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

Un circuito elettrico comprendente due elementi irriducibili di accumulo di energia è chiamato circuito del secondo ordine.
Alcuni esempi includono circuiti RLC, circuiti RC e RL con doppi condensatori e induttori, rispettivamente.
Questi circuiti sono caratterizzati da un'equazione differenziale del secondo ordine che correla i segnali di ingresso e di uscita del circuito.
I segnali di ingresso provengono generalmente da fonti di tensione o corrente, mentre l'uscita è solitamente la tensione attraverso il condensatore e/o la corrente attraverso l'induttore.
Si consideri un circuito RLC, in cui l'energia iniziale immagazzinata nel condensatore e nell'induttore aziona il circuito.
Applicando la legge della tensione di Kirchhoff al circuito e prendendo la derivata del tempo si ottiene un'equazione differenziale del secondo ordine.
I coefficienti di questa equazione, espressi in termini di resistenza, capacità e induttanza, sono noti come coefficiente di smorzamento e frequenza di risonanza.
Il coefficiente di smorzamento indica lo smorzamento dovuto al resistore e determina il tasso di dissipazione dell'energia nel sistema.
La frequenza di risonanza rappresenta la frequenza di oscillazione naturale del circuito quando l'energia viene scambiata tra l'induttore e il condensatore.

Nessun conflitto di interessi dichiarato.

circuiti di secondo ordine

Second-Order Circuits

L'integrazione di due elementi fondamentali di accumulo dell'energia nei circuiti elettrici, si traduce in circuiti di secondo ordine, che comprendono circuiti RLC e circuiti con doppi condensatori o induttori (circuiti RC e RL). I circuiti di secondo ordine sono identificati da equazioni differenziali di secondo ordine, che collegano i segnali di ingresso e di uscita.

I segnali di ingresso provengono tipicamente da sorgenti di tensione o corrente, con l'uscita che spesso rappresenta la tensione attraverso il condensatore e/o la corrente attraverso l'induttore. Ad esempio, in un circuito RLC, l'energia iniziale immagazzinata nel condensatore e nell'induttore, avvia il circuito. L'applicazione della legge sulla tensione di Kirchhoff e l'esecuzione di una derivata temporale, producono un'equazione differenziale di secondo ordine.

<img alt="Equation1" src="/files/ftp_upload/15097/15097_PT_Equation_1.svg" style="text-align: center; height: 36px;">

I suoi coefficienti, determinati da resistenza, capacità e induttanza, si manifestano come coefficiente di smorzamento e frequenza di risonanza.

<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;">

Il coefficiente di smorzamento gioca un ruolo critico in questi circuiti, indicando l'entità dello smorzamento causato principalmente dal resistore. Influenza direttamente il ritmo con cui l'energia si dissipa all'interno del sistema, controllando efficacemente il tasso di perdita di energia. D'altra parte, la frequenza di risonanza è una caratteristica chiave che rappresenta la frequenza di oscillazione innata del circuito. Misura la velocità con cui l'energia viene scambiata tra l'induttore e il condensatore nel circuito, illustrando la tendenza naturale del circuito a oscillare a una particolare frequenza.

Il coefficiente di smorzamento determina la velocità con cui l'energia viene persa nel sistema a causa della resistenza. Allo stesso tempo, la frequenza di risonanza, evidenzia la velocità di oscillazione intrinseca del circuito mentre l'energia si sposta tra l'induttore e il condensatore. Questi due fattori sono cruciali per comprendere e analizzare il comportamento dei circuiti di secondo ordine.

Circuiti di Secondo Ordine

Un circuito elettrico comprendente due elementi irriducibili di accumulo di energia è chiamato circuito del secondo ordine.
Alcuni esempi includono circuiti RLC e ...

Education

Didattica

JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

First and Second-Order Circuits

Circuiti del primo e del secondo ordine

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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...

design example: frog muscle response

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...

parallel rlc circuits

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...

second-order op amp circuits

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

Description

Transcript

Tags