1. The problem statement, all variables and given/known data 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. 2. Relevant equations 3. The attempt at a solution 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 suppose you could obtain one relationship between C and L2 from the resonance peak, since the imaginary part of the impedance should go to zero at ##\omega_o##. You've found values for all the other components, and ##\omega_o## can be read on the graph. Then you'd have to find another relationship from a point on the graph where all the components are contributing to the net impedance. The minimum after the resonance peak catches my eye, but off hand I don't know the formula for the minima there... Maybe the information can be gleaned from the Q of the circuit? Will the width of the peak give you the fractional bandwidth?
I think everything you've mentioned is heading in the correct direction. I want to follow through with these ideas and run with it. What formula are you reffering to? Does it have a name associated to it, or is it just a general formula for determining a minimum from a finite set of data points? Why would you select the minimum after the resonance peak? How I can relate that back to L2 and C2? When you say fractional bandwidth what are you referring to? (I've never heard that term before) Is it as such, [tex]Q = \frac{f_{o}}{\Delta f}[/tex] where, [tex]\Delta f[/tex] is the fractional bandwidth? If that is the case I could approximate, [tex]\Delta f[/tex] 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, [tex]Q = R \sqrt{\frac{C}{L}}[/tex] since it is modelled as a parallel RLC. I'd love to hear more of your thoughts. Cheers!
Not quite, at resonance you have R1 + L1 + R2 in series or (R1+R2, jX(L1)). Essentially true, however it's not that difficult to set up a complex expression that represents the whole network. With the proper Add-In, Excel can do complex arithmetic. Yes, it is. What do you know about how the Q of a circuit is defined and calculated? What is the relationship between X(C) and X(L2) at resonance? What is the relationship between R2 and X(C) or R2 and X(L) at resonance?
The minimum after the resonant peak is due to the series resonance of L1 with the parallel combination of C and L2 where the parallel combination has a negative reactance that conjugately matches the reactance of L1.
Is the method I described above with Q enough for determining L2 and C2 uniquely? (With some error of course)
Q is defined by the Δf of the half power frequencies, by either L/Z or C/Z for series circuits or Z/L and Z/C for parallel circuits. So how do we find the half power frequencies when we haven't defined the power. Since P = E^2/R then PR = E^2, so yes, you would select the frequencies at which the impedance is 1/2 the maximum value. So by dividing the frequency of peak impedance by the bandwidth between the two half impedance values, you can get the Q. At resonance the impedance seen is due almost entirely to R2 and X(C) and X(L2) are equal to R2/Q. Once you have X(C) and X(L2), you can easily calculate C and L2.