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

Der Frequenzgang einer Schaltung, der ihr Verhalten bei unterschiedlicher Signalfrequenz charakterisiert, kann mit der Übertragungs- oder Netzwerkfunktion analysiert werden.
Diese mathematische Funktion definiert das Verhältnis von Ausgang zu Eingang. Es wird in vier Typen eingeteilt: Spannungsverstärkung, Stromverstärkung, Übertragungsimpedanz und Übertragungsadmittanz.
Das Verhalten der Schaltung wird in der Laplace-Domäne mit Hilfe der komplexen Variablen 's' analysiert.
Die Übertragungsfunktion der Schaltung kann im Allgemeinen in Nullen und Polen dargestellt werden.
Hier sind Nullen die Wurzeln des Zählerpolynoms und Pole die Wurzeln des Nennerpolynoms. Der Wert der Übertragungsfunktion wird an den Polen unendlich und an den Nullstellen Null.
Stellen Sie sich eine Audio-Frequenzweiche mit zwei Kondensatoren und einer Induktivität vor, die hochfrequente Signale von einem Verstärker zu einem Hochtöner auswählt.
Die Kombination aus Induktivität, Widerständen und Kondensatoren sorgt für die Schaltungsimpedanz.
Bei Anwendung des Ohmschen Gesetzes ist die Knotenspannung proportional zur Eingangsspannung.
In ähnlicher Weise ist die Ausgangsspannung proportional zur Knotenspannung.
Schließlich wird die Übertragungsfunktion erhalten, die durch das Verhältnis von Ausgangs- zu Eingangsspannung dargestellt wird.

netzwerkfunktion eines schaltkreises

Die Frequenzganganalyse in elektrischen Schaltkreisen liefert wichtige Einblicke in das Verhalten eines Schaltkreises, wenn sich die Frequenz des Eingangssignals ändert. Die Übertragungsfunktion, ein mathematisches Werkzeug, ist entscheidend für das Verständnis dieses Verhaltens. Sie definiert die Beziehung zwischen Zeigerausgang und -eingang und ist in vier Typen erhältlich: Spannungsverstärkung, Stromverstärkung, Übertragungsimpedanz und Übertragungsadmittanz. Die kritischen Komponenten der Übertragungsfunktion sind die Pole und Nullstellen.

<img alt="Equation 1" src="/files/ftp_upload/15137/15137_Equation_1.svg" style="text-align: center; height: 32px;">

Hier sind Nullstellen Wurzeln des Zählerpolynoms und Pole Wurzeln des Nennerpolynoms.

Als veranschaulichendes Beispiel betrachten wir einen Audio-Crossover-Schaltkreis, der Hochfrequenzsignale für einen Hochtöner von einem Verstärker trennt. Die Analyse dieses Schaltkreises hinsichtlich der Signalfrequenz erfolgt im Laplace-Bereich, indem „s“ als komplexe Variable verwendet wird.

Unter Verwendung des Ohmschen Gesetzes bezieht sich die Eingangsspannung auf die Knotenspannung; Anschließend wird die Ausgangsspannung in Bezug auf die Eingangsspannung ausgedrückt. Diese Analyse leitet die Übertragungsfunktion ab, die das Verhältnis von Ausgangs- zu Eingangsspannung darstellt.

<img alt="Equation 2" src="/files/ftp_upload/15137/15137_Equation_2.svg" style="text-align: center; height: 32px;">

Die Frequenzganganalyse bietet umfangreiche Anwendungsmöglichkeiten, insbesondere in Kommunikations- und Steuerungssystemen. Elektrische Filter, die in verschiedenen Anwendungen wie Radio- und Fernsehtechnologien eingesetzt werden, verlassen sich auf diese Analyse, um die gewünschten Signale passieren zu lassen und unerwünschte Signale zu unterdrücken. Ein umfassendes Verständnis von Übertragungsfunktionen und Frequenzgängen ist für das Design und die Entwicklung von Schaltungen von entscheidender Bedeutung.

Als grundlegendes Werkzeug vereinfacht die Übertragungsfunktion die Bewertung des Schaltungsverhaltens über verschiedene Frequenzen in der Technik und Elektronik.

Netzwerkfunktion eines Schaltkreises

Der Frequenzgang einer Schaltung, der ihr Verhalten bei unterschiedlicher Signalfrequenz charakterisiert, kann mit Hilfe der Übertragungs- oder Netzwerksteuerung analysiert werden.

Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

Frequency Response

Frequenzgang

KEY TERMS AND CONCEPTS

ZENTRALE BEGRIFFE UND KONZEPTE

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 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

How Transfer Functions Shape Circuit Response

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