A (simple?) question on fractions

[t_n_p]

Homework Statement

Hi. Bit of a weird question here, was thinking about this and am stumped as to how the following is true.

I have a fraction here,

[PLAIN]http://img109.imageshack.us/img109/2280/math1.jpg

I know this equation is definitely true, I have checked it by graphing on my calculator.

My issue here is with the quadratic in the denominator. If I want to show this, I would have taken out 1/2500 like so.

[PLAIN]http://img258.imageshack.us/img258/5485/math2.jpg

As you can see the only difference now between my answer and the correct answer is the 1/2500. So what happened to this 1/2500!?

 

[danago]

Have you tried substituting the same value of 's' into both of them and numerically evaluating each expression?

 

[t_n_p]

hey your right.
If I sub s=1, the top two expressions are not the same, however 1 and 3 are the same.

this suggests that expression 1 does not equal expression 2.

HOWEVER,
expression 1 and expression 2 have the same roots!?

 

[danago]

hey your right.
If I sub s=1, the top two expressions are not the same, however 1 and 3 are the same.

this suggests that expression 1 does not equal expression 2.

HOWEVER,
expression 1 and expression 2 have the same roots!?

Because two expressions have the same roots does not mean that they are equivalent. As a very simple example, consider the following two functions:

f(x) = (x+1)(x+3)
g(x) = 50(x+1)(x+3)

The zeros for both are clearly x=-3 and x=-1, however it is far from true that f(x)=g(x).

Your example is very similar; your two expressions differ by a factor of 2500.

 

[t_n_p]

I think this problem has more to do with application.

fyi the question is in relation to engineering control systems, the equation is a transfer function and the denominator is analysed for poles.

I believe that's why the quadratic has simply been swapped. The expressions are not exactly the same "per se" but they have the same roots. Since we are interested in the poles, i.e. roots of the denominator I think that's why the 1/2500 is simply ignored.

 

[danago]

When i first saw the expression i did wonder if you were doing a course in control systems. I'm also currently doing a course in process control and am all too familiar with these transfer functions haha :P

 

[t_n_p]

lol, interesting coincidence.

Still one question remains though (not sure if you will be able to answer, its more related to the control systems end than the mathematical side).

generally, say you have a tf:

c/r = 10/(5s+5)

You will take out 5 from the denominator to get it into "s+1" form (for want of a better name), which makes it easy to get your corner frequencies. This leads to:

c/r = 10/5(s+1)

which then becomes:
c/r = 2/(s+1) which is important because the K' value is now 2, and is used in kb=20logk' to create an approximate bode plot.

so my question is then, if you take 1/2500 out, will that get absorbed into the k' value? It seems not, why, I don't know.

Not sure if you can follow, but thought its worth a shot explaining anyway...

 

[danago]

lol, interesting coincidence.

Still one question remains though (not sure if you will be able to answer, its more related to the control systems end than the mathematical side).

generally, say you have a tf:

c/r = 10/(5s+5)

You will take out 5 from the denominator to get it into "s+1" form (for want of a better name), which makes it easy to get your corner frequencies. This leads to:

c/r = 10/5(s+1)

which then becomes:
c/r = 2/(s+1) which is important because the K' value is now 2, and is used in kb=20logk' to create an approximate bode plot.

so my question is then, if you take 1/2500 out, will that get absorbed into the k' value? It seems not, why, I don't know.

Not sure if you can follow, but thought its worth a shot explaining anyway...

I haven't really covered drawing bode plots by hand in my class (we just did it in MATLAB), but is the idea behind your method to separate each factor of the transfer function, make a bode plot of each factor and then superposition them? If so, then it shouldn't matter if the factor of 2500 is considered as part of the constant term or as part of any of the other factors.

Im not sure if that is even what you mean, so i could have completely missed your question :P

 

[t_n_p]

if you havn't done it by hand you probably wouldn't know.

thanks a lot for the help anyway!

 

[danago]

No problem. Good luck with it!