Name: rlc circuit state equations
Uploaded: 2024-09-26T13:17:19.000Z
Duration: 1 min 27 s
Channel: JoVE

频域技术常用于分析和设计反馈控制系统，但仅适用于线性时不变系统。
时域或状态空间方法能够处理非线性、时变以及多输入多输出系统。
在状态空间表示中，通过选取系统变量的一个特定子集（称为状态变量），为 n 阶系统建立一组联立的一阶微分方程，这些方程被称为状态方程。
考虑一个RLC电路。由于该网络是二阶的，因此需要使用两个同时成立的一阶微分方程。
在储能元件（电感和电容）的导数方程中被微分的量是状态变量。
基尔霍夫电压定律和电流定律用于将非状态变量表示为状态变量和输入的线性组合。
将这些结果代入原始的导数方程，以得到状态方程。
输出方程是通过电阻的电流。
最后一步是将这些方程表示为向量-矩阵形式，以获得给定电路网络的状态空间表示。

状态空间表示

频域技术通常用于分析和设计反馈控制系统，对线性、时不变系统有效。然而，它在处理非线性、时变和多输入多输出系统时就显得不足了。时域或状态空间方法通过利用状态变量为 n 阶系统构建同时的一阶微分方程（称为状态方程），解决了这些限制。

考虑一个 RLC 电路，这是一种常见的二阶系统。要使用状态空间方法分析该电路，需要两个同时的一阶微分方程。在这种情况下，状态变量来自与储能元件（特别是电感器和电容器）相关的导数方程中微分的量。

采用基尔霍夫电压和电流定律来制定状态方程。基尔霍夫电压定律 (KVL) 指出，环路周围的所有电势差为零，而基尔霍夫电流定律 (KCL) 则断言，进入结点的电流总和等于离开结点的电流总和。这些定律使得非状态变量能够表示为状态变量和输入的线性组合。

在 RLC 电路中，状态变量是电容器两端的电压 VC 和流过电感器的电流 iL。基尔霍夫定律用 VC 和 iL 来表示电阻器电流和其他非状态变量。然后将这些表达式代入电路的原始微分方程中。

推导出状态方程后，最后一步是将这些方程表示为矢量矩阵形式，从而实现状态空间表示。对于 RLC 电路，这可能涉及定义状态向量 x、输入向量 u、输出向量 y 以及矩阵 A、B、C 和 D，以便：

<img alt="Equation1" src="/files/ftp_upload/16100/16100_PT%20EQUATION_1.svg" style="height: 16px;">
<img alt="Equation2" src="/files/ftp_upload/16100/16100_PT%20EQUATION_2.svg" style="height: 16px;">

此表示对于分析系统的动态行为和设计适当的控制策略至关重要。

总之，状态空间方法为处理复杂系统提供了一个强大的框架，通过适应非线性、时间变化和多个输入和输出，超越了频域技术的能力。

频域技术常用于分析和设计反馈控制系统，仅适用于线性时不变系统……

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

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 system like an RLC circuit, two first-order equations are needed. The state variables come from the quantities tied to the circuit’s energy-storage elements, which are the inductor and the capacitor.

Kirchhoff’s voltage law, or KVL, says that the total potential difference around a loop is zero. Kirchhoff’s current law, or KCL, says that the current entering a junction equals the current leaving it. These rules let engineers write non-state variables as linear combinations of the state variables and the input.

In an RLC circuit, the state variables are the capacitor voltage, VC, and the inductor current, iL. KVL and KCL are used to express the resistor current and other non-state variables in terms of VC and iL. Those expressions are then substituted into the circuit’s original differential equations.

After the state equations are found, they are written in vector-matrix form. This gives the state-space representation, with a state vector x, an input vector u, an output vector y, and matrices A, B, C, and D. This form is useful for studying system behavior and for designing control strategies.

RLC Circuit State Equations

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

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 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 an RLC circuit is written as state equations and converted into state-space form using KVL, KCL, and the key state variables VC and iL.

RLC Circuit State Equations | 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

RLC Circuit State Equations

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