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

A técnica de domínio da frequência, que é comumente usada para analisar e projetar sistemas de controle de feedback, é adequada apenas para sistemas lineares invariantes no tempo.
A abordagem no domínio do tempo ou no espaço de estados pode lidar com sistemas não lineares, variáveis no tempo e de múltiplas entradas e saídas.
Na representação do espaço de estados, um subconjunto particular de variáveis de sistema, identificadas como variáveis de estado, é usado para construir equações diferenciais simultâneas de primeira ordem para um sistema de enésima ordem, rotuladas como equações de estado.
Considere um circuito RLC. Dado que a rede é de segunda ordem, é necessário empregar duas equações diferenciais simultâneas de primeira ordem.
As quantidades que são diferenciadas na equação derivada para os elementos de armazenamento de energia - o indutor e o capacitor são as variáveis de estado.
As leis de tensão e corrente de Kirchhoff são usadas para expressar as variáveis não-estatais como combinações lineares das variáveis de estado e da entrada.
Esses resultados são substituídos nas equações derivadas originais para obter as equações de estado.
A equação de saída é a corrente através do resistor.
A etapa final envolve representar essas equações em uma forma de matriz vetorial para adquirir a representação no espaço de estados da rede elétrica fornecida.

representação do espaço de estado

A técnica de domínio de frequência, comumente usada na análise e no projeto de sistemas de controle de feedback, é eficaz para sistemas lineares e invariantes no tempo. No entanto, ela fica aquém ao lidar com sistemas não lineares, variáveis ​​no tempo e com múltiplas entradas e múltiplas saídas. A abordagem do domínio de tempo ou espaço de estado aborda essas limitações utilizando variáveis ​​de estado para construir equações diferenciais simultâneas de primeira ordem, conhecidas como equações de estado, para um sistema de ordem n.
Considere um circuito RLC, um sistema comum de segunda ordem. Para analisar esse circuito usando a abordagem do espaço de estado, duas equações diferenciais simultâneas de primeira ordem são necessárias. As variáveis ​​de estado neste contexto são derivadas das quantidades diferenciadas nas equações derivadas associadas aos elementos de armazenamento de energia, especificamente o indutor e o capacitor.

As leis de tensão e corrente de Kirchhoff são empregadas para formular as equações de estado. A lei dos nós de Kirchhoff (KVL) afirma que todas as diferenças de potencial elétrico ao redor de um loop são zero, enquanto a lei das malhas de Kirchhoff (KCL) afirma que a soma das correntes que entram em uma junção é igual à soma das correntes que saem. Essas leis permitem a expressão de variáveis ​​não estatais como combinações lineares de variáveis ​​de estado e entradas.

Em um circuito RLC, as variáveis ​​de estado são a tensão no capacitor VC e a corrente no indutor iL. As leis de Kirchhoff expressam a corrente do resistor e outras variáveis ​​não estatais em termos de VC e iL. Essas expressões são então substituídas de volta nas equações diferenciais originais do circuito.
Após derivar as equações de estado, a etapa final é representar essas equações na forma de matriz vetorial, alcançando a representação do espaço de estado. Para um circuito RLC, isso pode envolver a definição do vetor de estado x, o vetor de entrada u, o vetor de saída y e as matrizes A, B, C e D, de modo que:
<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;">
Esta representação é essencial para analisar o comportamento dinâmico do sistema e projetar estratégias de controle apropriadas.
Em resumo, a abordagem de espaço de estado fornece uma estrutura robusta para lidar com sistemas complexos, estendendo-se além das capacidades das técnicas de domínio de frequência, acomodando não linearidades, variações de tempo e múltiplas entradas e saídas.

Representação do Espaço de Estado

A técnica de domínio da frequência, que é comumente usada para analisar e projetar sistemas de controle de feedback, é adequada apenas para sistemas lineares e invariantes no tempo.

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

Unidade 1

Modeling in Time and Frequency Domain

Modelagem no Domínio do Tempo e da Frequência

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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...

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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

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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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