Resonance: RLC for Tweeter of Speaker

We are modeling the tweeter of speaker and its enclosure with the circuit shown in the figure attached.

We are also given an "magnitude of input impedance" curve as shown in the figure with various data points.

With access to these various data points we are able to determine the various components values within the equivalent circuit.

At low frequencies only R1 is visible, thus |Zin| = R1.

At the resonance frequency due to the enclose we will see R1 and R2 in series, thus the peak value in the |Zin| graph is R1 + R2. Since we know R1 already we can solve R2.

At high frequencies we will see R1 in series with L1, but due to the high frequency we can essentially neglect R1, thus |Zin| ≈ ωL1.

So the linear portion seen after the hump due to the resonance from the encloser can be thought to have an angle β, such that tanβ = slope = L1.

Thus we can use the data computes to compute the slope on this section of the curve and solve for L1.

Now that part I am stuck on is with regards to L2 and C2.

Is it possible to find L2 and C2 uniquely? If so, how?

Thanks again!

I thought resonance was where the magnitude of the impedance of C and L2 are equal so the load looks purely resistive.

L1 creates a "rolloff" so the speaker output should drop at about 20dB/decade as frequency goes up.

I don't think this problem can be solved without more information. You don't even have a frequency.

A tweeter usually has a sealed back so its enclosure has nothing to do with its resonant frequency. The highpass crossover network in the speaker usually rolls off above the resonant frequency because a tweeter is never played at its resonant frequency.

Audioguru said:

A tweeter usually has a sealed back so its enclosure has nothing to do with its resonant frequency. The highpass crossover network in the speaker usually rolls off above the resonant frequency because a tweeter is never played at its resonant frequency.

Click to expand...

This is indeed what we are implementing, a crossover network filtering out low frequencies for tweeter and high frequencies for the woofer, both with cutoff frequencies of about 3kHz.

Regardless, my main concerns are about L2 and C2.

Can they be determined uniquely? If so, how?

You may have inadvertently answered my question in you previous post, and if that's the case I'm still missing it.

(If this is indeed true could you dumb it down further?)

I don't think this problem can be solved without more information. You don't even have a frequency.

Click to expand...

We have various data points for the graph I mentioned, all of which include a frequency value.

Here is an article that explains everything about a speaker's passive crossover network including adding an RC Zobel network to flatten the woofer impedance and a tuned LCR circuit to null the tweeter's resonance peak:
http://www.sound.westhost.com/lr-passive.htm

\(Q = \frac{f_{o}}{\Delta f}\)

I should be able to approximate,

\(\Delta f\)

from the data points on the graph by finding two points on either side of the maximum such that |Zin| has a value that is half of this maximum value, occuring at the resonance frequency.

Once I have this I have Q becuase I know the resonant frequency at which the maximum occurs and Q will relate back to L2 and C2 as such,

\(Q = R \sqrt{\frac{C}{L}}\)

since it is modelled as a parallel RLC.

My guess is that L2 and C are mechanical in nature. L2 represents a spring and C represents a mass. R2 is a damper. The spring is a combination of the compressibility of the air and the deflection (mounting) of the speaker(s). (If you push gently on a speaker and release it, it will spring back.) The mass is probably mostly from the speaker coil and cone but probably some mass from the air. That would fit with the resonance of the speaker enclosure.

I'm not sure how simple it would be to find "unique" values of L2 and C. It would probably take some experimentation with different speakers and enclosure sizes. If the enclosure is ported, it will have an effect on the mechanical response too.

In mechanical terms, the spring constant "k" (force vs. deflection) is the reciprocal of L (or L2 in this case), and the mass "m" is equivalent to C. So if an electrical resonant frequency (in radians) is defined as the square root of 1/LC, then the mechanical resonant frequency is defined as the square root of k/m.