In this series, we have mainly discussed the Fourier transform (including Fourier series), but this time we will talk about the Laplace transform, which can be considered a related concept, and its applications.
The Fourier transform converts a time-domain signal into a frequency-domain signal (function), and is relatively clear from a physical standpoint. In contrast, the Laplace transform converts a real-time function into the complex world of s-domain, and its physical meaning is not always easy to understand. However, it can mechanically solve differential equations and is a very useful tool in engineering, and is frequently used in the fields of electrical circuits and control systems. The Fourier transform is a function of angular frequency ω (a real number) (complex number data), while the Laplace transform is a complex function of the complex number s, and a rigorous understanding requires knowledge of complex function theory. Here, we will ignore mathematical rigor and outline its characteristics.
As a review of the previous discussion on frequency analysis from the basics (4) - "Fourier transform," the Fourier transform F(ω) of a time signal f(t) is:
.................................（1）
Here, ω = 2πf (angular frequency), where f is the frequency.
(Note: Previously, the frequency variable was represented by f, but here we will use ω.)
This is the result. Here, equation (1) does not necessarily converge to infinity. For example,
f(t)=1 (DC component)
f(t) = t (ramp function)
f(t) = sinωt (trigonometric function)
Time functions such as these do not converge in the ordinary sense. (A generalized function, the delta function, is required.)
Here, in order to force convergence, we multiply f(t) by e − σ (σ>0) (where σ is a real number), and if we set f(t)=0 in the negative time region (t<0), then equation (1) becomes
.................................（2）
Thus, the usual function will converge. Here, the complex number (σ+jω)
s=σ+jω .................................（3）
If we set this, then equation (2) is
.................................（4）
Therefore, equation (4) is called the Laplace transform. In other words, the Laplace transform can be described as extending the jω (purely imaginary number) of the Fourier transform to the complex number s, so that the transform integral converges. Equation (4) transforms a real function f(t) of a real number t to a complex function F(s) of a complex number s.
I will omit the derivation, but the inverse Laplace transform is
.................................（5）
This is defined as the Bromitch integral. While calculating equation (5) requires knowledge of complex function theory (the residue theorem), in practice it can be derived from the Laplace transform table.
Now, the intuitive meaning of using the exponential function in equation (4) is that calculus is simplified, that is, the derivative of e at is ae at and the integral of e at is
This means that calculus can be replaced with multiplication and division. This is where it becomes powerful when solving differential equations.
Table 1 shows the Laplace transforms F(s) of commonly used time functions f(t).
Table 1: Laplace Transform Table of Typical Time Waveforms
| Time waveform | f(t) | F(s) |
 |
δ(t) Impulse |
1 |
 |
u(t) Step |
 |
 |
e-a exponential function |
|
 |
sinωt sine wave |
 |
 |
cosωt Cosine wave |
 |
 |
e-asinωt Damped sine wave |
 |
As an example, let's find the Laplace transform of a sine wave.
According to Euler's formula,
that's why
(Note) For the above equation to converge, the real part of the complex number s must be positive, i.e., Re(s) > 0.
As in this example, the Laplace transform is generally defined by an infinite integral, so a convergence condition is necessary. However, once F(s) is transformed into the complex domain (s domain), it will have a finite value in all regions of the complex plane (s plane), except for the point where the value is undefined (for example, s=0 in the step waveform in the second row of Table 1).
This characteristic is a major advantage of converting to complex functions. (It can be extended using a technique called analytic continuation in complex function theory.) Therefore, the convergence domain of the Laplace transform can be treated with little practical concern.
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Figure 1: Time domain (t domain) and complex domain (s domain)
Now, before solving the differential equation, it is necessary to find the Laplace transforms of the derivative and integral of the time function f(t). I will omit a detailed explanation, but they are as follows.
...................................(8) n-fold integral
-
Figure 2 1st-order low-pass filter
Let's try an example of solving a differential equation. In Figure 2, a step waveform is applied to the input x(t).
We will find the response time waveform y(t) when an input is received.
In Figure 2, if the current is i,
Ri+y(t)=x(t) .................................（9）
If you delete i
.................................（10）
Perform a Laplace transform (starting with an initial value of 0)
RCsY(s)+Y(s)=X(t) .................................（11）
More table
that's why
(Let T=RC)
.................................（12）
Refer to the Laplace transform table.
When a step waveform (upper part of Figure 3) is input to the filter in Table 1, the result is as shown in the lower part of Figure 3. This is called the step response (or individual response).
Furthermore, T (=RC) is called the time constant, and it is the time it takes for the output to reach 63.2% of the input. The smaller T, the faster the rise response and the faster the transient response decays.
In solving differential equations using Laplace transforms, the equations can be treated like algebraic operations within the transformed s-domain, and the solutions can be found almost mechanically using Laplace transform tables.
-
Figure 3: Step response of the filter in Figure 1
As this example shows, the transient response of electrical and mechanical systems can be determined by solving differential equations using Laplace transforms.
Next time, we'll talk about transfer functions and frequency response.
In conclusion, here's a summary.
- The Laplace transform is a tool that allows for the mechanical solution of differential equations and is frequently used in fields such as electrical circuits and control systems.
- The Laplace transform can be described as extending the Fourier transform's ωj to a complex number s, allowing the transform integral to converge.
- The Laplace transform and inverse Laplace transform can usually be easily calculated from a Laplace transform table.
- In solving differential equations using Laplace transforms, the equations can be treated like algebraic operations within the transformed s-domain, and the solutions can be found almost mechanically using Laplace transform tables.
- By solving differential equations using Laplace transforms, we can determine the transient responses of electrical and mechanical systems.
【keyword】
Fourier transform, Fourier series, Laplace transform, differential equations, angular frequency, complex functions, complex function theory, distributions, pure imaginary numbers, inverse Laplace transform, Bromwich integral, residue theorem, complex domain, s-domain, complex plane, s-plane, analytic continuation, convergence domain, step response, indical response, time constant, transient response
【reference】
Hiroshi Harashima and Yoichi Hori, "Laplace Transforms and Z-Transforms," Suuri Kogakusha (2004).
"Spectral Analysis" by Mikio Hino, Asakura Shoten (1977)
(Excerpt from the email newsletter issued on July 25, 2017)
