Electrical Engineering - Control Systems - 2cnd Order System

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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.

 

[NascentOxygen]

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.

 

[rude man]

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)?