Notes: Frequency Response Analysis | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 1


 
 
 
 
 
 
 
 
FREQUENCY RESPONSE ANALYSIS 
 
Whatis Frequency Response? 
The response of a system can be partitioned into both the transient response and the 
steady state response. We can find the transient response by using Fourier integrals. The 
steady state response of a system for an input sinusoidal signal is known as the frequency 
response. In this chapter, we will focus only on the steady state response. 
If a sinusoidal signal is applied as an input to a Linear Time-Invariant (LTI) system, then it 
produces the steady state output, which is also a sinusoidal signal. The input and output 
sinusoidal signals have the same frequency, but different amplitudes and phase angles. Let 
the input signal be 
 
Where, 
 A is the amplitude of the input sinusoidal signal. 
 ? 0 is angular frequency of the input sinusoidal 
signal. We can write, angular frequency ? 0 as shown 
Page 2


 
 
 
 
 
 
 
 
FREQUENCY RESPONSE ANALYSIS 
 
Whatis Frequency Response? 
The response of a system can be partitioned into both the transient response and the 
steady state response. We can find the transient response by using Fourier integrals. The 
steady state response of a system for an input sinusoidal signal is known as the frequency 
response. In this chapter, we will focus only on the steady state response. 
If a sinusoidal signal is applied as an input to a Linear Time-Invariant (LTI) system, then it 
produces the steady state output, which is also a sinusoidal signal. The input and output 
sinusoidal signals have the same frequency, but different amplitudes and phase angles. Let 
the input signal be 
 
Where, 
 A is the amplitude of the input sinusoidal signal. 
 ? 0 is angular frequency of the input sinusoidal 
signal. We can write, angular frequency ? 0 as shown 
 
 
 
 
 
 
? 0=2pf 0 
 
Here, f 0 is the frequency of the input sinusoidal signal. Similarly, you can follow the same 
procedure for closed loop control system. 
Frequency Domain Specifications 
The frequency domain specifications are 
 Resonant peak 
 Resonant frequency 
 Bandwidth. 
Consider the transfer function of the second order closed control system as 
 
 
Page 3


 
 
 
 
 
 
 
 
FREQUENCY RESPONSE ANALYSIS 
 
Whatis Frequency Response? 
The response of a system can be partitioned into both the transient response and the 
steady state response. We can find the transient response by using Fourier integrals. The 
steady state response of a system for an input sinusoidal signal is known as the frequency 
response. In this chapter, we will focus only on the steady state response. 
If a sinusoidal signal is applied as an input to a Linear Time-Invariant (LTI) system, then it 
produces the steady state output, which is also a sinusoidal signal. The input and output 
sinusoidal signals have the same frequency, but different amplitudes and phase angles. Let 
the input signal be 
 
Where, 
 A is the amplitude of the input sinusoidal signal. 
 ? 0 is angular frequency of the input sinusoidal 
signal. We can write, angular frequency ? 0 as shown 
 
 
 
 
 
 
? 0=2pf 0 
 
Here, f 0 is the frequency of the input sinusoidal signal. Similarly, you can follow the same 
procedure for closed loop control system. 
Frequency Domain Specifications 
The frequency domain specifications are 
 Resonant peak 
 Resonant frequency 
 Bandwidth. 
Consider the transfer function of the second order closed control system as 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Resonant Peak 
It is the peak (maximum) value of the magnitude of T(j?). It is denoted by 
M r. At u=ur, the Magnitude of T(j?) is - 
Page 4


 
 
 
 
 
 
 
 
FREQUENCY RESPONSE ANALYSIS 
 
Whatis Frequency Response? 
The response of a system can be partitioned into both the transient response and the 
steady state response. We can find the transient response by using Fourier integrals. The 
steady state response of a system for an input sinusoidal signal is known as the frequency 
response. In this chapter, we will focus only on the steady state response. 
If a sinusoidal signal is applied as an input to a Linear Time-Invariant (LTI) system, then it 
produces the steady state output, which is also a sinusoidal signal. The input and output 
sinusoidal signals have the same frequency, but different amplitudes and phase angles. Let 
the input signal be 
 
Where, 
 A is the amplitude of the input sinusoidal signal. 
 ? 0 is angular frequency of the input sinusoidal 
signal. We can write, angular frequency ? 0 as shown 
 
 
 
 
 
 
? 0=2pf 0 
 
Here, f 0 is the frequency of the input sinusoidal signal. Similarly, you can follow the same 
procedure for closed loop control system. 
Frequency Domain Specifications 
The frequency domain specifications are 
 Resonant peak 
 Resonant frequency 
 Bandwidth. 
Consider the transfer function of the second order closed control system as 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Resonant Peak 
It is the peak (maximum) value of the magnitude of T(j?). It is denoted by 
M r. At u=ur, the Magnitude of T(j?) is - 
 
 
 
 
 
 
 
 
 
 
Resonant peak in frequency response corresponds to the peak overshoot in the time 
domain transient response for certain values of damping ratio dd. So, the resonant peak 
and peak overshoot are correlated to each other. 
Bandwidth 
It is the range of frequencies over which, the magnitude of T(j?) drops to 70.7% from its zero 
frequency value. 
At ?=0, the value of u will be zero. 
Substitute, u=0 in M. 
Therefore, the magnitude of T(j?) is one at ?=0 
At 3-dB frequency, the magnitude of T(j?) will be 70.7% of magnitude of T(j?)) at ?=0 
 
Page 5


 
 
 
 
 
 
 
 
FREQUENCY RESPONSE ANALYSIS 
 
Whatis Frequency Response? 
The response of a system can be partitioned into both the transient response and the 
steady state response. We can find the transient response by using Fourier integrals. The 
steady state response of a system for an input sinusoidal signal is known as the frequency 
response. In this chapter, we will focus only on the steady state response. 
If a sinusoidal signal is applied as an input to a Linear Time-Invariant (LTI) system, then it 
produces the steady state output, which is also a sinusoidal signal. The input and output 
sinusoidal signals have the same frequency, but different amplitudes and phase angles. Let 
the input signal be 
 
Where, 
 A is the amplitude of the input sinusoidal signal. 
 ? 0 is angular frequency of the input sinusoidal 
signal. We can write, angular frequency ? 0 as shown 
 
 
 
 
 
 
? 0=2pf 0 
 
Here, f 0 is the frequency of the input sinusoidal signal. Similarly, you can follow the same 
procedure for closed loop control system. 
Frequency Domain Specifications 
The frequency domain specifications are 
 Resonant peak 
 Resonant frequency 
 Bandwidth. 
Consider the transfer function of the second order closed control system as 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Resonant Peak 
It is the peak (maximum) value of the magnitude of T(j?). It is denoted by 
M r. At u=ur, the Magnitude of T(j?) is - 
 
 
 
 
 
 
 
 
 
 
Resonant peak in frequency response corresponds to the peak overshoot in the time 
domain transient response for certain values of damping ratio dd. So, the resonant peak 
and peak overshoot are correlated to each other. 
Bandwidth 
It is the range of frequencies over which, the magnitude of T(j?) drops to 70.7% from its zero 
frequency value. 
At ?=0, the value of u will be zero. 
Substitute, u=0 in M. 
Therefore, the magnitude of T(j?) is one at ?=0 
At 3-dB frequency, the magnitude of T(j?) will be 70.7% of magnitude of T(j?)) at ?=0 
 
 
 
 
 
 
 
Bandwidth ?b in the frequency response is inversely proportional to the rise time tr in 
the time domain transient response. 
Bode plots 
The Bode plot or the Bode diagram consists of two plots - 
 
 Magnitude plot 
 Phase plot 
In both the plots, x-axis represents angular frequency (logarithmic scale). Whereas, yaxis 
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 -