Basic Concepts of Frequency Response | Control Systems - Electrical Engineering (EE) PDF Download

Introduction

The frequency response of a linear time-invariant (LTI) system describes how the system responds, in steady state, to sinusoidal inputs whose frequency is varied over a specified range. The input frequency is varied and the corresponding steady-state output amplitude and phase are recorded. The frequency response is normally expressed as a complex function T(jω) (or G(jω)), and it consists of two parts: the magnitude function and the phase function. Frequency response can be obtained for both open-loop and closed-loop systems.

• The transfer function evaluated on the imaginary axis is written as G(jω), where s = jω.
• The magnitude is written as |G(jω)| and the phase (angle) as ∠G(jω).
• When the system is driven by a sinusoid of frequency ω, the steady-state output is also sinusoidal at the same frequency; only the amplitude and phase change compared with the input.
• For example, for a first-order transfer function in time-constant form, G(s) = 1/(Ts + 1), substitution s = jω gives G(jω) = 1/(1 + jωT) (for the unity-gain case).
• More generally, with a DC gain K, the first-order form is G(s) = K/(1 + Ts) and hence G(jω) = K/(1 + jωT) on the imaginary axis.
• For the general first-order case G(jω) = K/(1 + jωT):

Magnitude:

|G(jω)| = K / √(1 + ω²T²)

Phase:

∠G(jω) = -tan^-1(ωT)

Thus, for an input A sin(ωt), the steady-state output has the form

Css(t) = A · |G(jω)| · sin(ωt + ∠G(jω)).

Sinusoidal transfer function

If the input to a control system is sinusoidal, the input is called a sinusoidal input, and the system's steady-state response is a sinusoidal response. The sinusoidal transfer function (often called the frequency response function) is defined as the ratio of the complex amplitude of the steady-state output to the complex amplitude of the sinusoidal input when the input frequency is ω. This ratio is evaluated by replacing s with jω in the transfer function.

• The sinusoidal transfer function is denoted by T(jω) or G(jω).
• Numerically, if input is Xin(t)=A·sin(ωt) and the corresponding steady-state output is Xout(t)=B·sin(ωt+ϕ), then
• The complex ratio is G(jω) = (B∠ϕ) / (A∠0), so |G(jω)| = B/A and ∠G(jω) = ϕ.
• When represented versus frequency ω, the sinusoidal transfer function gives direct information on gain (amplitude ratio) and phase shift introduced by the system at each frequency.

Advantages of frequency-response methods

The frequency-response approach offers several practical advantages for analysis and design of control systems. Key advantages are:

• Calculations and graphical techniques (such as Bode plots) are often simple and intuitive to apply.
• Design and tuning of controllers using frequency methods is straightforward and widely used in practice.
• Stability margins (gain margin and phase margin) and robustness can be obtained directly from frequency plots without computing time-domain solutions.
• In many cases the frequency response of a system can be measured experimentally without explicit knowledge of a detailed mathematical model.
• Frequency methods can be applied to systems with certain kinds of non-idealities, time delays or irrational transfer elements (for example, e^-2Ts).
• Frequency methods allow easier analysis of the effect of high-frequency noise and disturbance rejection properties.
• Apparatus required for experimental frequency response measurement is often simple and inexpensive.
• For complex control problems, the Nyquist plot (and related methods) provide a comprehensive route to determine closed-loop stability from open-loop data.

Disadvantages of frequency-response methods

Despite their strengths, frequency methods have limitations and are not always the most convenient approach:

• They are best suited to linear time-invariant systems; for strongly nonlinear systems the interpretation of frequency response may be misleading.
• Practical measurement of frequency response over a wide band can be time-consuming.
• Although there is a relationship between frequency response and time-domain responses (via Fourier or Laplace transforms), constructing exact time-domain behaviour from frequency data can require complex integrals and is not always easy.

Frequency-response plots

Frequency-response analysis is commonly carried out using graphical methods. The main graphical techniques are listed below and described briefly.

• Bode plot: Two separate plots versus logarithmic frequency: magnitude (in decibels) and phase (in degrees). Bode plots are convenient for controller design, show asymptotic behaviour clearly, and make it easy to read gain and phase margins, bandwidth and resonant peaks.
• Polar plot and Nyquist plot: A polar plot shows the complex value G(jω) as a locus in the complex plane as ω varies from 0 to ∞. The Nyquist plot is the extension that typically traces from ω = -∞ to +∞ (or uses the mapped contour) and is used directly in Nyquist stability criterion to determine closed-loop stability from open-loop data.
• M and N circles: These are graphical tools drawn on the Nyquist (or polar) diagram to obtain closed-loop magnitude and phase values from the open-loop plot. The technique was introduced by Albert C. Hall and is useful in design: intersections of the Nyquist locus with M and N circles give closed-loop gain and phase for feedback systems.
• Nichols chart: A plot of open-loop gain (in dB) versus phase (in degrees) on a single chart. Nichols charts include contours of constant closed-loop gain and phase (equivalent to M and N circles) and are convenient for loop-shaping design, showing how changes in open-loop gain or phase affect closed-loop performance and stability.
• Nichols plot: The Nichols plot (the actual trace of open-loop gain versus phase) is used with the Nichols chart to evaluate robustness and stability margins, and to design controllers that give the desired closed-loop response.

How plots relate to closed-loop behaviour

Given an open-loop transfer function G(jω)H(jω) (where H(jω) is the feedback path), frequency plots enable direct evaluation of closed-loop characteristics:

• Gain margin and phase margin indicate how much gain or phase can vary before the closed loop becomes unstable.
• Bandwidth from the magnitude plot approximates the frequency range where the closed loop has adequate gain for disturbance rejection and reference tracking.
• Resonant peak and resonant frequency found from the magnitude plot relate to overshoot and transient peaking in the time domain.

Worked example: steady-state sinusoidal response of a first-order system

Consider the first-order transfer function with DC gain K and time constant T:

G(s) = K / (1 + Ts)
Replace s by jω to get the frequency response:

G(jω) = K / (1 + jωT)
Compute the magnitude and phase:

Magnitude:
|G(jω)| = K / √(1 + ω²T²)

Phase:
∠G(jω) = -tan^-1(ωT)
If the input is xin(t) = A sin(ωt), the steady-state output is
css(t) = A · |G(jω)| · sin(ωt + ∠G(jω))
Substituting the magnitude and phase for the first-order system gives
css(t) = A · K · sin(ωt - tan^-1(ωT)) / √(1 + ω²T²)

Summary

Frequency-response methods provide a powerful and practical set of tools to analyse and design linear control systems. By evaluating the transfer function on the imaginary axis-G(jω)-we obtain the gain and phase at each frequency and can use graphical methods (Bode, Nyquist, Nichols, M and N circles) to assess stability, robustness and closed-loop performance. The first-order example illustrates how magnitude and phase combine to give the steady-state sinusoidal output for a given input frequency.