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Control Lecture 6

The document discusses the time domain analysis of second-order systems, focusing on their transfer functions characterized by natural frequency (ωn) and damping ratio (δ). It explains the relationship between the S-plane, ωn, and δ, illustrating how poles are positioned based on these parameters. Additionally, it addresses the effects of varying δ and ωn on the step response of second-order systems.

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mohamedalbialy312
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7 views18 pages

Control Lecture 6

The document discusses the time domain analysis of second-order systems, focusing on their transfer functions characterized by natural frequency (ωn) and damping ratio (δ). It explains the relationship between the S-plane, ωn, and δ, illustrating how poles are positioned based on these parameters. Additionally, it addresses the effects of varying δ and ωn on the step response of second-order systems.

Uploaded by

mohamedalbialy312
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Automatic Control Systems

Time Domain Analysis of 2nd order System


• General 2nd order system is characterized by the following transfer
function:

• Where:
• C(s) is the Laplace transform of the output signal, c(t)
• R(s) is the Laplace transform of the input signal, r(t)
• ωn is the natural frequency
• δ is the damping ratio.
Time Domain Analysis of 2nd order System
Relation between S-plane and ωn and δ
• The distance from the origin of s-plane to the pole is the natural undamped
frequency ωn in rad/sec.
• For example, if ωn = 3, the pole is located anywhere on a circle with radius 3.
• Therefore the s-plane is divided into Constant Natural Undamped Frequency (ωn)
circles.
Relation between S-plane and ωn and δ
• Cosine of the angle between the vector connecting origin to pole and the -ve real axis yields
damping ratio.
δ = cos(θ)
• For Undamped system θ = 90. So, δ = 0
• For Critically damped systems θ = 0. So, δ = 1
• The s-plane is divided into sections of constant damping ratio lines.
Step
Response
of Second
order
Systems
Effect of Changing δ on Step Response
Effect of Changing ωn on Step Response
Second Order System Time Domain
Specifications

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