Name: transfer function from input to output
Uploaded: 2024-09-26T13:17:18.000Z
Duration: 1 min 21 s
Channel: JoVE

전달 함수는 주파수 영역에서 가능한 각 입력에 대한 시스템의 출력을 설명하는 수학적 표현입니다.
일반적인 n차 선형 시불변 미분 방정식이 있다고 가정하겠습니다. 이 방정식은 한 변수가 입력을 나타내고 다른 변수가 출력을 나타내는 시스템을 특성화합니다.
이 방정식의 양쪽에 라플라스 변환을 적용하면 대수 표현식이 생성됩니다.
모든 초기 조건이 0이라고 가정하면 이 방정식은 더 단순화됩니다.
출력의 라플라스 변환과 입력의 라플라스 변환의 비율을 전달 함수라고 합니다.
전달 함수는 블록 다이어그램으로 표시되며, 입력은 왼쪽에, 출력은 오른쪽에 표시되며, 시스템 전달 함수는 블록 내부에 있습니다.
전달 함수의 분모는 미분 방정식의 특성 다항식과 동일합니다.
1계 미분 방정식을 생각해 보겠습니다. 이 방정식의 전달 함수는 초기 조건이 0이라고 가정하고 양쪽에서 라플라스 변환을 수행하여 계산됩니다.
단순화하면 결과는 주파수 영역의 입력에 대한 시스템의 응답을 나타내는 전달 함수입니다.

제어 시스템의 전달 함수

Transfer Function in Control Systems

전달 함수는 선형 시불변(LTI) 시스템의 분석 및 설계에서 기본적인 개념입니다. 이는 시스템이 주파수 영역에서 다양한 입력에 어떻게 반응하는지 이해하는 간결한 방법을 제공합니다. 이는 시스템 동역학을 설명하는 시역 미분 방정식과 더 쉬운 조작 및 분석을 용이하게 하는 주파수 영역 표현 사이의 다리 역할을 합니다.

전달 함수를 도출하기 위해 다음과 같은 형태의 일반적인 n차 선형 시불변 미분 방정식을 고려합니다.

<img alt="Equation1" src="/files/ftp_upload/16095/16095_PT%20EQUATION_1.svg" style="text-align: center; height: 40px;">

여기서 c(t)는 출력이고 r(t)는 입력이며 a_i와 b_i는 상수 계수입니다. 모든 초기 조건이 0이라고 가정하고 양쪽에 라플라스 변환을 적용하면 미분 방정식을 복소 주파수 변수인 s에 대한 대수 방정식으로 변환할 수 있습니다. 용어를 재배열하면 다음과 같습니다.

<img alt="Equation2" src="/files/ftp_upload/16095/16095_PT%20EQUATION_2.svg" style="text-align: center; height: 22px;">

전달 함수 H(s)는 출력 C(s)와 입력 R(s)의 비율로 정의됩니다.

<img alt="Equation3" src="/files/ftp_upload/16095/16095_PT%20EQUATION_3.svg" style="text-align: center; height: 40px;">

이 표현식은 전달 함수가 s의 유리 함수임을 보여줍니다. 분자는 입력 계수로 형성된 다항식이고 분모는 미분 방정식의 특성 다항식입니다.

이 전달 함수는 시스템의 출력 c(t)가 주파수 영역에서 입력 r(t)에 어떻게 반응하는지 나타냅니다. 전달 함수는 왼쪽에 입력 R(s), 오른쪽에 출력 C(s), 블록 내부에 전달 함수 H(s)가 있는 블록 다이어그램으로 표현할 수 있습니다. 이 시각화는 특히 더 복잡한 시스템을 다룰 때 시스템 동작을 이해하고 분석하는 것을 간소화합니다.

전달 함수는 주파수 영역에서 가능한 각 입력에 대한 시스템의 출력을 설명하는 수학적 표현입니다.
고려 ...

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

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 coefficients a_i and b_i remain fixed, which makes the system linear and time-invariant.

The transfer function is found by applying the Laplace transform to both sides of the differential equation. This step assumes zero initial conditions. The result is an algebraic equation in terms of s, the complex frequency variable.

After rearranging the terms, the transfer function H(s) is defined as C(s) divided by R(s), where C(s) is the output in the Laplace domain and R(s) is the input in the Laplace domain. This form shows that H(s) is a rational function of s. Its numerator comes from the input coefficients, and its denominator is the characteristic polynomial of the differential equation.

This representation shows how the output c(t) responds to an input r(t) in the frequency domain. It can also be drawn as a block diagram, with R(s) entering on the left, C(s) leaving on the right, and H(s) inside the block. That diagram makes system analysis easier, especially for more complex systems.

Transfer Function from Input to Output

transfer function in control systems

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

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 transfer functions connect a system’s input and output using Laplace transforms, zero initial conditions, and a simple block diagram.

Transfer Function from Input to Output | 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

Transfer Function from Input to Output

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