How to draw Bode plot

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Explanation of Bode Plot

A Bode plot is a graphical representation of a linear, time-invariant system's frequency response. Developed by Hendrik Wade Bode, it is widely used in electrical engineering and control theory. The Bode plot actually consists of two plots:
  1. Magnitude Plot: This plot shows how the magnitude (or amplitude) of the system's output signal varies with frequency. It is typically presented in decibels (dB).
  1. Phase Plot: This plot shows how the phase of the system's output signal varies with frequency, typically measured in degrees.

Key Features of Bode Plots:

  1. Logarithmic Scale for Frequency: The horizontal axis represents the frequency on a logarithmic scale, usually in radians per second (rad/s) or Hertz (Hz). This allows for a wide range of frequencies to be displayed compactly.
  1. Magnitude in Decibels: The magnitude plot uses a logarithmic scale for the amplitude. The magnitude in dB is calculated as \( 20 \log_{10} |G(j\omega)| \), where \( |G(j\omega)| \) is the magnitude of the system's transfer function at a given frequency \( \omega \).
  1. Phase in Degrees: The phase plot shows the phase angle of the transfer function, \( \angle G(j\omega) \), in degrees. The phase can be positive or negative, indicating a phase lead or lag, respectively.
  1. Slope of the Magnitude Plot: The slope of the magnitude plot is particularly important. In regions where the plot is flat, the system has little effect on the signal at those frequencies. Where the plot slopes downward, the system attenuates the signal. The slope is often expressed in dB per decade (or dB per octave).
  1. Breakpoints and Asymptotes: The Bode plot often includes breakpoints corresponding to the system's poles and zeros. The plot typically follows asymptotic lines, which change slope at these breakpoints.

Uses of Bode Plots:

  • Stability Analysis: Bode plots are used to assess the stability of control systems, particularly with the use of gain and phase margins.
  • System Design: They help in designing and tuning controllers, filters, and amplifiers.
  • Frequency Response Analysis: Bode plots provide a clear picture of how a system will respond to different frequencies, which is crucial in many applications like audio engineering, telecommunications, and vibration analysis.

Example:

For a simple RC low-pass filter, the Bode magnitude plot will show a constant magnitude at low frequencies, and then, beyond the cutoff frequency, the magnitude will decrease at a rate of -20 dB per decade. The phase plot will start at 0 degrees and asymptotically approach -90 degrees as frequency increases.
In summary, Bode plots are a fundamental tool in the analysis and design of linear systems, providing valuable insights into how systems behave across different frequencies.

How to Draw Bode Plot Manually

Drawing a Bode plot manually for a given transfer function involves a step-by-step process where you break down the transfer function into simpler components, analyze the magnitude and phase of each component across frequencies, and then combine these to get the overall plot. Let's go through the process using your transfer function as an example:

Step 1: Break Down the Transfer Function

Identify the individual components (poles, zeros, and gain) of the transfer function:
  1. Constant Gain: \( 10^7 \) (70 dB of gain).
  1. Zero at \( s = -10^4 \) (or \( \omega = 10^4 \) rad/s).
  1. Pole at \( s = 0 \) (integrator).
  1. Pole at \( s = -100 \) (or \( \omega = 100 \) rad/s).
  1. Second-order term: \( \frac{s^2}{10^{12}} + 2(0.2)\frac{s}{10^6} + 1 \).

Step 2: Sketch the Magnitude Plot

  1. Start with the Constant Gain: Plot a horizontal line at 70 dB.
  1. Add the Effect of the Zero: At \( \omega = 10^4 \) rad/s, the slope will increase by +20 dB/decade. Draw this slope change.
  1. Add the Effect of the Poles:
      • At \( \omega = 0 \), the slope decreases by -20 dB/decade due to the integrator.
      • At \( \omega = 100 \) rad/s, the slope decreases by another -20 dB/decade.
  1. Second-Order Term: This term will affect the plot around \( \omega = \sqrt{10^{12}} \) rad/s. If the system is underdamped, expect a resonant peak near this frequency.
  1. Combine the Slopes: Add or subtract the slopes from each other at their respective breakpoints.

Step 3: Sketch the Phase Plot

  1. Constant Gain: No phase shift.
  1. Zero at \( \omega = 10^4 \) rad/s: Phase begins to rise from 0 degrees to +90 degrees starting one decade before \( 10^4 \) rad/s and ending one decade after.
  1. Pole at \( \omega = 0 \): Phase starts at -90 degrees.
  1. Pole at \( \omega = 100 \) rad/s: Phase decreases from 0 degrees to -90 degrees starting one decade before \( 100 \) rad/s and ending one decade after.
  1. Second-Order Term: This will introduce an additional phase shift depending on the damping ratio.
  1. Combine the Phase Shifts: Add up the phase shifts from each component.

Step 4: Fine-Tuning

  • Refine the Plot: Adjust the plot for any subtle effects, especially near the breakpoints.
  • Check for Resonance: If the second-order term is underdamped, adjust the magnitude plot for any resonant peak.

Tips for Manual Drawing

  • Use Logarithmic Paper: Bode plots are easier to sketch on semi-logarithmic graph paper, where the x-axis (frequency) is logarithmic.
  • Straight-Line Approximations: For manual plotting, use straight-line approximations for each segment.
  • Accuracy: While manual plots are approximate, they can provide a good understanding of the system's behavior.
Drawing Bode plots manually is a skill that improves with practice. For complex systems, or when high accuracy is required, it's advisable to use computational tools like MATLAB or Python.

Meaning of the Increase of the Slope by +20

The increase in the slope of the magnitude plot in a Bode plot by +20 dB/decade when \( \omega = 10^4 \) rad/s is due to the presence of a zero in the transfer function at that frequency. To understand why this happens, let's consider the general behavior of zeros and poles in the context of Bode plots:

Zeros:

  • A zero in a transfer function can be represented as \( (s - z) \), where \( z \) is the zero's location in the complex plane. When \( s = j\omega \) (for Bode plot analysis), the magnitude of this term at high frequencies (far from the zero) is approximately proportional to \( \omega \).
  • The magnitude of a term like \( j\omega \) increases linearly with \( \omega \) on a logarithmic scale. This linear increase corresponds to a slope of +20 dB/decade. The reason for the "+20 dB/decade" is due to the logarithmic relationship between the magnitude in decibels and the frequency: \( 20 \log_{10}(\omega) \).

In Your Transfer Function:

  • Your transfer function has a term \( 10^{-4} s + 1 \), which introduces a zero at \( s = -10^4 \) (or \( \omega = 10^4 \) rad/s in terms of magnitude response).
  • At frequencies much lower than \( 10^4 \) rad/s, the \( 1 \) in \( 10^{-4} s + 1 \) dominates, and the zero has little effect.
  • As the frequency approaches \( 10^4 \) rad/s, the \( 10^{-4} s \) term becomes significant. Beyond \( 10^4 \) rad/s, the \( 10^{-4} s \) term dominates, and the magnitude plot starts to rise at a rate of +20 dB/decade.

Visualizing the Slope Change:

  • Before \( 10^4 \) rad/s: The slope of the magnitude plot is determined by other factors in the transfer function (other poles and zeros).
  • At \( 10^4 \) rad/s: The zero begins to influence the plot. The slope increases by +20 dB/decade compared to what it was just before \( 10^4 \) rad/s.
  • After \( 10^4 \) rad/s: The slope includes the +20 dB/decade effect of the zero.
In summary, the presence of a zero at \( \omega = 10^4 \) rad/s in your transfer function causes the magnitude plot to increase at a rate of +20 dB/decade beyond this frequency. This is standard behavior of zeros in Bode plot analysis and is a key concept in understanding how the frequency response of a system is shaped by its zeros and poles.

How to determine the slope rate of the graph

Determining the slope rate of a Bode plot graph involves understanding how poles and zeros of a transfer function affect the magnitude plot. In a Bode magnitude plot, the slope is typically measured in decibels per decade (dB/decade). Here's how you can determine it:

Rules for Determining Slope Rate:

  1. Effect of Poles and Zeros:
      • Each zero in the transfer function contributes a +20 dB/decade increase to the slope.
      • Each pole contributes a -20 dB/decade decrease to the slope.
  1. Identify the Frequency of Poles and Zeros:
      • Find the frequency at which each pole and zero occurs. This is where the slope will change.
      • For a zero at \( s = z \), the frequency is \( \omega = |z| \).
      • For a pole at \( s = p \), the frequency is \( \omega = |p| \).
  1. Calculate the Slope at Each Frequency Range:
      • Start from the lowest frequency range.
      • Initially, if there are no poles or zeros at zero frequency, the slope is 0 dB/decade.
      • As you pass each pole or zero, adjust the slope by ±20 dB/decade accordingly.

Example:

Consider your transfer function:
\[ G(s) = \frac{10^7(10^{-4} s + 1)}{s(0.01 s + 1)\left(\frac{s^2}{10^{12}} + 2(0.2)\frac{s}{10^6} + 1\right)} \]
  • Pole at \( s = 0 \) (from \( s \) in the denominator): Starts at 0 Hz (DC) and contributes -20 dB/decade.
  • Zero at \( s = -10^4 \) (from \( 10^{-4} s + 1 \)): At \( \omega = 10^4 \) rad/s, increases the slope by +20 dB/decade.
  • Pole at \( s = -100 \) (from \( 0.01 s + 1 \)): At \( \omega = 100 \) rad/s, decreases the slope by -20 dB/decade.
  • Second-order term: More complex to analyze, but generally, it will affect the slope around its natural frequency \( \omega_n = \sqrt{10^{12}} \) rad/s. The exact effect depends on the damping ratio.

Putting It Together:

  1. Start at 0 Hz: Slope is -20 dB/decade due to the integrator pole at \( s = 0 \).
  1. At \( \omega = 100 \) rad/s: Another pole decreases the slope to -40 dB/decade.
  1. At \( \omega = 10^4 \) rad/s: A zero increases the slope to -20 dB/decade.
  1. At higher frequencies: The second-order term will further influence the slope, depending on its damping ratio and natural frequency.

Note:

  • The actual Bode plot might be more complex, especially due to the second-order term. For precise analysis, especially with complex transfer functions, using computational tools like MATLAB is recommended.
  • The slope changes are typically approximated as straight lines on a logarithmic scale for simplicity, though the actual response might be curved, especially near the poles and zeros.
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