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

Ein elektrischer Stromkreis, der aus zwei irreduziblen Energiespeicherelementen besteht, wird als Schaltung zweiter Ordnung bezeichnet.
Einige Beispiele sind RLC-Schaltungen sowie RC- und RL-Schaltungen mit Doppelkondensatoren bzw. Induktivitäten.
Diese Schaltungen zeichnen sich durch eine Differentialgleichung zweiter Ordnung aus, die die Eingangs- und Ausgangssignale der Schaltung korreliert.
Eingangssignale stammen in der Regel von Spannungs- oder Stromquellen, während der Ausgang in der Regel die Spannung am Kondensator und/oder der Strom durch die Induktivität ist.
Stellen Sie sich eine RLC-Schaltung vor, bei der die im Kondensator und in der Induktivität gespeicherte Anfangsenergie die Schaltung antreibt.
Wendet man das Kirchhoffsche Spannungsgesetz auf die Schaltung an und nimmt man die Zeitableitung, erhält man eine Differentialgleichung zweiter Ordnung.
Die Koeffizienten dieser Gleichung, ausgedrückt in Bezug auf Widerstand, Kapazität und Induktivität, werden als Dämpfungskoeffizient und Resonanzfrequenz bezeichnet.
Der Dämpfungskoeffizient gibt die Dämpfung durch den Widerstand an und bestimmt die Rate der Energiedissipation im System.
Die Resonanzfrequenz stellt die Eigenschwingungsfrequenz der Schaltung dar, wenn Energie zwischen Induktivität und Kondensator ausgetauscht wird.

Es wurden keine Interessenkonflikte angegeben.

schaltkreise zweiter ordnung

Second-Order Circuits

Durch die Integration zweier grundlegender Energiespeicherelemente in elektrische Schaltkreise entstehen Schaltkreise zweiter Ordnung, darunter RLC-Schaltkreise und Schaltkreise mit Doppelkondensatoren oder Induktivitäten (RC- und RL-Schaltkreise). Schaltkreise zweiter Ordnung werden durch Differentialgleichungen zweiter Ordnung identifiziert, die Eingangs- und Ausgangssignale verknüpfen. Eingangssignale stammen typischerweise von Spannungs- oder Stromquellen, wobei der Ausgang häufig die Spannung am Kondensator und/oder den Strom durch die Induktivität darstellt. In einer RLC-Schaltung wird beispielsweise die im Kondensator und in der Induktivität gespeicherte Anfangsenergie initiiert. Die Anwendung des Kirchhoffschen Spannungsgesetzes und die Durchführung einer zeitlichen Ableitung ergibt eine Differentialgleichung zweiter Ordnung. <img alt="Equation1" src="/files/ftp_upload/15097/15097_PT_Equation_1.svg" style="text-align: center; height: 36px;"> Seine durch Widerstand, Kapazität und Induktivität bestimmten Koeffizienten manifestieren sich als Dämpfungskoeffizient und Resonanzfrequenz. <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;"> Der Dämpfungskoeffizient spielt in diesen Schaltkreisen eine entscheidende Rolle und gibt das Ausmaß der Dämpfung an, die hauptsächlich durch den Widerstand verursacht wird. Es beeinflusst direkt die Geschwindigkeit, mit der Energie innerhalb des Systems verloren geht, und steuert so effektiv die Energieverlustrate. Andererseits ist die Resonanzfrequenz ein Schlüsselmerkmal, das die inhärente Schwingungsfrequenz der Schaltung darstellt. Es misst, wie schnell Energie zwischen der Induktivität und dem Kondensator im Stromkreis ausgetauscht wird, und veranschaulicht die natürliche Tendenz des Stromkreises, bei einer bestimmten Frequenz zu schwingen. Der Dämpfungskoeffizient bestimmt, wie schnell Energie im System aufgrund des Widerstands verloren geht. Gleichzeitig verdeutlicht die Resonanzfrequenz die inhärente Schwingungsgeschwindigkeit des Schaltkreises bei Energieverschiebungen zwischen Induktor und Kondensator. Diese beiden Faktoren sind entscheidend für das Verständnis und die Analyse des Verhaltens von Schaltkreisen zweiter Ordnung.

Schaltkreise zweiter Ordnung

Ein elektrischer Stromkreis, der aus zwei irreduziblen Energiespeicherelementen besteht, wird als Schaltung zweiter Ordnung bezeichnet.
Einige Beispiele sind RLC-Schaltungen sowie ...

Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

First and Second-Order Circuits

Schaltungen erster und zweiter Ordnung

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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