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

線形システムは、個々の入力に対する応答を合計する重ね合わせと、スカラーを掛けた入力に対して応答を一貫してスケーリングする均質性を示します。
非線形システムは、小さな変化の操作点を中心に線形と見なすことができます。
テイラー級数展開は、関数の値を特定の点での導関数と、その点からの偏差にリンクします。小さな偏差に対して高次の項を無視すると、線形関係が得られます。
非線形抵抗を持つRL回路を考えてみましょう。非線形成分が存在するため、伝達関数を導出する前にシステムの線形化が必要です。
キルヒホッフの電圧法則を利用して、非線形微分方程式を導き出します。
定常状態の電流は、小信号源をゼロに設定することによって求められます。
式は、電流の平衡値に基づいて書き換えられます。非線形抵抗の特性を使用して、線形化された微分方程式を導き出します。
既知の値が代入され、ラプラス変換が初期条件ゼロで適用されます。
平衡点の周りのインダクタの両端の電圧の式は、ラプラス変換を受け、伝達関数を得るために簡略化されます。

周波数領域での線形近似

Linear Approximation in Frequency Domain

線形システムは、重ね合わせと均質性という 2 つの主な特性によって特徴付けられます。重ね合わせにより、複数の入力に対する応答は、各入力に対する応答の和になります。均質性により、入力をスカラーでスケーリングすると、応答が同じスカラーでスケーリングされます。

対照的に、非線形システムは本質的にこれらの特性を持ちません。ただし、動作点の周りの小さな偏差については、非線形システムを線形として近似できることがよくあります。この近似は、特定の点での導関数で関数を表現するテイラー級数展開によって実現されます。小さな偏差に対して高次の項を無視することで、線形関係が得られます。

非線形抵抗器を含む RL 回路を考えてみましょう。このシステムを分析するには、伝達関数を導出する前に線形化が必要です。

<img alt="Figure1" src="/files/ftp_upload/16099/16099_PT%20Figure_1.svg" style="height: 180px;">

まず、キルヒホッフの電圧法則を回路に適用し、システムを記述する非線形微分方程式を作成します。たとえば、電圧法則の方程式は次の形式になります。

<img alt="Equation1" src="/files/ftp_upload/16099/16099_PT%20EQUATION_1.svg" style="height: 20px;">

ここで、V(t) は印加電圧、L はインダクタンス、R は抵抗、E はバッテリ電圧を表します。

定常電流を求めるには、小信号源をゼロに設定し、平衡電流 i_0 を解きます。次に、この平衡からの偏差に基づいて非線形微分方程式を書き直します。

<img alt="Equation2" src="/files/ftp_upload/16099/16099_PT%20EQUATION_2.svg" style="height: 22px;">

非線形抵抗器の特性を使用して、線形化された微分方程式を導きます。電流の偏差が小さい場合、電圧方程式は次のように記述できます。

<img alt="Equation3" src="/files/ftp_upload/16099/16099_PT%20EQUATION_3.svg" style="height: 32px;">

この近似値を電圧法則方程式に代入すると、線形微分方程式が得られます。既知の値を代入し、初期条件がゼロであると仮定すると、ラプラス変換が適用され、微分方程式がラプラス領域の代数方程式に変換されます。

線形システムは、重ね合わせ、個々の入力に対する応答の合計、および均質性を示し、入力に応答を一貫してスケーリングし、...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Modeling in Time and Frequency Domain

時間領域および周波数領域におけるモデリング

KEY TERMS AND CONCEPTS

主要用語と概念

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

Linearizing Nonlinear RL Circuits

Description

Transcript

Tags