A Laplace transform is a generalization of a Fourier transform to complex eigenvalues for an LTI system.
Laplace transforms allow stability to be easily analyzed in LTI systems, or in systems with harmonic time-dependence in certain parameters.
Pole-zero analysis is a Laplace-domain technique that allows you to easily understand the transient and steady-state behavior of a system. This can be performed in many circuit simulators.
Stability in coupled systems can be described using a Laplace transform
If you’ve ever run a time-domain simulation for your circuits, you can quickly see how your system responds to different input signals. It’s easy to pick out the response amplitude and spot modulation at specific frequencies, but quantifying the transient behavior in terms of time constants requires some additional analysis. This is especially true in systems with multiple poles, such as higher order RLC networks.
Using a Laplace transform allows you to quickly convert between a general input function in a circuit and the output you would expect to see in the circuit. If this sounds like a Fourier Transform, it’s not too far off the mark; a Fourier transform is related to a Laplace transform, but they describe different types of behavior, particularly steady-state vs. transient behavior. In addition, working in the Laplace domain with linear systems provides a simple way to predict the transient response in your circuits. In this article, we discuss how this works and how you can quickly determine the transient and steady-state behavior in your system before running time-domain simulations.
What is a Laplace Transform?
A Laplace transform is one of many transform methods used to understand the behavior of a physical system in terms of a conjugate variable. In this case, the conjugate variable is a complex frequency, meaning it has an associated rate constant and a real-valued frequency that defines how the system behaves in time. The Laplace transform converts a time-domain function into a function of decay rate and frequency.
Laplace transform definition.
This extension into a conjugate variable domain simplifies analysis of linear circuits, or of nonlinear circuits operating in a linear regime. Laplace transform techniques are useful in the following types of circuits:
Electrically linear circuits—Any LTI system with linear components (i.e., all component values are linear functions of input voltage/current) can be examined, even if the input voltage/current in the system is a nonlinear function of time.
Time-variant circuits—Circuits with time-varying component values (e.g., a parametric amplifier or similar time-varying circuit) can be examined, including when the circuit values have nonlinear time dependence.
Coupled systems—Any system with coupling between circuit responses and multiple sources, as well as systems with feedback, can be examined using a Laplace transform.
There are many tables online and in textbooks that contain Laplace transforms of many common functions. There are also some useful relations you can find in many textbooks for converting combinations of functions between the Laplace and time domains. In the above equation, s is a complex number, and it's real and imaginary parts have specific physical meanings:
Re[s]: The real part of s is a rate constant, or the inverse of a time constant for an exponentially rising or falling response to an input stimulus.
Im[s]: The imaginary part of s is a frequency, which defines how the system oscillates during its transient response in the approach to the steady-state.
In other words, there will be a specific set of s values that form the time-domain response for a system; these s-values are called the poles and zeros of the system. In particular, these s-values will tell you the system’s transient response. Consider an underdamped RLC circuit; the s-values will tell you both the damped oscillation frequency (imaginary part) and the damping constant (real part). This should illustrate the value of a Laplace transform for circuit analysis over a Fourier transform.
Laplace Transform vs. Fourier Transform
These two techniques can be used to determine a transfer function for your system. A transfer function may normally be shown as a Bode plot, which shows specific frequencies giving zero response and resonances giving maximum response. The Fourier transform and Laplace transform of a system have very different meanings:
In a transfer function: A transfer function in the Fourier domain tells you how the system responds when driven with specific frequencies.
For a narrow-bandwidth input: Fourier transforms and Laplace transforms of input signals are two different ways to represent a signal in its conjugate domain. Similarly, the transfer function will be defined in the same conjugate domain:
Transfer function in the Laplace domain
As long as you know the Laplace or Fourier transform of the input and the transfer function, you can calculate the circuit’s response using the inverse Laplace or Fourier transform. The above definition is also used to determine a system’s transient behavior with pole-zero analysis.
Because the s-values are complex numbers, a plot of a transfer function in the Laplace domain is not a simple curve showing magnitude and phase. Instead, this would show a heat map in the complex plane. Although visualization is quite difficult, a Laplace transform is much more useful for coupled systems and systems with feedback where stability is very important.
Coupled and Uncoupled Linear Systems
When working with coupled systems, or with systems that have feedback, you can use a matrix technique to extract the system’s eigenvalues, which will tell you how the system responds to an impulse in the time domain. It also tells you qualitatively the stability of the system. This is determined by calculating the eigenvalues of the system’s characteristic system of first order equations:
Determining stability by calculating eigenvalues of a first order system of equations.
Note that, in the above equation, we are dealing with time-dependent functions, but this technique will give you the same results as working with a transfer function for your system in the Laplace domain. The eigenvalues of the above matrix equation will give you the same results you would find from pole-zero analysis.
Rather than using Laplace transform calculations by hand and running time-domain simulations of your circuits, a great circuit simulator can show you the poles and zeros of the system directly from the system’s transfer function. The front-end design features from Cadence and the PSpice Simulator gives you this functionality for circuit design and simulation. You can also run a variety of time-domain and frequency-domain simulations for your circuits.
If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.