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

오디오 신호에서 원치 않는 고주파 잡음을 제거하기 위해 2차 저역 통과 필터를 사용하는 오디오 시스템이 있다고 가정해 보겠습니다.
이 설정에는 전압 팔로워 구성의 2차 연산 증폭기 회로와 두 개의 저장 소자로 구성된 두 개의 노드가 포함됩니다.
이 노드에서 Kirchhoff의 현재 법칙을 적용하면 두 개의 미분 방정식이 생성됩니다.
또한 피드백 커패시터의 전압은 저항기의 전압에서 다른 커패시터의 전압을 뺀 값과 같습니다.
이러한 방정식은 첫 번째 노드의 KCL 방정식에서 커패시터 전압을 제거하는 데 사용할 수 있습니다.
결과는 2차 특성미분 방정식입니다.
이 미분 방정식의 해는 과도 응답과 정상 상태 응답을 모두 포함합니다.
과도 응답은 시간이 지남에 따라 소멸되며 과감쇠, 부족감쇠 및 임계 감쇠 시나리오에서 source-free 회로에 대한 솔루션과 유사합니다.
회로가 정상 상태에 도달하면 커패시터 또는 저항을 통해 전류가 흐르지 않으며 이상적인 연산 증폭기의 입력 단자는 전류 유입을 방지합니다. 그 결과, 정상 상태 응답은 소스 전압과 일치합니다.
입력 소스 전압을 제거하면 순전히 과도 응답이 발생합니다.

2차 연산 증폭기 회로

Second-order Op Amp Circuits

오디오 시스템에서 2차 저역 통과 필터를 구현하는 것은 바람직하지 않은 고주파 잡음을 제거하여 오디오 신호를 정제하는 데 중요합니다. 이러한 필터에는 일반적으로 별도의 저장 요소가 있는 두 개의 노드를 포함하는 전압 팔로워로 구성된 2차 연산 증폭기 회로가 포함됩니다.

이러한 회로의 분석은 2차 RLC 회로와 유사한 체계적인 접근 방식을 따릅니다. 실제 시나리오에서는 크기와 무게로 인해 부피가 큰 인덕터가 거의 사용되지 않습니다. 이는 여기서 초점이 필터 및 발진기와 같은 장치에 광범위하게 적용되는 RC 2차 연산 증폭기 회로에 주로 맞춰져 있음을 의미합니다.

키르히호프의 현행 법칙을 노드에 적용한 후 두 개의 미분 방정식이 나타납니다. 또한, 회로 구성 요소 간의 전압 관계를 관찰하여 2차 특성 미분 방정식을 추론합니다. 이 방정식은 과도 및 정상 상태 응답을 모두 구현합니다.

<img alt="Equation1" src="/files/ftp_upload/15103/15103_PT_Equation_1.svg" style="text-align: center; height: 34px;">

과도 응답은 시간이 지남에 따라 점차 감소하며 소스가 없는 회로에서 발견되는 솔루션과 유사성을 공유하며 과감쇠, 과소감쇠 및 임계 감쇠 시나리오의 특성을 나타냅니다. 회로가 정상 상태에 도달하면 축전기와 저항기는 더 이상 전류를 전도하지 않으며 이상적인 연산 증폭기 입력 단자는 전류 흐름을 차단합니다. 결과적으로 정상 상태 응답은 소스 전압과 일치합니다.

특히, 입력 소스 전압을 제거하면 순수한 과도 응답이 발생합니다. 이러한 2차 연산 증폭기 회로는 오디오 품질을 향상시키는 다양한 용도로 사용되며 다양한 오디오 처리 시스템에서 중추적인 역할을 합니다. 다양한 감쇠 시나리오에서의 동작을 이해하면 최적의 오디오 신호 개선을 달성하는 데 도움이 됩니다.

오디오 신호에서 원치 않는 고주파 잡음을 제거하기 위해 2차 저역 통과 필터를 사용하는 오디오 시스템이 있다고 가정해 보겠습니다.
이 설정은 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

First and Second-Order Circuits

1차 및 2차 회로

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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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