In a standard Bode plot used for frequency-response analysis, the phase angle Φ is plotted against angular frequency ω on what kind of graph paper or axis scaling?

Introduction / Context:
Bode plots are ubiquitous in control engineering for analyzing system stability and performance in the frequency domain. A complete Bode diagram consists of two plots: magnitude (in decibels) vs frequency and phase (in degrees) vs frequency. Understanding the axis scaling is essential to correctly interpret slopes, phase margins, and crossover frequencies.

Given Data / Assumptions:

• We consider standard Bode diagrams for linear time-invariant systems.
• Frequency axis is represented by angular frequency ω (or frequency f), spanning several decades.
• Phase Φ is plotted in degrees.

Concept / Approach:
In Bode plots, the horizontal axis (frequency) is logarithmic to accommodate multiple decades conveniently. Both the magnitude and phase plots share this log frequency axis. The vertical axis is linear (decibels for magnitude, degrees for phase). Hence, the phase plot is made on semi-log axis: log scale horizontally and linear scale vertically.

Step-by-Step Solution:
Identify axes: horizontal = frequency ω, vertical = Φ (degrees).Apply Bode convention: frequency axis is log-scaled to compress decades.Vertical axis for phase is linear; thus the overall paper is semi-log.

Verification / Alternative check:
Typical plotting software and textbooks show phase curves (e.g., arctangent-type for first-order systems) laid out over log-spaced ω to clearly display low-, mid-, and high-frequency asymptotes and transitions—confirming semi-log usage.

Why Other Options Are Wrong:
Log-log: Used for magnitude (in some contexts like Nyquist slopes) but the phase ordinate is not logarithmic.Ordinary (linear-linear): Would not compress frequency decades and is not standard for Bode plots.Triangular: Not a recognized plotting format in control theory.

Common Pitfalls:

• Confusing semi-log with log-log due to the decibel axis on the magnitude plot; decibels are linear in the vertical direction.
• Plotting frequency linearly, which obscures asymptotic behavior across decades.
• Mixing up Bode (semi-log) with Nichols or Nyquist which have different representations.

Final Answer:
semi-log