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

Тканевые электроды на электрокардиограммах (E-C-G) устанавливают проводящий путь для электрических токов между тканями и измерительными электродами, что позволяет наблюдать за сердечной деятельностью.
Динамика интерфейса электрод-ткань имеет модель цепи, охватывающую сопротивление электрода, емкость на границе электрод-ткань и сопротивление ткани. Разность потенциалов представляет собой разность напряжений между электродом и тканью.
В этом случае входное сопротивление равно сопротивлению ткани. Выходное сопротивление представляет собой сложение сопротивления ткани к параллельному сочетанию сопротивления и емкости на границе электрод-ткань.
Отношение выходного вектора к входному вектору, рассчитанное с использованием известных значений сопротивления и емкости, является передаточной функцией.
Это может быть аппроксимировано в трех различных частотных диапазонах.
График магнитуды Боде на полулогарифмическом графике отображает вычисленное логарифмическое усиление в децибелах в зависимости от частоты в радианах в секунду.
Низкочастотные и высокочастотные асимптоты представляют собой горизонтальные линии с постоянным усилением. В промежуточном диапазоне частот график асимптотической амплитуды является линейным с наклоном 20 децибел за десятилетие.

Частотная характеристика

График Боде является важным инструментом в анализе системы управления; он отображает частотную характеристику системы с помощью кривых величины и фазы, обе относительно логарифмической оси частот. Чтобы построить график Боде, рассмотрим передаточную функцию H(ω):
<img alt="Equation 1" src="/files/ftp_upload/15102/15102_Equation_1.svg" style="height: 32px;">
Она имеет постоянный коэффициент усиления, нули и полюса. После нормализации передаточная функция записывается как:
<img alt="Equation 1" src="/files/ftp_upload/15102/15102_Equation_2.svg" style="height: 32px;">
Очевидно влияние каждого члена этой нормализованной функции на график Боде. Член 10jω имеет постоянный коэффициент усиления 10 и ноль в начале координат jω. Каждый простой полюсный член (1+jω/ωi) вводит точку излома или угловую частоту ωi. Положительный наклон при нуле очевиден в начале координат, тогда как отрицательный наклон полюсов начинается с их угловых частот, ω=2 и ω=10, соответственно.
<img alt="Equation 1" src="/files/ftp_upload/15102/15102_Equation_3.svg" style="height: 36px;">
<img alt="Equation 1" src="/files/ftp_upload/15102/15102_Equation_4.svg" style="height: 28px;">
Как только определен индивидуальный вклад каждого члена, можно построить общий график Боде. Это предполагает наложение наклонов и фазовых изменений каждого фактора. На низких частотах график амплитуды начинается с 20 дБ (поскольку 20log10(10) =20) и представляет собой плато до первой угловой частоты. Фазовый график начинается с угла 90° из-за нуля в начале координат и начинает загибаться вниз перед первой угловой частотой.
Наклон графика амплитуды будет меняться на каждой угловой частоте, уменьшаясь на 20 дБ/декаду при ω=2 и снова при ω=10. Соответственно, фазовый график будет загибаться вниз, приближаясь к -90° на частоте, значительно превышающей ω=10.
Наконец, асимптотический график Боде, состоящий из этих прямых линий, можно изменить для более точного приближения к фактической частотной характеристике. Это предполагает добавление плавной кривой, пересекающей асимптотический график на каждой угловой частоте, что обычно приводит к небольшому выбросу вблизи угловых частот, известному как пик.

Тканевые электроды на электрокардиограммах (E-C-G) устанавливают проводящий путь для электрических токов между тканями и измерительными электродами, ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

Frequency Response

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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