Exp No: 7 FREQUENCY RESPONSE ANALYSIS OF SYSTEM
Aim:
To obtain the frequency response for the given system using Bode plot.
Apparatus Required:
S.No Equipment Range Type Quantity
1 Function Generator - - 1
2 Digital Storage Oscilloscope - - 1
3 Resistor 10KΏ - 1
4 Capacitor 100µF - 1
Theory:
Frequency response analysis is the technique that is used to analyze how a system
responds to a sinusoidally varying inputs of different frequencies. For a linear system, a
sinusoidal input of a particular frequency will generate in steady-state a sinusoidal output of the
same frequency. The output, however, may have a different amplitude than the input and the
output may be phase shifted as compared to the input. Fig.1 shows a system with transfer
function G(s) applied with a sine wave input.
Consider a system represented by the transfer function G(s),then the output is simply the
product of the transfer function and the input, Y(s) = G(s)R(s) . Therefore, the output can be
separated (via a partial fraction expansion) into a component with the poles of the transfer
function(representing the system's natural response) and a component with the poles of the input
signal. If the system is stable, then the natural response will die out resulting in a steady-state
output that has the same form (poles) as the input signal.
It is important to analyse a system's frequency response to know how the amplitude and
phase of the steady-state output vary compared to the sinusoidal input. One way to represent this
amplitude (magnitude) data and phase data is by using a Bode plot.
Fig.1System with sinusoidal input
The advantages of frequency response analysis are as follows:
● The absolute and relative stability of a closed loop system can be estimated from the
knowledge of their open loop frequency response.
● The practical testing of a system can be easily carried with available sinusoidal signal
generators and precise measurement equipments.
● The transfer function of complicated system can be obtained experimentally byfrequency
response test.
● The design and parameter adjustment of the open loop transfer function of a system for
specified closed loop performance is carried out more easily in frequency domain.
● When the system is designed by use of the frequency response analysis, the effect of
noise disturbance and parameters variations are relatively easy to visualize and
incorporate corrective measurements.
● The frequency response analysis and designs can be extended to certain nonlinear
systems as well.
The performance characteristics of a system in frequency domain are measured in terms
of frequency domain specifications. The requirements of a system to be designed are usually
specified in terms of these specifications.
The frequency domain specifications are,
1. Resonant Peak: Maximum value of the closed loop transfer function.
2. Resonant Frequency: Frequency at which resonant peak occurs.
3. Bandwidth: Range of frequencies for which the system normalized gain is more than
-3db.
4. Cut-off rate-It is the slop of the log-magnitude curve near the cut off frequency.
5. Gain Margin: The value of gain to be added to system in order to bring the system to the
verge of instability. It is given by the Reciprocal of the magnitude of open loop transfer
function at phase cross over frequency.
6. Phase Margin: Additional phase lag at the gain cross over frequency in order to bring the
system to the verge of instability.
Fig. 2 Bode plot
Circuit Diagram:
Fig. 3 RC Circuit
Bode Plot:
A Bode plot is a (semilog) plot of the transfer function magnitude and phase angle as a
function of frequency. It helps in identifying the stability of the system. Also, a simple method
for sketching an approximate log-magnitude curve is available. Bode plot is a frequency
response plot that contains 2 graphs;
● Magnitude plot
● Phase plot
In both the plots, x-axis represents angular frequency (logarithmic scale). Whereas, y axis
represents the magnitude (linear scale) of open loop transfer function in the magnitude plot and
the phase angle (linear scale) of the open loop transfer function in the phase plot.
The magnitude of the open loop transfer function in dB is
M=20log|G(jω)H(jω)| (1)
The phase angle of the open loop transfer function in degrees is
Φ=∠G(jω)H(jω) (2)
Fig.2 shows the magnitude and phase plot. The gain cross over frequency( gc) and phase
cross over frequency (pc) can be calculated using magnitude plot and phase plot
respectively.
Procedure:
1. Construct the RC circuit as given in the circuit diagram.
2. Apply a sinusoidal input signal that sweeps through a range of frequencies.
3. Obtain the frequency response of the system.
4. Also generate the theoretical Bode plot for the constructed RC circuit using
the MATLAB command bode ().
5. Obtain the gain margin, phase margin, gain cross over frequency and
phase cross over frequency using margin () function.
6. Analyze the obtained frequency response to decide on the stability of the system.
Result:
Thus the frequency response analysis for the given system is carried out.
