1. G(s) Input (sinusoid) Time Output Ti me InputOutput A linear, time-invariant single input and single output (SISO) system. The input to this system is a sinusoid signal.

2. Consider a system. Calculate the ratio of output to the input for each input frequency. Go on increasing the frequency from zero. We have now many gain values corresponding to each frequency. If we plot this curve, i.e., frequency versus gain curve, then the resulting plot called frequency response curve which provides much information about the system.

3. frequency Maximum gain -3dB (c) frequency Maximum gain -3dB (d) Maximum gain -3dB (e) Gain Frequency Maximum gain -3dB Gain Frequency Maximum gain -3dB (a) (b) Gain Typical Frequency response curves of various systems

4. First Order System Input Output The transfer function describes the internal properties of the systems. It is essentially a ratio of output to the input of the system, however, from engineering point of view it is preferred to define the transfer function as a ratio of Laplace transform of the output to the Laplace transform of the input, assuming initial conditions are zero. Figure shows the schematic diagram of a first order system as far as transfer function is concerned.

5. First Order System Input Output A Schematic diagram of a second order system showing transfer function

6. First order system Input (sinusoid) Time Output Ti me InputOutput In a typical first order system the output lags input frequency response can be written by substituting in place of the complex variable ‘s’ in the transfer function.

7. 10 0 -10 -20 -30 Gain in dB 0.1 1 10 100 Frequency in radians per second 45 0 -45 -90 -135 Phase angle in degrees 0.1 1 10 100 Frequency in radians per second (a) (b) Actual curve Asymptotes A Bode diagram, helps to quantify how well the output, follows the input by showing the relationships between input and the output. Bode's method consists of plotting two curves. The gain and phase versus frequency respectively. Typical Bode Diagram of a first order system; (a) Frequency versus gain curve; (b) Frequency versus phase angle curve

8. 20 10 -0 -10 -20 -30 -40 -50 -60 Gain in dB 1 2 4 3 0.05 0.1 1 10 Normalised frequency Asymptotes Typical Bode Diagram of a first order system; Frequency versus gain curve; (b) Frequency versus phase angle curve

9. 90 45 -0 --45 --90 --135 -180 -50 -60 Phase angle in degrees 0.05 0.1 1 10 Normalised frequency 1 2 3 4 Typical Bode Diagram of a first order system; Frequency versus phase angle curve

10. Real part Imaginary part S-plane -4 -2 Pole-Zero plots are another way of analysing the system. Poles are defined as the complex frequencies that make the overall gain of the system to infinite. Zeros are complex frequencies that make the overall gain of the transfer function zero. The pole-zero locations offer much information about the system.