What is the Bode amplitude plot of a first-order lowpass system?

Explanation:

Bode Amplitude Plot for a First-Order Lowpass System

Definition: A first-order lowpass system is a type of linear system that allows low-frequency signals to pass through while attenuating high-frequency signals. The system is characterized by a single pole in its transfer function, and the Bode plot is used to graphically represent the frequency response of the system.

The Bode amplitude plot typically consists of two regions:

• A flat region at low frequencies (where the system behaves as a constant gain).
• A sloped region at high frequencies (where the system attenuates the signal).

Working Principle: The transfer function of a first-order lowpass system can be expressed mathematically as:

H(s) = K / (1 + s/ω_c)

Where:

• K is the DC gain of the system.
• s is the Laplace variable.
• ω_c is the cutoff angular frequency of the system.

The frequency response of this transfer function can be evaluated by substituting s = jω, where ω is the frequency in radians per second. The magnitude of the frequency response is given by:

|H(jω)| = K / √(1 + (ω/ω_c)²)

Bode Amplitude Plot Analysis:

• Low Frequencies: When ω << ω_c, the term (ω/ω_c)² becomes negligible, and the magnitude of the frequency response approximates to |H(jω)| ≈ K. On the Bode plot, this appears as a horizontal line parallel to the frequency axis with a constant gain in decibels.
• High Frequencies: When ω >> ω_c, the term (ω/ω_c)² dominates, and the magnitude of the frequency response approximates to |H(jω)| ≈ K / (ω/ω_c). In decibels, this corresponds to a slope of -20 dB/decade, indicating that the amplitude decreases by 20 dB for every tenfold increase in frequency.

Correct Option Analysis:

The correct option is:

Option 2: Line with slope -20 dB/decade.

This option correctly describes the behavior of the Bode amplitude plot of a first-order lowpass system. At frequencies much greater than the cutoff frequency, the amplitude decreases at a rate of -20 dB/decade, which is characteristic of a first-order system.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Line with slope -40 dB/decade.

This description is incorrect. A slope of -40 dB/decade is characteristic of a second-order system, not a first-order system. In a second-order system, the amplitude decreases at a faster rate (two poles contribute to the attenuation), whereas in a first-order system, the slope is limited to -20 dB/decade.

Option 3: Line with slope +20 dB/decade.

This option is incorrect because it represents a system where the amplitude increases with frequency, which is not the behavior of a first-order lowpass system. A first-order lowpass system attenuates high-frequency signals rather than amplifying them.

Option 4: Straight line parallel to the frequency axis.

This option is partially correct but incomplete. While the amplitude plot of a first-order lowpass system is indeed a straight line parallel to the frequency axis at low frequencies, this description fails to account for the sloped region at higher frequencies where the amplitude decreases at a rate of -20 dB/decade.

Option 5: None of the above.

This option is incorrect because Option 2 accurately describes the behavior of the Bode amplitude plot for a first-order lowpass system.

Conclusion:

Understanding the frequency response and Bode plot characteristics of a first-order lowpass system is crucial for analyzing its behavior in various applications. The amplitude plot consists of a flat region at low frequencies and a sloped region with a slope of -20 dB/decade at high frequencies. This characteristic distinguishes first-order systems from higher-order systems, which exhibit steeper slopes in their amplitude plots.