Bode plots are graphical representations of the frequency response of a system. They’re used to investigate the soundness and efficiency of management programs, and to design filters and different sign processing circuits. Second-order linear time-invariant (LTI) programs are a standard kind of system that may be analyzed utilizing Bode plots. On this article, we are going to present you how one can graph a second-order LTI system on a Bode plot.
To graph a second-order LTI system on a Bode plot, you have to to know the system’s pure frequency and damping ratio. The pure frequency is the frequency at which the system would oscillate if there have been no damping. The damping ratio is a measure of how shortly the system’s oscillations decay. As soon as you realize the system’s pure frequency and damping ratio, you should utilize the next steps to graph the system on a Bode plot:
1. Plot the system’s magnitude response. The magnitude response is the ratio of the output amplitude to the enter amplitude. For a second-order LTI system, the magnitude response is given by the next equation:
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|H(f)| = frac{1}{sqrt{1 + (2zeta f/ω_n)^2 + (f/ω_n)^4}}
“`
the place:
* f is the frequency
* ω_n is the pure frequency
* ζ is the damping ratio
2. Plot the system’s section response. The section response is the distinction between the output section and the enter section. For a second-order LTI system, the section response is given by the next equation:
“`
∠H(f) = -arctan(2ζ f/ω_n) – arctan(f/ω_n)^2
“`
How To Graph 2nd Order Lti On Bode Plot
A second-order linear time-invariant (LTI) system is a system that may be described by a second-order differential equation. The switch perform of a second-order LTI system is given by:
$$H(s) = frac{Okay omega_n^2}{s^2 + 2zeta omega_n s + omega_n^2}$$
the place:
* $Okay$ is the acquire of the system
* $omega_n$ is the pure frequency of the system
* $zeta$ is the damping ratio of the system
To graph the Bode plot of a second-order LTI system, we have to discover the magnitude and section of the switch perform at totally different frequencies.
Magnitude
The magnitude of the switch perform is given by:
$$|H(jomega)| = frac{Okay omega_n^2}{sqrt{(jomega)^2 + 2zeta omega_n jomega + omega_n^2}}$$
We are able to simplify this expression by utilizing the next substitutions:
$$u = jomega$$
$$a = omega_n$$
$$b = 2zeta omega_n$$
This offers us:
$$|H(jomega)| = frac{Okay a^2}{sqrt{-u^2 + bu + a^2}}$$
We are able to now graph the magnitude of the switch perform by plotting $|H(jomega)|$ as a perform of $omega$.
Section
The section of the switch perform is given by:
$$angle H(jomega) = -arctanleft(frac{2zeta omega_n jomega}{omega_n^2 – j^2 omega^2}proper)$$
We are able to simplify this expression by utilizing the next substitutions:
$$u = jomega$$
$$a = omega_n$$
$$b = 2zeta omega_n$$
This offers us:
$$angle H(jomega) = -arctanleft(frac{bu}{-a^2 – u^2}proper)$$
We are able to now graph the section of the switch perform by plotting $angle H(jomega)$ as a perform of $omega$.
Individuals Additionally Ask About How To Graph 2nd Order Lti On Bode Plot
What’s the distinction between a Bode plot and a Nyquist plot?
A Bode plot is a graphical illustration of the frequency response of a system. It exhibits the magnitude and section of the system’s switch perform at totally different frequencies. A Nyquist plot is a graphical illustration of the system’s stability. It exhibits the system’s poles and zeros within the complicated aircraft.
How can I exploit a Bode plot to design a filter?
A Bode plot can be utilized to design a filter by selecting the suitable cutoff frequencies and beneficial properties. The cutoff frequencies are the frequencies at which the filter’s magnitude response drops by 3 dB. The beneficial properties are the elements by which the filter amplifies the sign at totally different frequencies.
What’s the relationship between the Bode plot and the Laplace rework?
The Bode plot is said to the Laplace rework by the next equation:
$$H(s) = mathcal{L}^{-1}left{H(jomega)proper}$$
the place:
* $H(s)$ is the Laplace rework of the switch perform
* $H(jomega)$ is the frequency response of the switch perform
