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Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE) PDF Download

What is 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 −
r(t) = A sin(ω0t)
The open loop transfer function will be −
G(s) = G(jω)
We can represent G(jω) in terms of magnitude and phase as shown below.
G(jω) = |G(jω)|∠G(jω)
Substitute, ω = ω0 in the above equation.
G(jω0) = |G(jω0)|∠G(jω0)
The output signal is
c(t) = A|G(jω0)|sin(ω0t + ∠G(jω0))

  • The amplitude of the output sinusoidal signal is obtained by multiplying the amplitude of the input sinusoidal signal and the magnitude of G(jω) at ω = ω0.
  • The phase of the output sinusoidal signal is obtained by adding the phase of the input sinusoidal signal and the phase of G(jω) at ω = ω0.

Where,

  • A is the amplitude of the input sinusoidal signal.
  • ω0 is angular frequency of the input sinusoidal signal.

We can write, angular frequency ωas shown below.

  • ω0 = 2πf0

Here, f0 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 and bandwidth. Consider the transfer function of the second order closed loop control system as,
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Substitute, s = jω in the above equation.
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Let, ω/ωn = u Substitute this value in the above equation.

Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Magnitude of T(jω) is -
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Phase of T(jω) is -
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)

Resonant Frequency
It is the frequency at which the magnitude of the frequency response has peak value for the first time. It is denoted by ωr. At ω = ωr, the first derivate of the magnitude of T(jω) is zero.
Differentiate M with respect to u.

Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)

Substitute, u = ur and dM/du == 0 in the above equation.
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
⇒ 4ur(Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)−1 + 2δ2) = 0
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)−1 + 2δ2 = 0
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE) = 1−2δ2

Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)

Substitute, ur = ωrn in the above equation.

Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)

Resonant Peak
It is the peak (maximum) value of the magnitude of T(jω). It is denoted by Mr.
At u = ur, the Magnitude of T(jω) is -
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Substitute, Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)and 1−Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)=2δ2 in the above equation.

Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Resonant peak in frequency response corresponds to the peak overshoot in the time domain transient response for certain values of damping ratio δ. 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(jω) drops to 70.7% from its zero frequency value.
At ω=0, the value of u will be zero.
Substitute, u = 0 in M.
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Therefore, the magnitude of T(jω) is one at ω=0.
At 3-dB frequency, the magnitude of T(jω) will be 70.7% of magnitude of T(jω) at ω = 0.
i.e., at ω = ωB, M = 0.707(1) = 1/√2
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Let, Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)= x
⇒ 2 = (1 − x)2 + (2δ)2x
⇒ x2 + (4δ− 2)x − 1 = 0
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)

Consider only the positive value of x.
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Substitute, Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)

Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE)
Bandwidth ωb in the frequency response is inversely proportional to the rise time tr in the time domain transient response.

The document Frequency Response Analysis of Second Order Control System | Control Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Control Systems.
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