Name: bode plot slope and phase changes
Uploaded: 2025-05-22T09:06:14.000Z
Duration: 1 min 23 s
Channel: JoVE

Considere a função de transferência em sua forma padrão, com pólos e zeros
Para uma função de transferência com um zero simples, o ganho de magnitude em pequenos valores de frequência é uma linha reta com inclinação zero e a fase se aproxima de zero.
Na frequência do canto, a magnitude assintótica se desvia da linha de inclinação zero e a fase se aproxima de 45 graus.
Em frequências mais altas, o gráfico de magnitude forma uma linha de +20 dB/década e a fase é de 90 graus.
Um pólo simples é o recíproco de um zero simples. Isso significa que os gráficos de Bode baseados em pólos espelham o gráfico zero simples, refletido sobre o eixo horizontal.
Considere uma função de transferência de pólo quadrática.
Em frequências mais baixas, o ganho e o ângulo de fase se aproximam de zero.
Na frequência do canto, o desvio de magnitude assintótica depende do fator de amortecimento e o ângulo de fase é de quase -90 graus.
Em frequências mais altas, o gráfico de magnitude forma uma linha reta com uma inclinação de -40 dB/década e uma fase de -180 graus.
Para mais de um pólo quadrático, a inclinação da linha e a mudança de fase são multiplicadas pelo número de pólos.

resposta de frequência

Na forma padrão, a função de transferência é mostrada em ganho constante, pólos/zeros na origem, pólos/zeros simples e pólos/zeros quadráticos; cada um contribuindo exclusivamente para a resposta geral do sistema. O termo representa a magnitude do zero simples:
<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="height: 36px;">
O gráfico de magnitude de Bode permanece plano em baixas frequências (aproximando-se de 0 dB) e começa a subir a 20 dB/década após uma frequência específica conhecida como frequência de canto ou quebra, ω_1. Esta é a frequência em que a inclinação do gráfico de magnitude muda e a resposta real começa a desviar-se da aproximação em linha reta. Este desvio é quantificado como 3 dB em ω=ω_1.
O ângulo de fase ϕ, expresso como:
<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="height: 60px;">
O ângulo de fase começa em 0° e se aproxima de 90° assintoticamente à medida que a frequência aumenta. Para frequências muito inferiores à frequência de canto (ω≪ω_1), o termo jω/ω_1 é muito pequeno, então a magnitude é insignificante e a fase é essencialmente zero. À medida que a frequência se aproxima de ω_1, levando a um ponto de -3 dB em magnitude e um ângulo de fase de 45°. Para frequências muito superiores a ω_1 (ω≫ω_1), a inclinação da magnitude muda para 20 dB/década e a fase se estabiliza em 90°.
<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="height: 160px;">
Pólo quadrático/zero:
A magnitude e o ângulo de fase de um pólo quadrático são:
<img alt="Equation 3" src="/files/ftp_upload/15142/15142_Equation_3.svg" style="height: 42px;">
<img alt="Equation 4" src="/files/ftp_upload/15142/15142_Equation_4.svg" style="height: 60px;">
O gráfico de amplitude para um pólo quadrático tem duas partes: uma resposta plana abaixo da frequência natural ω_n, e uma inclinação de -40 dB/década acima de ω_n, com o pico real do gráfico variando com o fator de amortecimento ζ_2. O gráfico de fase para um pólo quadrático diminui linearmente com uma inclinação de -90° por década, começando em um décimo da frequência natural e terminando em dez vezes isso, influenciado pelo fator de amortecimento ζ_2.

Resposta de Frequência

Considere a função de transferência em sua forma padrão, com pólos e zeros
Para uma função de transferência com um zero simples, o ganho de magnitude em pequenos ...

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bode plot slope and phase changes

Bode plots show how a transfer function changes with frequency. The standard form includes constant gain, poles and zeros at the origin, simple poles and zeros, and quadratic poles and zeros. Each part shapes the system response in a different way.

A simple zero sets the magnitude trend first. The Bode magnitude plot stays flat at low frequencies, near 0 dB, and then rises at 20 dB per decade after the corner or break frequency, ω1. That corner frequency is where the straight-line approximation starts to differ from the actual curve, and the error is 3 dB at ω = ω1.

The phase angle for a simple zero also changes with frequency. It starts at 0° and rises toward 90° as frequency increases. When ω is much smaller than ω1, the term jω/ω1 is very small, so the magnitude is negligible and the phase is close to zero. At ω = ω1, the magnitude is at the -3 dB point and the phase is 45°. When ω is much larger than ω1, the magnitude slope is 20 dB per decade and the phase settles near 90°.

Quadratic poles have a different shape on Bode plots. Their magnitude is flat below the natural frequency, ωn, and drops at -40 dB per decade above ωn. The actual peak depends on the damping factor, ζ2. Their phase plot decreases linearly at -90° per decade from one-tenth of ωn to ten times ωn, and that shape is also affected by ζ2.

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Bode Plot Slope and Phase Changes

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