Name: rc circuit charging after a dc step
Uploaded: 2024-04-11T12:48:43.000Z
Duration: 1 min 15 s
Channel: JoVE

当突然将直流电源施加到RC电路时，其电压可用单位阶跃函数表示。
电容器两端的电压是阶跃响应。
由于电容器两端的电压不能瞬时改变，因此开关动作后其值与开关动作前瞬间的值相同。
在时间等于零时应用基尔霍夫电流定律，重新整理各项，并将方程改写为时间大于零的形式，可得到一个一阶微分方程。
对公式进行积分，代入上下限，并对两边取指数，可得到电容器在大于零时刻的阶跃响应。
将电容器的初始电压作为零时刻之前的电路响应，即可得到 RC 电路的完整响应。
随着时间推移，电容器的电压呈指数增长并逐渐接近电源电压。
如果电容器初始未充电，则其完全响应会相应发生变化。由此可确定通过电容器的电流，并观察到该电流随时间呈指数衰减。

带源的电阻电容电路

当直流源突然施加到电阻电容（RC）电路中时，电压可以表示为单位阶跃函数。位于电容器两端的电压（称为阶跃响应）体现了电路对突然变化的输入所产生的反应。 由于电容器的固有特性可以得知，其电压不能发生瞬时变化。这意味着在开关闭合后，电容器的电压立即保持与开关闭合之前相同的状态。 通过应用开关闭合时刻（t=0）的基尔霍夫电流定律，同时对各项进行重新排列，并在t>0时对方程进行重写，从而能够得到一阶微分方程。该方程描述了在突然施加直流电源后，通过电路的电流是如何随时间发生变化的。 该微分方程通过积分、应用极限并在两边取指数来进行求解。这便产生了时间t>0时电容器中所发生的阶跃响应。将此阶跃响应与电容器两端的初始电压（t<0）相结合，便能够得出电阻电容电路中所产生的完整响应。 随着时间的推移，电容器两端的电压呈指数的形式增加并接近于其所施加的源电压。这个过程代表了电容器的充电过程。 如果电容器在最初没有未充电，则电路的完整响应会产生相应的修改。根据这个修改后的响应可以计算出通过电容器的电流。由于观察到该电流会随时间的变化呈指数的趋势下降，那么便将其表示为电容器是处于逐渐充电的状态，直至其达到源电压为止。 总之，了解电阻电容电路的阶跃响应可以为了解这些电路是如何响应输入电压的突然变化来提供宝贵的见解。这些知识对于设计和分析电子电路来说是至关重要的，特别是在信号处理等应用中，其中的电容器被广泛的应用在过滤或整形信号中。

当直流电源突然施加到RC电路时，其电压可表示为单位阶跃函数。  
电容器两端的电压为阶跃...

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

First and Second-Order Circuits

一阶与二阶电路

KEY TERMS AND CONCEPTS

关键术语与概念

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 rewritten for times t&gt;0. This produces a first-order differential equation for the circuit current.

The differential equation is solved by integration. After applying the limits and taking the exponential form, the capacitor’s step response is found for t&gt;0. When this result is combined with the capacitor voltage for t&lt;0, it gives the complete response of the RC circuit. If the capacitor starts uncharged, the response is adjusted accordingly.

As time passes, the capacitor voltage rises exponentially and approaches the source voltage. This is the charging process of the capacitor. The current through the capacitor can also be calculated from the modified response. It decreases exponentially with time as the capacitor gradually charges until it reaches the source voltage.

Understanding the step response of an RC circuit helps explain how the circuit reacts to a sudden DC input. This is useful for analyzing and designing electronic circuits. It is especially important in signal processing, where capacitors are often used to filter or shape signals.

RC Circuit Charging After a DC Step

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

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

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...

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

Study how an RC circuit responds to a sudden DC step, including capacitor charging, the exponential voltage rise, and the decreasing current over time.

RC Circuit Charging After a DC Step | 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

RC Circuit Charging After a DC Step

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