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The roots of the characteristic equation of the second order system in which real and imaginary part represents the :
- a)Damped frequency and damping
- b)Damping and damped frequency
- c)Natural frequency and damping ratio
- d)Damping ratio and natural frequency
Correct answer is option 'B'. Can you explain this answer?
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The roots of the characteristic equation of the second order system in...
Real part represents the damping and imaginary part damped frequency.
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Understanding the Characteristic Equation
In control systems, the characteristic equation of a second-order system is crucial for analyzing system dynamics. The typical form of the characteristic equation is:
s² + 2ζω_ns + ω_n² = 0
where:
- s represents the complex roots,
- ζ is the damping ratio,
- ω_n is the natural frequency.
Roots of the Characteristic Equation
The roots of this equation can be expressed as:
s = -ζω_n ± jω_d
where:
- j represents the imaginary unit,
- ω_d is the damped frequency.
Real and Imaginary Parts
The roots consist of two components:
- The real part: -ζω_n (indicates the damping effect)
- The imaginary part: ± jω_d (indicates the oscillatory behavior)
Explanation of Option B
The correct identification of these components leads to understanding why option B is correct.
- Damping and Damped Frequency:
- The real part (-ζω_n) corresponds to the system’s damping. It determines how oscillations in the system decay over time.
- The imaginary part (± jω_d) corresponds to the damped frequency. It represents the frequency of oscillation when damping is present.
Conclusion
Thus, understanding the roots of the characteristic equation clarifies that the real part indicates the damping aspect while the imaginary part reflects the damped frequency, validating option B as the correct choice.
In control systems, the characteristic equation of a second-order system is crucial for analyzing system dynamics. The typical form of the characteristic equation is:
s² + 2ζω_ns + ω_n² = 0
where:
- s represents the complex roots,
- ζ is the damping ratio,
- ω_n is the natural frequency.
Roots of the Characteristic Equation
The roots of this equation can be expressed as:
s = -ζω_n ± jω_d
where:
- j represents the imaginary unit,
- ω_d is the damped frequency.
Real and Imaginary Parts
The roots consist of two components:
- The real part: -ζω_n (indicates the damping effect)
- The imaginary part: ± jω_d (indicates the oscillatory behavior)
Explanation of Option B
The correct identification of these components leads to understanding why option B is correct.
- Damping and Damped Frequency:
- The real part (-ζω_n) corresponds to the system’s damping. It determines how oscillations in the system decay over time.
- The imaginary part (± jω_d) corresponds to the damped frequency. It represents the frequency of oscillation when damping is present.
Conclusion
Thus, understanding the roots of the characteristic equation clarifies that the real part indicates the damping aspect while the imaginary part reflects the damped frequency, validating option B as the correct choice.
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The roots of the characteristic equation of the second order system in which real and imaginary part represents the :a)Damped frequency and dampingb)Damping and damped frequencyc)Natural frequency and damping ratiod)Damping ratio and natural frequencyCorrect answer is option 'B'. Can you explain this answer?
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