Bode Plot

• Source: Internet Page | Downladed PDF

• Amplitude:

  • Real zero: $+20\ast \text{zero’s multiplicity}\frac{\text{dB}}{\text{decade}}$
  • Real pole: $-20\ast \text{pole’s multiplicity}\frac{\text{dB}}{\text{decade}}$
  • Imaginary zero:
    • $+40\ast \text{zero’s multiplicity}\frac{\text{dB}}{\text{decade}}$
    • Peak at $w=w_0\sqrt{1-2{\zeta }^2}$
  • Imaginary pole:
    • $-40\ast \text{pole’s multiplicity}\frac{\text{dB}}{\text{decade}}$
    • Peak at $w=w_0\sqrt{1-2{\zeta }^2}$

• Phase:

  • Starting Phase*:
    • $+0°$ if $K>0$ ($K$ is the gain)
    • $\pm 180°$ if $K<0$ ($K$ is the gain)
  • Real zero: $+90°\ast \text{zero’s multiplicity}$
  • Real pole: $-90°\ast \text{zero’s multiplicity}$
  • Imaginary zero: $+180°\ast \text{zero’s multiplicity}$
  • Imaginary pole: $-180°\ast \text{pole’s multiplicity}$

• Frequency Domain Response (Bode Form):

  • For a single-pole system, the transfer function in frequency domain is represented as: $H(j\omega )=\frac{A\_DC}{1+j\frac{\omega }{\omega \_T}}$, where $A_{DC}$ is the DC gain and ${\omega }_T$ is the cutoff frequency.
  • Cutoff frequency ($f_T$) or $-3,\text{dB}$ frequency is related to the time constant $\tau$ as: ${\omega }_T=\frac{1}{\tau }$.

• Bode Plots:

  • The module of $H(j\omega )$ is represented in dB as $20\log |H(j\omega )|$.
  • The DC gain ($A_{DC}$) is the value of $H(j\omega )$ when $\omega =0$.
  • The cutoff frequency ($f_T$) corresponds to $|H(j{\omega }_T)|=\frac{1}{\sqrt{2}}$ in dB.

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Gain

• A real zero means $+20\cdot \mu \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• A real pole means $-20\cdot \mu \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• An imaginary zero
  • means $+40\cdot \mu \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
  • In the case of an imaginary zero there is a also peak is at $w=w_0\sqrt{1-2{\zeta }^2}$
• An imaginary pole:
  • means $+40\cdot \mu \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
  • There is a also peak is at $w=w_0\sqrt{1-2{\zeta }^2}$
  • ~Ex.:
    

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Phase

• Starting Phase*:
  • $+0°$ if $K>0$ ($K$ is the gain)
  • $\pm 180°$ if $K<0$ ($K$ is the gain)
• Real zero: $+90°\cdot \mu \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• Real pole: $-90°\cdot \mu \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• Couple of conjugate complex zero: $+180°\cdot \mu \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• Couple of conjugate complex pole: $-180°\cdot \mu \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity

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