Question

Bode diagram consists of two graphs:

• A. \(\log(AR)\) versus \(\log(\omega)\) and phase angle (\(\phi\)) versus \(\log(\omega)\)
• B. \(\log(AR)\) versus \(\omega\) and \(\log(\phi)\) versus \(\omega\)
• C. \(\log(\omega)\) versus \(AR\) and \(\phi\) versus \(\log(\omega)\)
• D. \(\omega\) versus \(AR\) and \(\phi\) versus \(\omega\)

Hint:

In Bode magnitude plots, the y-axis can be expressed in Decibels (dB):
\[ \text{Decibels (dB)} = 20 \log_{10}(AR) \].
Using decibels turns the log-log magnitude plot into a semi-log plot where both graphs share a common linear-log format.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks for the standard graphical representation of a Bode diagram (or Bode plot).
Bode diagrams are used to analyze the frequency response of linear time-invariant (LTI) systems by showing how amplitude and phase change across a range of frequencies.

Step 2: Key Formula or Approach:
A Bode diagram consists of two separate plots:
1. Magnitude Plot: Plotted on a log-log scale. The y-axis represents the logarithm of the Amplitude Ratio ($\log(AR)$) or magnitude in decibels ($20 \log_{10}(AR)$), and the x-axis represents the logarithm of the frequency ($\log(\omega)$).
2. Phase Plot: Plotted on a semi-log scale. The y-axis represents the phase angle $\phi$ (linearly in degrees or radians), and the x-axis represents the logarithm of the frequency ($\log(\omega)$).

Step 3: Detailed Explanation:
Let us examine why these specific logarithmic scales are chosen:
1. Frequency Scale ($\log(\omega)$): Frequencies of interest in control systems often span several orders of magnitude (e.g., from $0.01$ rad/s to $100$ rad/s). A logarithmic scale allows a wide range of frequencies to be clearly displayed on a single graph.
2. Amplitude Scale ($\log(AR)$): Using logarithms converts multiplication of transfer function magnitudes into simple addition:
\[ |G_1(j\omega) \cdot G_2(j\omega)| = |G_1(j\omega)| \cdot |G_2(j\omega)| \implies \log |G_1 \cdot G_2| = \log |G_1| + \log |G_2| \] This makes it very easy to construct composite Bode plots of complex systems by simply adding individual component curves.
3. Phase Scale ($\phi$): Phase angles add directly: \[ \angle (G_1 \cdot G_2) = \angle G_1 + \angle G_2 \] So phase is plotted linearly against the logarithmic frequency $\log(\omega)$.
This matches option (A) perfectly.

Step 4: Final Answer
Thus, the Bode diagram consists of $\log(AR)$ vs $\log(\omega)$ and phase angle $\phi$ vs $\log(\omega)$, corresponding to option (A).