Name: reading frequency response with bode plots
Uploaded: 2024-07-08T13:20:02.000Z
Duration: 1 min 24 s
Channel: JoVE

Gewebeelektroden im Elektrokardiogramm (E-C-G) stellen einen leitenden Weg für elektrische Ströme zwischen dem Gewebe und den Messelektroden her und ermöglichen so die Beobachtung der Herzaktivität.
Die Dynamik der Elektroden-Gewebe-Grenzfläche hat ein Schaltungsmodell, das den Elektrodenwiderstand, die Kapazität an der Elektroden-Gewebe-Grenzfläche und den Gewebewiderstand umfasst. Die Potentialdifferenz stellt die Spannungsdifferenz zwischen der Elektrode und dem Gewebe dar.
Hier entspricht die Eingangsimpedanz dem Gewebewiderstand. Die Ausgangsimpedanz ist die Addition des Gewebewiderstands zur parallelen Kombination von Widerstand und Kapazität an der Elektroden-Gewebe-Grenzfläche.
Das Verhältnis des Ausgangsphasors zum Eingangsphasor, berechnet aus den bekannten Widerstands- und Kapazitätswerten, ist die Übertragungsfunktion.
Dies kann über drei verschiedene Frequenzbereiche approximiert werden.
Das Bode-Magnitudendiagramm in einem Semilog-Diagramm stellt den berechneten logarithmischen Gewinn in Dezibel gegenüber der Frequenz im Bogenmaß pro Sekunde dar.
Die tief- und hochfrequenten Asymptoten sind horizontale Linien mit einer konstanten Verstärkung. Im mittleren Frequenzbereich ist das asymptotische Magnitudendiagramm linear mit einer Steigung von 20 Dezibel pro Dekade.

frequenzgang

Das Bode-Diagramm ist ein wichtiges Werkzeug bei der Analyse von Steuerungssystemen. Es bildet den Frequenzgang eines Systems durch ein Betragsdiagramm und ein Phasendiagramm ab, beide gegenüber einer logarithmischen Frequenzachse. Um ein Bode-Diagramm zu erstellen, betrachten Sie die Übertragungsfunktion H(ω):

<img alt="Equation 1" src="/files/ftp_upload/15102/15102_Equation_1.svg" style="text-align: center; height: 32px;">

Sie hat eine konstante Verstärkung, Nullstellen und Pole. Nach der Normalisierung wird die Übertragungsfunktion wie folgt geschrieben:

<img alt="Equation 2" src="/files/ftp_upload/15102/15102_Equation_2.svg" style="text-align: center; height: 32px;">

Jeder Term in dieser normalisierten Funktion beeinflusst das Bode-Diagramm deutlich. Der Term 10jω hat eine konstante Verstärkung von 10 und Null am Ursprung jω. Jeder einfache Polterm (1+jω/ω_i) führt einen Knickpunkt oder eine Eckfrequenz ω_i ein. Die positive Steigung der Nullstelle ist vom Ursprung aus ersichtlich, während die negativen Steigungen der Pole bei ihren Eckfrequenzen beginnen, ω=2 bzw. ω=10.

<img alt="Equation 3" src="/files/ftp_upload/15102/15102_Equation_3.svg" style="text-align: center; height: 36px;">

<img alt="Equation 4" src="/files/ftp_upload/15102/15102_Equation_4.svg" style="text-align: center; height: 28px;">

Sobald der individuelle Beitrag jedes Terms identifiziert ist, kann das gesamte Bode-Diagramm erstellt werden. Dabei werden die Steigungen und Phasenänderungen jedes Faktors überlagert. Bei niedrigen Frequenzen beginnt das Betragsdiagramm bei 20 dB (da 20log_10(10) =20) und behält eine flache Reaktion bis zur ersten Eckfrequenz bei. Das Phasendiagramm beginnt aufgrund der Null am Ursprung bei 90° und beginnt seinen Abstieg vor der ersten Eckfrequenz.

Die Steigung des Betragsdiagramms ändert sich bei jeder Eckfrequenz und nimmt bei ω=2 und erneut bei ω=10 um 20 dB/Dekade ab. Entsprechend biegt sich das Phasendiagramm nach unten und nähert sich bei einer Frequenz, die viel höher ist als ω=10, -90°.

Schließlich kann das asymptotische Bode-Diagramm, das aus diesen geraden Linien besteht, angepasst werden, um die tatsächliche Frequenzreaktion genauer anzunähern. Dabei wird eine glatte Kurve hinzugefügt, die den asymptotischen Verlauf an jeder Eckfrequenz schneidet. Dies führt typischerweise zu einem leichten Überschwingen in der Nähe der Eckfrequenzen, auch Peaking genannt.

Frequenzgang

Gewebeelektroden in Elektrokardiogrammen (E-C-G) stellen einen leitenden Weg für elektrische Ströme zwischen dem Gewebe und den Messelektroden her, ...

Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

Frequency Response

KEY TERMS AND CONCEPTS

ZENTRALE BEGRIFFE UND KONZEPTE

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 log10(10) = 20. The term 10jω also creates a zero at the origin, so the magnitude has a positive slope from the start. Each simple pole term, such as 1 + jω/ωi, adds a breakpoint or corner frequency ωi.

The zero at the origin gives the phase plot an initial value of 90°. The phase then begins to fall before the first corner frequency. The poles at ω = 2 and ω = 10 add negative slope changes to the magnitude plot, each reducing the slope by 20 dB per decade at its corner frequency.

The full Bode plot comes from adding the effects of each factor together. At low frequencies, the magnitude stays flat until the first corner frequency. The phase curve bends downward and approaches -90° at a frequency much higher than ω = 10.

The first result is an asymptotic Bode plot made of straight-line segments. A smoother curve can then be added to match the actual frequency response more closely. This adjustment often crosses the asymptotic lines near each corner frequency and can show a small peak, called peaking.

Reading Frequency Response with Bode Plots

reading-frequency-response-with-bode-plots

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

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 Bode plots track magnitude and phase on a log frequency axis, including gain, zeros, poles, corner frequencies, and peaking.

Reading Frequency Response with Bode Plots | 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

Reading Frequency Response with Bode Plots

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