Is this equation from my lecture notes wrong? (RE: Transfer Functions)

[CraigH]

Mistake 1.

In the lecture slides from my university it says that:

"The response of a stable first-order transfer function to a unit sine wave input is:"

Y(s)=\frac{1}{s+a}*\frac{\omega}{s^2+\omega^2}

Isn't this missing an a in the numerator since the standard form of a first order transfer function is:

H(s) = \frac{1}{\tau s +1} = \frac{a}{s+a}

where \tau=1/a

and the laplace transform of the sine wave input is:

\frac{\omega}{s^2+\omega^2}

Mistake 2.

The lecture slides also say that:

"The response of a stable second-order transfer function to a unit sine wave input is:"

Y(s)=\frac{1}{s^2+2\zeta\omega_n+\omega_n^2}*\frac{\omega}{s^2+\omega^2}

Similarly, isn't this missing an \omega_n^2 in the numerator as the standard form of a second order transfer function is:

H(s) = \frac{\omega_n^2}{s^2+2 \zeta \omega_n s + \omega_n^2}

 

[milesyoung]

For your first two cases, I wouldn't necessarily call them mistakes. Your quotes do not mention anything about standard form. The systems do not have unity DC-gain, but there's nothing inherently wrong with that.

I can't really guess as to what's going on in your last case without seeing some context.

 

[CraigH]

Thanks for your answer. The last case was just the same equation split into partial fractions. I misread the addition for multiplication. I've deleted that part of the question now.

As for the first two transfer functions, I'm still really confused. I don't even know what questions to ask, I'm really struggling in this module. I'll just ask this though:

In the equation for "The response of a stable second-order transfer function to a unit sine wave input" is the natural undamped frequency \omega_n still given by the \omega_n in the equation, even though it is not in standard form? and likewise for the damping coefficient \zeta?

Could this equation be re-written in standard form so that it has \omega_n^2 in the numerator so that the actual natural undamped frequency can be found?

If the numerator in the given transfer function represents the DC-gain, then what represents the DC gain in the standard form equation?

Thanks again!

 

[milesyoung]

In the equation for "The response of a stable second-order transfer function to a unit sine wave input" is the natural undamped frequency \omega_n still given by the \omega_n in the equation, even though it is not in standard form? and likewise for the damping coefficient \zeta?

Since your systems don't have any zeros, their dynamics is determined fully by their pole locations. The poles are the roots of the denominator of the transfer functions of your systems, and since you aren't altering them in any way, nothing changes in terms of the dynamics. All that changes is the (frequency-dependent) gain.

Could this equation be re-written in standard form so that it has \omega_n^2 in the numerator so that the actual natural undamped frequency can be found?

You don't have to rewrite it.

If the numerator in the given transfer function represents the DC-gain, then what represents the DC gain in the standard form equation?

A system represented in the standard form always has unity DC-gain. You can easily determine this yourself using the final value theorem. Edit: The numerator alone doesn't represent its DC-gain.