Name: circuit component scaling for filters
Uploaded: 2024-07-08T13:20:03.000Z
Duration: 1 min 26 s
Channel: JoVE

Skalowanie wielkości i częstotliwości upraszcza projektowanie filtrów poprzez kalibrację reakcji filtra i parametrów obwodu tak, aby pasowały do użytecznych zakresów.
Skalowanie wielkości polega na wzmocnieniu wszystkich impedancji w sieci o współczynnik, utrzymując spójne pasma przenoszenia.
Skalowane wartości impedancji i częstotliwości są następnie wyrażane za pomocą współczynnika skalowania.
Proces ten nie zmienia częstotliwości rezonansowej obwodu, współczynnika jakości, szerokości pasma ani funkcji przenoszenia.
Skalowanie częstotliwości przesuwa charakterystykę częstotliwościową sieci w górę lub w dół osi częstotliwości, pozostawiając impedancję bez zmian. Osiąga się to poprzez pomnożenie częstotliwości przez współczynnik skalowania.
Skalowanie częstotliwości wpływa na impedancje elementów reaktywnych zależnych od częstotliwości, ale rezystor pozostaje nienaruszony.
Współczynnik skalowania zmienia częstotliwość rezonansową i szerokość pasma, ale współczynnik jakości pozostaje taki sam.
Ogólne wyrażenie na impedancję i skalowanie częstotliwości obejmuje jednoczesne skalowanie wielkości i częstotliwości.
W przypadkach, gdy skalowanie wielkości nie jest stosowane, współczynnik skalowania wielkości jest ustawiany na jeden. Podobnie, bez skalowania częstotliwości, współczynnik skalowania częstotliwości jest ustawiony na jeden.

odpowiedź na częstotliwość

Podczas projektowania i analizowania filtrów, obwodów rezonansowych lub ogólnie analizy obwodów praca ze standardowymi wartościami, takimi jak om, henr lub farad, może być wygodna przed skalowaniem tych wartości do bardziej realistycznych wielkości. To podejście jest szeroko stosowane, ponieważ w wielu przykładach i problemach nie wykorzystuje się realistycznych wartości elementów; upraszcza analizę obwodów masteringowych dzięki wygodnym wartościom komponentów. W ten sposób zmniejsza się złożoność obliczeń, przy założeniu, że techniki skalowania mogą później dostosować te wartości do bardziej praktycznych poziomów.

Skalowanie obwodu można przeprowadzić na dwa sposoby: skalowanie wielkości (lub impedancji) i skalowanie częstotliwości. Metody te dostosowują elementy obwodu do pracy w praktycznych zakresach. Skalowanie wielkości zmienia rozmiar komponentów bez wpływu na reakcję obwodu na różne częstotliwości. Z drugiej strony skalowanie częstotliwości przesuwa odpowiedź obwodu na wyższe lub niższe częstotliwości widma.

Skalowanie wielkości:

Skalowanie wielkości polega na dostosowywaniu rozmiarów elementów obwodu (takich jak rezystory, cewki indukcyjne i kondensatory) o określony współczynnik, ale bez zmiany sposobu, w jaki obwód reaguje na różne częstotliwości. Impedancje obwodu wyrażają się w postaci rezystorów (R), cewek indukcyjnych (L) i kondensatorów (C) w obwodzie. Kiedy stosuje się skalowanie wielkości K_m, składowe te przekształca się w następujący sposób:

<img alt="Equation 1" src="/files/ftp_upload/15150/15150_Equation_1.svg" style="height: 12px;">
<img alt="Equation 2" src="/files/ftp_upload/15150/15150_Equation_2.svg" style="height: 36px;">

Skalowanie częstotliwości:

Skalowanie częstotliwości przesuwa charakterystykę częstotliwościową obwodu wzdłuż osi częstotliwości w górę lub w dół, bez zmiany poziomów impedancji. Osiąga się to poprzez pomnożenie częstotliwości przez współczynnik skalowania, oznaczony przez K_f. Nowe wartości indukcyjności i pojemności wyznaczane są ze wzoru:

<img alt="Equation 3" src="/files/ftp_upload/15150/15150_Equation_3.svg" style="height: 36px;">
<img alt="Equation 4" src="/files/ftp_upload/15150/15150_Equation_4.svg" style="height: 36px;">

Jeżeli obwód jest skalowany jednocześnie dla obu parametrów – wielkości i częstotliwości, wówczas:

<img alt="Equation 5" src="/files/ftp_upload/15150/15150_Equation_5.svg" style="height: 36px;">
<img alt="Equation 6" src="/files/ftp_upload/15150/15150_Equation_6.svg" style="height: 36px;">

Jeśli współczynniki skalowania wielkości i częstotliwości są równe, nie następuje skalowanie wielkości ani częstotliwości.

Odpowiedź na częstotliwość

Skalowanie wielkości i częstotliwości upraszcza projektowanie filtrów poprzez kalibrację reakcji filtra i parametrów obwodu tak, aby pasowały do użytecznych zakresów.
Wielkość...

Education

JoVE Core

Electrical Engineering

Elektrotechnika

Unit 1

Jednostka 1

Frequency Response

Charakterystyka częstotliwościowa

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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. The idea is to learn the analysis first and then adjust the component values to fit real-world use.

There are two main ways to scale a circuit: magnitude scaling, also called impedance scaling, and frequency scaling. Magnitude scaling changes the size of the components without changing how the circuit responds to different frequencies. Frequency scaling shifts the circuit response higher or lower on the frequency axis without changing the impedance levels.

Magnitude scaling uses a factor called K m to change resistors, inductors, and capacitors. Frequency scaling uses a factor called K f to move the frequency response and to determine new inductance and capacitance values. When both magnitude and frequency scaling are applied together, the circuit can be adjusted in both size and response range. If the magnitude and frequency scaling factors are equal, neither type of scaling occurs.

Circuit Component Scaling for Filters

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

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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 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

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...

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 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

Learn how magnitude scaling and frequency scaling adjust resistors, inductors, capacitors, and circuit response for easier filter design and analysis.

Circuit Component Scaling for Filters | 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

Circuit Component Scaling for Filters

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