Name: transfer function gain and origin effects
Uploaded: 2024-07-08T13:20:02.000Z
Duration: 1 min 19 s
Channel: JoVE

Рассмотрим передаточную функцию системы, выраженную в стандартной форме.
Передаточная функция имеет постоянное усиление, ноль и полюс в начале координат, простой ноль, простой полюс, квадратичный полюс и квадратичный ноль.
Графики Боде для отдельных факторов строятся отдельно и объединяются графически.
Для положительного члена с постоянным коэффициентом усиления амплитуда постоянна, а фазовый угол равен нулю и не зависит от частоты.
Для члена с отрицательным постоянным усилением амплитуда остается прежней, а фазовый угол составляет ±180°.
Для передаточной функции с нулем в начале координат график амплитуды представляет собой прямую линию с положительным наклоном 20 дБ/десятилетие. Фазовый график показывает постоянный фазовый угол 90°.
Для полюса в начале координат наклон диаграммы магнитуды составляет -20 дБ/декаду, а фазовый сдвиг равен -90°.
Если в начале координат имеется более одного полюса или нуля, наклон графика магнитуды и сдвиг фазы умножаются на количество полюсов или нулей, присутствующих в начале координат.

Частотная характеристика

Передаточная функция, представленная в своей стандартной форме, объединяет постоянный коэффициент усиления элементов, нули и полюса в начале координат, простые нули и полюса, а также квадратичные полюса и нули. Передаточную функцию можно записать как H(ω):
<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="height: 40px;">
Передаточная функция H(ω), часто выражаемая в стандартной форме, получается путем нормализации полиномиальных коэффициентов передаточной функции. Полюса (jω) и нули (jω) представляют собой критические частоты, при которых величина и фаза выходного сигнала системы претерпевают значительные изменения.
Усиление, К:
Передаточная функция имеет постоянный член K с величиной 20 log_10K и фазовым углом 0° для положительного значения K. И величина, и фаза постоянны в зависимости от частоты. Если K отрицателен, величина остается неизменной, но фаза составляет ±180°. Когда коэффициент усиления K =1, величина станет нулевой вместе с фазовым углом. В терминах графика Боде величина выражается в децибелах (дБ) как 20log_10K, а ее фаза остается равной 0∘ или 180∘, в зависимости от знака K.
Полюс/ноль в начале координат:
Полюс или ноль в начале координат оказывают определяющее влияние на график. Ноль (jω)^+1 в начале координат имеет величину 20 log_10 ω, а фаза равна 90°. Наклон графика амплитуды составляет 20 дБ/декаду, а фаза постоянна в зависимости от частоты.
Для полюса (jω)^-1 в начале координат величина составляет -20 дБ/декаду, а фаза -90°. Обычно для (jω)^N, где N представляет собой целое число, наклон графика величины будет составлять -20 НдБ/декада, а фаза равна 90°.
В случае постоянной величины K, полюсы/нули в начале координат имеют фазовый угол, который не меняется с частотой. В случае нулей/полюсов фазовый угол изменяется, когда количество полюсов/нулей изменяется в начале координат.

Рассмотрим передаточную функцию системы, выраженную в стандартной форме.
Передаточная функция имеет постоянное усиление, ноль и полюс в начале координат, ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

Frequency Response

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 is 0° when K is positive. If K is negative, the magnitude stays the same, but the phase becomes ±180°. When K = 1, the magnitude is 0 dB and the phase angle is 0°.

A zero at the origin has a strong effect on the Bode plot. For a zero at the origin, the magnitude is 20 log 10 ω, and the phase is 90°. The magnitude rises with a slope of 20 dB/decade, and the phase stays constant as frequency changes. A pole at the origin acts in the opposite way. Its magnitude slope is -20 dB/decade, and its phase is -90°.

More generally, a factor of (jω)^N changes the plot by an integer amount. For a zero at the origin, the magnitude slope becomes 20N dB/decade and the phase is 90N°. For a pole at the origin, the magnitude slope becomes -20N dB/decade, and the phase is also 90N°. In both cases, the constant gain K and the poles or zeros at the origin keep a phase angle that does not change with frequency. The phase changes when the number of poles or zeros at the origin changes.

Transfer Function Gain and Origin Effects

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network function of a circuit

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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...

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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

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...

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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...

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Series Resonance in RLC Circuits

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characteristics of series resonant circuit

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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...

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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...

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Op Amp Gain and Bode Plot Basics

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passive filters

Audio Frequency Control with Passive Filters

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design example

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key terms and concepts

Explore how constant gain, poles, and zeros at the origin affect transfer functions and Bode plots, including dB slope and phase changes.

Transfer Function Gain and Origin Effects | 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

Transfer Function Gain and Origin Effects

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