Name: second-order op amp low-pass filters
Uploaded: 2024-04-11T12:48:44.000Z
Duration: 1 min 19 s
Channel: JoVE

考虑一个音频系统，该系统采用二阶低通滤波器来消除音频信号中不需要的高频噪声。
该装置包含一个二阶运算放大器电路，采用电压跟随器配置，并具有两个包含两个储能元件的节点。
对这些节点应用基尔霍夫电流定律可得到两个微分方程。
此外，反馈电容器两端的电压等于电阻两端的电压减去另一个电容器两端的电压。
这些方程可用于从第一个节点的基尔霍夫电流定律（KCL）方程中消去电容电压。
结果是一个二阶特征微分方程。
该微分方程的解包含瞬态响应和稳态响应。
瞬态响应随时间逐渐消失，在过阻尼、欠阻尼和临界阻尼情况下，其行为类似于无源电路的解。
当电路达到稳态时，电容器或电阻器中没有电流流过，理想运算放大器的输入端子阻止电流流入。因此，稳态响应与源电压相匹配。
消除输入源电压将导致纯粹的瞬态响应。

二阶运算放大器电路

Second-order Op Amp Circuits

在音频系统中实现二阶低通滤波器对于通过消除不需要的高频噪声来细化音频信号是至关重要的。这些滤波器通常会涉及到电路中含有电压跟随器的二阶运算放大器电路，其中包含了具有不同存储元件的两个节点。 此类电路的分析会遵循系统方法，这类似于二阶 RLC 电路。在实际的场景中，由于体积和重量的原因，通常会很少使用笨重的电感器。这意味着这里的重点主要是 RC 二阶运算放大器电路，该电路在滤波器和振荡器等设备中具有广泛的应用。 在节点处应用基尔霍夫电流定律后，便会得到两个微分方程。此外，通过观察电路元件上的电压关系能够推导出二阶特征微分方程。该方程体现了瞬态响应和稳态响应。 <img alt="Equation1" src="/files/ftp_upload/15103/15103_PT_Equation_1.svg" style="text-align: center; height: 34px;"> 瞬态响应会随着时间的推移而逐渐减弱，并且与无源电路中的解决方案有相似之处，从而表现出过阻尼、欠阻尼和临界阻尼场景的特征。一旦电路达到稳定状态，电容器和电阻器便不再传导电流，并且理想运算放大器的输入端子会阻止电流的流动。因此，稳态响应与源电压是匹配的。 值得注意的是，消除输入源的电压会导致其出现纯粹的瞬态响应。这些二阶运算放大器电路在增强音频质量方面具有多种应用，并且在各种音频处理系统中发挥着至关重要的作用。了解它们在不同阻尼情况下的现象是否有助于实现最佳音频信号细化的效果。

考虑一个利用二阶低通滤波器消除音频信号中不希望存在的高频噪声的音频系统。  
该装置……

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

First and Second-Order Circuits

一阶与二阶电路

KEY TERMS AND CONCEPTS

关键术语与概念

second-order op amp low-pass filters

Second-order op-amp low-pass filters remove unwanted high-frequency noise from audio signals. They are used in audio systems to refine sound quality. These filters are often built as voltage followers and have two nodes with storage elements.

The circuit analysis follows the same basic method used for second-order RLC circuits. In practice, bulky inductors are usually avoided because of their size and weight. For that reason, the focus here is on RC second-order op-amp circuits, which are common in filters and oscillators.

Applying Kirchhoff's current law at the nodes gives two differential equations. A second-order characteristic differential equation is then found from the voltage relationships across the circuit parts. That equation describes both the transient response and the steady-state response.

The transient response fades over time. It can show overdamped, underdamped, or critically damped behavior, much like the solutions for source-free circuits. When the circuit reaches steady state, the capacitors and resistors no longer carry current, and the ideal op-amp input terminals block current flow. As a result, the steady-state response matches the source voltage.

If the input source voltage is removed, the circuit shows a pure transient response. These second-order op-amp circuits are important in audio processing systems because they help improve audio quality. Knowing how the circuit behaves under different damping conditions helps explain how the filter shapes the signal.

Second-Order Op Amp Low-Pass Filters

first-order circuits

First-Order Circuits in Flashing Lamps

First-order circuits use a resistor and one energy storage element, either a capacitor or an inductor. They are common in electronic systems and are described by a first-order differential equation that links the input and output signals. One important example is the RC circuit, which uses a resistor and a capacitor. In a relaxation oscillator such as a neon lamp circuit, the capacitor starts charging when voltage is applied. At first, the lamp behaves like an open circuit. When the capacitor...

rc circuit without source

RC Capacitor Discharge After Power Loss

An RC circuit discharges after the DC source is removed. If the capacitor was fully charged first, its initial voltage, V0, provides the energy that drives the source-free circuit. Kirchhoff’s current law is applied at the top node of the circuit. The current terms for the resistor and capacitor are then substituted, which leads to a first-order differential equation. Rearranging, integrating, and taking the exponential gives the natural response of the circuit. The integration constant in...

rc circuit with source

RC Circuit Charging After a DC Step

An RC circuit with a DC source shows how a capacitor charges after a sudden voltage step. The applied source is treated as a unit step function, and the capacitor voltage is the step response of the circuit. This response describes how the circuit reacts right after the input changes. A capacitor cannot change its voltage instantly. So, immediately after the switch closes at t=0, the capacitor voltage stays at its initial value. Kirchhoff’s current law is then applied at that moment and...

rl circuit without source

RL Circuit Decay After Source Removal

An RL circuit begins to decay after the DC source is removed. At that moment, the circuit becomes source-free. If the inductor has an initial current, I0, it also stores initial energy. That stored energy drives the natural response of the circuit. Kirchhoff’s voltage law is applied around the loop. The inductor voltage and resistor voltage are substituted into the equation. This gives a first-order differential equation for the current. Rearranging the terms produces a logarithmic form, which...

rl circuit with source

RL Step Response Under a DC Source

An RL circuit connected to a DC source has a step response that changes over time. The response has two parts: the transient response and the steady-state response. The transient response is the short-lived reaction that happens right after the source is applied. During the transient phase, the current changes exponentially. If the circuit starts with current in the inductor, that current decays toward zero as time goes on. At this stage, the inductor acts like a short circuit, so the source...

design example: frog muscle response

RL Circuit and Frog Leg Current Response

An RL circuit is used to model the frog leg’s response to electrical stimulation. In this setup, the student studies how current changes in a resistor-inductor circuit can trigger muscle contraction. The circuit provides a way to control and measure the electrical impulses involved in the experiment. When the switch is closed, the frog’s leg shows a brief contraction. At steady state, the inductor acts like a short circuit. That lets current bypass the resistor and produces a mild, short...

second-order circuits

RLC Damping and Resonant Frequency

Second-order circuits use two energy storage elements, such as a capacitor and an inductor, to shape how current and voltage change over time. These circuits include RLC circuits, along with RC and RL circuits that have two capacitors or two inductors. Their behavior is described by second-order differential equations that connect the input signal to the output signal. The input usually comes from a voltage source or a current source. The output is often the voltage across the capacitor, the...

series rlc circuit without source

Natural Response of a Source-Free RLC Circuit

A source-free RLC circuit is a series circuit with a resistor, inductor, and capacitor, but no external energy source. It begins with the energy already stored in the capacitor and inductor. That stored energy drives the circuit at first. The circuit is described by a second-order differential equation. This equation shows how the resistor, inductor, and capacitor interact over time. The resistor removes energy from the circuit, so the solution has an exponential form. When the exponential...

types of responses of series rlc circuits

types-of-responses-of-series-rlc-circuits

Damping in Series RLC Circuit Responses

A source-free series RLC circuit shows three response patterns based on damping. The circuit is described by a second-order differential equation. Its complete solution is made from two distinct solutions. Those solutions depend on the roots, which are written in terms of the damping factor and the resonant frequency. When the damping factor is greater than the resonant frequency, both roots are real and negative. This gives an overdamped response. The circuit output decays slowly over time. <br...

series rlc circuit with source

Forced Response in a Series RLC Circuit

A series RLC circuit with a DC source can model an automobile ignition system. The circuit shows how a battery can produce the high voltage needed for a spark. In this case, the input DC voltage acts like a forcing step function. Applying Kirchhoff's voltage law leads to a second-order differential equation. A second-order differential equation is an equation with a term for the second derivative. The form of this equation is very similar to the one used for a source-free RLC circuit, so the...

parallel rlc circuits

Parallel RLC Step Response in Surge Protectors

Parallel RLC circuits help explain how a surge protector works in a street lamp. The circuit model uses a DC input source that creates a step response when the switch is turned on. This setup shows how the lamp’s components can be protected from sudden voltage spikes. When the switch closes, Kirchhoff’s current law is applied to the parallel RLC circuit. That leads to a second-order differential equation. The solution has two parts: a transient response and a steady-state response. The...

second-order op amp circuits

Second-order op-amp low-pass filters remove unwanted high-frequency noise from audio signals. They are used in audio systems to refine sound quality. These filters are often built as voltage followers and have two nodes with storage elements. The circuit analysis follows the same basic method used for second-order RLC circuits. In practice, bulky inductors are usually avoided because of their size and weight. For that reason, the focus here is on RC second-order op-amp circuits, which are...

design example: underdamped parallel rlc circuit

design-example-underdamped-parallel-rlc-circuit

Oscillator Design with a Parallel RLC Circuit

A parallel RLC circuit can be designed to work as an oscillator with underdamped behavior. This example targets a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. A fixed resistance of 200 Ω is used as the starting point for the design. The first step is to find the capacitance. The damping factor is the reciprocal of twice the product of resistance and capacitance. Using that relationship, the needed capacitance value can be calculated from the given resistance...

key terms and concepts

Learn how second-order op-amp low-pass filters reduce audio noise, form characteristic equations, and show damping behavior in RC circuits.

Second-Order Op Amp Low-Pass Filters | 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

Second-Order Op Amp Low-Pass Filters

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