Bode Plot.md

• Source: Internet Page | [[Rules for Drawing Bode Diagrams 1.pdf|Downladed PDF]]

• Amplitude:

  • Real zero: $+20 * \text{zero's multiplicity} \kern3px {\text{dB}\over\text{decade}}$
  • Real pole: $-20 * \text{pole's multiplicity} \kern3px {\text{dB}\over\text{decade}}$
  • Imaginary zero:
    • $+40 * \text{zero's multiplicity} \kern3px {\text{dB}\over\text{decade}}$
    • Peak at $w = w_0 \sqrt{1 - 2 \zeta^2}$
  • Imaginary pole:
    • $-40 * \text{pole's multiplicity} \kern3px {\text{dB}\over\text{decade}}$
    • Peak at $w = w_0 \sqrt{1 - 2 \zeta^2}$

• Phase:

  • Starting Phase*:
    • $+0°$ if $K \gt 0$ ($K$ is the gain)
    • $\pm180°$ if $K \lt 0$ ($K$ is the gain)
  • Real zero: $+90° * \text{zero's multiplicity}$
  • Real pole: $-90° * \text{zero's multiplicity}$
  • Imaginary zero: $+180° * \text{zero's multiplicity}$
  • Imaginary pole: $-180° * \text{pole's multiplicity}$

• 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 $+20 \cdot \mu \kern5px \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• A real pole means $-20 \cdot \mu \kern5px \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• An imaginary zero
  • means $+40 \cdot \mu \kern5px \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
  • In the case of an imaginary zero there is a also peak is at $w = w_0 \sqrt{1 - 2 \kern2px \zeta^2}$
• An imaginary pole:
  • means $+40 \cdot \mu \kern5px \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
  • There is a also peak is at $w = w_0 \sqrt{1 - 2 \kern2px \zeta^2}$
  • ~Ex.:
    ![[Pasted image 20230418162842.png]]

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Phase

• Starting Phase*:
  • $+0°$ if $K \gt 0$ ($K$ is the gain)
  • $\pm180°$ if $K \lt 0$ ($K$ is the gain)
• Real zero: $+90° \cdot \mu \kern5px \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• Real pole: $-90° \cdot \mu \kern5px \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• Couple of conjugate complex zero: $+180° \cdot \mu \kern5px \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity
• Couple of conjugate complex pole: $-180° \cdot \mu \kern5px \frac{\text{dB}}{\text{dec}}$, where $\mu$ is its multiplicity

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