Name: inductive circuit frequency response
Uploaded: 2024-07-08T13:20:02.000Z
Duration: 1 min 20 s
Channel: JoVE

考虑时域中的一个电感电路。通过将电阻和电感替换为其阻抗，可得到其等效的频域电路。
该电路连接至一个频率可变的输入端，输出则取自电感两端。
 输出电压与输入电压的分压比决定了传递函数。
传递函数的表达式给出了由其导出的幅值和相位。
电感与电阻的比值等于电感电路的时间常数。
可以用时间常数来表示幅值和相位。
当频率趋近于零时，传递函数趋近于零，相位差近似为 π 除以 2。
当频率等于时间常数的倒数时，振幅达到其最大值的约 0.707 倍，相位差为 π/4。
同样，在较高频率下，传递函数趋近于1，相位偏移趋于零。
这些点以及其他若干点，通过幅值（或幅度）响应和相位响应图来描绘频率响应。

电路的频率响应

Frequency Response of a Circuit

电感电路给电气工程带来了引人入胜的挑战，尤其是在从时域到频域的转换过程中。这种转换过程通常会将电感器转换为阻抗，并用相位来进行表示。
传递函数在描述这些电路如何对不同频率做出反应方面发挥着至关重要的作用，这有助于深入了解它们的行为。其中一个重要的参数便是时间常数，这一常数能够用来表示电路中的电感与电阻之比。
将时间常数的倒数代入到积分后传递函数方程中，便能够得出传递函数的幅度。此外，这种方法还可以用来计算相移，并以此来表示输出正弦波和输入正弦波之间的相位角差异。
<img alt="Equation 1" src="/files/ftp_upload/15138/15138_Equation_1.svg" style="text-align: center; height: 32px;">
<img alt="Equation 2" src="/files/ftp_upload/15138/15138_Equation_2.svg" style="text-align: center; height: 32px;">
绘制幅度和相位响应可以直观地来表示电路的频率特性，从而有效地传达了电路性能的基本方面。随着对频谱的不断探索，独特的趋势便出现了。在较低的频率下，传递函数趋近于零，同时还伴有大约 π/2 的相移。相反，在较高的频率下，传递函数将会趋于统一，而相移则会趋近于零。
最重要的是需要确定半功率频率，它能够用来表示时间常数的倒数。在这个特定的频率下能够准确的评估传递函数的幅度和相位，从而能够进一步全面的了解电路的响应特性。

考虑一个时域中的电感电路。通过将电阻和电感分别替换为……，可得到其等效的频域电路。

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Frequency Response

频率响应

KEY TERMS AND CONCEPTS

关键术语与概念

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 ratio of inductance to resistance in the circuit. The magnitude of the transfer function is found by using the inverse of the time constant in the transfer function equation.

This approach also gives the phase shift between the output and input sinusoids. The phase angle shows how far one signal leads or lags the other. Graphs of amplitude response and phase response help show the circuit’s frequency behavior in a clear way.

The response changes across the frequency spectrum. At low frequencies, the transfer function approaches zero, and the phase shift is about π/2. At high frequencies, the transfer function approaches unity, and the phase shift moves toward zero.

The half-power frequency is an important point in this behavior. It is the inverse of the time constant. At this frequency, the magnitude and phase of the transfer function can be determined accurately, giving a complete view of the circuit’s response.

Inductive Circuit 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 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 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...

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

Explore how inductive circuits change with frequency, using impedance, phasors, transfer functions, phase shift, and the half-power frequency.

Inductive Circuit 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

Inductive Circuit Frequency Response

Description

Transcript