This version (28 Feb 2013 16:12) was approved by dmercer.The Previously approved version (14 Feb 2013 20:10) is available.

# Activity: Measuring a Loudspeaker Impedance Profile

## Objective:

The objective of this lab activity is to measure the impedance profile and the resonate frequency of a permanent magnet loudspeaker.

## Background:

The chief electrical characteristic of a dynamic loudspeaker is its electrical impedance as a function of frequency. It can be visualized by plotting it as a graph, called the impedance curve.

The most common type of loudspeaker is an electro-mechanical transducer using a voice coil connected to a diaphragm or cone. The voice coil in moving coil loudspeakers is suspended in a magnetic field provided by a permanent magnet. As electric current flows through the voice coil, from an audio amplifier, the electro-magnetic field created by the current in the coil reacts against the permanent magnet's fixed field and moves the voice coil (also the cone). Alternating current will move the cone back and forth. The movement of the cove vibrates the air producing the sound.

The moving system of the loudspeaker, including the cone, cone suspension, spider and the voice coil, has a certain mass and compliance. This is most commonly modeled as a simple mass suspended by a spring that has a certain resonant frequency at which the system will vibrate most freely.

This frequency is known as the “free-space resonance” of the speaker and is designated by FS. At this frequency, since the voice coil is vibrating with the maximum peak-to-peak amplitude and velocity, the back-emf generated by coil motion in a magnetic field is also at its maximum. This causes the effective electrical impedance of the speaker to be at its maximum at FS, known as ZMAX. For frequencies just below resonance, the impedance rises rapidly as the frequency approaches FS and is inductive in nature. At resonance, the impedance is purely resistive and beyond it, as the impedance drops, it looks capacitive. The impedance reaches a minimum value, ZMIN, at some frequency where the behavior is mostly (but not perfectly) resistive over some range of frequencies. A speaker's rated or nominal impedance, ZNOM, is derived from this ZMIN value.

Knowing the resonate frequency and the minimum and maximum impedances are important when designing cross over filter networks for multiple driver speakers and the physical enclosure the speakers are mounted in.

### Loudspeaker Impedance Model

To help understand the measurements we are about to make, a simplified electrical model of a loudspeaker is shown in figure 1.

Figure 1. Loudspeaker Impedance Model

The circuit in figure 1 has a dc resistance placed in series with a lossy parallel resonant circuit made up of L, R, and C, which models the dynamic impedance of the speaker over the frequency range of interest.

• Rdc is the dc resistance of the loudspeaker as measured with a DC ohm meter. The dc resistance is often referred to as the DCR in a speaker/subwoofer data sheet. The dc resistance measurement is usually less than the driver's nominal impedance ZNOM. Rdc is typically less than the specified loudspeaker impedance and the novice loudspeaker enthusiast may be fearful that the driver amplifier will be overloaded. However, because the inductance (L) of a speaker increases with an increase in frequency, it is unlikely that the driver amplifier actually sees the dc resistance as its load.

• L is the voice coil inductance usually measured in millihenries (mH). Typically, the industry standard is to measure the voice coil inductance at 1000 Hz. As frequencies increase above 0 Hz, there is a rise in impedance above the Rdc value. This is because the voice coil acts as an inductor. Consequently, the overall impedance of a loudspeaker is not a constant impedance, but can be represented as a dynamic profile that changes with input frequency as we will see when we make measurements. Maximum impedance, ZMAX, of the loudspeaker occurs at the resonant frequency, FS, of the loudspeaker.

• FS is the resonant frequency of a loudspeaker. The impedance of a loudspeaker is a maximum at FS. The resonant frequency is the point at which the total mass of the moving parts of the loudspeaker become balanced with the force of the speaker suspension when in motion. The resonant frequency information is important to prevent an enclosure from ringing. In general, the mass of the moving parts and the stiffness of the speaker suspension are the key elements that affect the resonant frequency. A vented enclosure (bass reflex) is tuned to FS so that the two work in unison. As a rule, a speaker with a lower FS is better for low-frequency reproduction than a speaker with a higher FS.

• R represents the mechanical resistance of a driver's suspension losses.

## Materials:

Analog Discovery Lab Hardware
1 - 100Ω Resistor (or any similar value)
1 - Loudspeaker, it is best if the speaker is one with a cone diameter larger than 4 inches such that is has a relatively low resonant frequency.

## Directions :

Build the circuit shown in figure 2, preferably using your solder-less breadboard. The loudspeaker can be in an enclosure or not.

Figure 2 Speaker measurement set up

Connect waveform generator 1, and the two scope channels to the loudspeaker circuit as shown.

## Procedure:

Start the Waveforms software. On the More Instruments menu select the Network Analyzer instrument. Under the Scope Channels controls unselect the Use Channel 1 as Reference. Set the start frequency to 10Hz and the end frequency to 1KHz. Set the amplitude to 2 Volts and the offset to 0V. Set the max gain to 1X. Under the Settings drop down tab open the options window and set the settle time to 40 and the FFT window to cosine.

A few words on why these setting should be adjusted. As the frequency is swept the AWG output is stopped briefly between frequency steps and the signal driving the speaker will be turned off. The speaker is a mechanical system with resonance and this step change in the driving signal will cause it to ring at the resonate frequency. In order to make an accurate measurement at the driving frequency we must wait for the ringing to die out. The amount of time needed will depend on the particular speaker being measured. The 40 mSec suggested above was the correct value for the speaker used in this example. Your results may vary depending on your particular speaker. Switching to the cosine window function gives a more accurate amplitude result.

Hit the green Run Single button. You should see the frequency response of the voltage across the loudspeaker and the current through the speaker (by measuring the voltage across the 100 ohm resistor). The data on the screen is plotted in dB so the vertical scale is not in volts. An example plot is shown in figure 3. Your speaker will probably look much different than this.

Figure 3 Example sweep

You can now Export the data, as gain not in dB to make the math easier, to a comma separated values file and load it into a spreadsheet program such as Excel.

By saving the data as gain the signal generator amplitude (in volts) falls out of the equation.

Where:
G1 is the channel 1 gain ( voltage across speaker )
G2 is the channel 2 gain ( voltage across 100Ω )
A is the AWG amplitude

You can calculate the magnitude of the speaker impedance is by dividing the channel 1 voltage gain by the channel 2 voltage gain all multiplied by the 100Ω resistor. An example plot is shown in figure 4. Your speaker will probably look much different than this.

Figure 4 Calculated example impedance plot

The speaker impedance is very close to 8Ω in the linear region but is much higher at the resonance frequency FS.

## Questions:

Based on your measured data extract the L C and R for the speaker electrical model shown in figure 1 for the speaker you used. You can measure Rdc with a DC ohm meter if you have one available. Ignore LINPUT as it will be small compared to L. Enter these values in to a circuit simulation schematic of the model and generate a frequency response sweep from 10 Hz to 1 KHz and compare your model to the data you measured in the lab.