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

Linear systems exhibit superposition, summing responses to individual inputs, and homogeneity, scaling responses consistently to an input multiplied by a scalar.
A nonlinear system can be considered linear about an operating point for small changes.
The Taylor series expansion links a function&#39;s value to its derivatives at a specific point and deviations from that point. Neglecting higher-order terms for small deviations gives a linear relationship.
Consider an RL circuit with a non-linear resistor. The presence of the nonlinear component necessitates system linearization before deriving the transfer function.
Kirchhoff&#39;s voltage law is utilized to derive a nonlinear differential equation.
The steady-state current is found by setting the small-signal source to zero.
The expression is rewritten in terms of the current&#39;s equilibrium value. The characteristics of the non-linear resistor are used to derive the linearized differential equation.
The known values are substituted, and the Laplace transform is applied with zero initial conditions.
The expression for the voltage across the inductor around the equilibrium point is subjected to the Laplace transform and simplified to obtain the transfer function.

linear approximation in frequency domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear. This approximation is achieved through the Taylor series expansion, which expresses a function in terms of its derivatives at a specific point. By neglecting higher-order terms for small deviations, a linear relationship is obtained.
Consider an RL circuit containing a nonlinear resistor. To analyze this system, linearization is necessary before deriving the transfer function.
<img alt="RLC circuit diagram; equations: v(t), i(t), L, R, displaying current-voltage relationships." src="/files/ftp_upload/16099/16099_PT%20Figure_1.svg" style="text-align: center; height: 180px;" data-altupdatedat="1762173717000" data-alt="Figure1" />
The first step involves applying Kirchhoff's voltage law to the circuit, resulting in a nonlinear differential equation that describes the system. For instance, the voltage law equation might take the form:
<img alt="Electrical circuit equation v(t)=vL(t)+vR(t)&#8722;E, featuring time-dependent voltage analysis." src="/files/ftp_upload/16099/16099_PT%20EQUATION_1.svg" style="text-align: center; height: 20px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
Where V(t) is the applied voltage, L is the inductance, R is the resistance, and E represents the battery voltage.
To find the steady-state current, we set the small-signal source to zero and solve for the equilibrium current i0. The nonlinear differential equation is then rewritten in terms of deviations from this equilibrium:
<img alt="Mathematical equation: i = i&#8320; + &#948;i, describing signal fluctuation analysis." src="/files/ftp_upload/16099/16099_PT%20EQUATION_2.svg" style="text-align: center; height: 22px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
The characteristics of the nonlinear resistor are used to derive the linearized differential equation. For small deviations in current, the voltage equation can be written as:
<img alt="Mathematical formula, v(t)=L d(i&#8320;+&#948;i)/dt+1/N ln 1/M(i&#8320;+&#948;i)-E, dynamic response analysis." src="/files/ftp_upload/16099/16099_PT%20EQUATION_3.svg" style="text-align: center; height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
Substituting this approximation into the voltage law equation, we obtain a linear differential equation. With known values substituted and assuming zero initial conditions, the Laplace transform is applied to convert the differential equation into an algebraic equation in the Laplace domain.

Linear Approximation in Frequency Domain

Linear systems exhibit superposition, summing responses to individual inputs, and homogeneity, scaling responses consistently to an input multiplied by a ...

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KEY TERMS AND CONCEPTS

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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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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