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

首先用电阻、电容和电感这三个无源线性元件组成的等效电路来表示电路网络。
这些元件被组合成电路，确定输入和输出后，利用基尔霍夫定律求得传递函数。
在RLC电路中，可通过基尔霍夫电压定律推导出电容器两端电压与输入电压之间的传递函数。
这将得到一个关于该网络的积分-微分方程，假设初始条件为零。
首先将变量从电流转换为电荷，然后应用电容器的电压-电荷关系。
对该方程进行拉普拉斯变换并加以简化，可得到该电路的传递函数。
阻抗是一种类似于电阻的传递函数，但适用于电容器和电感器。
也可以使用基尔霍夫电流定律，通过节点分析来获得传递函数。
该系统中的电流由流经电容器的电流以及流经串联电阻和电感的电流组成。
简化后，得到相同的传递函数结果。

电气系统

Electrical Systems

在电气工程中，对由无源线性元件（电阻器 (R)、电容器 (C) 和电感器 (L)）组成的网络进行分析是基础。这些元件被组织成电路，可以使用传递函数分析输入和输出之间的关系。RLC 电路的传递函数（将电容器两端的电压与输入电压联系起来）可以使用基尔霍夫定律推导出来。

要推导传递函数，请考虑将元件串联连接的 RLC 电路。基尔霍夫电压定律 (KVL) 指出，闭环周围所有电压的总和为零。对于 RLC 电路，这可转换为以下积分微分方程，假设初始条件为零：

<img alt="Equation1" src="/files/ftp_upload/16096/16096_PT%20EQUATION_1.svg" style="height: 32px;">

此处，V(t) 为输入电压，VR(t)=i(t)R 为电阻两端的电压，VC(t) = q(t)/C 为电容两端的电压，其中 q(t) 为电容上的电荷，VL(t) = L(di(t)/dt) 为电感两端的电压。

电流 i(t) 与电荷 q(t) 的关系为 i(t)=dq(t)/dt 。将这些关系代入 KVL 方程可得出：

<img alt="Equation2" src="/files/ftp_upload/16096/16096_PT%20EQUATION_2.svg" style="height: 48px;">

将方程写成电压形式，对两边进行拉普拉斯变换，假设所有初始条件均为零，可得到以下方程：

<img alt="Equation3" src="/files/ftp_upload/16096/16096_PT%20EQUATION_3.svg" style="height: 30px;">

重新排列项以求解传递函数：

<img alt="Equation4" src="/files/ftp_upload/16096/16096_PT%20EQUATION_4.svg" style="height: 46px;">

此传递函数表示响应频域中的输入电压而跨电容器的电压。阻抗类似于电阻，但适用于电容器和电感器，在定义传递函数方面起着关键作用。此外，基尔霍夫电流定律 (KCL) 也可用于通过节点分析得出传递函数。该定律指出，进入节点的电流总和必须等于离开节点的电流总和。在 RLC 电路中，总电流是通过电容器的电流与通过电阻器和电感器的串联组合的电流之和。通过应用 KCL 并进行简化，可以实现相同的传递函数。

KVL 和 KCL 是电路分析中的强大工具，可以推导出简洁地描述频域中电气网络动态行为的传递函数。这些方法对于各种应用中复杂电路的设计和分析至关重要。

首先，用电阻、电容和电感这三个无源线性元件构成的等效电路来表示电路网络。

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Modeling in Time and Frequency Domain

时域与频域建模

KEY TERMS AND CONCEPTS

关键术语与概念

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...

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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

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Finding a Transfer Function from State Space

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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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