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

선형 시스템은 중첩, 개별 입력에 대한 응답 합산 및 균질성을 나타내며, 스칼라를 곱한 입력에 대해 응답을 일관되게 스케일링합니다.
비선형 시스템은 작은 변화에 대한 작동점을 기준으로 선형으로 간주될 수 있습니다.
테일러 급수 확장은 함수의 값을 특정 지점에서 도함수와 해당 지점에서 벗어나는 도함수에 연결합니다. 작은 편차에 대해 고차 항을 무시하면 선형 관계가 생성됩니다.
비선형 저항이 있는 RL 회로를 고려하십시오. 비선형 구성요소가 존재하기 때문에 전달 함수를 도출하기 전에 시스템 선형화가 필요합니다.
Kirchhoff의 전압 법칙은 비선형 미분 방정식을 유도하는 데 사용됩니다.
정상 상태 전류는 소신호 소스를 0으로 설정하여 구합니다.
이 표현은 전류의 평형 값으로 다시 작성됩니다. 비선형 저항기의 특성은 선형화된 차동 방정식을 도출하는 데 사용됩니다.
알려진 값이 대체되고 라플라스 변환이 0의 초기 조건으로 적용됩니다.
평형점 주위의 인덕터를 통과하는 전압에 대한 표현은 라플라스 변환을 적용받고 전달 함수를 얻기 위해 단순화됩니다.

 주파수 영역에서의 선형 근사

Linear Approximation in Frequency Domain

선형 시스템은 중첩과 균질성이라는 두 가지 주요 속성이 특징입니다. 중첩은 여러 입력에 대한 응답이 각 개별 입력에 대한 응답의 합이 되도록 합니다. 균질성은 입력을 스칼라로 스케일링하면 응답이 동일한 스칼라로 스케일링됨을 보장합니다.

반면, 비선형 시스템은 본질적으로 이러한 속성을 가지고 있지 않습니다. 그러나 작동 지점 주변의 작은 편차의 경우 비선형 시스템은 종종 선형으로 근사될 수 있습니다. 이 근사는 특정 지점에서 미분으로 함수를 표현하는 테일러 급수 확장을 통해 달성됩니다. 작은 편차에 대한 고차 항을 무시함으로써 선형 관계가 얻어집니다.

비선형 저항을 포함하는 RL 회로를 고려합니다. 이 시스템을 분석하려면 전달 함수를 도출하기 전에 선형화가 필요합니다.

<img alt="Figure1" src="/files/ftp_upload/16099/16099_PT%20Figure_1.svg" style="text-align: center; height: 180px;">

첫 번째 단계는 회로에 키르히호프의 전압 법칙을 적용하여 시스템을 설명하는 비선형 미분 방정식을 만드는 것입니다. 예를 들어, 전압 법칙 방정식은 다음과 같은 형태를 가질 수 있습니다.

<img alt="Equation1" src="/files/ftp_upload/16099/16099_PT%20EQUATION_1.svg" style="text-align: center; height: 20px;">

여기서 V(t)는 인가 전압, L은 유도계수, R는 저항, E는 배터리 전압을 나타냅니다.

정상 상태 전류를 찾기 위해 소신호 소스를 0으로 설정하고 평형 전류 i_0를 구합니다. 그런 다음 비선형 미분 방정식을 이 평형으로부터의 편차를 기준으로 다시 작성합니다.

<img alt="Equation2" src="/files/ftp_upload/16099/16099_PT%20EQUATION_2.svg" style="text-align: center; height: 22px;">

비선형 저항의 특성을 사용하여 선형화된 미분 방정식을 도출합니다. 전류의 작은 편차에 대해 전압 방정식은 다음과 같이 작성할 수 있습니다.

<img alt="Equation3" src="/files/ftp_upload/16099/16099_PT%20EQUATION_3.svg" style="text-align: center; height: 32px;">

이 근사값을 전압 법칙 방정식에 대입하면 선형 미분 방정식을 얻습니다. 알려진 값을 대입하고 초기 조건이 0이라고 가정하면 라플라스 변환을 적용하여 미분 방정식을 라플라스 도메인의 대수 방정식으로 변환합니다.

선형 시스템은 중첩, 개별 입력에 대한 응답을 합산하고 균질성을 나타내며 입력에 대한 응답을 일관되게 스케일링합니다.

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Modeling in Time and Frequency Domain

시간 및 주파수 영역 모델링

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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