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

Os sistemas lineares exibem superposição, somando respostas a entradas individuais, e homogeneidade, dimensionando respostas consistentemente para uma entrada multiplicada por um escalar.
Um sistema não linear pode ser considerado linear em torno de um ponto de operação para pequenas mudanças.
A expansão da série de Taylor vincula o valor de uma função às suas derivadas em um ponto específico e desvios desse ponto. Negligenciar termos de ordem superior para pequenos desvios fornece uma relação linear.
Considere um circuito RL com um resistor não linear. A presença do componente não linear requer a linearização do sistema antes de derivar a função de transferência.
A lei de tensão de Kirchhoff é utilizada para derivar uma equação diferencial não linear.
A corrente de estado estacionário é encontrada definindo a fonte de sinal pequeno como zero.
A expressão é reescrita em termos do valor de equilíbrio da corrente. As características do resistor não linear são usadas para derivar a equação diferencial linearizada.
Os valores conhecidos são substituídos e a transformada de Laplace é aplicada com zero condições iniciais.
A expressão da tensão através do indutor em torno do ponto de equilíbrio é submetida à transformada de Laplace e simplificada para obter a função de transferência.

aproximação linear no domínio da frequência

Linear Approximation in Frequency Domain

Os sistemas lineares são caracterizados por duas propriedades principais: superposição e homogeneidade. A superposição permite que a resposta a múltiplas entradas seja a soma das respostas a cada entrada individual. A homogeneidade garante que a escala de uma entrada por um escalar resulta na resposta sendo escalada pelo mesmo escalar.
Em contraste, os sistemas não lineares não possuem inerentemente essas propriedades. No entanto, para pequenos desvios em torno de um ponto operacional, um sistema não linear pode frequentemente ser aproximado como linear. Essa aproximação é obtida por meio da expansão da série de Taylor, que expressa uma função em termos de suas derivadas em um ponto específico. Ao negligenciar termos de ordem superior para pequenos desvios, uma relação linear é obtida.

Considere um circuito RL contendo um resistor não linear. Para analisar esse sistema, a linearização é necessária antes de derivar a função de transferência.
<img alt="Figure1" src="/files/ftp_upload/16099/16099_PT%20Figure_1.svg" style="height: 180px;">
O primeiro passo envolve aplicar a lei de tensão de Kirchhoff ao circuito, resultando em uma equação diferencial não linear que descreve o sistema. Por exemplo, a equação da lei de tensão pode assumir a forma:
<img alt="Equation1" src="/files/ftp_upload/16099/16099_PT%20EQUATION_1.svg" style="text-align: center; height: 20px;">
Onde V(t) é a tensão aplicada, L é a indutância, R é a resistência e E representa a tensão da bateria.
Para encontrar a corrente de estado estacionário, definimos a fonte de pequeno sinal como zero e resolvemos para a corrente de equilíbrio i_0. A equação diferencial não linear é então reescrita em termos de desvios deste equilíbrio:
<img alt="Equation2" src="/files/ftp_upload/16099/16099_PT%20EQUATION_2.svg" style="height: 22px;">
As características do resistor não linear são usadas para derivar a equação diferencial linearizada. Para pequenos desvios na corrente, a equação de voltagem pode ser escrita como:
<img alt="Equation3" src="/files/ftp_upload/16099/16099_PT%20EQUATION_3.svg" style="height: 32px;">
Substituindo essa aproximação na equação da lei de voltagem, obtemos uma equação diferencial linear. Com valores conhecidos substituídos e assumindo condições iniciais zero, a transformada de Laplace é aplicada para converter a equação diferencial em uma equação algébrica no domínio de Laplace.

Aproximação Linear no Domínio da Frequência

Os sistemas lineares exibem superposição, somando respostas a entradas individuais, e homogeneidade, dimensionando respostas consistentemente para uma entrada multiplicada por um ...

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

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