Name: rlc circuit transfer functions
Uploaded: 2024-09-26T13:17:18.000Z
Duration: 1 min 21 s
Channel: JoVE

전기 네트워크는 먼저 세 가지 수동 선형 구성 요소(저항기, 커패시터 및 인덕터)로 구성된 등가 회로로 표시됩니다.
이러한 구성 요소는 회로로 결합되고 입력과 출력이 결정되며 전달 함수는 Kirchhoff의 법칙을 사용하여 구됩니다.
RLC 회로에서 커패시터 양단의 전압을 입력 전압과 관련시키는 전달 함수는 Kirchhoff의 전압 법칙을 사용하여 파생 될 수 있습니다.
이는 초기 조건이 0이라고 가정하고 네트워크에 대한 적분 미분 방정식을 생성합니다.
변수는 처음에 전류에서 충전으로 변경된 다음 커패시터에 대한 전압-충전 관계를 적용합니다.
이 방정식의 라플라스 변환을 취하고 단순화하면 이 회로에 대한 전달 함수가 생성됩니다.
임피던스는 저항과 유사하지만 커패시터와 인덕터에 적용할 수 있는 전달 함수입니다.
전달 함수는 노드 분석을 사용하여 Kirchhoff의 현재 법칙을 사용하여 얻을 수도 있습니다.
시스템의 전류는 커패시터를 통해 흐르는 전류와 직렬 저항기 및 인덕터를 통해 순환하는 전류로 구성됩니다.
단순화 후에는 전달 함수에 대해 동일한 결과를 얻을 수 있습니다.

전기 시스템

Electrical Systems

전기 공학에서 수동 선형 구성 요소(저항(R), 축전기(C), 유도자(L))로 구성된 네트워크 분석은 기본입니다. 이러한 구성 요소는 입력과 출력 간의 관계를 전달 함수를 사용하여 분석할 수 있는 회로로 구성됩니다. 커패시터 전압을 입력 전압과 연관시키는 RLC 회로의 전달 함수는 키르히호프의 법칙을 사용하여 도출할 수 있습니다.

전달 함수를 도출하려면 구성 요소가 직렬로 연결된 RLC 회로를 고려합니다. 키르히호프의 전압 법칙(KVL)은 폐쇄 루프 주변의 모든 전압의 합이 0이라고 명시합니다. RLC 회로의 경우, 이는 초기 조건이 0이라고 가정할 때 다음의 적분-미분 방정식으로 변환됩니다.

<img alt="Equation1" src="/files/ftp_upload/16096/16096_PT%20EQUATION_1.svg" style="text-align: center; height: 32px;">

여기서 V(t)는 입력 전압이고, VR(t)=i(t)R은 저항기 전압이고, VC(t)=q(t)/C는 커패시터 전압입니다. 여기서 q(t)는 커패시터의 전하이고, VL(t) = L(di(t)/dt)는 인덕터 전압입니다.

전류 i(t)는 i(t)=dq(t)/dt에 의해 전하 q(t)와 관련이 있습니다. 이러한 관계를 KVL 방정식에 대입하면 다음과 같습니다.

<img alt="Equation2" src="/files/ftp_upload/16096/16096_PT%20EQUATION_2.svg" style="text-align: center; height: 48px;">

방정식을 전압 형태로 작성하고 양쪽의 라플라스 변환을 취하고 모든 초기 조건이 0이라고 가정하면 다음 방정식이 얻어집니다.

<img alt="Equation3" src="/files/ftp_upload/16096/16096_PT%20EQUATION_3.svg" style="text-align: center; height: 30px;">

전달 함수를 풀기 위해 항을 재배열합니다.

<img alt="Equation4" src="/files/ftp_upload/16096/16096_PT%20EQUATION_4.svg" style="text-align: center; height: 46px;">

이 전달 함수는 주파수 영역에서 입력 전압에 대한 축전기 전압을 나타냅니다. 저항과 유사하지만 축전기와 유도자에 적용할 수 있는 온저항은 전달 함수를 정의하는 데 중요한 역할을 합니다. 또한 키르히호프의 전류 법칙(KCL)을 사용하여 노드 분석을 통해 전달 함수를 도출할 수도 있습니다. 이 법칙은 노드에 들어오는 전류의 합은 노드에서 나가는 전류의 합과 같아야 한다고 명시합니다. RLC 회로에서 총 전류는 축전기를 통과하는 전류와 저항과 유도자의 직렬 조합을 통과하는 전류의 합입니다. KCL을 적용하고 단순화하면 동일한 전달 함수를 얻을 수 있습니다.

KVL과 KCL은 회로 분석에서 강력한 도구로, 주파수 영역에서 전기 네트워크의 동적 동작을 간결하게 설명하는 전달 함수를 도출할 수 있습니다. 이러한 방법은 다양한 응용 분야에서 복잡한 회로를 설계하고 분석하는 데 필수적입니다.

전기 네트워크는 먼저 세 가지 수동 선형 구성 요소(저항기, 커패시터 및 인덕터)로 구성된 등가 회로로 표시됩니다.
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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

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. Using that rule, the circuit can be written as an integro-differential equation when the initial conditions are zero.

In the equation, the input voltage is V(t). The resistor voltage is VR(t) = i(t)R, where i(t) is the current. The capacitor voltage depends on the charge q(t), and the inductor voltage depends on the current change in the loop. Since current and charge are related, these expressions can be substituted into the KVL equation.

The same circuit can also be written in voltage form and converted with the Laplace transform. With all initial conditions set to zero, this step gives an equation in the frequency domain. Rearranging the terms produces the transfer function for the capacitor voltage response.

Impedance plays an important role in this result. It works like resistance, but it applies to capacitors and inductors as well. The transfer function shows how the capacitor voltage responds to the input voltage in the frequency domain.

Kirchhoff's Current Law, or KCL, can also be used to reach the same result through nodal analysis. KCL says that the sum of currents entering a node must equal the sum of currents leaving it. In the RLC circuit, the total current is the current through the capacitor plus the current through the series resistor and inductor, and simplifying this relationship gives the same transfer function.

RLC Circuit Transfer Functions

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 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 KVL, KCL, and the Laplace transform are used to find the transfer function of a series RLC circuit and track capacitor voltage in the frequency domain.

RLC Circuit Transfer Functions | 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

RLC Circuit Transfer Functions

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