Name: rlc circuit state equations
Uploaded: 2024-09-26T13:17:19.000Z
Duration: 1 min 27 s
Channel: JoVE

La técnica del dominio de la frecuencia, que se usa comúnmente para analizar y diseñar sistemas de control de retroalimentación, solo es adecuada para sistemas lineales invariantes en el tiempo.
El enfoque de dominio del tiempo o de espacio de estados puede manejar sistemas no lineales, variables en el tiempo y de múltiples entradas y múltiples salidas.
En la representación del espacio de estados, se utiliza un subconjunto particular de variables del sistema, identificadas como variables de estado, para construir ecuaciones diferenciales simultáneas de primer orden para un sistema de n-ésimo orden, etiquetadas como ecuaciones de estado.
Considere un circuito RLC. Dado que la red es de segundo orden, es necesario emplear dos ecuaciones diferenciales simultáneas de primer orden.
Las cantidades que se diferencian en la ecuación derivada para los elementos de almacenamiento de energía: el inductor y el condensador son las variables de estado.
Las leyes de voltaje y corriente de Kirchhoff se utilizan para expresar las variables no estatales como combinaciones lineales de las variables de estado y la entrada.
Estos resultados se sustituyen en las ecuaciones derivadas originales para obtener las ecuaciones de estado.
La ecuación de salida es la corriente a través de la resistencia.
El paso final consiste en representar estas ecuaciones en forma de matriz vectorial para adquirir la representación del espacio de estados de la red eléctrica dada.

representación en el espacio de estados

La técnica del dominio de frecuencia, que se utiliza habitualmente para analizar y diseñar sistemas de control por retroalimentación, es eficaz para sistemas lineales e invariantes en el tiempo. Sin embargo, no es suficiente cuando se trata de sistemas no lineales, variables en el tiempo y con múltiples entradas y múltiples salidas. El enfoque del dominio del tiempo o del espacio de estados aborda estas limitaciones utilizando variables de estado para construir ecuaciones diferenciales simultáneas de primer orden, conocidas como ecuaciones de estado, para un sistema de orden n.
Consideremos un circuito RLC, un sistema común de segundo orden. Para analizar este circuito utilizando el enfoque del espacio de estados, son necesarias dos ecuaciones diferenciales simultáneas de primer orden. Las variables de estado en este contexto se derivan de las cantidades diferenciadas en las ecuaciones derivadas asociadas con los elementos de almacenamiento de energía, específicamente el inductor y el capacitor.
Se emplean las leyes de tensión y corriente de Kirchhoff para formular las ecuaciones de estado. La ley de tensión de Kirchhoff (KVL) establece que todas las diferencias de potencial eléctrico alrededor de un bucle son cero, mientras que la ley de corriente de Kirchhoff (KCL) afirma que la suma de las corrientes que ingresan a una unión es igual a la suma de las corrientes que salen. Estas leyes permiten la expresión de variables no estatales como combinaciones lineales de variables de estado y entradas.
En un circuito RLC, las variables de estado son el voltaje a través del capacitor VC y la corriente a través del inductor iL. Las leyes de Kirchhoff expresan la corriente del resistor y otras variables no estatales en términos de VC e iL. Luego, estas expresiones se sustituyen nuevamente en las ecuaciones diferenciales originales del circuito.
Después de derivar las ecuaciones de estado, el paso final es representarlas en forma de matriz vectorial, logrando la representación en el espacio de estados. En el caso de un circuito RLC, esto podría implicar definir el vector de estado x, el vector de entrada u, el vector de salida y y las matrices A, B, C y D de manera que:
<img alt="Equation1" src="/files/ftp_upload/16100/16100_PT%20EQUATION_1.svg" style="text-align: center; height: 16px;">
<img alt="Equation2" src="/files/ftp_upload/16100/16100_PT%20EQUATION_2.svg" style="text-align: center; height: 16px;">
Esta representación es esencial para analizar el comportamiento dinámico del sistema y diseñar estrategias de control adecuadas.
En resumen, el enfoque de espacio de estados proporciona un marco sólido para manejar sistemas complejos, que se extiende más allá de las capacidades de las técnicas de dominio de frecuencia al acomodar no linealidades, variaciones de tiempo y múltiples entradas y salidas.

Representación en el espacio de estados

La técnica del dominio de la frecuencia, que se utiliza habitualmente para analizar y diseñar sistemas de control de retroalimentación, sólo es adecuada para sistemas lineales, invariantes en el tiempo ...

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

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 system like an RLC circuit, two first-order equations are needed. The state variables come from the quantities tied to the circuit’s energy-storage elements, which are the inductor and the capacitor.

Kirchhoff’s voltage law, or KVL, says that the total potential difference around a loop is zero. Kirchhoff’s current law, or KCL, says that the current entering a junction equals the current leaving it. These rules let engineers write non-state variables as linear combinations of the state variables and the input.

In an RLC circuit, the state variables are the capacitor voltage, VC, and the inductor current, iL. KVL and KCL are used to express the resistor current and other non-state variables in terms of VC and iL. Those expressions are then substituted into the circuit’s original differential equations.

After the state equations are found, they are written in vector-matrix form. This gives the state-space representation, with a state vector x, an input vector u, an output vector y, and matrices A, B, C, and D. This form is useful for studying system behavior and for designing control strategies.

RLC Circuit State Equations

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 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.

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 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 an RLC circuit is written as state equations and converted into state-space form using KVL, KCL, and the key state variables VC and iL.

RLC Circuit State Equations | 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 State Equations

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