Name: linearizing nonlinear rl circuits
Uploaded: 2024-09-26T13:17:19.000Z
Duration: 1 min 26 s
Channel: JoVE

线性系统具有叠加性，即对各个输入的响应可相加；同时具有齐次性，即当输入乘以一个标量时，其响应也按相同比例缩放。
对于微小变化，非线性系统可在工作点附近近似为线性系统。
泰勒级数展开将函数的值与其在某一点的导数以及偏离该点的偏差关联起来。当偏差较小时，忽略高阶项可得到线性关系。
考虑一个带有非线性电阻的RL电路。由于非线性元件的存在，在推导传递函数之前必须对系统进行线性化处理。
利用基尔霍夫电压定律推导出一个非线性微分方程。
通过将小信号源置零来求得稳态电流。
该表达式被重写为用电流的平衡值表示的形式。利用非线性电阻的特性推导出线性化微分方程。
将已知值代入，并在零初始条件下应用拉普拉斯变换。
对电感器两端电压在平衡点附近的表达式进行拉普拉斯变换，并加以简化，以得到传递函数。

频域中的线性近似

Linear Approximation in Frequency Domain

线性系统具有两个主要特性：叠加和同质性。叠加允许对多个输入的响应是每个单独输入的响应之和。同质性确保用标量缩放输入会导致响应被同一标量缩放。

相比之下，非线性系统本身并不具备这些特性。但是，对于工作点周围的小偏差，非线性系统通常可以近似为线性。这种近似是通过泰勒级数展开实现的，该展开用特定点的导数来表示函数。通过忽略小偏差的高阶项，可以得到线性关系。

考虑包含非线性电阻器的 RL 电路。要分析此系统，在推导传递函数之前必须进行线性化。

<img alt="Figure1" src="/files/ftp_upload/16099/16099_PT%20Figure_1.svg" style="height: 180px;">

第一步是将基尔霍夫电压定律应用于电路，从而得到描述系统的非线性微分方程。例如，电压定律方程可能采用以下形式：

<img alt="Equation1" src="/files/ftp_upload/16099/16099_PT%20EQUATION_1.svg" style="height: 20px;">

其中 V(t) 是施加的电压，L 是电感，R 是电阻，E 代表电池电压。

为了找到稳态电流，我们将小信号源设置为零并求解平衡电流 i_0。然后根据与该平衡的偏差重写非线性微分方程：

<img alt="Equation2" src="/files/ftp_upload/16099/16099_PT%20EQUATION_2.svg" style="height: 22px;">

非线性电阻器的特性用于推导线性化微分方程。对于电流的小偏差，电压方程可以写成：

<img alt="Equation3" src="/files/ftp_upload/16099/16099_PT%20EQUATION_3.svg" style="height: 32px;">

将此近似值代入电压定律方程，我们得到一个线性微分方程。在已知值替换并假设零初始条件的情况下，应用拉普拉斯变换将微分方程转换为拉普拉斯域中的代数方程。

线性系统具有叠加性，即对各个输入的响应可叠加，以及齐次性，即当输入乘以一个……时，响应会相应地按比例缩放

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Modeling in Time and Frequency Domain

时域与频域建模

KEY TERMS AND CONCEPTS

关键术语与概念

linearizing nonlinear rl circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount.

Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a Taylor series expansion, which writes a function using its derivatives at one point. When higher-order terms are ignored for small deviations, the result is a linear relationship.

An RL circuit with a nonlinear resistor is a common example. The circuit is first described with Kirchhoff's voltage law, which gives a nonlinear differential equation. In that equation, V(t) is the applied voltage, L is the inductance, R is the resistance, and E is the battery voltage.

To analyze the circuit, the steady-state current is found first. The small-signal source is set to zero, and the equilibrium current i0 is solved for. The nonlinear equation is then rewritten in terms of small deviations from that equilibrium.

Next, the resistor’s characteristics are used to build the linearized equation. For small changes in current, the voltage can be approximated with a linear expression. Substituting that approximation into the voltage law gives a linear differential equation. With known values and zero initial conditions, the Laplace transform converts the equation into an algebraic form in the Laplace domain.

Linearizing Nonlinear RL Circuits

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

electrical systems

RLC Circuit Transfer Functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

mechanical systems

Transfer Functions in Mechanical Motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

electro-mechanical systems

DC Motor Modeling in Electromechanical Systems

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

Linearizing Nonlinear State-Space Models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how a nonlinear RL circuit is linearized with Taylor series, steady-state current, Kirchhoff's voltage law, and the Laplace transform.

Linearizing Nonlinear RL Circuits | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Linearizing Nonlinear RL Circuits

Description

Transcript

Tags