Series RLC - Resonant Circuits

Discussion in 'Homework Help' started by jegues, Dec 4, 2010.

1. jegues Thread Starter Well-Known Member

Sep 13, 2010
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43
See figure attached for problem statement and my work.

I am asked to derive the transfer function for the circuit shown in the figure and plot its magnitude characteristic versus frequency.

I'm working on the transfer function part and I've almost got everything in a pretty form expect for the complex part in the denominator is giving me problems.

As indicated in the figure, my cheatsheet says that for Series RLC I should get something of that form. (See figure)

I just don't see how I can transform something like this into that form. I have a feeling it's not possible to put it into that form because the Vo is being measured across different nodes than usual.

Is this the case? What am I missing/not seeing?

Is the form I have my transfer function in sufficient enough to do the bode plot and determine what type of filter it is?

Once I can get these things cleared up I'll try to work through the rest of the question.

Thanks again!

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2. Georacer Moderator

Nov 25, 2009
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What you have built there is a band-blocking filter. You can see that by finding H(w) for zero frequency and for very high (infinite) frequency. You can safely presume that in the mid-frequencies Z2 will be very small, grounding the signal.

I don't know the RLC filter TF by heart, but the one you are trying to end to doesn't seem right. It doesn't have any zeros (at w=w0). The expression you have come to, does.

Finding the exact values of the poles however, requires to solve a 2nd grade polynomial and is more easilly done if you stay at complex frequency plane (s) rather than at circular frequency (w).

3. jegues Thread Starter Well-Known Member

Sep 13, 2010
735
43
Okay I'm looking at my transfer function before I tried to work it into that form and I can clearly see that low and high frequencies,

$H(w) = 1$

From here I know I'm going to be looking at a notch filter correct?

***Now all I have to do is figure out whether $|H(w)|$ increases in mid-frequencies or decreases at mid-frequencies.***

How do I make this distinction?

Once I do, I will know whether or not the filter is passing a smaller range of frequencies or eliminating those frequencies.

From this distinction I will also be able to configure the resonance frequency and bandwidth.

The 2nd part of the question says to Choose L and C such that the resonance frequency and bandwidth are 60 Hz and 1 Hz.

Again, If I can figure out how I can make the conclusion questions in stars (***) then I should be able to finish off the question.

Someone care to explain/comment?

4. mik3 Senior Member

Feb 4, 2008
4,846
63
Just to remind to that the magnitude of a complex number a+jb is:

sqrt(a^2+b^2)

Use this in your denominator to find the magnitude of H vs frequency.

5. jegues Thread Starter Well-Known Member

Sep 13, 2010
735
43
I'm still looking for how I can determine whether the mid-range frequencies increase or decrease the magnitude.

There's got to be a better way than simply computing the expression for the magnitude of H(w) and testing an w in the middle.

Any ideas?

6. mik3 Senior Member

Feb 4, 2008
4,846
63
It is the resonant frequency:

$f_{res}=\fra{1}{2\cdot{pi}\cdot\sqrt{LC}}$

Last edited: Dec 4, 2010
7. jegues Thread Starter Well-Known Member

Sep 13, 2010
735
43
Hmmm... So I have two equations then,

$1Hz = \frac{1}{L}$

and

$60Hz = \frac{1}{2\pi \sqrt{LC}}$

Then I can sub $\omega _{o}=60Hz$

And throw into my expression for the magnitude of H(w) and see whether it decreases or increases, correct?

8. mik3 Senior Member

Feb 4, 2008
4,846
63
You want:

wo=60Hz
BW=1Hz

$wo=\frac{1}{\sqrt{LC}}$

$BW=\frac{wo}{Q}$

$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$

Use these equations to get the values of L and C, provided you know R.

http://en.wikipedia.org/wiki/RLC_circuit

9. jegues Thread Starter Well-Known Member

Sep 13, 2010
735
43
Since R=1ohm.

I solve L and C being the following,

C = 278 uF,

L = 1H.

Is this correct?

I'm still confused on how I determine whether the magnitude is increasing around the resonance frequency or decreasing around the resonance frequency.

Someone care to explain?

10. mik3 Senior Member

Feb 4, 2008
4,846
63
Ohh, a stupid mistake I have just realised.

wo is not 60Hz but it is 2pi60 rad/s

Assign values to wo or draw the bode plot.

11. jegues Thread Starter Well-Known Member

Sep 13, 2010
735
43
Assign values to wo? We are given wo to be 60 hz.

Draw the bode plot? That's why I want to figure out whether it's increasing of decreasing at mid-frequencies so I can draw the bode plot.

Am I missing something?

12. mik3 Senior Member

Feb 4, 2008
4,846
63
I meant assign values to w, another mistake.

If you assign values to w, which are close to wo, then you will see how it behaves around wo. Otherwise you should know that the output decreases for this particular filter for w values near wo.

Another idea, is to use calculus to find the local minimums or maximums of the transfer function. Never tried this but it is worth doing it.

13. jegues Thread Starter Well-Known Member

Sep 13, 2010
735
43
Ya that calculus approach might be just what I'm looking for. I hate having to plug in values and use my calculator to see what the transfer function is doing at mid range frequencies. I'll play around with it and see what I get.

14. mik3 Senior Member

Feb 4, 2008
4,846
63
Let us know the results.

15. jegues Thread Starter Well-Known Member

Sep 13, 2010
735
43
I simply took the derivative of my transfer function and things are way to messy to get any "easy" answers out of it. I guess I have to test a point in mid-range frequencies to determine wheter its increasing or decreasing around $
\omega_{o}$