# Best RLC circuit for Low Frequency underdamped resonance

1. Oct 25, 2012

### swraman

Hi all,

I am tring to design an RLC circuit with a low (<3Khz) resonance out of RLC circuit elements. The circuit must be underdamped. The lower the damping the better.

Originally I planned (and built) a RLC circuit with R=.1, C=100uF, L=100uH. On paper this gives a resonance at 1590 and a damping factor of .05, perfect. But, when I am actually driving this thing my source is current limited to 10mA. The Current/Voltage transfer function of a series RLC circuit with these values has a gain of ~10 at resonance, meaning I have a max of 1mV driving voltage when operating at resonance. Too low. My driving voltage will have to be at least 50mv, preferably more.

The Current/Voltage transfer function is scaled by changing the resistor value. A resistor value of 10Ohm gives a good gain (.1 at peak), however the system then becomes severely overdamped. To compensate I would have to up the inductance by the same factor of 100, but 10000uH inducters are hard to come by.

Is there any other RLC circuit design that I can use that gives me an underdamped resonance below 3kHz, and doesn't require more than 10mA current from a reasonable (>10mV) drive voltage?

I've looked at some other designs (R || L || C) or ( R - [L || C] ) but the current/voltage transfer functions of these seem to have infinite gain at some frequencies (unless I made an error in my math).

thanks

2. Oct 25, 2012

### AlephZero

Unless you must have a real inductor for some reason, you might be better building a virtual inductor from an op-amp, a capacitor and a couple of resistors. This building block is widely used in filter design etc. See "simulated inductor" at the bottom of this page: http://en.wikipedia.org/wiki/Gyrator

3. Oct 26, 2012

### swraman

Interesting, I have never heard of this. I will check it out.

Also, if it helps my goal is to simulate a mechanical resonance. Both input and output must be voltage measurements (no ammeter).

Last edited: Oct 26, 2012
4. Oct 26, 2012

### swraman

I tried this out, it doesn't seem to work. I think my fake/desired "inductance" value is too high (1000H), and am probably running into some issue with that. Is this a likely possibility?

I was looking for a solution possibly with a different circuit design. Any other ideas?

5. Oct 26, 2012

### AlephZero

Hm... you must be doing something wrong. Google will find plenty of circuits using this idea that do work.

FWIW you seem to be having problems with units (or typos). In your OP you said
C = 100 uF and L = 100 uH gives 1500 Hz resonance, which is off by a factor of 105. Now you switched to 1000H

1H and 100 uF seems more like it to me.

Rememeber that if you are using electrolytic caps, the leakage will reduce the Q factor, and you may need non-polarized electrolytics in this type of circuit. So it might be better to reduce the C values to avoid electrolytics altogether.

6. Oct 26, 2012

### Staff: Mentor

Another alternative for the caps is to use two electrolytic caps back-to-back to make a non-polar capacitor of half the value. It's a common trick used in audio circuits and link-powered control networks...

7. Oct 29, 2012

### swraman

C = 100 uF and L = 100 uH gives a resonance of 10000rad/sec (1/sqrt(LC)), or 1592hz. Am I incorrect in this first step?

These values required that we have a tiny resistor in order for the resonance to be underdamped, which required too much current from my source.

Can this "gyrator" inductor "simulator" be used for abnormally large inductance values, on the order of 1000H?

8. Oct 29, 2012

### AlephZero

OOOPS. I forgot fhe square root. C = 100 uF and L = 100 uH does indeed gives a resonance of 10000rad/sec

You should be able to make a simulated 1000H inductor, but why not make the impedances bigger by making C a lot smaller?

C = 0.01uF, L = 1H, R = 1k gives a damping factor of 0.05 for a series RLC circuit.
http://www.calctool.org/CALC/eng/electronics/RLC_circuit

Your simulated inductor L = Rl.R.C could then say RL = 1k (acting as the series resistor) R = 10k and C = 0.1uF.

Those values also get rid of the issues with electrolytic caps.

9. Nov 1, 2012

### swraman

Thanks, I will try that out. I didn't try using a lower C because I didn't think it was possible to get an inductor with L>1mH.

Thanks - Ill let you know how this goes.

10. Nov 29, 2012

### swraman

AlphaZero,

thanks again for the help. But I have not been able to get it to work. I have checked my Op amp and it is working properly. Even with the values suggested I see no resonance in the response measurements. It seems like a pretty flat response from 0Hz past the resonance, with a gain of about .1.

Any other suggestions or things I may be overlooking?