III.5 Frequency Response Analysis 
When a linear system is subjected to sinusoidal input perturbation, its ultimate response after a long time also becomes a sinusoidal wave, however with different amplitude and a phase shift. This characteristic constitutes the basis of frequency response analysis. One needs to study how the amplitude and phase shift change with the frequency of the input perturbation.
III.5.1 Response of a First-Order System to a Sinusoidal Input 
Consider a simple first order process, 
(III.125) 
Let the sinusoidal input u(t) = A sin ωt perturb the system. Then the output of the process will be 
(III.126) 
Computing the constants and  and taking inverse Laplace Transform of the above equation we obtain, 
(III.127) 
After sufficiently long time , the first term disappears as  Hence, using the identity eq. (III.91) we obtain,


(III.128) 
Hence we observe that 
•  Sinusoidal output wave has the same frequency as that of input sinusoid 
•  Amplitude Ratio between the output wave and input wave is 
•  Output wave lags behind the input wave with a phase difference of 
Fig. III.13 shows the Input and Output wave profile for a frequency response analysis. 
Fig. III.13: Input and Output wave profile for a frequency response analysis


III.5.2 Complex Plane and Frequency Response Analysis 
Consider a complex number 
Fig. III.14: Complex Plane of number 
The modulus (or absolute value or magnitude) of W is  and the argument (or phase angle) is . Let us put  in the transfer function of the first order process as 
(III.129) 
As  is now a complex number, the modulus and argument can be calculated as, 


The last two relationships indicate the amplitude ratio and phase lag for the ultimate response of the first order process. Hence the observations in the last subsection can also be stated in the light of the above results as follows: 
•  Sinusoidal output wave has the same frequency as that of input sinusoid 
•  Amplitude Ratio between the output wave and input wave is 
•  Output wave lags behind the input wave with a phase difference of 
III.5.3 Example of frequency response of a second order system 
The process is 
(III.131) 
Put and  calculate 
(III.132) 
Then, 
(III.133) 
and, 


III.6 Bode Diagram 
The Bode diagrams are a convinient way of representing the frequency response characteristics of a system. A Bode diagram consists of a pair of plots that show how the amplitude ratio and phase shift vary with frequency of the signal that perturbs a process. In order to cover a large range of frequency, a logarithmic scale of representation is employed.
III.6.1 Bode Diagram of a first order process 
The first order process has the following characteristics:
(III.135) 
The Bode Diagram of a first order process is shown in the following figure.
Fig. III.15: Bode Diagram of a first order process
(III.134) 
Expressing AR with logarithmic representation, 
(III.136) 
We encounter the following two terminal situations: 
•  As  and hence . That means at very low frequency, the profile of AR approaches a constant value (equal to gain K of the system) with slope zero. This is termed as low frequency asymptote . 
•  As  and hence . That means at very high frequency, the profile of AR approaches a value which is inverse of the frequency. This is termed as high frequency asymptote . 
•  The two asymptotes meet at a point where . The frequency  is termed as corner frequency . 
•  The profile of amplitude ratio transits from one asymptote to the other and the deviation of the true value of AR is maximum from its asymptote(s) at the corner frequency. 
The profile of phase shift can be analyzed in the similar manner: 
•  As  then 
•  As  then 
•  As  then 
Note that the gain does not have any effect on the phase shift.



III.6.2 Bode Diagram of a second order process 
The second order process has the following characteristics: 
(III.137) 
The Bode Diagram of a second order process is shown in the following figure. 
Fig. III.16: Bode Diagram of a second order process 

Expressing with logarithmic representation, 
(III.138) 

We encounter the following two terminal situations: 
•  As  and hence . That means at very low frequency, the profile of AR approaches a constant value with slope zero. This is low frequency asymptote . 
•As  and hence
 .That means at very high frequency, the profile of approaches a value which is inverse of the square of the frequency. This is high frequency asymptote . 
•  The two asymptotes meet at corner frequency 
•  The profile of amplitude ratio will have three different shapes that depends upon the value of the damping coefficient. It is understood that
•  For overdamped process ,
where  is a non-negative quantity. Hence,  or  is always less than one. 
•  For critically damped process , . Hence  is always less than 1 
•  For underdamped process , . Hence there will be some values of frequency where  is greater than 1 
The profile of phase shift can be analyzed in the similar manner: 
•  As  then 
•  As  then 
•  As  then 
Note that the phase shift leaps by a full  as the order of the process increases by one.

III.6.3 Bode Diagram of pure time delay 
It is worth to study the frequency response characteristics of pure time delay  that has  and . The following figure shows the Bode Diagram of pure time delay. 
Fig. III.17: Bode Diagram of pure time delay process 

Note that  as .

III.6.4 Bode Diagram for processes in series Let  and  are two processes in series. Let signals  and  be the input and output of the process  and  signals  and  be the input and output of the process .(see the following figure). 
Fig. III.18: Processes in series
The amplitude ratio of  is  and the amplitude ratio of  is . The overall amplitude ratio of the combined process  should be . Hence,
(III.141) 
or,
(III.142) 
Similarly the overall phase shift for the combined process is
(III.143) 
The processes in series can be studied individually and their individual characteristics can be combined as per the equations above. Note that the individual processes will have their own asymptotes along with the corresponding corner frequencies.

III.7 Nyquist Plot 
A Nyquist plot is an alternative way of representing the frequency response characteristics. It uses Cartesian coordinates in two dimensions whose ordinate represents the imaginary axis and abscissa represents the real axis. A specific value of frequency defines a point on the table with as ordinate and  as abscissa. The distance vector of the point from origin has the magnitude  which is equal to the amplitude ratio and the angle set by this vector with the real axis is phase shift. The following figure is an example of Nyquist plot for first order process.
Fig. III.19: Nyquist plot of first order process