RC circuit time constants and coefficients

[evol_w10lv]

Homework Statement

I have got this circuit:

I have to find coefficients k1 and k2, and time constants T1 and T2. Actually, we can see that T1 =R1*C1 and T2=R2*C2, but is it possible in this circuit or it's just theoretically time constants? I'm not sure, but as there is no amplifier in that circuit, capacitors interacts between each other, that's why solution of time constants not so "clear" as we would like to have. Or is it absolutely wrong?

Bode plot, using transfer function H(s), which I calculated before:
http://www.bildites.lv/images/c54l1249hb71uqesw4mj.png

Homework Equations

The Attempt at a Solution

This is how I found transfer function:

But I guess, circuit suggests, there are series of two ''inertial differential sections''. Then theoretically:

When I compare these transfer functions:

I'm quite sure, that my first transfer function H(s) should be correct. Maybe second one is more theoretical? Where am I wrong with my solution?

 

[NascentOxygen]

When you cascade blocks, no longer does each 'see' itself driven by an ideal voltage source, nor does each block 'see' a high resistance at its output imposing negligible loading---the blocks will now interact to some extent.

You can multiply the individual transfer functions to obtain the overall transfer function ONLY where each block is isolated from its two adjacent blocks by buffer amplifiers. If there are no buffer amplifiers, then you have to determine anew the transfer function of the network as a whole.

 

[evol_w10lv]

Yes, just as I thought about amplifier..
This is transfer function for whole network:

From Bode plot I can see, that there are two corner frequencies, so it means that there are two time constants.
Time constants are roots from denominator? And then coefficient k is numerator? But can I determine them in this case, when I haven't got buffer amplifiers in my circuit?

 

[NascentOxygen]

See whether you can factorize the denominator of your general transfer function. Yes, the roots of the denominator, providing they are real.

(s + a)(s+b) will give you the two poles/time constants as algebraic expressions

 

[The Electrician]

Notice that C2 is 1000 times smaller than C1. This means that the R2 C2 section doesn't load the R1 C1 section very much. The overall transfer function is almost what you would get if there were a buffer amplifier between the sections.

Is it possible that the problem wants you to recognize this, and get approximate results using the fact that C1>>C2?

If there were a buffer, the (R1 C2 s) term in the denominator vanishes, and it becomes trivial to factor the denominator.

Otherwise, since the denominator is a quadratic, the factorization with the (R1 C2 s) term present using the quadratic formula is fairly easy.

 

[evol_w10lv]

I have to determine k, T1 and T2 for a whole network.
So I did some calculations:

Is it correct?

And what about the numerator C1C2R1R2 (red color) Is this coefficient k?

 

[The Electrician]

That is the coefficient K.

Note that since k = k1*k2, you can factor k in more than one way.

For example, you could let k1 and k2 both be √(C1 C2 R1 R2).

I would let k1 = R1 C1, and k2 = R2 C2. This also gives you correct units, and it is the same k1 and k2 you would have gotten if there were a buffer between the sections.

 

[NascentOxygen]

A useful check is to compare the two corner frequencies with what you'd find for each stage if it were independent. You'd expect these should not be too dissimilar.

 

[evol_w10lv]

Thanks for your help. Task is completed. :)