Name: bode plot slope and phase changes
Uploaded: 2025-05-22T09:06:14.000Z
Duration: 1 min 23 s
Channel: JoVE

전달 함수(transfer function)를 극점(poles)과 영점(zeros)이 있는 표준 형식으로 생각해 보십시오
단순 0을 갖는 전달 함수의 경우, 작은 주파수 값에서의 크기 이득은 기울기가 0인 직선이고 위상은 0에 가깝습니다.
코너 주파수에서 점근 크기는 0 기울기 선에서 벗어나고 위상은 45도에 접근합니다.
더 높은 주파수에서 크기 플롯은 +20dB/decade 선을 형성하고 위상은 90도입니다.
단순 극점은 단순 0의 역수입니다. 이는 극 기반 보드 플롯이 수평 축에 대해 반영된 단순 0 플롯을 반영한다는 것을 의미합니다.
2차 극점 전달 함수가 있다고 가정하겠습니다.
더 낮은 주파수에서는 이득 및 위상각이 0에 가깝습니다.
코너 주파수에서 점근 크기 편차는 감쇠 계수에 따라 달라지며 위상각은 거의 -90도입니다.
더 높은 주파수에서 크기 플롯은 기울기가 -40dB/decade이고 위상이 -180도인 직선을 형성합니다.
하나 이상의 2차 극점에 대해, 선의 기울기와 위상 변이에 극의 수를 곱합니다.

주파수 응답

표준 형식에서 전달 함수는 상수 이득, 원점의 극점/영점, 단순 극점/영점 및 2차 극점/0으로 표시됩니다. 각각은 시스템의 전반적인 응답에 고유하게 기여합니다. 이 항은 단순 0의 크기를 나타냅니다.
<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="height: 36px;">
보드 크기 플롯은 낮은 주파수(0dB에 접근)에서 평탄한 상태를 유지하고 코너 또는 브레이크 주파수(ω_1)로 알려진 특정 주파수 이후 20dB/10년에서 상승하기 시작합니다. 이는 크기 플롯의 기울기가 변경되고 실제 응답이 직선 근사에서 벗어나기 시작하는 주파수입니다. 이 편차는 ω_=ω_1에서 3dB로 정량화됩니다.
위상각 ϕ 는 다음과 같이 표현됩니다.
<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="height: 60px;">
위상각은 0°에서 시작하여 주파수가 증가함에 따라 점근적으로 90°에 가까워집니다. 코너 주파수(ω_&lt;ω_1)보다 훨씬 낮은 주파수의 경우 jω_/ω_1 항은 매우 작으므로 크기는 무시할 수 있고 위상은 본질적으로 0입니다. 주파수가 ω_1에 접근하면 크기가 -3dB 지점이 되고 위상각이 45°가 됩니다. ω_1(ω_&gt;&gt;ω_1)보다 훨씬 높은 주파수의 경우 크기의 기울기는 20dB/decade로 변경되고 위상은 90°로 고정됩니다.
<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="height: 160px;">
2차 극점/영점:
2차 극의 크기와 위상각은 다음과 같습니다.
<img alt="Equation 3" src="/files/ftp_upload/15142/15142_Equation_3.svg" style="height: 42px;">
<img alt="Equation 4" src="/files/ftp_upload/15142/15142_Equation_4.svg" style="height: 60px;">
2차 극점에 대한 진폭 플롯은 두 부분으로 구성됩니다. 즉, 고유 주파수 ω_n 아래의 평탄한 응답과 ω_n 위의 -40dB/10년 기울기로 구성되며, 실제 플롯의 피크는 감쇠 계수 ζ_2에 따라 달라집니다. 2차 극점에 대한 위상 플롯은 감쇠 계수 ζ_2의 영향을 받아 고유 주파수의 1/10에서 시작하여 그 10배에서 끝나며 10년당 -90°의 기울기로 선형적으로 감소합니다.

전달 함수(transfer function)를 극점(poles)과 영점(zeros)이 있는 표준 형식으로 생각해 보십시오
단순 0을 갖는 전달 함수의 경우, 작은 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Frequency Response

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 approximation starts to differ from the actual curve, and the error is 3 dB at ω = ω1.

The phase angle for a simple zero also changes with frequency. It starts at 0° and rises toward 90° as frequency increases. When ω is much smaller than ω1, the term jω/ω1 is very small, so the magnitude is negligible and the phase is close to zero. At ω = ω1, the magnitude is at the -3 dB point and the phase is 45°. When ω is much larger than ω1, the magnitude slope is 20 dB per decade and the phase settles near 90°.

Quadratic poles have a different shape on Bode plots. Their magnitude is flat below the natural frequency, ωn, and drops at -40 dB per decade above ωn. The actual peak depends on the damping factor, ζ2. Their phase plot decreases linearly at -90° per decade from one-tenth of ωn to ten times ωn, and that shape is also affected by ζ2.

Bode Plot Slope and Phase Changes

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 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...

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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 simple zeros and quadratic poles shape Bode plot magnitude and phase, including corner frequency, 3 dB shift, and damping effects.

Bode Plot Slope and Phase Changes | 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

Bode Plot Slope and Phase Changes

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