Name: linearizing nonlinear rl circuits
Uploaded: 2024-09-26T13:17:19.000Z
Duration: 1 min 26 s
Channel: JoVE

Los sistemas lineales exhiben superposición, sumando respuestas a entradas individuales, y homogeneidad, escalando las respuestas de manera consistente a una entrada multiplicada por un escalar.
Un sistema no lineal puede considerarse lineal alrededor de un punto de operación para cambios pequeños.
La expansión de la serie de Taylor vincula el valor de una función a sus derivadas en un punto específico y las desviaciones de ese punto. Descuidar los términos de orden superior por pequeñas desviaciones da una relación lineal.
Considere un circuito RL con una resistencia no lineal. La presencia del componente no lineal requiere la linealización del sistema antes de derivar la función de transferencia.
La ley de voltaje de Kirchhoff se utiliza para derivar una ecuación diferencial no lineal.
La corriente de estado estacionario se encuentra poniendo a cero la fuente de señal pequeña.
La expresión se reescribe en términos del valor de equilibrio de la corriente. Las características de la resistencia no lineal se utilizan para derivar la ecuación diferencial lineal.
Se sustituyen los valores conocidos y se aplica la transformada de Laplace sin condiciones iniciales.
La expresión del voltaje a través del inductor alrededor del punto de equilibrio se somete a la transformada de Laplace y se simplifica para obtener la función de transferencia.

aproximación lineal en el dominio de la frecuencia

Linear Approximation in Frequency Domain

Los sistemas lineales se caracterizan por dos propiedades principales: superposición y homogeneidad. La superposición permite que la respuesta a múltiples entradas sea la suma de las respuestas a cada entrada individual. La homogeneidad garantiza que al escalar una entrada por un escalar, la respuesta se escale por el mismo escalar.
En cambio, los sistemas no lineales no poseen estas propiedades de manera inherente. Sin embargo, para pequeñas desviaciones alrededor de un punto de operación, un sistema no lineal a menudo se puede aproximar como lineal. Esta aproximación se logra a través de la expansión de la serie de Taylor, que expresa una función en términos de sus derivadas en un punto específico. Al descuidar los términos de orden superior para pequeñas desviaciones, se obtiene una relación lineal.
Considere un circuito RL que contiene una resistencia no lineal. Para analizar este sistema, es necesaria la linealización antes de derivar la función de transferencia.
<img alt="Figure1" src="/files/ftp_upload/16099/16099_PT%20Figure_1.svg" style="text-align: center; height: 180px;">
El primer paso consiste en aplicar la ley de tensión de Kirchhoff al circuito, lo que da como resultado una ecuación diferencial no lineal que describe el sistema. Por ejemplo, la ecuación de la ley de voltaje podría adoptar la forma:
<img alt="Equation1" src="/files/ftp_upload/16099/16099_PT%20EQUATION_1.svg" style="text-align: center; height: 20px;">
Donde V(t) es la tensión aplicada, L es la inductancia, R es la resistencia y E representa la tensión de la batería.
Para encontrar la corriente de estado estable, establecemos la fuente de pequeña señal en cero y calculamos la corriente de equilibrio i_0. Luego, la ecuación diferencial no lineal se reescribe en términos de desviaciones de este equilibrio:
<img alt="Equation2" src="/files/ftp_upload/16099/16099_PT%20EQUATION_2.svg" style="text-align: center; height: 22px;">
Las características de la resistencia no lineal se utilizan para derivar la ecuación diferencial linealizada. Para pequeñas desviaciones de la corriente, la ecuación de voltaje se puede escribir como:
<img alt="Equation3" src="/files/ftp_upload/16099/16099_PT%20EQUATION_3.svg" style="text-align: center; height: 32px;">
Sustituyendo esta aproximación en la ecuación de la ley de tensión, obtenemos una ecuación diferencial lineal. Con los valores conocidos sustituidos y suponiendo condiciones iniciales cero, se aplica la transformada de Laplace para convertir la ecuación diferencial en una ecuación algebraica en el dominio de Laplace.

Aproximación lineal en el dominio de la frecuencia

Los sistemas lineales exhiben superposición, sumando respuestas a entradas individuales, y homogeneidad, escalando las respuestas de manera consistente a una entrada multiplicada por un ...

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

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 Taylor series expansion, which writes a function using its derivatives at one point. When higher-order terms are ignored for small deviations, the result is a linear relationship.

An RL circuit with a nonlinear resistor is a common example. The circuit is first described with Kirchhoff's voltage law, which gives a nonlinear differential equation. In that equation, V(t) is the applied voltage, L is the inductance, R is the resistance, and E is the battery voltage.

To analyze the circuit, the steady-state current is found first. The small-signal source is set to zero, and the equilibrium current i0 is solved for. The nonlinear equation is then rewritten in terms of small deviations from that equilibrium.

Next, the resistor’s characteristics are used to build the linearized equation. For small changes in current, the voltage can be approximated with a linear expression. Substituting that approximation into the voltage law gives a linear differential equation. With known values and zero initial conditions, the Laplace transform converts the equation into an algebraic form in the Laplace domain.

Linearizing Nonlinear RL Circuits

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linear approximation in frequency domain

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...

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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 a nonlinear RL circuit is linearized with Taylor series, steady-state current, Kirchhoff's voltage law, and the Laplace transform.

Linearizing Nonlinear RL Circuits | 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

Linearizing Nonlinear RL Circuits

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