See Impedance and Loudspeaker Parameter Measurement (pdf file).
E1. Impedance Measurement. Use the soundcard probe with amplifier and a Matlab script to measure the inductance L and the resistance R of the "air core" inductor shown in the following picture.
The equivalent circuit of this inductor is shown in the figure below and the goal is to estimate the values of R L from the measurement of Z over a whole frequency range, e.g., from 20 Hz to 20 kHz.
Use the following circuit to measure the voltage across the inductor (element E) and current through the inductor simultaneously using the left (for vS(t)) and right (for vO(t)) channels of the soundcard probe with amplifier.
Values for the reference resistor RM that is used to convert current to voltage range from about 10 to 47 ohms. Use a value such that both the left and the right channel of the wav file that you record have about the same amplitude. Make a note of which value you used as you will need this for the processing in Matlab. Record vS(t) and vO(t), sweeping a sinusoid in logarithmic mode over the frequency range from 20 Hz to 20 kHz in about 5 seconds.
To process the resulting wav file, start from the fresp_14.m script (which uses the functions avgT.m and shift90.m). Make a new Matlab script file, e.g., called Zwplot14.m and import the beginning of fresp_14.m up to and including the statement
vO = y(:,2)'; %Output vO(t) is right channel
Then you need to compute vE(t) = vS(t) - vO(t) and iE(t) = vO(t)/RM. To compute the magnitude of Z you need the amplitude AvE of vE(t) and the amplitude AiE of iE(t). Then you can use the Matlab statement
magZ = abs(AvE./AiE); %Magnitude of Z
To determine the phase of Z you can start from computing
vEcos = vE./AvE; %Normalized cosine from vE(t)
ord = 1000; %Filter order
vEsin = shift90(vEcos,Fs,ord); %Shift vEcos by -90 deg
where vE is vE(t) and AvE is the amplitude of vE(t). Then you can compute
a = avgT(2*iE.*vEcos,Fs,T); %a = AiE*cos(phi)
b = avgT(2*iE.*vEsin,Fs,T); %b = -AiE*sin(phi)
phi = -atan2(-b,a); %Phase phi of Z mod 2*pi
where phi is the phase of Z in radians. If necessary use the unwrap command to unwrap the phase and then convert it to degrees for plotting.
To determine w0 = R/L it is easier to use plots of |Z| and angle(Z) versus a linear frequency axis (as opposed to the logarithmic one used in fresp_14.m). Thus, use the plot command in Matlab to plot magZ and phi (in degrees) versus the (estimated) frequency axis in ffest. An example that uses the data in RLs.wav with RM = 47 ohms is shown in the graphs below.
Once you obtain the correct graphs for the magnitude and the phase of Z, you can compute R and L using:
Verify the result for R by measuring the dc resistance of the inductor with a multimeter. Check the result for L by looking at |Z(w)|/w for w > w0. You may also want to measure the known 15 mH inductor from the LC box to make sure your hardware and software for impedance measurements works correctly.
E2. Parameter Estimation of Resonant Circuit. The general goal of this experiment is to determine R, L, and C of a resonant circuit (either series or parallel resonant circuit) from two-terminal impedance measurements over a range of frequencies.
(a) Use the air core inductor in series with the 6.8 uF capacitor from the LC box as a series resonant circuit with equivalent circuit as shown in the figure below. You can either use a separate resistor for Rs (with value in the range of about 10...100 ohms) or assume that Rs is the resistance of the inductor wire.
Make a plot of Z versus omega=2*pi*f for f = 20 Hz ... 20 kHz. Then use the formulas
and
to determine Rs, Ls and Cs. Compare your findings with the actual (known) element values. If you see any big discrepancies, you need to find out where they come from before you continue with the other experiments.
(b) To measure a parallel resonant circuit and identify its elements from a two-terminal impedance measurement, use the circuit below with Lp = 15 mH and Cp = 6.8 uF from the LC box and Rp = 150 ohm.
Use a sinusoidal sweep from 20 Hz ... 20 kHz (5 sec sweep time, logarithmic sweep) to measure Z. Plot the result in Matlab and estimate the (known) values of Rp, Lp and Cp using the formulas
and
(c) The schematic below shows a parallel resonant circuit with a series resistor Rs and a series inductor Ls. Use the same elements and values for the parallel resonant circuit as in (b). Use the air core inductor for Ls and make Rs = 100 ohm.
Measure and plot the impedance Z of the circuit shown above with Ls and the circuit shown below without Ls, both in the frequency range from 20 Hz ... 20 kHz.
Over which range of frequencies is Z(w) (approximately) the same for the two circuits? Can you explain this result?
Use the following formulas to determine Rs, Rp, Lp, and Cp from the impedance graph that you obtained for the above circuit.
Note also that
(d) Finally, use the air core inductor and Rs = 100 ohm to implement the circuit in the next figure
Measure and plot Z(w) for the same frequency range as in (c) and compare the outcome to the two graphs (magnitude and phase each) you obtained in (c). Over which range of frequencies is your new graph similar to one of the graphs from (c)? Why? Can you determine Ls (and Rs) from the region where the graphs are similar? How?
E3. Equivalent Circuit of Speaker. The first goal of this experiment is to find the main parameters in the electrical equivalent circuit of an electrodynamic loudspeaker. The second goal is to find out how these parameters change when the speaker is enclosed in a box or if the cone is loaded with additional weight.
(a) The loudspeaker that is analyzed in this experiment is a small alnico magnet speaker with a nominal impedance of 8 ohm and a power rating of 0.25 W. A picture of the speaker is shown below.
A (simplified) equivalent electrical circuit of the speaker is shown in the following schematic.
The series resistor Re and the series inductor Le model the resistance and the inductance of the voice coil. The three parallel elements are electrical equivalents for the mechanical losses (Res), the compliance (Les), and the moving mass (Ces) of the speaker system. Compliance is the inverse of stiffness and, in open air, Les is an inverse measure of the stiffness of the speaker suspension. If the speaker is mounted in a closed box, then Les also includes the (inverse of the) stiffness of the air trapped in the box. The capacitance Ces is the electrical equivalent of the mass of the cone, the voice coil, and the air mass that is moved by the cone.
Use the experience and insights you got from the impedance measurements and element parameter estimations in experiment 2 to determine Re, Le, Res, Les, and Ces for the speaker shown above in free air. The example below (made for a much larger speaker) gives you some idea of what to expect. Note that, depending on what you use the impedance curve for, you may choose to use either a linear or a logarithmic frequency axis.
Example: The file woofer.wav is a recording of vS(t) (left channel) and vO(t) (right channel) that was made from a 6.5" woofer with a resonance frequency of about 70 Hz in free air. A reference resistor with value RM = 47 ohm was used to convert current to voltage. The graphs below show the measured magnitude and phase of the speaker impedance. The dashed purple lines show the (computed) impedance of the electrical equivalent circuit for the estimated element values shown in the title of the lower graph.
(b) Repeat (a) for the speaker "in a closed box" as shown in the picture below.
Note that it does not matter on which side of the speaker the box is. The main point is that the front and the back of the speaker are acoustically isolated from each other. When you make the measurement of the speaker in the box, make sure that the speaker fits exactly onto the plastic tube and you get a reasonably tight air seal. Look at how the impedance curve and the element values change. In particular, the resonant frequency is going to change. From the description of what the electrical equivalents correspond to in the mechanical speaker system, do you expect the resonant frequency to become larger or smaller? Why?
(c) Repeat (a) for the speaker in free air, but with the cone "loaded" with one or more Super Hero stamps as shown in the picture below.
When you perform this measurement, keep in mind that this is a rather small speaker and its ability to handle extra weight is very limited, so start with one or two of the Super Hero stamps. Again, you should observe a (rather small this time) change in the resonant frequency. Should this be a change up or down? why?
E4. Frequency Response of Speaker.
An important characterization of a loudspeaker is its frequency response. One way to measure the frequency response is to bring the speaker to a fully anechoic chamber (i.e., a soundproof room in which there are no echoes) and use a calibrated measuring microphone. A much simpler (but somewhat less precise) method is to make use of the fact that a loudspeaker is a reversible transducer. That is, a speaker which is designed to convert electrical to acoustic energy can also be used to convert acoustic to electrical energy, thereby acting as a microphone. If two speakers of the same type are connected together acoustically as shown in the following schematic, then a measurement from electrical input to electrical output will essentially correspond to measuring H2(w) where H(w) is the frequency response of the speaker.
The figure below shows two speakers mounted on each side of a 1" piece of plastic tube. Connect one of the speakers and the left input of the soundcard probe to the waveform generator and sweep a sinusoid from 20 Hz to 20 kHz in about 5 seconds using the logarithmic sweep mode. Terminate the second speaker, which is used as microphone, with a 10 ohm load resistor and record the voltage across this resistor with the right input of the soundcard probe. You may have to use the ×10 setting for this input.
Plot the magnitude and the phase of the frequency response H(w) (using a slight modification of the fresp_14.m script).
There are some shortcomings to this method of determining the frequency response, the most obvious one (after experiment 3) being the fact that the mechanical resonance frequency of the speaker is moved significantly when the speaker is enclosed in a small box like the 1" pipe used above. This effect can be reduced by using a box with a larger volume, such as the 2" pipe setup shown below.
Make another frequency response measurement when the two speakers are acoustically coupled using the 2" pipe. Compare the result to the graphs you obtained from the 1" coupling setup. You will see some bumps in the frequency response, especially at higher frequencies, that change as the pipe length changes and thus characterize the measurement setup rather than the speakers themselves. Try to explain where some of these bumps might come from. Hint: Not all of the sound energy that arrives at the speaker which is used as a microphone is converted to electrical energy. Some of the sound energy is reflected back. For your explanation make use of the fact that the speed of sound is 344 m/s at room temperature and the wavelength lambda of an acoustic sinewave is computed as
speed of sound [m/s]
lambda = -------------------- [m]
frequency [Hz]
Once you understand the limitations imposed by the frequency response measurement method used here, use a combination of the two H(w) measurements that you made to determine the (approximate) range of frequencies for which the magnitude of the frequency response of the speaker is within ±6 dB.
E5. Crossover Network Design for Woofer/Tweeter Speaker System. (Extra credit, optional experiment.)
To obtain extra credit, attach an additional sheet to your lab worksheet with the results and findings from this experiment.
(a) Measure the free-air impedances of the woofer and the tweeter shown in the following figures.
Determine the values of Re, Le, Res, Les, and Ces for both speakers and identify the useable frequency range (above the resonant frequency and below the region where the contribution from Le becomes signifcant).
(b) From the measurements in (a) determine a good value for the crossover frequency of the crossover network. Then, assuming a constant value (of about 8 ohms) for the woofer and tweeter impedance in the range of interest, design a 2'nd order crossover network using the following information to get started.
The goal is to obtain a LPF (for the woofer) and a HPF (for the tweeter) with overlapping transition bands such that the amplifier sees a constant impedance when the two networks are connected in parallel (i.e., when ZW and ZT are connected in parallel).
(c) Use PSpice to verify your crossover network design, first using constant impedance values for the woofer and the tweeter and then using the actual equivalent circuits that you found in (a). Over which range of frequencies is the impedance Z seen by the amplifier approximately constant?
Lab worksheet in PDF format: lws04.pdf
Note: Each student needs to turn in a lab worksheet. The raw measured data for each student in a lab team will be the same, but the conclusions drawn from it are individual to each partner in the team.
©2003-2007, P. Mathys. Last revised: 3-12-07, PM.