Name: active filter design and frequency response
Uploaded: 2025-05-22T09:06:15.000Z
Duration: 1 min 25 s
Channel: JoVE

某些无源滤波器包含昂贵的电感器，其设计不会超过单位增益。这些滤波器在低于音频范围的频率下可能会表现出降低的衰减效果。
有源滤波器由运算放大器、电阻器和电容器组成。
这些滤波器用于音频处理、电信和电源单元，分为一阶低通、一阶高通、带通和带阻滤波器。
考虑低通和高通有源滤波器的传递函数和频率响应。
与无源滤波器不同，低通有源滤波器中的有限低频增益和高通有源滤波器中的高频增益取决于反馈与输入电阻比。
低通滤波器会衰减高于转角频率的频率，而高通滤波器会抑制低于转角频率的频率。
低通和高通滤波器的组合会产生带通滤波器，而带阻滤波器将两种类型的滤波器与求和放大器组合在一起。
带通滤波器在转折频率之间传输频率并抑制转折频率之外的频率;带阻滤波器阻止范围内的频率并允许外部频率。

频率响应

有源滤波器是使用运算放大器（op-amps）、电阻器和电容器从信号中滤除不需要的频率成分中的电子电路。一阶低通有源滤波器旨在通过频率低于某一截止频率的信号，并衰减高于该截止频率的频率。一阶低通有源滤波器的传递函数为：
<img alt="Equation 1" src="/files/ftp_upload/15149/15149_Equation_1.svg" style="text-align: center; height: 40px;">
其中，A_f 为低频增益（也称为直流增益），s 为复频率，ω_c 为转折频率或截止频率。一阶高通有源滤波器允许高于特定截止频率的信号通过，同时还会衰减低于该频率的信号。其中的传递函数可以表示为： 
<img alt="Equation 2" src="/files/ftp_upload/15149/15149_Equation_2.svg" style="text-align: center; height: 36px;">
其中，A_f 为高频增益，ω_c 是截止频率，当低于该频率时，输出就会显著降低。带通滤波器兼具低通滤波器和高通滤波器的特性。它只允许一定范围内的频率通过，同时还会衰减这一范围之外的频率。带通滤波器的传递函数具有以下表示形式：
<img alt="Equation 3" src="/files/ftp_upload/15149/15149_Equation_3.svg" style="text-align: center; height: 42px;">
在这里，ω_0 是中心频率，Q 是品质因数。带阻滤波器（也称为陷波滤波器）的作用是相反的，除了衰减特定范围的频率之外，它会允许大多数频率通过。带阻滤波器的标准传递函数为：
<img alt="Equation 4" src="/files/ftp_upload/15149/15149_Equation_4.svg" style="text-align: center; height: 42px;">
在有源滤波器中，通带内的增益可以独立于截止频率来进行设置，增益是由反馈电阻与输入电阻的比值所决定的，其中的设计参数（例如电阻和电容值）能够用来设置截止频率并形成滤波器的整体频率响应，并以此来满足特定要求。

某些无源滤波器包含昂贵的电感器，其设计不会超过单位增益。这些滤波器在 ...

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Frequency Response

KEY TERMS AND CONCEPTS

关键术语与概念

Active Filters

active filter design and frequency response

Active filters use op-amps, resistors, and capacitors to shape a signal's frequency response. They remove unwanted frequency components while keeping the parts of the signal that are needed. The design depends on the choice of resistor and capacitor values, which set the cutoff points and help define the overall response.

A first-order low-pass active filter passes signals below a cutoff frequency and reduces signals above it. Its transfer function includes the low-frequency gain, also called the dc gain, along with the complex frequency s and the corner or cutoff frequency ωc. A first-order high-pass active filter does the opposite. It lets signals above a cutoff frequency pass and reduces signals below that frequency.

Active filters can also be built as band-pass filters. These filters combine low-pass and high-pass behavior and allow only a certain range of frequencies to pass. Their transfer function includes the center frequency ω0 and the quality factor Q. A band-reject filter, also called a notch filter, works in the opposite way. It passes most frequencies but attenuates a specific range.

In active filters, the gain in the passband can be set independently of the cutoff frequency. The gain is determined by the ratio of the feedback resistor to the input resistor. By choosing resistor and capacitor values carefully, engineers can set the cutoff frequencies and shape the frequency response to meet specific needs.

Active Filter Design and Frequency Response

active-filter-design-and-frequency-response

network function of a circuit

How Transfer Functions Shape Circuit Response

Transfer functions show how a circuit responds as signal frequency changes. They describe the link between input and output in frequency response analysis. This makes them useful for studying how circuits behave in electronics and engineering. A transfer function is a mathematical tool that relates phasor output to phasor input. It can be written as voltage gain, current gain, transfer impedance, or transfer admittance. The poles and zeros are the most important parts of the function. Zeros...

frequency response of a circuit

Inductive Circuit Frequency Response

Inductive circuit frequency response shows how a circuit with an inductor reacts to different frequencies. The analysis starts in the time domain and then moves into the frequency domain. In that step, the inductor is replaced by an impedance, and the circuit is written with phasors, which are sinusoidal signals shown with magnitude and phase. The transfer function describes how the output changes compared with the input at each frequency. A key parameter is the time constant, which is the...

gain

Circuit Gain and Phase Shift

Circuit gain and phase shift describe how a linear circuit changes a sinusoidal input voltage or current. These properties matter in circuits with reactive elements, because the response depends on the input frequency. As the frequency changes, the gain and phase shift also change. Gain compares the size of the output sinusoid to the size of the input sinusoid. If Vin is the input and Vout is the output, the gain K is the ratio of the output amplitude to the input amplitude. A gain greater...

bode plots

Reading Frequency Response on Bode Plots

Bode plots show how a circuit or system responds across frequency. They use a logarithmic frequency scale on the x-axis and gain in decibels on the y-axis. This format makes it easier to display a wide range of frequencies and to see how circuit parts affect behavior over that range. A network function gives the ratio of output to input for a system. Its complex form provides both magnitude and phase angle. The logarithmic gain on a Bode plot comes from the magnitude, calculated by multiplying...

transfer function and bode plots-i

Transfer Function Gain and Origin Effects

Transfer function gain and poles or zeros at the origin shape a system's Bode plot. The transfer function is often written in standard form after the polynomial coefficients are normalized. In that form, the constant gain, zeros, poles, and quadratic terms all appear clearly. The poles and zeros mark critical frequencies where the output magnitude and phase change noticeably. The constant gain K creates a fixed magnitude and phase term. Its magnitude is 20 log 10 K in decibels, and its phase...

transfer function and bode plots-ii

Bode Plot Slope and Phase Changes

Bode plots show how a transfer function changes with frequency. The standard form includes constant gain, poles and zeros at the origin, simple poles and zeros, and quadratic poles and zeros. Each part shapes the system response in a different way. A simple zero sets the magnitude trend first. The Bode magnitude plot stays flat at low frequencies, near 0 dB, and then rises at 20 dB per decade after the corner or break frequency, ω1. That corner frequency is where the straight-line...

bode plots construction

Reading Frequency Response with Bode Plots

Bode plots show the frequency response of a system with a magnitude plot and a phase plot. Both plots use a logarithmic frequency axis. In control system analysis, this makes it easier to see how a transfer function changes with frequency. To build a Bode plot, start with the transfer function H(ω). After normalization, the function includes constant gain, zeros, and poles. Each term affects the plot in a different way. A constant gain of 10 gives a starting magnitude of 20 dB, since 20...

series resonance

Series Resonance in RLC Circuits

Series resonance in an RLC circuit happens when the inductive reactance and capacitive reactance are equal. An RLC circuit contains a resistor, an inductor, and a capacitor. The impedance of the circuit is the ratio of the supply voltage to the circuit current. At resonance, the imaginary part of the impedance becomes zero. The frequency at which this occurs is called the resonant frequency. It depends on the inductance (L) and capacitance (C) of the circuit. Mathematically, the resonant...

characteristics of series resonant circuit

characteristics-of-series-resonant-circuit

Series Resonance and RLC Filter Behavior

Series resonance in an RLC circuit happens when the inductor, capacitor, and resistor are connected in series. At the resonant frequency, the inductive reactance and capacitive reactance are equal in size but opposite in sign, so they cancel each other. As a result, the circuit impedance becomes very small and is mainly set by the resistance. The resonant frequency of a series RLC circuit can be calculated from its inductance and capacitance. At this frequency, the current reaches its maximum...

parallel resonance

Parallel RLC Tuning at Resonance

A parallel RLC circuit uses a resistor, inductor, and capacitor connected to the same nodes. Because they share the same voltage, the circuit is analyzed with admittance, which describes how easily current can flow. In a parallel RLC circuit, admittance helps show how the circuit responds as the frequency changes. Resonance happens when the net reactance is zero. At that point, the capacitive and inductive effects cancel each other out. The resonant frequency is found from this condition. At...

frequency response of op amp circuits

Op Amp Gain and Bode Plot Basics

Operational amplifiers are used for signal conditioning, filtering, and mathematical tasks such as addition, subtraction, integration, and differentiation. Their frequency response shows how amplifier gain changes as the input frequency changes. This makes the gain of an op-amp a key idea in circuit design. At low frequencies, an op-amp can hold a constant gain called the dc gain, A0. As frequency rises, the gain starts to drop after the corner frequency, or break frequency, ω1. In other...

passive filters

Audio Frequency Control with Passive Filters

Passive filters shape the frequency spectrum of electrical signals. They use only passive parts such as resistors, inductors, and capacitors. Because they do not need an external power source, they can let selected frequency ranges pass or block others in many applications. Low-pass filters pass signals below the cutoff frequency, ωc, and reduce signals above it. In audio systems, they can send bass frequencies to woofers or subwoofers. For an RC low-pass filter, the transfer function H(s)...

active filters

Active filters use op-amps, resistors, and capacitors to shape a signal's frequency response. They remove unwanted frequency components while keeping the parts of the signal that are needed. The design depends on the choice of resistor and capacitor values, which set the cutoff points and help define the overall response. A first-order low-pass active filter passes signals below a cutoff frequency and reduces signals above it. Its transfer function includes the low-frequency gain, also called...

scaling

Circuit Component Scaling for Filters

Circuit component scaling helps turn simple circuit values into practical ones for filter design and circuit analysis. Engineers often start with standard values like 1 ohm, 1 henry, or 1 farad because they make examples and problems easier to work through. Later, those values can be scaled to more realistic levels. This method is useful in filters, resonant circuits, and many other circuit analysis problems. It reduces calculation complexity while keeping the circuit behavior easy to study.

design example

Touch-Tone Filter Design Example

Touch-tone telephony uses dual-tone multi-frequency (DTMF) signaling and a filter design example to show how keypad tones are separated. It replaced the old rotary dial with a matrix-style keypad arranged in four rows and three columns. Each of the 12 buttons produces a unique signal pair. Every key press creates two sinusoidal tones at the same time. One tone comes from a low-frequency group between 697 and 941 Hz. The other comes from a high-frequency group between 1209 and 1477 Hz. This...

key terms and concepts

Learn how active filters use op-amps, resistors, and capacitors to shape frequency response with low-pass, high-pass, band-pass, and notch designs.

Active Filter Design and Frequency Response | 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

Active Filter Design and Frequency Response

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