Name: circuit component scaling for filters
Uploaded: 2024-07-08T13:20:03.000Z
Duration: 1 min 26 s
Channel: JoVE

振幅と周波数のスケーリングにより、使用可能な範囲内に収まるようにフィルタ応答と回路パラメータを較正することにより、フィルタの設計が簡素化されます。
振幅スケーリングでは、ネットワーク内のすべてのインピーダンスを係数で増幅し、一貫した周波数応答を維持します。
スケーリングされたインピーダンスと周波数の値は、スケーリング係数を使用して表されます。
このプロセスでは、回路の共振周波数、品質係数、帯域幅、または伝達関数は変更されません。
周波数スケーリングは、インピーダンスを変化させないまま、ネットワークの周波数応答を周波数軸の上下にシフトします。これは、周波数にスケーリング係数を掛けることによって実現されます。
周波数スケーリングは、周波数依存の無効要素のインピーダンスに影響を与えますが、抵抗は影響を受けません。
スケーリング係数は共振周波数と帯域幅を変更しますが、品質係数は同じままです。
インピーダンスと周波数のスケーリングの一般的な式は、振幅と周波数の同時スケーリングに対応します。
マグニチュード スケールが適用されない場合、マグニチュード スケール ファクターは 1 に設定されます。同様に、周波数スケーリングを使用しない場合、周波数スケーリング係数は 1 に設定されます。

周波数応答

フィルタ、共振回路、または回路解析全般の設計と解析では、1 オーム、1 ヘンリー、1 ファラドなどの標準的な要素値​​を使用してから、これらの値をより現実的な数値にスケーリングすると便利です。このアプローチは、多くの例や問題で現実的な要素値を使用しないことで広く利用されています。これにより、便利なコンポーネント値を使用して回路解析を習得しやすくなります。加えて、スケーリング手法によって後でこれらの値をより実用的なレベルに調整できることを理解すれば、計算の複雑さが軽減されます。

回路のスケーリングは、振幅 (またはインピーダンス) スケーリングと周波数スケーリングの 2 つの方法で実行できます。これらの方法では、回路のコンポーネントを実用的な範囲内で動作するように調整します。振幅スケーリングでは、異なる周波数に対する回路の応答に影響を与えることなく、コンポーネントのサイズが変更されます。一方、周波数スケーリングでは、回路の応答がスペクトル上のより高い周波数またはより低い周波数に移動します。

振幅スケーリング:

振幅スケーリングでは、回路コンポーネント (抵抗器、インダクタ、コンデンサなど) のサイズを特定の係数で調整しますが、回路がさまざまな周波数に応答する方法は変更しません。回路のインピーダンスは、回路内の抵抗器 (R)、インダクタ (L)、コンデンサ (C) で表されます。振幅スケーリング K_m を適用すると、これらのコンポーネントは次のように変換されます:

<img alt="Equation 1" src="/files/ftp_upload/15150/15150_Equation_1.svg" style="height: 12px;">
<img alt="Equation 2" src="/files/ftp_upload/15150/15150_Equation_2.svg" style="height: 36px;">

周波数スケーリング:
周波数スケーリングでは、インピーダンスレベルを変更せずに、回路の周波数応答を周波数軸に沿って上下にシフトします。これは、周波数に K_f で示されるスケーリング係数を乗算することで実現されます。インダクタンスと静電容量の新しい値は、次の式で決定されます:

<img alt="Equation 3" src="/files/ftp_upload/15150/15150_Equation_3.svg" style="height: 36px;">
<img alt="Equation 4" src="/files/ftp_upload/15150/15150_Equation_4.svg" style="height: 36px;">

回路が振幅と周波数の両方のパラメータに対して同時にスケーリングされる場合、次のようになります:

<img alt="Equation 5" src="/files/ftp_upload/15150/15150_Equation_5.svg" style="height: 36px;">
<img alt="Equation 6" src="/files/ftp_upload/15150/15150_Equation_6.svg" style="height: 36px;">

振幅と周波数のスケーリング係数が等しい場合、振幅も周波数もスケーリングされません。

振幅と周波数のスケーリングにより、使用可能な範囲内に収まるようにフィルタ応答と回路パラメータを較正することにより、フィルタの設計が簡素化されます。
マグニチュード。。。

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

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. The idea is to learn the analysis first and then adjust the component values to fit real-world use.

There are two main ways to scale a circuit: magnitude scaling, also called impedance scaling, and frequency scaling. Magnitude scaling changes the size of the components without changing how the circuit responds to different frequencies. Frequency scaling shifts the circuit response higher or lower on the frequency axis without changing the impedance levels.

Magnitude scaling uses a factor called K m to change resistors, inductors, and capacitors. Frequency scaling uses a factor called K f to move the frequency response and to determine new inductance and capacitance values. When both magnitude and frequency scaling are applied together, the circuit can be adjusted in both size and response range. If the magnitude and frequency scaling factors are equal, neither type of scaling occurs.

Circuit Component Scaling for Filters

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 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 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 magnitude scaling and frequency scaling adjust resistors, inductors, capacitors, and circuit response for easier filter design and analysis.

Circuit Component Scaling for Filters | 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

Circuit Component Scaling for Filters

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