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

フィードバック制御システムの解析と設計に一般的に使用される周波数領域手法は、線形の時間不変システムにのみ適しています。
時間領域または状態空間アプローチでは、非線形、時間変動、および複数入力、複数出力システムを処理できます。
状態空間表現では、状態変数として識別されるシステム変数の特定のサブセットを使用して、状態方程式としてラベル付けされた n 次システムの同時 1 次微分方程式を作成します。
RLC回路について考えてみましょう。ネットワークが 2 次であることを考えると、2 つの同時 1 次微分方程式を使用する必要があります。
エネルギー貯蔵要素の微分方程式で微分される量 - インダクタとコンデンサは状態変数です。
キルヒホッフの電圧法則と電流法則は、非状態変数を状態変数と入力の線形結合として表すために使用されます。
これらの結果は、元の微分方程式に代入されて、状態方程式が得られます。
出力式は、抵抗を流れる電流です。
最後のステップでは、これらの方程式をベクトル行列形式で表現して、特定の電気ネットワークの状態空間表現を取得します。

状態空間表現

フィードバック制御システムの解析と設計でよく使用される周波数領域手法は、線形で時間不変のシステムには効果的です。ただし、非線形で時間変動の、複数入力複数出力のシステムを扱う場合には不十分です。時間領域または状態空間アプローチは、状態変数を使用して n 次システムの状態方程式と呼ばれる同時一次微分方程式を構築することで、これらの制限に対処します。

一般的な二次システムである RLC 回路を考えてみましょう。状態空間アプローチを使用してこの回路を解析するには、2 つの同時一次微分方程式が必要です。この文脈での状態変数は、エネルギー貯蔵要素、具体的にはインダクタとコンデンサに関連する微分方程式で微分された量から導出されます。

状態方程式を定式化するために、キルヒホッフの電圧法則と電流法則が使用されます。キルヒホッフの電圧法則 (KVL) は、ループの周りのすべての電位差がゼロであることを示しています。一方、キルヒホッフの電流法則 (KCL) は、接合部に入る電流の合計が接合部から出る電流の合計に等しいことを示しています。これらの法則により、非状態変数を状態変数と入力の線形結合として表現できます。

RLC 回路では、状態変数はコンデンサ VC の両端の電圧とインダクタ iL を流れる電流です。キルヒホッフの法則は、抵抗器の電流とその他の非状態変数を VC と iL で表します。これらの式は、回路の元の微分方程式に代入されます。

状態方程式を導出した後、最後のステップは、これらの方程式をベクトル行列形式で表し、状態空間表現を実現することです。 RLC 回路の場合、状態ベクトル x、入力ベクトル u、出力ベクトル y、および行列 A、B、C、D を次のように定義する必要があります。

<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;">

この表現は、システムの動的動作を分析し、適切な制御戦略を設計するために不可欠です。

要約すると、状態空間アプローチは、非線形性、時間変動、および複数の入力と出力に対応することで、周波数領域手法の能力を超えて、複雑なシステムを処理するための強力なフレームワークを提供します。

フィードバック制御システムの解析と設計に一般的に使用される周波数領域技術は、線形で時間不変の...

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

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

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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...

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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...

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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 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

RLC Circuit State Equations

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