Frequency Response Concept

The frequency response of a control system describes how the system responds to sinusoidal inputs at different frequencies. It is one of the most important analytical tools in control engineering because it reveals stability margins, bandwidth, and disturbance rejection capability directly from the open-loop transfer function without solving the full time-domain differential equations.

Core Concept Explanation

When a sinusoidal signal is applied to a linear time-invariant (LTI) system that has reached steady state, the output is also sinusoidal at the same frequency. However, the output differs from the input in two ways: its amplitude is scaled by a factor and its phase is shifted by an angle. Both the scaling factor and the phase shift depend on the frequency of the input. This frequency-dependent behavior is what is collectively called the frequency response of the system.

To compute the frequency response, the Laplace-domain transfer function G(s) is evaluated on the imaginary axis by substituting s = j times omega, where omega is the angular frequency in radians per second. The resulting complex function G(j omega) has a magnitude and an angle at each frequency. The magnitude |G(j omega)| gives the ratio of output amplitude to input amplitude, and the angle of G(j omega) gives the phase lead or lag of the output with respect to the input.

This substitution is valid because a stable LTI system excited by sin(omega t) produces a steady-state output equal to |G(j omega)| times sin(omega t + angle of G(j omega)). The transient part of the response, arising from initial conditions and poles of G(s), decays to zero for stable systems, leaving only this sinusoidal steady state.

The frequency response provides a complete picture of system behavior across all frequencies. At low frequencies, many systems pass signals with unit gain and zero phase shift. At high frequencies, practical systems roll off in magnitude due to the dominance of poles, and phase increasingly lags behind the input. The frequency at which magnitude drops to 0.707 times the DC value (that is, 3 dB below) is called the bandwidth of the system.

Mathematical Expression

For a transfer function G(s), the frequency response is obtained by substituting s = j omega. The result is a complex number written as G(j omega) = M(omega) times e raised to the power j phi(omega), where M is the magnitude and phi is the phase. In polar form, this is expressed as:

• Magnitude: |G(j omega)| = square root of [real part squared plus imaginary part squared]
• Phase: angle of G(j omega) = arctan (imaginary part / real part)
• In decibels: magnitude in dB = 20 log base 10 of |G(j omega)|

For a first-order system G(s) = 1 / (1 + s tau), the magnitude is 1 / sqrt(1 + omega squared tau squared) and the phase is minus arctan(omega tau). At omega = 1/tau, the magnitude drops to 1/sqrt(2) = 0.707, confirming this as the corner frequency or break frequency of the system.

Practical Understanding

Frequency response analysis allows engineers to design controllers without solving differential equations. By examining the magnitude and phase of the open-loop transfer function across frequencies, one can determine gain margin and phase margin, which are direct indicators of how close a feedback system is to instability. A system with large gain margin and phase margin tolerates significant parameter variations before becoming unstable.

In communication systems, frequency response is used to characterize filters, amplifiers, and transmission channels. In power electronics, it determines how well a converter tracks a reference signal across operating frequencies. The concept of frequency response therefore spans not just control theory but all areas of electronics and signal processing.

Given:
Transfer function: G(s) = 10 / (s + 5)
Frequency: omega = 5 rad/s

Why this formula applies:
Substitute s = j*omega to find steady-state sinusoidal response.

Formula:
G(j*omega) = 10 / (j*omega + 5)

Substitution:
G(j5) = 10 / (j5 + 5) = 10 / (5 + j5)

Calculation:
|G(j5)| = 10 / sqrt(5^2 + 5^2) = 10 / sqrt(50) = 10 / 7.07 = 1.414
Angle of G(j5) = 0 - arctan(5/5) = -arctan(1) = -45 degrees

Final Answer:
Magnitude = 1.414 (approx 3 dB above unity)
Phase = -45 degrees
Output is: 1.414 * A * sin(5t - 45 degrees) for input A*sin(5t)

Exam Tip: To find frequency response from G(s), simply replace s with j*omega. Then compute magnitude as |numerator|/|denominator| and phase as angle(numerator) - angle(denominator). This direct substitution method is faster than separating real and imaginary parts for simple transfer functions.

Mechanism: How Frequency Response Characterizes a System

• At very low frequencies, the magnitude approaches the DC gain of the system and the phase is approximately zero for minimum-phase systems.
• At the bandwidth frequency, magnitude falls to 1/sqrt(2) times the DC gain, corresponding to -3 dB in the logarithmic scale.
• Each pole in the transfer function contributes a -20 dB/decade slope in the magnitude plot and a -90 degree phase shift at frequencies well above the pole location.
• Gain margin is measured in dB at the phase crossover frequency (where phase equals -180 degrees), and phase margin is the additional phase needed to reach -180 degrees at the gain crossover frequency.
• A higher bandwidth generally means faster transient response, but also greater susceptibility to high-frequency noise.

Quick Revision

• Frequency response: evaluate G(s) at s = j*omega to get magnitude and phase at each frequency.
• Magnitude = |G(j omega)|; Phase = angle of G(j omega) in degrees.
• Bandwidth: frequency where magnitude drops to 0.707 (= -3 dB) of DC value.
• Each pole adds -20 dB/dec to slope and -90 degrees to phase at high frequency.
• Gain margin and phase margin are read from the frequency response and determine closed-loop stability robustness.
• Exam trap: Phase angle calculation — remember arctan is negative for lag, meaning the output lags the input for poles.
• Frequency response is only valid for LTI stable systems in their sinusoidal steady state.