Polar & Power Response & Sensitivity

2b. Polar response, Beamwidth, Directivity & Q.

Figure 6: Polar response

Purpose: To determine the amplitude response of the subwoofer at various angles along a common plane, referenced to the on-axis amplitude response and measurement position.

Value: In addition to providing a wealth of information regarding the off-axis amplitude response characteristics of the subwoofer a variety of other performance specifications can be derived from the measurement data, such as beam width (BW), Q, directivity index (DI) and power response. Note that Q for directivity bears no relation to the quality factors Qms, Qes, and Qts. mentioned in Section 1b.

Method of Measurement: Ground-plane measurement lends itself particularly well to collecting polar response data as it requires only a bare minimum of hardware resources. Essentially, the sub is set up as for a standard on-axis measurement session. The reference point is established (the point of intersection between the reference axis with the reference plane) and the on-axis measurement is made. (If it is practical to raise the sub above the ground any response plot discrepancies owing to mutual coupling or baffle-size increase owing to actual-virtual system interaction can be minimized, however minimal they may be). The sub is then rotated, in angular increments, typically on the order of 5°, 10° or 15°. Below left shows a collection of such plots, in this case the measurements were made at 15° increments.

Once all the measurements have been collected, the curves are then normalized to the 0°on-axis curve by dividing each curve into the 0°on-axis curve. In doing so, the resulting plotted response at each point along the measurement path is relative to the on-axis response. Below right shows the result of normalizing the data plots at left. Though the data curves presented below right are normalized polar amplitude response curves, they are not usually presented in Cartesian format. Rather, they are more commonly seen as presented above in Figure 6.

Regardless of how or where you locate your reference measurement point, for polar response measurement purposes, the reference point and the point of rotation must be one and the same. As such, it is essential to keep the distance from the reference or rotation point to the measurement microphone identical at each angular position used during the measurement sequence. Keep in mind, too, that holding the sub in one position and moving the microphone from measurement point to point produces results just as valid as rotating the sub and holding the measurement mic in one position. Marking the rotation point and drawing radii of equivalent length from it to each angular measurement position with a piece of chalk is a dirt-cheap way to set up for the polar amplitude response measurement sequence. Once your artwork is complete all you need do is place the sub so that the reference measurement point intersects an imaginary axis (shown in blue in the graphic below), normal to the ground, and drawn through the previously marked point of rotation. You’re now ready to measure! (If your initial measurement set was taken in the horizontal plane, vertical plane measurements can be subsequently be taken by tipping the sub on its side and repeating the sequence.

Figure 7: Horizontal plane polar response plot measurement layout

2b.II: Beam width (BW), Q, Directivity index (DI) and Power Response

Referring now to Figure 6, Beamwidth (also referred to as “coverage angle”) is that angle formed by drawing 2 radii, located either side of the reference axis, where the amplitude response, for a given frequency has decreased by 6dB, with respect to the 0° on-axis reference value. In Figure 6, the BW at 320 Hz is 180°.

Q is the numerical expression of the directionality of a system’s response. Strictly speaking, directivity is the ratio of axial intensity of the actual source to the intensity that would be produced by a point source emitting an equivalent amount of power. A source with equal output at a given frequency in every direction would have a Q = 1. With increasing directionality the Q value increases. A quick, first-order approximation of Q at a particular frequency (assuming axial symmetry) can be had by the following formula:

Q = 360°/BW°
where BW is specified in degrees.

The Directivity Index is simply the Q value specified above, expressed in dB. Specifically:

DI = 10 * Log (Q) (dB)

Just as with on-axis measurements, polar response data should be presented with information such as drive level, measurement distance, etc, providing sufficient detail depth & breadth to allow the measurement sequence to be reproduced by anyone wishing to do so.

2c. Power response.

Figure 8: Red curve: system on-axis response; blue curve: system power response

Purpose: To determine a subwoofer’s power response, expressed in dB spl.

Value: The axial amplitude response measurement, illustrated in Section 2a, has traditionally been a mainstay of the objective assessment process. However, as important as it is, it does not provide -in and of itself - sufficient information to convey as complete a picture as is needed in working up an objective assessment of a subwoofer. Where the axial amplitude response measurement resents a view of the sub’s direct sound response characteristics as measured at a single point in space, the power response gives an overall view of the amplitude response characteristics of the sound field generated by the sub, as measured at several points in space.

Method of Measurement: The approach here is similar to that for capturing polar response data (see section 2b). And just as with polar response measurement, groundplane measurement lends itself nicely to the power response measurement process.

Common practice is to take measurements at 15° intervals, in both the horizontal (below, left) and vertical plane (below, right). (A common alternate is 15° intervals in the forward hemisphere, 30° intervals for the rear hemisphere; various factors may dictate less or more widely spaced intervals and measurement at intermediate angles). The resultant plots are then averaged to produce a good approximation of what is commonly referred to as the power response of the system.

Just as with on-axis & and polar amplitude vs. angle measurements power response data should be presented with information such as drive level, measurement distance,etc, providing sufficient detail depth & breadth to allow the measurement sequence to be reproduced by anyone wishing to do so.

2d. Sensitivity

Figure 2: Sensitivity

Purpose: To determine the sound pressure level produced at 1m, on-axis, when 1 watt (power sensitivity) or 2.828 Vrms (voltage sensitivity) is applied to the subwoofer.

Value: Indicates the actual acoustical output, in terms of sound pressure level, that will be produced by a subwoofer, given an electrical input signal of a specified voltage or wattage, the latter determined by the nominal system impedance value (1W = V^2/Znom). Knowing a system’s sensitivity assists in matching multiple systems. Also, dB-SPL acoustical output can be calculated given a known amplifier wattage.

Approach: Sensitivity can be either measured or calculated (See Engineers Note #1 for the latter or graphic at right).

Signal Used: Swept sine wave (320 Hz to 10 Hz), capturing sufficient data points to ensure post-processing accuracy, displayed on a semi-log plot, charting both magnitude and phase. The test signal is delivered at a pre-determined voltage, typically 2.828Vrms or calculated using 1W = V^2/Znom. Subsequently scaling the amplitude response plots to 1m, the sensitivity of the system can then be determined.

Metric specification: XY dB spl, X-Oct averaged SAS, 1W or 2.828 Vrms/1m/4π/Znom

Σngineers Note #4… How do I calculate the mid-band acoustic output of a sub if I know the sensitivity of the unit, given a known electrical input, in Watts?

Where:

dB-SPL Out = Mid-band, On-axis, acoustical output at 1m (dB-SPL)
Sensitivity = Measured or calculated sensitivity of the device (dB-SPL)
W = Input electrical power (Watts)

Worked Example

The 1m, on-axis, sensitivity of a subwoofer is 87 dB - spl. The sub is a powered system and features an amplifier rated at 300 Watts. What is the mid-band, on-axis dB-SPL this sub can theoretically produce at 1m when fed an input at 300W?

dB-SPL Out = 87.50 dB-spl + 10 * Log(300.0)
dB-SPL Out = 112.1 dB-SPL, at 1m, On-axis, 2 –π SR

When calculating efficiency, sensitivity and so forth, indicate if the values calculated are for a 2 or 4π-SR environment. “SR” stands for “steradian”, the mathematical expression for a solid angle. 4π or 2π -SR is thus the mathematical expression for full or half-space, respectively. 4π or 2π –SR is also sometimes expressed as simply “4π” or “2π”.