems, which may exhibit unstable transient behavior. Stability analysis uses a range of techniques to predict conditions under which a system will have a stable transient response.
Parameter extraction: This class of regression techniques is used to determine parameters in an analytical model. This is done by comparing model predictions with measured data, followed by applying statistical techniques to judge how well the predicted and experimental data align.

Empirical modeling: One approach to summative transient analysis is to build empirical models from experimental or field solver data. Empirical models are useful for quickly predicting the behavior of a system or finding general relationships between different models of varying complexity. These models have even found their way into IPC standards.

The graph below shows some typical experimental data for the output voltage from an LDO regulator when the load draws a current pulse, such as a switching digital circuit. The dip in the output voltage could be examined using transient analysis, with the goal of modifying the circuit to provide more stable output voltage.

Time domain transient analysis LDO regulator

Example transient response in an LDO regulator showing a voltage droop and recovery

Some possible points to examine in transient analysis include the settling time (or recovery time) and how this relates to various parameters in the regulator. In many systems, a small number of specific parameters will be responsible for determining the transient response in the system, but there may not be an analytical model that describes such a relationship.

Parameter Extraction

Oftentimes, you have a theoretical model for a system and you need to extract parameters in the model from measured data. There are a number of ways to do this, depending on the form of the model and whether an analytical solution is available. These methods generally fall into two areas:

Regression: If an analytical model describing the system is known, then it can be used in regression to determine any unknown model parameters.
Statistical comparison: Sometimes, an analytical model for the system is not known, but there may be a numerical model, such as from a field solver. In this case, the field solver results can be generated for various system parameter values, and the two data sets can be compared statistically to determine the most likely system parameter value.

Stability Analysis

Laplace transforms and functions in the Laplace domain are often used to describe the transient behavior of a physical system. Stability analysis involves a more complete set of techniques that build on the Laplace domain. In linear time-invariant (LTI) systems, including in coupled systems, stability analysis methods generate the same results as Laplace transform solution techniques.

In stability analysis of linear systems, the goal is to solve the following eigenvalue equation:

Eigenvalue equation

Principal eigenvalue equation used for stability analysis and transient analysis

The real and imaginary parts of the eigenvalues (λ) appear as complex conjugate pairs and will tell you the poles and zeros of the system. From here, you will immediately know whether the system has a stable response, whether there will be a transient oscillation (limit cycle) while approaching the steady state, and how quickly the system approaches the steady state. The table below summarizes expected transient behavior for various values of λ in a linear system.

Real λ < 0	Unconditionally stable behavior; the solution approaches the steady state without oscillation.
Real λ > 0	Unstable behavior; solution grows without bound. If there is a second critical point, the behavior may change to limit cycle behavior as this is approached.
Imaginary λ	This result corresponds to a purely undamped oscillation without decay; there is no steady state.
Complex λ, with Real λ < 0	This corresponds to stable limit cycle behavior, where the system oscillates as it approaches the steady state (e.g., underdamped oscillation).
Complex λ, with Real λ > 0	The system exhibits an unstable (growing) response while also oscillating at a definite frequency.

For nonlinear systems, there is not a single method for determining stability, and a range of possible methods can be applied to different systems. Brute-force time domain simulations can give some useful results, and this is the primary method used in SPICE solvers for nonlinear circuits. This remains an active area of research, and stability methods often need to be tailored to specific nonlinear systems to get useful insights.

Empirical Modeling

This is the most general form of transient analysis, as it simply involves fitting a proposed model equation to your experimental data using linear or nonlinear regression. If you have MS Excel or another spreadsheet program, you can create an empirical model for your system and use it in any other analysis. The best electronics circuit simulators can build a model directly from your data and create a SPICE subcircuit from the results, which you can then incorporate into larger circuits. Doing empirical modeling for electronics takes the right design and simulation application to help automate transient analysis.

When you need to perform any of these time domain transient analysis techniques, make sure you use the best set of PCB design and analysis tools you can find. Cadence provides powerful software that automates many important tasks in systems analysis, including transient analysis in circuit simulations, power/signal integrity simulations, and 3D field solver simulations.

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