Looking over the problem again, I think we are both wrong.
When one of the two Tau goes to 0 (and the other doesn't) the result is seen immediately from the expression itself: the terms containing the Tau that vanishes disappear from the numerator and from the denominator. Whether it is Tau1=0 or Tau2=0 is irrelevant.
Te book states "This approximation works best when one time constant is significantly bigger that the other". Then shows that it works better for ratio 6.5 (error <7%) than for ratio 1 (error <15%).
The case in which one of the Tau's is 0 is not of interest.
I tried to show that your assertion that the situations tau1<<tau2 and tau2<<tau1 are different is wrong.
Your assertion that the book is wrong is unfair too.
I don't understant why you say that I'm wrong too. Please explain. Thanks
So it does, but that is not what we wanted to prove.
Yes, you proved that the reduction formula was not related to Tau1 or Tau2 alone, but at that time, you did not state that the relationship was related to the sum of Tau1 and Tau2 or Tau.
At the time neither of us noticed that the book stated the reduced expression in terms of Tau, not Tau1 or Tau2. On that point, we were both wrong.
But why do calculus operations when the reduction formula can be discerned by algebraic operations and limits calculatons?
Also, the book already gave the relationship Tau = Tau1 + Tau2.
Ratch
What we wanted to prove (or even better, to deduce) is that a good approximation to the second-order response (4.11) is a single-pole response with tau=tau1+tau2.
Please read my post #8.
It was not shown in this thread (nor in the page of the book) a rationale for the approximation Tau=Tau1+Tau2 using algebraic operations and limits calculatons.
Do you dislike calculus?
Originally Posted by Ratch
Also, the book already gave the relationship Tau = Tau1 + Tau2.
Ratch
But this is precisely what we try to prove!!!
Please see post #1.
Hi.
Please help me prove the approximation formula below given in my book. Thanks.
**broken link removed**
I did. It shows that Tau = Tau1+Tau2, which the book already states is true.
But, the formula with Tau only was not derived.
No, the request was to prove the reduction formula, not Tau = Tau1 + Tau2.
The book does not show why.
Do you call "reduction formula" the relationship (4.11) VDD*exp(-t/Tau) ?
It doesn't need to be proved.
The book approximates the second-order circuit as a first-order system with a single time constant. Its response has the form VDD*exp(-t/Tau), and he fact that it is a suitable approximation is evident.
This is a model with one parameter that has to be determined (Tau).
It would be possible to propose a different approximation, for instance a linear-piecewise one, or a polynomial one if its range of validity can be restricted. In that cases, we would need to estimate the parameters (slopes, coefficients, etc) in terms of Tau1 and Tau2.
(For the purposes of the book, the single-pole model is more suitable than the two above mentioned examples.)
zorro,
Yes, the book does show why it is true in equation (4.12) as I already explained in my last post. Furthermore, it shows that Tau is a easily obtained constant
Ratch
As stated by the book, the equation 4.12 is too complicated and it has to be simplified. If you want to reduce it to the sum of the two time constants, you should prove that the term under the square root is negligible.
But this doesn't mean that tau1+tau2 is the time constant that well approximate the behaviour of a second oprder system, unless you can prove it
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