Name: rlc damping and resonant frequency
Uploaded: 2024-04-11T12:48:44.000Z
Duration: 1 min 17 s
Channel: JoVE

Электрическая цепь, состоящая из двух неприводимых накопителей энергии, называется цепью второго порядка.
Некоторые примеры включают схемы RLC, а также RC и RL схемы с двойными конденсаторами и катушками индуктивности соответственно.
Эти схемы характеризуются дифференциальным уравнением второго порядка, которое коррелирует входные и выходные сигналы цепи.
Входные сигналы обычно исходят от источников напряжения или тока, в то время как выходом обычно является напряжение на конденсаторе и/или ток через катушку индуктивности.
Рассмотрим схему RLC, в которой начальная энергия, накопленная в конденсаторе и катушке индуктивности, приводит в движение цепь.
Применение закона напряжения Кирхгофа к цепи и взятие производной по времени дает дифференциальное уравнение второго порядка.
Коэффициенты этого уравнения, выраженные в терминах сопротивления, емкости и индуктивности, известны как коэффициент демпфирования и резонансная частота.
Коэффициент демпфирования указывает на демпфирование за счет резистора и определяет скорость рассеивания энергии в системе.
Резонансная частота представляет собой собственную частоту колебаний цепи при обмене энергией между катушкой индуктивности и конденсатором.

Конфликт интересов не декларируется.

Цепи второго порядка

Second-Order Circuits

Интеграция двух фундаментальных элементов накопления энергии в электрические схемы приводит к созданию схем второго порядка, включающих в себя схемы RLC и схемы с двойными конденсаторами или индукторами (цепи RC и RL). Цепи второго порядка идентифицируются дифференциальными уравнениями второго порядка, которые связывают входные и выходные сигналы.

Входные сигналы обычно исходят от источников напряжения или тока, а выходные сигналы часто представляют собой напряжение на конденсаторе и/или ток через катушку индуктивности. Например, в схеме RLC начальная энергия, запасенная в конденсаторе и катушке индуктивности, инициирует цепь. Применяя закон напряжений Кирхгофа и вычисляя производную по времени, получаем дифференциальное уравнение второго порядка.

<img alt="Equation1" src="/files/ftp_upload/15097/15097_PT_Equation_1.svg" style="text-align: center; height: 36px;" />

Его коэффициенты, определяемые сопротивлением, электроёмкостью и индуктивностью, представляют собой коэффициент затухания и резонансную частоту.

<img alt="Equation2" src="/files/ftp_upload/15097/15097_PT_Equation_2.svg" style="text-align: center; height: 34px;" />

<img alt="Equation3" src="/files/ftp_upload/15097/15097_PT_Equation_3.svg" style="text-align: center; height: 36px;" />

Коэффициент затухания, или демпфирования, играет решающую роль в этих схемах, указывая степень демпфирования, вызванного, прежде всего, резистором. Он напрямую влияет на скорость рассеивания энергии внутри системы, эффективно контролируя скорость потери энергии. С другой стороны, резонансная частота является ключевой характеристикой, которая представляет внутреннюю частоту колебаний схемы. Она показывает, насколько быстро происходит обмен энергией между катушкой индуктивности и конденсатором в цепи, иллюстрируя естественную тенденцию цепи колебаться с определенной частотой.

Коэффициент затухания определяет, насколько быстро энергия теряется в системе из-за сопротивления. В то же время резонансная частота подчеркивает собственную скорость колебаний схемы при перемещении энергии между катушкой индуктивности и конденсатором. Эти два фактора имеют решающее значение для понимания и анализа поведения схем второго порядка.

Электрическая цепь, состоящая из двух неприводимых накопителей энергии, называется цепью второго порядка.
Некоторые примеры включают в себя RLC-схемы, а также ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

First and Second-Order Circuits

Схемы первого и второго порядка

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

rlc damping and resonant frequency

Second-order circuits use two energy storage elements, such as a capacitor and an inductor, to shape how current and voltage change over time. These circuits include RLC circuits, along with RC and RL circuits that have two capacitors or two inductors. Their behavior is described by second-order differential equations that connect the input signal to the output signal.

The input usually comes from a voltage source or a current source. The output is often the voltage across the capacitor, the current through the inductor, or both. In an RLC circuit, energy already stored in the capacitor and inductor starts the circuit response.

To analyze an RLC circuit, Kirchhoff's voltage law is applied and then differentiated with respect to time. This step produces a second-order differential equation. The equation's coefficients depend on resistance, capacitance, and inductance.

Those coefficients show up as the damping coefficient and the resonant frequency. The damping coefficient describes how much the resistor slows the circuit down. It also shows how quickly energy is lost from the system.

The resonant frequency is the circuit's natural oscillation frequency. It shows how fast energy moves back and forth between the inductor and capacitor. Together, the damping coefficient and resonant frequency are key to understanding how second-order circuits behave.

RLC Damping and Resonant Frequency

first-order circuits

First-Order Circuits in Flashing Lamps

First-order circuits use a resistor and one energy storage element, either a capacitor or an inductor. They are common in electronic systems and are described by a first-order differential equation that links the input and output signals. One important example is the RC circuit, which uses a resistor and a capacitor. In a relaxation oscillator such as a neon lamp circuit, the capacitor starts charging when voltage is applied. At first, the lamp behaves like an open circuit. When the capacitor...

rc circuit without source

RC Capacitor Discharge After Power Loss

An RC circuit discharges after the DC source is removed. If the capacitor was fully charged first, its initial voltage, V0, provides the energy that drives the source-free circuit. Kirchhoff’s current law is applied at the top node of the circuit. The current terms for the resistor and capacitor are then substituted, which leads to a first-order differential equation. Rearranging, integrating, and taking the exponential gives the natural response of the circuit. The integration constant in...

rc circuit with source

RC Circuit Charging After a DC Step

An RC circuit with a DC source shows how a capacitor charges after a sudden voltage step. The applied source is treated as a unit step function, and the capacitor voltage is the step response of the circuit. This response describes how the circuit reacts right after the input changes. A capacitor cannot change its voltage instantly. So, immediately after the switch closes at t=0, the capacitor voltage stays at its initial value. Kirchhoff’s current law is then applied at that moment and...

rl circuit without source

RL Circuit Decay After Source Removal

An RL circuit begins to decay after the DC source is removed. At that moment, the circuit becomes source-free. If the inductor has an initial current, I0, it also stores initial energy. That stored energy drives the natural response of the circuit. Kirchhoff’s voltage law is applied around the loop. The inductor voltage and resistor voltage are substituted into the equation. This gives a first-order differential equation for the current. Rearranging the terms produces a logarithmic form, which...

rl circuit with source

RL Step Response Under a DC Source

An RL circuit connected to a DC source has a step response that changes over time. The response has two parts: the transient response and the steady-state response. The transient response is the short-lived reaction that happens right after the source is applied. During the transient phase, the current changes exponentially. If the circuit starts with current in the inductor, that current decays toward zero as time goes on. At this stage, the inductor acts like a short circuit, so the source...

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RL Circuit and Frog Leg Current Response

An RL circuit is used to model the frog leg’s response to electrical stimulation. In this setup, the student studies how current changes in a resistor-inductor circuit can trigger muscle contraction. The circuit provides a way to control and measure the electrical impulses involved in the experiment. When the switch is closed, the frog’s leg shows a brief contraction. At steady state, the inductor acts like a short circuit. That lets current bypass the resistor and produces a mild, short...

second-order circuits

Second-order circuits use two energy storage elements, such as a capacitor and an inductor, to shape how current and voltage change over time. These circuits include RLC circuits, along with RC and RL circuits that have two capacitors or two inductors. Their behavior is described by second-order differential equations that connect the input signal to the output signal. The input usually comes from a voltage source or a current source. The output is often the voltage across the capacitor, the...

series rlc circuit without source

Natural Response of a Source-Free RLC Circuit

A source-free RLC circuit is a series circuit with a resistor, inductor, and capacitor, but no external energy source. It begins with the energy already stored in the capacitor and inductor. That stored energy drives the circuit at first. The circuit is described by a second-order differential equation. This equation shows how the resistor, inductor, and capacitor interact over time. The resistor removes energy from the circuit, so the solution has an exponential form. When the exponential...

types of responses of series rlc circuits

types-of-responses-of-series-rlc-circuits

Damping in Series RLC Circuit Responses

A source-free series RLC circuit shows three response patterns based on damping. The circuit is described by a second-order differential equation. Its complete solution is made from two distinct solutions. Those solutions depend on the roots, which are written in terms of the damping factor and the resonant frequency. When the damping factor is greater than the resonant frequency, both roots are real and negative. This gives an overdamped response. The circuit output decays slowly over time. <br...

series rlc circuit with source

Forced Response in a Series RLC Circuit

A series RLC circuit with a DC source can model an automobile ignition system. The circuit shows how a battery can produce the high voltage needed for a spark. In this case, the input DC voltage acts like a forcing step function. Applying Kirchhoff's voltage law leads to a second-order differential equation. A second-order differential equation is an equation with a term for the second derivative. The form of this equation is very similar to the one used for a source-free RLC circuit, so the...

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Parallel RLC Step Response in Surge Protectors

Parallel RLC circuits help explain how a surge protector works in a street lamp. The circuit model uses a DC input source that creates a step response when the switch is turned on. This setup shows how the lamp’s components can be protected from sudden voltage spikes. When the switch closes, Kirchhoff’s current law is applied to the parallel RLC circuit. That leads to a second-order differential equation. The solution has two parts: a transient response and a steady-state response. The...

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Second-Order Op Amp Low-Pass Filters

Second-order op-amp low-pass filters remove unwanted high-frequency noise from audio signals. They are used in audio systems to refine sound quality. These filters are often built as voltage followers and have two nodes with storage elements. The circuit analysis follows the same basic method used for second-order RLC circuits. In practice, bulky inductors are usually avoided because of their size and weight. For that reason, the focus here is on RC second-order op-amp circuits, which are...

design example: underdamped parallel rlc circuit

design-example-underdamped-parallel-rlc-circuit

Oscillator Design with a Parallel RLC Circuit

A parallel RLC circuit can be designed to work as an oscillator with underdamped behavior. This example targets a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. A fixed resistance of 200 Ω is used as the starting point for the design. The first step is to find the capacitance. The damping factor is the reciprocal of twice the product of resistance and capacitance. Using that relationship, the needed capacitance value can be calculated from the given resistance...

key terms and concepts

Learn how second-order circuits use capacitors, inductors, and resistors. Explore damping coefficient, resonant frequency, and RLC circuit response.

RLC Damping and Resonant Frequency | 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

RLC Damping and Resonant Frequency

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