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

ضع في اعتبارك وظيفة النقل لنظام معبرا عنه في شكل قياسي.
دالة النقل لها كسب ثابت ، صفر وقطب في الأصل ، صفر بسيط ، قطب بسيط ، قطب تربيعي ، وصفر تربيعي.
يتم رسم مخططات Bode للعوامل الفردية بشكل منفصل ودمجها بيانيا.
بالنسبة لحد الكسب الثابت الموجب ، يكون كسب الحجم ثابتا ، وزاوية الطور صفرا ومستقلة عن التردد.
بالنسبة لحد الكسب الثابت السالب ، يظل كسب الحجم كما هو ، وزاوية الطور ±180 درجة.
بالنسبة لدالة النقل ذات الصفر عند الأصل ، يمثل مخطط الحجم خطا مستقيما بميل موجب قدره 20 ديسيبل / عقد. يصور مخطط الطور زاوية طور ثابتة تبلغ 90 درجة.
بالنسبة للقطب في الأصل ، يكون ميل مخطط الحجم -20 ديسيبل / عقد ، وتحول الطور هو -90 درجة.
إذا كان هناك أكثر من قطب واحد أو صفر في الأصل ، فإن ميل مخطط الحجم وتحول الطور يتم ضربه في عدد الأقطاب أو الأصفار الموجودة في الأصل.

الاستجابة الترددية

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تدمج دالة النقل المقدمة في شكلها القياسي الكسب الثابت للعناصر، والأصفار والأقطاب عند الأصل، والأصفار والأقطاب البسيطة، والأقطاب التربيعية والأصفار. يمكن كتابة وظيفة النقل كـ H(ω):


<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="height: 40px;">

يتم اشتقاق دالة النقل، H(ω)، التي يتم التعبير عنها غالبًا بالشكل القياسي عن طريق تطبيع المعاملات متعددة الحدود لدالة النقل. تعد الأقطاب (jω) والأصفار (jω) من الترددات الحرجة حيث يتعرض حجم ومرحلة مخرجات النظام لتغيرات كبيرة.


الكسب، ك:


دالة النقل لها حد ثابت K، بحجم 20 log_10K وزاوية طور قدرها 0° لقيمة موجبة K. كل من الحجم والطور ثابتان مع التردد. إذا كانت K سالبة، يظل الحجم دون تغيير، لكن الطور يكون ±180°. عندما يكون الكسب K = 1، يصبح الحجم صفرًا مع زاوية الطور. في مصطلحات مخطط بود، يتم التعبير عن الحجم بالديسيبل (ديسيبل) كـ 20log_10K، ويظل مرحلته عند 0∘ أو 180∘ اعتمادًا على إشارة K.


القطب/الصفر عند الأصل:


القطب أو الصفر عند الأصل له تأثير محدد على المؤامرة. الصفر (jω)^+1 عند الأصل يبلغ حجمه 20 log_10 ω والطور هو 90 درجة. يبلغ ميل مخطط الحجم 20 ديسيبل / عقد، والطور ثابت مع التردد.


بالنسبة للقطب (jω)^-1 عند الأصل، يكون الحجم -20dB/عقد، والطور هو -90°. بشكل عام، بالنسبة لـ (jω)^N، حيث N عدد صحيح، سيكون لمخطط الحجم ميل قدره -20NdB/عقد بينما يكون الطور 90N°.


الحجم الثابت K، الأقطاب/الأصفار عند الأصول لها زاوية الطور التي لا تتغير مع التردد. في حالة الأصفار/الأقطاب، تتغير زاوية الطور عندما يتغير عدد الأقطاب/الأصفار عند نقطة الأصل.


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ضع في اعتبارك وظيفة النقل لنظام معبرا عنه في شكل قياسي.
دالة النقل لها كسب ثابت ، صفر وقطب في الأصل ، ...

Education

JoVE Core

Electrical Engineering

الهندسة الكهربائية

Unit 1

الوحدة الأولى

Frequency Response

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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

transfer-function-gain-and-origin-effects

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 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...

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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)...

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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...

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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

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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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