# Electrical Engineering - Control Systems - 2cnd Order System

• GreenPrint
In summary, when graphing a bode plot for an overdamped system, the arccosine of the over damped value will be greater than one.
GreenPrint

## Homework Statement

Given a electrical circuit as one below

One can find the transfer function

For second order systems this can be written as

I know that the damping coefficient can be found using the following formula

In the over damped case ζ>1 can easily occur depending on the values of the components above.

My question is how would you plot a bode plot when ζ>1? I know that when 0<=ζ<=1 Θ can be found using, Θ=arccos(ζ). In the over damped case you would get the arccosine of a value greater than one. In which case Θ is imaginary so how would you go about plotting it such a angle? I know how to solve such a equation such as arccosine(2), but am unsure of the physical conclusions from such a equation. I'm just intrigued by learning physical representations of the trigonometric functions that fall out side of their real domains. I would like to learn more on this subject.

Thanks for any help.

You replace ##s## by ##jω## and this gives an expression with a complex number in the denominator. Determine the magnitude of the expression, and its angle.

GreenPrint said:

## Homework Statement

Given a electrical circuit as one below

One can find the transfer function

For second order systems this can be written as

I know that the damping coefficient can be found using the following formula

In the over damped case ζ>1 can easily occur depending on the values of the components above.

My question is how would you plot a bode plot when ζ>1? I know that when 0<=ζ<=1 Θ can be found using, Θ=arccos(ζ). In the over damped case you would get the arccosine of a value greater than one. In which case Θ is imaginary so how would you go about plotting it such a angle? I know how to solve such a equation such as arccosine(2), but am unsure of the physical conclusions from such a equation. I'm just intrigued by learning physical representations of the trigonometric functions that fall out side of their real domains. I would like to learn more on this subject.

Thanks for any help.

When the system is overdamp[ed the roots of the characteristic equation both lie on the real negative axis. So that's actually much easier to Bode-plot than for an underdamped system.

What are the Bode plots of ab/(s+a)(s+b)?

## 1. What is a second order system in electrical engineering?

A second order system in electrical engineering is a system that can be described by a second order differential equation. This type of system has two energy storage elements, such as capacitors and inductors, and can be represented by a transfer function with a second order polynomial in the denominator.

## 2. What are the characteristics of a second order system?

The characteristics of a second order system include a natural frequency, damping ratio, and settling time. The natural frequency is a measure of how fast the system can oscillate without any external forces. The damping ratio describes the system's ability to resist oscillations and is related to the system's stability. The settling time is the time it takes for the system to reach its steady-state response after a disturbance.

## 3. How is a second order system analyzed in control systems?

A second order system is typically analyzed using the root locus and Bode plot techniques. The root locus method plots the roots of the system's characteristic equation as a function of a parameter, such as the gain or controller parameters. The Bode plot is a graph of the system's frequency response, which shows how the system responds to different input frequencies.

## 4. What is the difference between a first and second order system in control systems?

The main difference between a first and second order system is the number of energy storage elements. A first order system has one energy storage element, while a second order system has two. This results in different characteristics, such as a higher natural frequency and faster response for a second order system compared to a first order system.

## 5. How are second order systems used in real-life applications?

Second order systems are used in a variety of real-life applications, such as electronic circuits, mechanical systems, and control systems. They are commonly used in the design of controllers for systems that require precise and stable control, such as aircraft autopilots, robotic arms, and industrial machinery. They are also used in the design of filters and amplifiers in electronic circuits.

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