Frequency Domain Analysis | Frequency Response

By frequency domain analysis, we mean the steady state response of a system when subjected to a sinusoidal input. In frequency domain methods, the frequency of the input signal is varied over a certain range and study the resulting response.

This is shown in Figure 1.

frequency response analysis, Frequency Domain Analysis in control system
Figure 1: System subjected to different frequencies

Frequency domain analysis methods, developed by Nyquist and Bode in the 1930’s are older than the root locus method which was discovered by Evans in 1948.

Frequency domain analysis method yields a new way of analysing feedback control systems.

Consider a linear system shown in Figure 2. In frequency domain analysis, as stated earlier, we subject this linear system to a sinusoidal input.

r (t) = A sin (ωt)

Under steady state, the output of the system c (t) will be of the form

c (t) = B sin (ωt +φ)

Frequency Domain Analysis | Frequency Response
Figure 2: Output of the system for sinusoidal input

In other words, when a sinusoidal input is fed to a linear system, the output of the system is also a sinusoidal signal of the same frequency but different in magnitude and phase.

The magnitude and phase relationship between the sinusoidal input and the steady state output is known as the frequency domain.

Methods Used in Frequency Domain Analysis

The frequency domain characteristics have two plots:

  • Magnitude function
  • Phase function.

In both these functions the variable is w, the angular frequency. How we plot the magnitude and phase values is what differentiates one frequency domain technique from the other.

The following techniques are used widely:

Bode Plots: There are two separate plots: (i) Magnitude (ii) Phase, both versus frequencyωin logarithmic value.

Polar Plots and Nyquist plots: It is a single plot of magnitude versus phase angle asωis varied from zero to infinity in polar co-ordinates.

Advantages of Frequency Domain Analysis

  1. It is easy to get a frequency domain test in a laboratory with good accuracy. This is useful if transfer function is difficult to be obtained by analytical techniques.
  2. Design of open loop transfer function for specified closed loop performance is easier in frequency domain than time domain.
  3. Effects of noise disturbance and parameter variations are easy to visualize in frequency domain.
  4. Frequency domain tests are simple to perform and can be performed accurately using signal generators and accurate / precise measuring instruments.
  5. The apparatus required for measuring frequency domain are easy to use, simple and inexpensive.
  6. Those systems which do not have rational transfer function, frequency domain can be precisely applied to them also.

The relations between time and frequency domain performance are indirect. Once these relations are understood, frequency domain may be adopted with advantage.

Time Domain and Frequency Domain Analysis

  • In time domain all variables are plotted with respect to time. The output can be easily plotted on Y-axis with time on X-axis.
  • In frequency domain, the independent variable is frequency. In this, the frequency is varied and the behaviour of the output is analysed. Here the X-axis has frequency and Y-axis has the output.
  • Control system have inputs which are usually a sudden change (step), acceleration, impulse, etc. No system is applied a sine wave as it’s input. Still frequency domain analysis is used.
  • This is because:
    1. The response and specification of frequency domain can be related to time domain response and specifications.
    2. Understanding how a system responds at different frequencies also give us information about the system stability.

Disadvantages of Frequency Domain Methods

  1. To obtain frequency domain practically, it consumes fair amount of time.
  2. The methods which are considered for obtaining frequency domain are outdated as compared to extensive methods developed for modelling and digital computer simulation.
  3. Frequency domain methods are basically applied to linear systems.

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