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

Die Skalierung von Helligkeit und Frequenz vereinfacht das Filterdesign, indem Filterreaktionen und Schaltungsparameter so kalibriert werden, dass sie in nutzbare Bereiche passen.
Bei der Größenskalierung werden alle Impedanzen in einem Netzwerk um einen Faktor verstärkt, wobei konsistente Frequenzgänge beibehalten werden.
Die skalierten Impedanz- und Frequenzwerte werden dann mit dem Skalierungsfaktor ausgedrückt.
Durch diesen Vorgang werden die Resonanzfrequenz, der Qualitätsfaktor, die Bandbreite oder die Übertragungsfunktionen der Schaltung nicht verändert.
Bei der Frequenzskalierung verschiebt sich der Frequenzgang eines Netzwerks auf der Frequenzachse nach oben oder unten, während die Impedanz unverändert bleibt. Dies wird erreicht, indem die Frequenz mit einem Skalierungsfaktor multipliziert wird.
Die Frequenzskalierung beeinflusst die Impedanzen von frequenzabhängigen Blindelementen, der Widerstand bleibt jedoch davon unberührt.
Der Skalierungsfaktor ändert die Resonanzfrequenz und die Bandbreite, aber der Qualitätsfaktor bleibt gleich.
Ein allgemeiner Ausdruck für Impedanz- und Frequenzskalierung ermöglicht die gleichzeitige Größen- und Frequenzskalierung.
In Fällen, in denen die Magnitudenskalierung nicht angewendet wird, wird der Magnitudenskalierungsfaktor auf eins festgelegt. Analog dazu wird ohne Frequenzskalierung der Frequenzskalierungsfaktor auf eins festgelegt.

frequenzgang

Beim Entwerfen und Analysieren von Filtern, Resonanzkreisen oder bei der Schaltungsanalyse im Allgemeinen kann es praktisch sein, mit Standardelementwerten wie 1 Ohm, 1 Henry oder 1 Farad zu arbeiten, bevor diese Werte auf realistischere Werte skaliert werden. Dieser Ansatz wird häufig verwendet, indem in zahlreichen Beispielen und Problemen keine realistischen Elementwerte verwendet werden. Er vereinfacht die Beherrschung der Schaltungsanalyse durch praktische Komponentenwerte. Die Komplexität der Berechnungen wird dadurch reduziert, da Skalierungstechniken diese Werte später auf praktischere Werte anpassen können.

Die Skalierung einer Schaltung kann auf zwei Arten erfolgen: Größenskalierung (oder Impedanzskalierung) und Frequenzskalierung. Diese Methoden passen die Komponenten der Schaltung an, damit sie innerhalb praktischer Bereiche arbeiten. Die Größenskalierung ändert die Größe der Komponenten, ohne die Reaktion der Schaltung auf unterschiedliche Frequenzen zu beeinflussen. Die Frequenzskalierung hingegen verschiebt die Reaktion der Schaltung auf höhere oder niedrigere Frequenzen im Spektrum.

Größenskalierung:

Bei der Größenskalierung werden die Größen der Schaltungskomponenten (wie Widerstände, Induktoren und Kondensatoren) um einen bestimmten Faktor angepasst, ohne jedoch die Art und Weise zu ändern, wie die Schaltung auf unterschiedliche Frequenzen reagiert. Die Impedanzen der Schaltung werden in Form von Widerständen (R), Induktoren (L) und Kondensatoren (C) in einer Schaltung ausgedrückt. Wenn die Größenskalierung K_m angewendet wird, werden diese Komponenten wie folgt transformiert:

<img alt="Equation 1" src="/files/ftp_upload/15150/15150_Equation_1.svg" style="text-align: center; height: 12px;">

<img alt="Equation 2" src="/files/ftp_upload/15150/15150_Equation_2.svg" style="text-align: center; height: 36px;">

Frequenzskalierung:

Bei der Frequenzskalierung wird die Frequenzantwort einer Schaltung entlang der Frequenzachse nach oben oder unten verschoben, ohne die Impedanzpegel zu ändern. Dies wird erreicht, indem die Frequenz mit einem Skalierungsfaktor multipliziert wird, der als K_f bezeichnet wird. Die neuen Werte der Induktivität und Kapazität werden wie folgt bestimmt:

<img alt="Equation 3" src="/files/ftp_upload/15150/15150_Equation_3.svg" style="text-align: center; height: 36px;">

<img alt="Equation 4" src="/files/ftp_upload/15150/15150_Equation_4.svg" style="text-align: center; height: 36px;">

Wenn ein Schaltkreis gleichzeitig für beide Parameter – Betrag und Frequenz – skaliert wird, dann gilt:

<img alt="Equation 5" src="/files/ftp_upload/15150/15150_Equation_5.svg" style="text-align: center; height: 36px;">

<img alt="Equation 6" src="/files/ftp_upload/15150/15150_Equation_6.svg" style="text-align: center; height: 36px;">

Wenn die Skalierungsfaktoren für Betrag und Frequenz gleich sind, findet weder eine Skalierung für Betrag noch für Frequenz statt.

Frequenzgang

Die Skalierung von Helligkeit und Frequenz vereinfacht das Filterdesign, indem Filterreaktionen und Schaltungsparameter so kalibriert werden, dass sie in nutzbare Bereiche passen.
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Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

Frequency Response

KEY TERMS AND CONCEPTS

ZENTRALE BEGRIFFE UND KONZEPTE

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

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bode plots

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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

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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

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parallel resonance

Parallel RLC Tuning at Resonance

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Op Amp Gain and Bode Plot Basics

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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

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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 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

Circuit Component Scaling for Filters

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