Power response of a speaker system is one of the most important aspects towards the perceived system sound. It is the total acoustic output of the speakers into the room. It interacts with the room due to reflections that ultimately impact the response at the listening position. Determining the true power response would require an integration of all points on the surface of a sphere that encloses the speaker system. That is not practical nor is measuring a large number of points on that surface for an approximation for large systems. I am aware of at least one large loudspeaker manufacturer of large venue systems that was working on a robotic measurement system capable of rotating a large system through 360 degrees on two axes that will provide the measured power response, but this is certainly rare. Short of a system such as that, other means of estimating the power response can be used.

WinPCD originally only made an equal-weighted average of the front hemisphere points. That is now updated to use a weighted average for a true power response calculation.

Power response can be determined either from measuring a pre-determined set of off-axis measurement points or from modeling the response at a number of off-axis points. A weighting scheme is then applied to each of these sample points, whether measured or modeled. The weighting is used to approximate the response of a sector of a virtual sphere enclosing the system. The sum of these sectors comprises the estimated power response. This is often done for the front hemisphere only due to difficulty of measuring the rear hemisphere due to the measurement environment for physical measurements or due to limitations of modeling the rear hemisphere.

Graphics of the concept behind the alternate methods are shown below. These are found in the paper "On the Calculation of Full and Partial Directivity Indices" with a complete description of the scheme. The paper is devoted primarily to determing the directivity index (DI), but is useful for the power weighting.

The calculation in WinPCD uses a modifed version of the weighting table in the Princeton paper. It provides data from the full-sphere. This was modified for the front hemisphere only.

The Princeton paper describes a method of measurements that are relatively easy to make for most speaker systems that can then be used for comparing different systems. The measurements are made in 5° increments and only use two planes, the vertical and horizontal planes that intersect at the origin of the coordinate system and are perpendicular to each other. This is apparently the same scheme defined in ANSI/CTA-2034A: "Standard Method of Measurement for In-Home Loudspeakers". It provides for a common reference between manufacturers, but both are limited measurements points. They also differ in the weighting tables. I cannot find a reason for the discrepency.

The images below show how the surface of the virtual sphere is divided into sectors coupled with the weighting scheme to approximate the contribution of each sector to the power response.

Coordinate System

Sample Points

On-Axis Disk

First Ring

Second Ring

The calculation in WinPCD was designed to average the front hemisphere in rings of 5° increments in both horizontal and vertical rotations. Starting from the on-axis reference (hereafter called the mic), the mic position is rotated five degrees for each move about an axis. Sample points are calculated starting from that initial z-axis mic position. The first rotation is 5 degrees around the vertical axis (y-axis). Each measurement ring starts on the zx plane. Successive samples are then made by rotating the mic position, also by 5° increments, counter-clockwise around the z-axis. This results in 72 sample points for each z-axis rotation. This continues until the final y-axis rotation to ninety degrees, positioning the mic on x-axis of the xy plane for the last ring. The end result is 1297 (on-axis reference point + all rotational) sample points.

The Princeton paper and ANSI/CTA-2034A define a set of orbits, meaning rotations on only the vertical or horizontal axes with sample points all within either the zy (vertical) or zx (horizontal) planes.

The Princeton paper provides a weighting table used to estimate the impact of the sample points within each orbit. This table was modified for use in WinPCD to account for the much higher number of sample points generated per orbit. When the calculations were being coded, the output was compared to the power response provided in John Kreskovsky's tech study paper "Consideration of Power Response in Speaker Design: Crossover and Setup" for a similar theoretical 2-way. Correlation was very good.

When WinPCD is making the calculations for each vector of the software spinorama the vector graph shows each point as it progresses through the full hemisphere. The angles displayed at the top show the vector just completed. The left angle is y-axis rotation. The right angle is the z-axis rotation.

When the calcluations are done the graph looks like concentric circles, but because it's 2D, it actually would be the appearance of viewing the surface of a sphere from the initial z-axis mic position. It defines the surface of the hemisphere of vector points. The odd-looking gap in the fourth quadrant is a graphing anomaly. It stops at the last calculation for the current ring (355 degrees) to start at the next ring (0 degrees).

Since developing this change to WinPCD, I have learned that the Dr. Floyd Toole had developed the "spinorama" measurement scheme and continued it at Harman International, the reason I changed the name of the graph in WinPCD. This was eventually incorporated into ANSI/CTA-2034A. The number of sample points used in WinPCD, however, is far higher that is described in the Princeton paper or ANSI/CTA-2034A and encompasses the full front hemisphere. It is not limited to the horizontal and vertical planes through the origin as developed by Toole. The number of samples specified for a hemisphere is 72 vs 1297 in WinPCD as stated previously. I may add a separate calculation that matches the overall power standard for comparison.

One difference in WinPCD is that all calculations are made relative to the origin (0, 0, 0) as set in WinPCD. This origin is wherever the user sets it to be, often the tweeter center. The Audio Engineering Society AES56-2008 standard is the center of the (upright) cabinet horizontally, parallel to the ground and prependicular to a specified radiatinig surface. However, diagrams in ANSI/CTA-2034A do not specify it this way, another discrepency I cannot explain. Since WinPCD is for modeling crossovers, there is no cabinet reference. The user may set the origin as desired. That is, if the user wants the cabinet to be center, then the drivers must set relative to that point. This may mean positive y-axis for the tweeter with negative y-axis for one or more other drivers.

I was aware of Floyd Toole's position that the best speaker systems have a good "listening window" response. That is a subset of the full front hemisphere measurements. He designated this window to be +/- 30 degrees horizontal by +/- 10 degrees vertical, relative to the on-axis response. Since the WinPCD calclulations were all stored for the calculation of power, it was easy to calculate this window response for display on the power response graph. That turns out to also be part of ANSI/CTA-2034A.

Another similar scheme used for testing is that of Joe D'Appolito. It can be read in the Theory Articles at audioXpress titled "Testing Loudspeakers: Which Measurements Matter". The specific quote is:

The first arrival response is just that — the first sound you hear from a loudspeaker. It is the primary source of localization and imaging in the case of stereo sound reproduction. This response is free of any room reflections. You may not always be able to listen on-axis, so the listening window response is an average response over a range of seating locations. It is still free of room reflections and as such represents what listeners experience in a typical seating arrangement. It also balances out subtle variations in on- and off-axis responses in both the horizontal and vertical planes.

Except for a slight rolloff at the higher frequencies, this response should look pretty much like the on-axis response. To determine listening window response I average on-axis response with off-axis responses in 5° increments from 25° left to 25° right and between 10° up and 10° down.

This seems to be closer to the calculation in WinPCD with 5° increments in the window, although unclear, it appears to be done in horizontal increments rather than rotational increments and doesn't address any weighting scheme to be applied to the meaurements.

There is one other difference, I think, among the measurement schemes. I can't say what it is for D'Appolito's scheme for the full set, but he does mention measuring on the tweeter axis. As mentioned previously, for the and the AES56-2008 standard the on-axis for measurements is specified to be the center of the speaker, not the tweeter axis which is the more typical listening axis.

WinPCD allows the user to set the axis to be used, but the tweeter axis is probably the most often used and should be, in my opinion, the reference axis for all measurements because it is most often the listening axis, rather than the center of the speaker system, wherever that is considered to be by the system designer. D'Appolito refers to the first arrival. This should be the primary listening axis, at least for design purposes. This should affect what is considered the listening window response as well.

After reading Part 2 of the D'Appolito article, one very interesting comment relates to the power response:

The power response is obtained by measuring responses at many locations over a spherical volume. This can only be done accurately in an anechoic chamber or, alternatively, in a totally reverberant enclosure. Because neither venue is available to me, I cannot show the power response for my example loudspeaker.

If Joe D'Appolito cannot measure the power response absent what he considers the necessary venue, how can someone in DIY accurately measure it? This provides some indirect support for using modeling techniques for determining power response by the DIY community.