Help with finding a transfer function

[7emeraldempre]

Homework Statement

---Resistor1=47K ohm----------------------------------------------
| | |
+ | | |
V1 capacitor1= .1micro farad | | +
- | c2=.002 micro Farad Load Resistor = 100K
| | | - R2= 1K | |
| | |
-----------------------------------------------------------------------


Add another resistance, R3 to the circuit so that the high-frequency magnitude asymptote
is flat instead of a ¡20dB=decade. Find the minimum values of R1 and R2 so that the
phase lag between V2 and V1 at any frequency does not exceed 50±. Write the new analytic
transfer function, H(s) = V2=V1, sketch the magnitude and phase of H(s) and label the
salient features.

I think if I add a resistor in series with c2 I will put the magnitude flat.


I figured an equation for this circuit (without the extra resistor) and the actual circuit is in the attachment. I could not copy and paste it into the window so I just put it into the attachment. I know that I am suppossed to be able to change the transfer funtion into a simplier form to get the poles and zeos but I get stumped there. I ask my professor but he said "that is what grading is for" and laughed. He's the type that gives students impossible problems and is amazed that we cannot do them right way. I like him and everything, just I do not flourish under his teaching style, other than that he is a good teacer.

I know this is an algebra problem but it would really help me out to see math and not just and answer.

 

[7emeraldempre]

Sorry, i did not know the circuit was going to come out that way, but what it is supposed to look like is in the attachment.

 

[The Electrician]

It looks like you got the rather complicated expression for the transfer function correct, but you made a mistake somewhere when you tried to simplify it.

See the first attached image showing the correct simplified expression, and also the expression for the case where an additional resistor, R3, is in series with C2.

The second image shows the log magnitude and phase of the transfer function. I've got a value of R3=1000 there, but it's not used in this plot.

The third image shows the same for the case of R3=30 ohms in series with C2.

The transfer function denominator only involves the variable s to the second power, so you can use the quadratic formula to determine the poles of the denominator, which are the zeros of the transfer function. The poles of the numerator are obvious by inspection. Knowing the poles and zeros, you can plot the asymptotic response (the Bode plot).

I don't understand the bit about the phase lag limits being 50+-, since the phase shift only goes negative. I suppose one way to get the values of R1 and R2 for some specified phase shift would be to manually tweak the values and repeatedly plot phase shift.

But, if your instructor wants precise values, you're going to have to differentiate the expression for phase shift, find the two frequencies of maximum negative phase shift and using those expressions, set them both equal to 50 degrees (or whatever your instructor wants), and solve for R1 and R2.