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

I sistemi lineari mostrano sovrapposizione, sommando le risposte ai singoli input, e omogeneità, scalando le risposte in modo coerente a un input moltiplicato per uno scalare.
Un sistema non lineare può essere considerato lineare rispetto a un punto di funzionamento per piccole modifiche.
L'espansione in serie di Taylor collega il valore di una funzione alle sue derivate in un punto specifico e alle deviazioni da quel punto. Trascurando i termini di ordine superiore per piccole deviazioni si ottiene una relazione lineare.
Si consideri un circuito RL con un resistore non lineare. La presenza della componente non lineare richiede la linearizzazione del sistema prima di derivare la funzione di trasferimento.
La legge della tensione di Kirchhoff viene utilizzata per derivare un'equazione differenziale non lineare.
La corrente in stato stazionario si trova impostando la sorgente del piccolo segnale su zero.
L'espressione viene riscritta in termini di valore di equilibrio della corrente. Le caratteristiche del resistore non lineare vengono utilizzate per derivare l'equazione differenziale linearizzata.
I valori noti vengono sostituiti e la trasformata di Laplace viene applicata con zero condizioni iniziali.
L'espressione per la tensione attraverso l'induttore attorno al punto di equilibrio è sottoposta alla trasformata di Laplace e semplificata per ottenere la funzione di trasferimento.

approssimazione lineare nel dominio della frequenza

Linear Approximation in Frequency Domain

I sistemi lineari sono caratterizzati da due proprietà principali: la sovrapposizione e l’omogeneità. La sovrapposizione consente alla risposta a più input di essere la somma delle risposte ad ogni singolo input. L'omogeneità assicura che il ridimensionamento di un input di uno scalare comporti il ridimensionamento della risposta dello stesso scalare.

Al contrario, i sistemi non lineari non possiedono intrinsecamente queste proprietà. Tuttavia, per piccole deviazioni attorno a un punto operativo, un sistema non lineare può spesso essere approssimato come lineare. Questa approssimazione è ottenuta grazie all'espansione in serie di Taylor, che esprime una funzione in termini delle sue derivate in un punto specifico. Trascurando i termini di ordine superiore per le piccole deviazioni, si ottiene una relazione lineare.

Consideriamo un circuito RL contenente un resistore non lineare. Per analizzare questo sistema, è necessaria la linearizzazione prima di derivare la funzione di trasferimento.

<img alt="Figure1" src="/files/ftp_upload/16099/16099_PT%20Figure_1.svg" style="text-align: center; height: 180px;">

Il primo passaggio consiste nell'applicare la legge di Kirchhoff sulla tensione al circuito, ottenendo un'equazione differenziale non lineare che descrive il sistema. Per esempio, l'equazione della legge sulla tensione potrebbe assumere la forma:

<img alt="Equation1" src="/files/ftp_upload/16099/16099_PT%20EQUATION_1.svg" style="text-align: center; height: 20px;">

Dove V(t) è la tensione applicata, L è l'induttanza, R è la resistenza ed E rappresenta la tensione della batteria.

Per trovare la corrente in stato stazionario, impostiamo la sorgente di piccolo segnale su 0 e risolviamo per la corrente di equilibrio i_0. L'equazione differenziale non lineare viene quindi riscritta in termini di deviazioni da questo equilibrio:

<img alt="Equation2" src="/files/ftp_upload/16099/16099_PT%20EQUATION_2.svg" style="text-align: center; height: 22px;">

Le caratteristiche del resistore non lineare vengono usate per derivare l'equazione differenziale linearizzata. Per piccole deviazioni di corrente, l'equazione della tensione può essere scritta come:

<img alt="Equation3" src="/files/ftp_upload/16099/16099_PT%20EQUATION_3.svg" style="text-align: center; height: 32px;">

Sostituendo questa approssimazione nell'equazione della legge della tensione, otteniamo un'equazione differenziale lineare. Con valori noti sostituiti e supponendo condizioni iniziali uguali a 0, la trasformata di Laplace viene applicata per convertire l'equazione differenziale in un'equazione algebrica nel dominio di Laplace.

Approssimazione lineare nel dominio della frequenza

I sistemi lineari mostrano sovrapposizione, sommando le risposte ai singoli input, e omogeneità, scalando le risposte in modo coerente a un input moltiplicato per un ...

Education

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JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

Modeling in Time and Frequency Domain

Modellazione nel dominio del tempo e della frequenza

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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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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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