What are time constants and where are they found?
时间常数是什么，它们在哪里可以找到？

Time constants are everywhere. Since (almost) nothing happens instantaneously, but rather with a delay, processes are said to have time constants associated with them.
时间常数无处不在。由于（几乎）没有什么是瞬时发生的，而是有延迟地发生，因此过程被认为与时间常数相关联。

In electrical systems, purely resistive circuits are static (not dynamic) as there are no energy storing elements. Resistors do not store energy but rather dissipate it. However, consider these circuits:
在电气系统中，纯电阻电路是静态的（而非动态的），因为没有能量储存元件。电阻器不储存能量，而是消耗能量。然而，考虑这些电路：

• Resistor-capacitor circuits have time constants (e.g. utilized in low-pass filters)
  电阻-电容电路具有时间常数（例如，应用于低通滤波器）
• Resistor-inductor circuits have time constants (e.g. it takes some time for a motor winding to achieve the rated current).
  电阻-电感电路具有时间常数（例如，电动机绕组达到额定电流需要一些时间）。

In mechanical systems:  在机械系统中：

• Acceleration - either linear or rotary - it is necessary to provide energy to a system to overcome the effect of inertia. E.g. our cars require torque over time to achieve the desired speed.
  加速度——无论是线性还是旋转——都需要为系统提供能量以克服惯性效应。例如，我们的汽车需要在一段时间内提供扭矩以达到所需的速度。

In thermal systems:  在热系统中：

• Making ice cubes for a party takes non-zero time. Here comes the time constant.
  制作冰块以备派对需要非零时间。这就是时间常数的作用。
• Power electronics equipment also heats up due no non-zero resistances. Thermal constants of device packaging are measured in milliseconds; thermal constants of heatsinks are measured in seconds or minutes.
  电力电子设备由于存在非零电阻也会发热。器件封装的热时间常数以毫秒为单位测量；散热器的热时间常数以秒或分钟为单位测量。

Psychology:  心理学：

• How much of History 101 (after taking the final exam) do you remember as a function of time?
  在参加完期末考试后，你对历史 101 的记忆随着时间的推移还剩下多少？

Measuring Time Constants   测量时间常数

When would one want to measure the time constant of a system?
何时需要测量系统的时间常数？

• What is the bandwidth of a low-pass filter?
  低通滤波器的带宽是多少？
• How long will it take for a HVAC unit for heat up the room?
  一个暖通空调单元加热房间需要多长时间？

It should be noted that systems typically have more than one time constant. However, it is very typical for a system to have its time constants far apart from each other, resulting in a separation between fast transient responses and the dominant response (which is the slowest one).
需要注意的是，系统通常具有多个时间常数。然而，系统的时间常数之间相距较远是非常典型的，这导致快速瞬态响应与主导响应（即最慢的响应）之间的分离。

E.g. a power electronics switch (say a silicon-carbide MOSFET) contains a junction within a package. The package is mounted on a heatsink. The switch junction temperature rises almost immediately with increased current since its time constant is very short; the package (middle picture) might take a few seconds. The heatsink will have a time constant on the order of minutes.
例如，一个电力电子开关（比如碳化硅 MOSFET）包含一个封装内的结。该封装安装在散热器上。随着电流的增加，开关结温几乎立即上升，因为它的时间常数非常短；而封装（中间图片）可能需要几秒钟。散热器的时间常数大约在分钟级别。

Picture is owned by GE
图片归 GE 所有

We will examine properties of time-domain response that will allow you to obtain an estimate of a time constant.
我们将研究时域响应的特性，这将使您能够估算时间常数。

This write-up focuses only on time-domain. See the Further Reading section for links to relevant articles in frequency domain analysis.
本报告仅关注时域。有关频域分析相关文章的链接，请参见进一步阅读部分。

A very standard first-order system response is as below:
一个非常标准的一阶系统响应如下：

Such shape looks like an impulse response of a system.
这种形状看起来像是一个系统的脉冲响应。

How does impulse response work?
脉冲响应是如何工作的？

Impulse response is a parameter of a system that dictates how the system reacts to its inputs. For example, when we take out a frozen fish from the refrigerator, the ambient temperature changes from 30F to 70F. The equations below show how the output of a system is related to the input and the impulse response (h(t) = impulse response, x(t) = input):
脉冲响应是一个系统的参数，它决定了系统如何对输入做出反应。例如，当我们从冰箱中取出一条冷冻鱼时，环境温度从 30 华氏度变化到 70 华氏度。下面的方程显示了系统的输出与输入和脉冲响应之间的关系（h(t) = 脉冲响应，x(t) = 输入）：

The figure above shows both the impulse response and the output signal when the input is a step function, which is quite often the case. The shape and duraration of the response is described with the time constant parameter "tau".
上面的图显示了当输入为阶跃函数时的脉冲响应和输出信号，这种情况是相当常见的。响应的形状和持续时间用时间常数参数“tau”来描述。

When time "t" is zero, no decay has happened yet and the signal is equal to y(0).
当时间“t”为零时，尚未发生衰减，信号等于 y(0)。

At time "t" equal to "tau", exactly one time constant has elapsed. The signal is now equal to:
当时间“t”等于“tau”时，正好经过了一个时间常数。此时信号等于：

Now that we have established the notion of time constant and the time-domain waveform, let's see how to obtain time constant from experimental data.
现在我们已经建立了时间常数和时域波形的概念，让我们看看如何从实验数据中获得时间常数。

There are three simple methods to estimate the time constant from time-domain data:
从时域数据中估计时间常数有三种简单的方法：

1. The 37% method   37% 方法
2. The initial slope method
   初始斜率方法
3. The logarithmic method   对数方法

The 37% Method   37% 方法

The 37% method is a widely used method for finding a time constant. It is quite simple:
37% 方法是一种广泛使用的寻找时间常数的方法。它非常简单：

1. Determine the initial value of the signal (y(0) as above).
   确定信号的初始值（y(0)如上所述）。
2. Determine the time when the signal has decayed to 37% of the initial value.
   确定信号衰减到初始值的 37%时的时间。
3. The elapsed time is the time constant.
   经过的时间就是时间常数。

E.g., for this system, the initial value is 5. 37% of 5 is 1.85, which happens at approximately 1 second. Hence, the time constant tau is 1 second.
例如，对于这个系统，初始值是 5。5 的 37%是 1.85，这大约发生在 1 秒时。因此，时间常数τ是 1 秒。

However, there are some restrictions to this method:
然而，这种方法有一些限制：

• The response needs to die out to zero. If the response does not converge to zero, then the only the "transient" part should be considered for the calculation of signal decay.
  响应需要衰减到零。如果响应没有收敛到零，那么在信号衰减的计算中只应考虑“瞬态”部分。
• Further, this method needs only two data points. In the graph above, the complete signal is shown but only two points are used for the calculation.
  此外，这种方法只需要两个数据点。在上面的图中，完整的信号被显示，但仅使用两个点进行计算。

The first restriction is explained in more detail here:
第一个限制在这里有更详细的解释：

• The time-domain response does not always decay to zero, such as in the plot below. Instead, it obviously converges to the value of 3. This is very typical in the nature. E.g., the velocity of a car after passing another car and merging back in the correct lane might look as below- the speed does not decay to zero. The calculation would go as follows: y(0) = 8, max decay = 8 - 3 (steady state) = 5, 37% of 5 = 1.85, look for 3 + 1.85 = 4.85, y(tau) = y(1) = 4.85. Hence, the time constant is also one.
  时域响应并不总是衰减到零，如下图所示。相反，它显然收敛到 3 的值。这在自然界中是非常典型的。例如，一辆车在超越另一辆车并重新并入正确车道后，其速度可能如下所示——速度并没有衰减到零。计算过程如下：y(0) = 8，最大衰减 = 8 - 3（稳态）= 5，5 的 37% = 1.85，寻找 3 + 1.85 = 4.85，y(τ) = y(1) = 4.85。因此，时间常数也是 1。

Alternatively, the temperature of chilled water in a glass converges to the ambient temperature of the environment. Exactly the same procedure as shown above can be applied to this situation (the time constant is equal to 1 second).
或者，玻璃中冷却水的温度会趋向于环境的温度。与上述情况完全相同的过程可以应用于这种情况（时间常数等于 1 秒）。


Therefore, it is not necessary to have a response that decays to zero.
因此，不必要求响应衰减到零。

The Initial Slope Method
初始斜率法

This method is based on the initial derivative of the transient response. Let's start from the equation shown above:
该方法基于瞬态响应的初始导数。我们从上面显示的方程开始：


The slope at time 0 is given by:
时间 0 的斜率由以下公式给出：


Hypothetically, the signal would decay linearly if it followed the initial slope:
假设如果信号遵循初始斜率，它将线性衰减：


Continuing the thought experiment, if the signal did decay at this slope, it would hit its steady state value (either 0 or a non-zero value) exactly at time "t" = "tau". In such case, the time term "t" in the numerator would cancel out with the denominator term "tau".
继续这个思想实验，如果信号确实以这个斜率衰减，它将在时间“t” = “tau”时恰好达到其稳态值（要么是 0，要么是非零值）。在这种情况下，分子中的时间项“t”将与分母中的“tau”项相抵消。

1. Graphically determine the initial slope at y(0).
   图形上确定 y(0)处的初始斜率。
2. Determine the time when the variable would have decayed to zero if the initial rate of decay (slope) had continued.
   确定如果初始衰减速率（斜率）持续下去，变量将衰减到零的时间。
3. That time is the measured time constant.
   该时间就是测量的时间常数。

E.g., the time constant "tau" would be equal to 2 seconds in the figure below.
例如，图中的时间常数“tau”将等于 2 秒。


The method can be applied to signals that do not decay to zero, such as the one shown below. The time constant would be measured on the steady-state line (dashed).
该方法可以应用于不衰减至零的信号，例如下面所示的信号。时间常数将在稳态线（虚线）上进行测量。


This method does not utilize any data but the initial point and the initial slope. Also, this method is used less often than the 37% method since measured waveforms typically contain noise.
该方法仅使用初始点和初始斜率，不利用任何其他数据。此外，由于测量波形通常包含噪声，因此该方法的使用频率低于 37%方法。

The Logarithmic Method   对数法

The last method does not rely on one or two data points as the previous two methods but rather on the complete dataset. This property comes in handy when the data set contains a lot of noise. Again, let's consider the basic decaying exponential:
最后一种方法并不依赖于前两种方法中的一个或两个数据点，而是依赖于完整的数据集。当数据集包含大量噪声时，这一特性非常有用。再一次，我们考虑基本的衰减指数：


Let's take the natural logarithm of the response curve:
我们取响应曲线的自然对数：


Note that this is an equation of a straight line when plotted against time "t".
请注意，当与时间“t”绘制时，这是一条直线的方程。

This is the strategy:
这是策略：

1. Obtain data at various times. The data need to eventually decay to zero although these data points do not need to be captured. If there is a non-zero steady-state offset, remove it from the captured data.
   在不同时间获取数据。这些数据最终需要衰减到零，尽管不需要捕获这些数据点。如果存在非零的稳态偏移量，请将其从捕获的数据中移除。
2. Take the natural logarithm of the data points.
   取数据点的自然对数。
3. Use linear fit to obtain the slope and intercept of the log data.
   使用线性拟合获得对数数据的斜率和截距。
   • The intercept should be the natural log of "y0".
     截距应为“y0”的自然对数。
   • The slope is the inverse of time constant "tau".
     斜率是时间常数“tau”的倒数。

An example is shown below:
下面给出了一个例子：

• The estimated time constant is 3.3 seconds while the actual time constant is 3.0 seconds. This is caused by the measurements noise (see the attached Python code).
  估计的时间常数为 3.3 秒，而实际的时间常数为 3.0 秒。这是由于测量噪声造成的（请参见附带的 Python 代码）。
• The figure shows data with noise (top figure), the log of the data (straight line) and the linear fit (dashed line) in the bottom figure.
  图中显示了带有噪声的数据（上图）、数据的对数（直线）以及下图中的线性拟合（虚线）。
• Note that the intercept in the second figure is the natural log of the initial condition y0.
  请注意，第二个图中的截距是初始条件 y0 的自然对数。
• The slope of the line fit is 1/tau.
  拟合线的斜率为 1/tau。
• The signal must decay to zero. If it does not, the steady-state value must be removed from the data.
  信号必须衰减到零。如果没有，稳态值必须从数据中去除。


Python Code


#=============================
# Logarithmic method

import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import curve_fit

def linear_fit(x, A, B): # this is your 'straight line' y=f(x)
    return A*x + B

# data parameters
tau = 3;
y0 = 25;
noise_amp = 2;

# generate time vector
y = y0*np.exp(-t/tau) + noise_amp*np.random.rand(len(t));

# take the natural logarithm
y_ln = np.log(y)

# do curve fit on ln(y)
A,B = curve_fit(linear_fit, t, y_ln)[0] # your data x, y to fit

# generate a line based on A, B
y_ln_fit = linear_fit(t,A,B)

fig = plt.figure(num=None, figsize=(5, 5), dpi=80, facecolor='w', edgecolor='k')

f, ax = plt.subplots(2, sharex=True)

ax[0].cla()
ax[0].plot(t, y, c='green', label='y1',linewidth=3.0)
ax[0].grid(1)
ax[0].set_ylabel('y(t)')

plt.xlabel('Time [s]')
plt.grid(1)

ax[1].plot(t, y_ln, c='green', label='y1',linewidth=3.0)
ax[1].set_ylabel('ln(y(t)')
line2, = ax[1].plot(t, y_ln_fit, c='green', label='y1',linewidth=3.0)
dashes = [10, 10]
line2.set_dashes(dashes)
line2.set_color('red')

# save the picture
plt.savefig('csa0008_2.png')

# Estimate time constant
tau_est = -1/A
print("Estimated time constant is ", tau_est, "s")

References  参考文献

This page is a modification of a now non-existent page located at:
此页面是对现已不存在的页面的修改，原页面位于：
1. http://www.facstaff.bucknell.edu:80/mastascu/econtrolhtml/courseindex.html

Version  版本

Version of this article is 10/1/2017.
本文的版本是 2017 年 10 月 1 日。

Author(s)  作者

TS

Further Reading   进一步阅读



Proportional Controller Implementation
比例控制器实现

In MatLab, DSPs, and FPGAs.
在 MatLab、DSP 和 FPGA 中。

.



Control System Block Diagram
控制系统框图

The fundamentals of signal flow.
信号流的基本原理。

 



System Modeling With Transfer Functions
使用传递函数进行系统建模

Introduction to dynamic systems.
动态系统简介。

 



Fourier Series Demo  傅里叶级数演示

It is all sine waves.
一切都是正弦波。