Name: rlc circuit transfer functions
Uploaded: 2024-09-26T13:17:18.000Z
Duration: 1 min 21 s
Channel: JoVE

Las redes eléctricas se representan primero mediante circuitos equivalentes que constan de tres componentes lineales pasivos: resistencias, condensadores e inductores.
Estos componentes se combinan en circuitos, con entrada y salida decididas, y la función de transferencia se encuentra utilizando las leyes de Kirchhoff.
En un circuito RLC, la función de transferencia que relaciona el voltaje a través del condensador con el voltaje de entrada se puede derivar utilizando la ley de voltaje de Kirchhoff.
Esto produce una ecuación integro-diferencial para la red, asumiendo cero condiciones iniciales.
Las variables se cambian inicialmente de corriente a carga, seguido de la aplicación de la relación voltaje-carga para un condensador.
Tomando la transformada de Laplace de esta ecuación y simplificándola se obtiene la función de transferencia para este circuito.
La impedancia es una función de transferencia similar a la resistencia pero aplicable a condensadores e inductores.
Las funciones de transferencia también se pueden obtener utilizando la ley actual de Kirchhoff, utilizando el análisis nodal.
Las corrientes en el sistema están compuestas por la corriente que fluye a través del condensador y la corriente que circula a través de la resistencia en serie y el inductor.
Después de la simplificación, se logra el mismo resultado para la función de transferencia.

sistemas eléctricos

Electrical Systems

En ingeniería eléctrica, el análisis de redes compuestas por componentes pasivos lineales (resistencias [R], capacitores [C] e inductores [L]) es fundamental. Estos componentes se organizan en circuitos donde la relación entre la entrada y la salida se puede analizar mediante funciones de transferencia. La función de transferencia de un circuito RLC, que relaciona la tensión a través de un condensador de la tensión de entrada, se puede derivar utilizando las leyes de Kirchhoff.
Para derivar la función de transferencia, considere un circuito RLC con los componentes conectados en serie. La Ley de Tensión de Kirchhoff (KVL) establece que la suma de todos las tensiones alrededor de un bucle cerrado es cero. Para un circuito RLC, esto se traduce en la siguiente ecuación integrodiferencial, suponiendo que las condiciones iniciales son cero:
<img alt="Equation1" src="/files/ftp_upload/16096/16096_PT%20EQUATION_1.svg" style="text-align: center; height: 32px;">
Aquí, V(t) es la tensión de entrada, VR(t)=i(t)R es la tensión a través del resistor, VC(t) =q(t)/C es el voltaje a través del condesador, donde q(t) es la carga en el capacitor y VL(t) = L(di(t)/dt) es la tensión a través del inductor.
La corriente i(t) está relacionada con la carga q(t) por i(t)=dq(t)/dt. Sustituyendo estas relaciones en la ecuación KVL se obtiene:
<img alt="Equation2" src="/files/ftp_upload/16096/16096_PT%20EQUATION_2.svg" style="text-align: center; height: 48px;">
Escribiendo la ecuación en forma de tensión, tomando la transformada de Laplace de ambos lados y suponiendo que todas las condiciones iniciales son cero, se obtiene la siguiente ecuación:
<img alt="Equation3" src="/files/ftp_upload/16096/16096_PT%20EQUATION_3.svg" style="text-align: center; height: 30px;">
Reordenando los términos para resolver la función de transferencia:
<img alt="Equation4" src="/files/ftp_upload/16096/16096_PT%20EQUATION_4.svg" style="text-align: center; height: 46px;">
Esta función de transferencia representa el voltaje a través del condensador en respuesta a la tensión de entrada en el dominio de la frecuencia. La impedancia, análoga a la resistencia pero aplicable a condensadores e inductores, juega un papel clave en la definición de la función de transferencia. Además, la Ley de Corriente de Kirchhoff (KCL) también se puede utilizar para derivar funciones de transferencia a través del análisis nodal. Esta ley establece que la suma de las corrientes que ingresan a un nodo debe ser igual a la suma de las corrientes que salen del nodo. En el circuito RLC, la corriente total es la suma de la corriente a través del condensador y la corriente a través de la combinación en serie del resistor y el inductor. Al aplicar la LCK y simplificar, se puede lograr la misma función de transferencia.
La LCK y la LCK son herramientas poderosas en el análisis de circuitos, que permiten la derivación de funciones de transferencia que describen sucintamente el comportamiento dinámico de las redes eléctricas en el dominio de la frecuencia. Estos métodos son esenciales para el diseño y análisis de circuitos complejos en diversas aplicaciones.

Sistemas eléctricos

Las redes eléctricas se representan primero mediante circuitos equivalentes que constan de tres componentes lineales pasivos: resistencias, condensadores e inductores.
...

Education

Educación

JoVE Core

Electrical Engineering

Ingeniería Eléctrica

Unit 1

Unidad 1

Modeling in Time and Frequency Domain

Modelado en el dominio del tiempo y de la frecuencia

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

rlc circuit transfer functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage.

A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero. Using that rule, the circuit can be written as an integro-differential equation when the initial conditions are zero.

In the equation, the input voltage is V(t). The resistor voltage is VR(t) = i(t)R, where i(t) is the current. The capacitor voltage depends on the charge q(t), and the inductor voltage depends on the current change in the loop. Since current and charge are related, these expressions can be substituted into the KVL equation.

The same circuit can also be written in voltage form and converted with the Laplace transform. With all initial conditions set to zero, this step gives an equation in the frequency domain. Rearranging the terms produces the transfer function for the capacitor voltage response.

Impedance plays an important role in this result. It works like resistance, but it applies to capacitors and inductors as well. The transfer function shows how the capacitor voltage responds to the input voltage in the frequency domain.

Kirchhoff's Current Law, or KCL, can also be used to reach the same result through nodal analysis. KCL says that the sum of currents entering a node must equal the sum of currents leaving it. In the RLC circuit, the total current is the current through the capacitor plus the current through the series resistor and inductor, and simplifying this relationship gives the same transfer function.

RLC Circuit Transfer Functions

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

electrical systems

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

mechanical systems

Transfer Functions in Mechanical Motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

electro-mechanical systems

DC Motor Modeling in Electromechanical Systems

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

Linearizing Nonlinear State-Space Models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how KVL, KCL, and the Laplace transform are used to find the transfer function of a series RLC circuit and track capacitor voltage in the frequency domain.

RLC Circuit Transfer Functions | 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 Circuit Transfer Functions

Description

Transcript

Tags