Name: how transfer functions shape circuit response
Uploaded: 2024-07-08T13:20:01.000Z
Duration: 1 min 25 s
Channel: JoVE

回路の周波数応答は、信号周波数が変化する場合の動作を特徴付け、転送関数またはネットワーク関数を使用して分析できます。
この数学関数は、出力と入力の比率を定義します。電圧ゲイン、電流ゲイン、トランスファーインピーダンス、トランスファーアドミタンスの4種類に分類されます。
回路の動作は、複素変数 's' を使用してラプラス領域で解析されます。
一般に、回路の伝達関数はゼロと極で表すことができます。
ここで、ゼロは分子多項式の根であり、極は分母多項式の根です。伝達関数の値は、極で無限大になり、ゼロでゼロになります。
2つのコンデンサと、アンプからツイーターへの高周波信号を選択するインダクタを備えたオーディオクロスオーバー回路について考えてみましょう。
インダクタ、抵抗、およびコンデンサの組み合わせにより、回路インピーダンスが得られます。
オームの法則を適用すると、ノード電圧は入力電圧に比例します。
同様に、出力電圧はノード電圧に比例します。
最後に、出力電圧比で表される伝達関数が得られます。

回路のネットワーク機能

電気回路の周波数応答解析は、入力信号の周波数が変化するときの回路の動作に関する重要な洞察を提供します。数学的ツールである伝達関数は、この動作を理解するのに役立ちます。伝達関数は位相器の出力と入力の関係を定義し、電圧ゲイン、電流ゲイン、伝達インピーダンス、伝達アドミタンスの 4 つのタイプがあります。伝達関数の重要なコンポーネントは、極とゼロです。

<img alt="Equation 1" src="/files/ftp_upload/15137/15137_Equation_1.svg" style="height: 32px;">

ここで、ゼロは分子多項式の根であり、極は分母多項式の根です。

説明例として、ツイーターの高周波信号をアンプから分離するオーディオのクロスオーバー回路を考えてみましょう。信号周波数に関するこの回路の解析は、複素変数として「s」を使用することでラプラス領域で行われます。

オームの法則を利用すると、入力電圧はノード電圧に関連します。その後、出力電圧は入力電圧で表されます。この解析により、出力と入力の電圧比を表す伝達関数が導出されます。
<img alt="Equation 2" src="/files/ftp_upload/15137/15137_Equation_2.svg" style="height: 32px;">
周波数応答解析は、特に通信システムや制御システムで広く使用されています。ラジオやテレビ技術などのさまざまなアプリケーションで利用される電気フィルターは、この解析を利用して、必要な信号を通過させながら不要な信号を抑制します。伝達関数と周波数応答を包括的に理解することは、回路設計とエンジニアリングにおいて極めて重要です。
伝達関数は基本的なツールであり、エンジニアリングと電気におけるさまざまな周波数での回路動作の評価を簡素化します。

回路の周波数応答は、信号周波数が変化する場合の動作を特徴付け、転送またはネットワークを使用して分析できます...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Frequency Response

周波数特性

KEY TERMS AND CONCEPTS

主要用語および概念

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 are the roots of the numerator polynomial, and poles are the roots of the denominator polynomial.

An audio crossover circuit gives a clear example of this idea. It separates high-frequency signals for a tweeter from an amplifier. To analyze this circuit, engineers work in the Laplace domain and use s as a complex variable. Ohm's Law is then used to connect the input voltage to the node voltage, and the output voltage is written in terms of the input voltage.

This process produces the transfer function as the output-to-input voltage ratio. Frequency response analysis is widely used in communication and control systems. Electric filters in radio and television also depend on it to let desired signals pass and reduce unwanted ones. A strong grasp of transfer functions helps with circuit design and with understanding how circuits act at different frequencies.

How Transfer Functions Shape Circuit Response

how-transfer-functions-shape-circuit-response

network function of a circuit

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 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

Learn how transfer functions describe circuit behavior across frequency, including poles, zeros, and an audio crossover example used in filters and electronics.

How Transfer Functions Shape Circuit Response | JoVE K12

JoVE Core 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

How Transfer Functions Shape Circuit Response

Description

Transcript