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

שקול את פונקציית ההעברה של מערכת המבוטאת בצורה סטנדרטית.
לפונקציית ההעברה יש רווח קבוע, אפס וקוטב במקור, אפס פשוט, קוטב פשוט, קוטב ריבועי ואפס ריבועי.
חלקות בודה עבור גורמים בודדים משורטטות בנפרד ומשולבות באופן גרפי.
עבור טווח הרווח הקבוע החיובי, רווח הגודל הוא קבוע, וזווית הפאזה היא אפס ובלתי תלויה בתדר.
עבור טווח הרווח הקבוע השלילי, רווח הגודל נשאר זהה, וזווית הפאזה היא ±180°.
עבור פונקציית העברה עם אפס במקור, תרשים הגודל מייצג קו ישר עם שיפוע חיובי של 20dB/עשור. תרשים הפאזה מתאר זווית פאזה קבועה של 90°.
עבור קוטב במוצא, שיפוע חלקת הגודל הוא -20dB לעשור, ושינוי הפאזה הוא -90°.
אם יש יותר מקוטב אחד או אפס במקור, השיפוע של חלקת הגודל ושינוי הפאזה מוכפל במספר הקטבים או האפסים הקיימים במקור.

תגובת התדר

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פונקציית התמסורת המוצגת בצורתה הסטנדרטית משלבת את ההגבר הקבוע של האלמנטים, האפסים והקטבים בנקודת המוצא, אפסים וקטבים פשוטים, וקטבים ואפסים ריבועיים. ניתן לכתוב את פונקציית התמסורת כ-H(ω):

<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="text-align: center; height: 40px;">

פונקציית התמסורת, H(ω), המבוטאת לרוב בצורתה הסטנדרטית, נגזרת על ידי נרמול המקדמים הפולינומיים של פונקציית התמסורת. הקטבים (jω) והאפסים (jω) הם תדרים קריטיים שבהם הגודל והפאזה של היציאה של המערכת חווים שינויים משמעותיים.

רווח, K:

לפונקציית התמסורת יש איבר קבוע K, עם גודל של 20 log10K וזווית פאזה של 0° עבור ערך חיובי של K. גם הגודל וגם הפאזה קבועים בתדירות. אם K הוא שלילי, הגודל נשאר ללא שינוי, אך הפאזה היא ±180°. כאשר ההגבר K =1, הגודל יהפוך לאפס יחד עם זווית הפאזה. במונחים של עקומת בודה, הגודל מתבטא בדציבלים (dB) כ-20log10K, והפאזה שלו נשארת ב-0∘ או 180∘ בהתאם לסימן של K.

קטב/אפס בנקודת המוצא:

קטב או אפס בנקודת המוצא יש השפעה מכרעת על העקומה. אפס (jω)+1 בנקודת המוצא הוא בעל גודל של 20 log10 והפאזה היא 90°. השיפוע של עקומת הגודל הוא 20dB/דקאדה, והפאזה קבועה עם התדר.

עבור קטב (jω)-1 בנקודת המוצא, הגודל הוא -20dB/דקאדה, והפאזה היא -90°. באופן כללי, עבור (jω)N, כאשר N הוא מספר שלם, לעקומת הגודל יהיה שיפוע של -20NdB/דקאדה בעוד הפאזה היא 90N°.

לגודל הקבוע K, קטבים/אפסים בנקודת המוצא יש את זווית הפאזה שאינה משתנה עם התדר. במקרה של אפסים/קטבים, זווית הפאזה משתנה כאשר מספר הקטבים/אפסים משתנה בנקודת המוצא.
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שקול את פונקציית ההעברה של מערכת המבוטאת בצורה סטנדרטית.
לפונקציית ההעברה יש רווח קבוע, אפס וקוטב במקור, ...

Education

JoVE Core

Electrical Engineering

הנדסה חשמלית

Unit 1

יחידה 1

Frequency Response

תגובת תדר

KEY TERMS AND CONCEPTS

מונחים ומושגי מפתח

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

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...

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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...

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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

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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

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...

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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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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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