Name: root locus rules for oscillator feedback
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 23 s
Channel: JoVE

Um oscilador Hartley, um sistema de feedback positivo, mantém a oscilação alimentando a saída de volta para a entrada em fase, reforçando o sinal.
Os sistemas de feedback positivo podem ser interpretados como sistemas de feedback negativo com sinais de feedback invertidos.
O locus raiz nesses sistemas inclui todos os pontos do plano s onde o ângulo da função de transferência do sistema é igual a 360 graus.
As ramificações e regras de simetria desses sistemas refletem as dos sistemas de feedback negativo.
No entanto, o locus raiz existe no eixo real à esquerda de um número par de pólos finitos de malha aberta e/ou zeros.
À medida que o locus se aproxima do infinito, ele aparece como linhas retas, semelhantes a assíntotas, com diferentes inclinações, mas interceptações idênticas no eixo real.
Os sistemas de feedback negativo com ganho negativo também seguem esses princípios, formando um sistema de feedback positivo equivalente, deslocando o ganho negativo para a direita, além do ponto de pick-off.
O locus raiz, à esquerda de uma contagem par de pólos reais e finitos de malha aberta e / ou zeros, abrange toda a extensão positiva do eixo real em intervalos designados.
As interseções e ângulos são então determinados para completar o esboço do locus raiz.

root loci para sistemas de feedback positivo

O oscilador Hartley é um sistema de feedback positivo que sustenta oscilações ao realimentar a saída de volta para a entrada em fase, reforçando assim o sinal. Os sistemas de feedback positivo podem ser vistos como sistemas de feedback negativo com sinais de feedback invertidos. Nesses sistemas, o root locus abrange todos os pontos no plano s onde o ângulo da função de transferência do sistema é igual a 360º.


 As regras de construção para o root locus em sistemas de feedback positivo são semelhantes às dos sistemas de feedback negativo. No entanto, uma diferença fundamental é que o root locus para feedback positivo existe no eixo real à esquerda de um número par de polos de malha aberta finitos e/ou zeros.


 À medida que o root locus se aproxima do infinito, ele segue assíntotas, que são linhas retas com inclinações variadas, mas interceptações idênticas no eixo real. Essas assíntotas guiam os caminhos dos loci à medida que se estendem ao infinito.


 Em sistemas com feedback negativo e ganho negativo, os princípios do root locus permanecem consistentes. Ao deslocar o ganho negativo para a direita, passando pelo ponto de coleta, o sistema efetivamente se comporta como um sistema de feedback positivo. Consequentemente, o lugar geométrico raiz fica à esquerda de um número par de polos ou zeros de malha aberta finitos e reais e abrange toda a extensão positiva do eixo real em intervalos especificados.


 Para esboçar com precisão o lugar geométrico raiz, é preciso determinar as interseções e ângulos nos quais os lugares geométricos partem ou se aproximam do eixo real. Esses pontos e ângulos são calculados com base nos polos e zeros do sistema. O diagrama do lugar geométrico raiz resultante fornece uma representação visual da estabilidade do sistema e das características de resposta à medida que o ganho varia.


 Em resumo, o oscilador Hartley exemplifica como os sistemas de feedback positivo sustentam oscilações por meio de feedback alinhado à fase. O método do lugar geométrico raiz, adaptado para feedback positivo, envolve a construção de lugares geométricos à esquerda de um número par de polos ou zeros de malha aberta no eixo real e estendendo-se em direção ao infinito ao longo de assíntotas. Essa análise é crucial para entender e projetar sistemas oscilatórios estáveis ​​e para prever seu comportamento sob diferentes condições de ganho.

Root Loci para sistemas de feedback positivo

Um oscilador Hartley, um sistema de feedback positivo, mantém a oscilação alimentando a saída de volta para a entrada em fase, reforçando o sinal.
Positivo...

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

Unidade 1

Root-Locus Method

Método do Lugar das Raízes

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

Root Loci for Positive-Feedback Systems

root locus rules for oscillator feedback

Root locus rules help describe how oscillator feedback systems behave as gain changes. The Hartley oscillator is one example of a positive feedback system. It keeps oscillating by feeding the output back to the input in phase, which reinforces the signal.

Positive feedback systems can also be treated as negative feedback systems with the feedback signal inverted. In root locus analysis, the locus is the set of points on the s-plane where the transfer function angle equals 360 degrees. The construction rules are similar to those used for negative feedback systems.

A key difference is the real-axis rule. For positive feedback systems, the root locus appears on the real axis to the left of an even number of finite open-loop poles and/or zeros. As the locus moves toward infinity, it follows asymptotes. These are straight lines with different slopes but the same real-axis intercept.

The same idea can apply when a system has negative feedback and negative gain. If the negative gain is shifted rightward past the pick-off point, the system behaves like a positive feedback system. The root locus then lies to the left of an even number of real, finite open-loop poles or zeros and spans the positive part of the real axis at the required intervals.

To sketch the root locus correctly, the departure and arrival points on the real axis must be found. Their angles are determined from the system poles and zeros. The finished root locus diagram shows how stability and response change as gain varies. The Hartley oscillator is a useful example of how positive feedback and root locus analysis work together in stable oscillatory design.

Root Locus Rules for Oscillator Feedback

control system problem

Root Locus and Closed-Loop Stability

Root locus and closed-loop stability are key ideas in control systems. A basic thermostat is an open-loop system, so its transfer function poles affect the response but do not set stability. When feedback is added, the system becomes closed-loop, like an advanced thermostat that changes heating based on room temperature. In a closed-loop system, stability depends on the new poles of the closed-loop transfer function. If those poles move into the unstable region, the system can start to...

vector representation of complex numbers

Complex Numbers as Vectors in Polar Form

Complex numbers can be shown as vectors, not just as points in Cartesian coordinates. In polar form, a vector is described by its magnitude, or length, and its angle. This makes complex numbers easier to compare by size and direction. The same idea also helps describe complex functions. When a complex number goes into a function, the output is another complex number. The function can be viewed as having a zero point, and the vector picture starts from that point. For a function written as a...

root-locus method

Cruise Control Stability with Root Locus

Root locus helps show how cruise control stays stable as conditions change. In a car, cruise control keeps a set speed by adjusting the gas pedal automatically. It keeps checking the car’s speed and makes small pedal changes to hold that target speed. This system can be drawn as a block diagram, and its transfer function gives the math model. The model is useful when the car goes uphill, downhill, or faces strong wind resistance. To find the system poles for different pedal forces, the...

properties of the root locus

Root Locus Angle and Gain Rules

Root locus angle and gain rules help analyze higher-order control systems. The method works without factoring the denominator of the transfer function. It is especially useful for unity feedback systems, where the transfer function can be studied by tracking how poles and zeros shape the response. A pole is a point where the characteristic polynomial in the denominator of the transfer function equals zero. To check whether a point lies on the root locus, use the angle criterion. Add the angles...

construction of root locus

Root Locus Paths and Asymptotes

Root locus paths and asymptotes show how closed-loop poles move as gain changes. This method helps students track system stability by following the pole locations on a graph. It also shows where the poles start, where they may end, and how they travel in between. The number of root locus branches equals the number of closed-loop poles. The plot is symmetric about the real axis. For positive gain, the root locus lies on the real axis to the left of an odd number of finite open-loop poles or...

plotting and calibrating the root locus

Root Locus Breakaway and Stability Points

Root locus analysis tracks how system poles move as gain changes. It shows when poles leave the real axis, move into the complex plane, and affect stability. For high school students studying control systems, the key ideas are breakaway points, break-in points, and jω-axis crossings. Breakaway points are locations where the root locus leaves the real axis. Break-in points are locations where it returns to the real axis. At either type of point, the branches meet the real axis at an angle of...

root loci for positive-feedback systems

Root locus rules help describe how oscillator feedback systems behave as gain changes. The Hartley oscillator is one example of a positive feedback system. It keeps oscillating by feeding the output back to the input in phase, which reinforces the signal. Positive feedback systems can also be treated as negative feedback systems with the feedback signal inverted. In root locus analysis, the locus is the set of points on the s-plane where the transfer function angle equals 360 degrees. The...

key terms and concepts

Learn how positive feedback, the Hartley oscillator, and root locus rules work together to show stability and oscillation as gain changes in control systems.

Root Locus Rules for Oscillator Feedback | 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

Root Locus Rules for Oscillator Feedback

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