Bode Plot of First Order System

The Bode plot of a first-order system is the foundational building block for understanding Bode analysis of any general transfer function. A first-order system has exactly one pole (or one zero), and its frequency response behavior sets the pattern that repeats in more complex systems. Understanding the magnitude rolloff, phase behavior, and corner frequency of a first-order system is essential for GATE and for practical control design.

Core Concept Explanation

A first-order system has a transfer function of the form G(s) = 1 / (1 + sT), where T is the time constant in seconds. The single pole of this system is located at s = -1/T on the real axis. In the frequency domain, substituting s = j omega gives G(j omega) = 1 / (1 + j omega T). This is a complex number whose magnitude decreases and whose phase lags as frequency increases.

The corner frequency, also called the break frequency, is defined as omega_c = 1/T radians per second. Below this frequency, the magnitude is approximately 1 (0 dB) and the phase is approximately 0 degrees. The system behaves like a pure gain at low frequencies, passing signals with no significant distortion. Above the corner frequency, the magnitude rolls off at -20 dB per decade and the phase approaches -90 degrees asymptotically.

The asymptotic Bode approximation for this system uses two straight lines for the magnitude: a flat line at 0 dB for frequencies below omega_c, and a line with slope -20 dB/decade for frequencies above omega_c. These two lines meet exactly at the corner frequency. The actual magnitude at the corner frequency is 1/sqrt(2) = 0.707, which corresponds to -3 dB, a well-known correction that should be applied at omega_c for precise results.

For the phase plot, the asymptotic approximation uses three line segments: a flat line at 0 degrees for omega below 0.1 * omega_c, a line decreasing from 0 to -90 degrees between 0.1 * omega_c and 10 * omega_c (spanning two decades), and a flat line at -90 degrees above 10 * omega_c. The actual phase at omega_c is exactly -45 degrees, which is a key value to remember.

Mathematical Expression

For G(s) = 1/(1 + sT), the frequency response is G(j omega) = 1 / (1 + j omega T). The magnitude and phase are:

• Magnitude: |G(j omega)| = 1 / sqrt(1 + omega squared T squared)
• Magnitude in dB: = -10 log(1 + omega squared T squared) = -20 log sqrt(1 + omega squared T squared)
• Phase: angle G(j omega) = - arctan(omega T) in degrees
• At omega = omega_c = 1/T: magnitude = 1/sqrt(2) = -3.01 dB, phase = -45 degrees

The slope of -20 dB per decade in the magnitude plot directly follows from the mathematical form. For omega much greater than omega_c, the term 1 inside the square root becomes negligible compared to (omega T) squared, so the magnitude approximates to 1/(omega T). A tenfold increase in frequency (one decade) gives a tenfold decrease in magnitude, which is exactly -20 dB.

Practical Understanding

The first-order system model applies to a wide range of physical systems: an RC low-pass filter has a time constant T = RC; a single-pole operational amplifier has a dominant pole that causes first-order rolloff; a thermal system approximates a first-order response. In each case, the corner frequency represents the boundary between the frequency region where the system responds well and the region where it attenuates signals.

In control systems, the first-order pole from a plant or controller introduces a known phase lag. When designing a feedback controller, the phase lag contributed by each pole must be accounted for to ensure sufficient phase margin. Recognizing the -20 dB/dec slope and -90 degree phase from a first-order pole is the starting point for all loop shaping procedures.

Given:
G(s) = 1 / (1 + 0.1s)
Time constant T = 0.1 s, Corner frequency omega_c = 1/0.1 = 10 rad/s
Find: magnitude and phase at omega = 10 rad/s and omega = 100 rad/s

Why this formula applies:
|G(j*omega)| = 1/sqrt(1 + (omega*T)^2), phase = -arctan(omega*T)

Formula:
|G(j*omega)| = 1 / sqrt(1 + omega^2 * T^2)
Angle = -arctan(omega * T)

Substitution at omega = 10:
|G(j10)| = 1 / sqrt(1 + (10*0.1)^2) = 1/sqrt(1+1) = 1/sqrt(2)
Phase = -arctan(10*0.1) = -arctan(1) = -45 degrees

Substitution at omega = 100:
|G(j100)| = 1/sqrt(1+(100*0.1)^2) = 1/sqrt(101) = 0.0995
Magnitude in dB = 20*log(0.0995) = -20.04 dB
Phase = -arctan(100*0.1) = -arctan(10) = -84.3 degrees

Final Answer:
At omega=10 rad/s: |G| = 0.707 (-3dB), Phase = -45 degrees
At omega=100 rad/s: |G| = 0.0995 (-20dB approx), Phase = -84.3 degrees

Exam Tip: For a first-order system G(s) = K/(1+sT), the DC gain is K (not 1), which shifts the entire magnitude plot up by 20*log(K) dB. The corner frequency remains 1/T regardless of K. A common GATE mistake is treating K as if it changes the corner frequency.

Quick Revision

• First-order transfer function: G(s) = 1/(1+sT) with pole at s = -1/T.
• Corner frequency: omega_c = 1/T rad/s — this is the break point on the Bode plot.
• Magnitude: flat at 0 dB below omega_c; rolls off at -20 dB/decade above omega_c.
• At corner frequency: actual magnitude = -3 dB (0.707); asymptotic value = 0 dB (error = 3 dB).
• Phase: 0 degrees at low frequency; -45 degrees at omega_c; -90 degrees at high frequency.
• Phase transitions from 0 to -90 degrees over a span of two decades (0.1*omega_c to 10*omega_c).
• Exam trap: The 3 dB error at the corner frequency is the most common asymptotic Bode correction tested in GATE.