Consideration of Power Response in Speaker Design:
Crossover Choice and Setup.
where the limits of integration are 0 to Pi for Theta and 0 to 2Pi for Phi.
With the pressure at a given R defined for all theta and phi we can evaluate the power integral and look at the power. We begin with
two omnidirectional point sources with a crossover frequency of 2K Hz separated by 6.5". This would be typical of a 2-way speaker
with 6" woofer and 1" dome tweeter with a 4" face plate. The separation corresponds to approximately 1 wave length. Note that all
crossover simulated sum flat on axis. 1st, 2nd, 3rd, 4th, 7th and 8th order crossover are examined and presented below.
Several observations are in order. First, in all cases the vertical polar response at the crossover frequency is dependent on whether the crossover
is odd order (Butterworth) or even order (Linkwitz/Riley), but not on the order. However, there will be differences in the polar response above and
below the crossover frequency which are dependent on order due to the varying degree of overlap. Second, the power response for the odd order
Butterworth crossovers is constant in all cases. Third, the power response for the Linkwitz/Riley crossovers shows a notch at the crossover point.
Fourth, the characteristics of this notch are dependent on the order of the crossover and the separation of the drivers (not shown) and the depth of
the notch can actually be greater than 3dB. This last point is of interest since -3dB is the power level which would be obtained from two uncorrelated
sources with amplitudes of -6dB. In fact, when the driver spacing is 0.707 wave lengths (at the crossover frequency) the power response of the
Linkwitz/Riley crossovers will have a minimum of -4dB. It should also be recognized that as the driver spacing becomes small compared to the wave
length at the crossover frequency all crossovers that sum flat on axis will also have flat power response. Thus, crossovers between woofers and
midrange drivers at frequencies below 300 Hz or so generally do not introduce irregularities in the power response.
Note: The flipping of the polar response for the Butterworth crossover is normal and can be inverted by altering the polarity of one driver.
1. R. Walker, Controlled Image Design: The management of stereophonic image quality, BBC RD 1995/4.
2. L. L. Beranek, Acoustics, McGraw-Hill Book Company, 1954.
The majority of discussion on crossovers used in loudspeaker intended for home use centers the on
axis response and the vertical/horizontal polar radiation pattern. Little attention is paid to power
response. When in an open space, such as outdoors, this is a reasonable consideration since the
sound radiated in a direction other than towards a listener in the audience has a limited effect on the
over all sound. There may be other considerations, such as limiting the sound energy propagating in
directions other than towards the audience which represents a waste of power and efficiency. However,
in a typical home environment the situation is some what different since the loudspeaker is radiating
into a closed environment. Sound is reflected from the walls, floor and ceilings and these reflections,
and the uniformity of the radiated power with frequency, can have a significant effect of the perceived
sound. The ratio of direct to reflected sound is also important in the perception of detail and localization
of sound sources, and the creation of the stereophonic illusion. Since the listening position for
optimal stereophonic illusion is fairly limited, and since the majority of critical listening is performed with
the listener at a relatively fix position, typically seated, the question arises, is too much attention be
placed on character of the polar response while ignoring the consequences on power response?
Indeed, if we accept the premise that listening height is relatively fixed, how significant is it if the vertical
polar response is symmetrical, as with a Linkwitz/Riley crossover, or asymmetric as with a Butterworth
crossover? Even in home theater applications it would seem that horizontal dispersion and uniformity
may play a more important roll than vertical polar response.
In any event, the goal of the present discussion is to examine the behavior of the power response of
conventional speaker systems using different order crossovers. We will begin by looking at a crossover
between two omnidirectional point sources. We will then expand the discussion to drivers of finite
diameter and include driver off axis effects. We will then include the effects of the baffle step. Finally,
we well look at the combined power from a stereo pair of "ideal" loudspeakers in a typical setup. We
should note, however, the results presented here do not include the effects of the proximity of the
speaker to any reflecting surface such as a floor, wall or ceiling. The effect of floor interaction are
addressed in the discussion of midrange and woofer power matching.
Before starting we need to present some analysis. To obtain the total radiated acoustic power we must
integrate the pressure over the surface of a sphere enclosing the source.