**The problem**

At high frequencies, a loudspeaker radiates sound directly
forwards. At low frequencies, the sound "leaks" around the sides
and rear of the enclosure, and becomes omnidirectional.
This transition manifests itself as a loss of output on the
forward axis at low frequencies, and in an anechoic chamber the
loss is 6dB, or a halving of SPL. With a spherical enclosure the
transition approximates a gentle first order step, with its centre
at a frequency f_{3} which is related to the baffle
dimensions (Olson, 1969), but with most speaker boxes diffraction from
the edges of the baffle tends to add wiggles to the response which
can be as large as the step.

In a system which uses drivers in a modest-sized enclosure, when the drivers themselves are optimised for flat response in a significantly larger baffle, there will therefore be a low frequency loss in the the axial response. Unless compensation is applied for the baffle step, the speakers will sound subjectively thin. By the way, the step is typically at rather higher frequency than those where "room gain" can be used.

John Murphy has an excellent
article
on the baffle step on his site.
John's approach is based on Harry Olson's work on diffraction
from free-standing enclosures of various shapes and sizes, and
models the diffraction loss as a first-order 6dB step, which is
3dB down from the low-frequency asymptote at some frequency
f_{b}, related to the baffle size by
f_{b}=115/W_{b}, where W_{b} is the baffle
dimension in metres.

The 6dB transition theory assumes that all of the energy that is not beamed into the forward half-space is lost for ever, where common sense tells me that a large part of it will be reflected back from the rear and side walls of a normal listening room. An alternative solution is to compensate for half the subjective low frequency loss, or 3dB.

Some speakers are designed for "boundary reinforcement" - in other words, they are meant to be used right against the wall. These include the Naim Audio range and some of Linn's speakers, and don't take the baffle step into account. Many others are intended for placement well away from the walls, and so have at least partial compensation for the loss of rear radiation. Most commercial loudspeakers are placed within a metre or two from the rear wall, so that the effective low-frequency loss is less than 6dB; some thoughtful designers use a mid-bass driver with a response which slopes up gently with rising frequency, so that this response trend partly compensates for a 6dB baffle step compensation and gives an overall axial response which is flat in the room.

In general it is almost impossible to engineer in the exact physical working environment that the speaker is to be used (unless you design both the speaker and the listening room at the same time!). For instance, speakers will sound very different with their backs to a brick or stone wall than they will against a stud wall. For a simple two-way with a passive crossover there's not much you can do about the problem, and it's simply a matter of moving the speakers around until you get the sound you like. However, if the speaker is designed from the start to be used with an active crossover we have plenty of flexibility to manipulate the frequency response, and here I describe a practical solution which allows the speaker to be used in a wide range of situations.

**The circuit**

I used the following circuit, which has unity gain at high frequencies and an adjustable gain at low frequencies:

The response is

,

and the magnitude of the response is

.

If we define the magnitude of the low-frequency limit
as A_{0}, we can calculate the ratio of the two resistors:

At some transition frequency f_{0,}
let the desired response be A_{1},
which we choose to be some sensible value between unity and the DC limit
A_{0}. I chose the geometric mean, equal to
the square root of A_{0}, which gives the
midpoint of the step on a decibel scale. Then we can calculate the values
of both the resistors:

where w_{0}=2 pi x f_{0}.

**6dB step**

This is required to compensate for the diffraction loss of a speaker in free space.

If A_{0} = 2 ~ 6dB and A_{1} = sqrt(2) ~ 3dB then

If C = 116nF (two 47nF and one 22nF in parallel) and f_{0
}= 450Hz (for a 25cm baffle), R_{1
}= R_{2 }= 4311 ohms (4K3 will be perfectly suitable).
For other step frequencies simply scale the capacitance so that
C = 116nF X 450Hz / f.

John Murphy suggests the following model response for the effect of diffraction for a baffle of width Wb:

where w_{3}=2 pi x f_{3} and f_{3} is the
frequency where there is 3dB of loss relative to the half-space
high-frequency limit, related to the baffle dimensions by John
Murphy's formula

This is exactly compensated for by my circuit when
A_{0 }= 2, and f_{3} = f_{0}.

**3dB step**

In a real-life listening room, some of the low frequency
diffracted sound will be reflected back towards the listener. In this case
the loss will be less than 6dB, and will generally not be a smooth function
of frequency. We can still attempt to compensate for the change in
directivity, however, by simulating a broad loss at low frequencies.
If we choose A_{0} = sqrt(2) ~ 3dB and
A_{1} = 2^{1/4} ~ 1.5dB then

and

Using C = 116nF and f_{0} = 450Hz,
R_{1} = 8753 ohms and R_{2} = 3626 ohms.

Since it is almost impossible to know exactly what fraction
of the diffracted wave will find its way to the listening position,
and since the effective baffle dimension is also hard to calculate
(particularly for a truncated pyramid sitting on top of a
rectangular box), it is useful to be able to experiment with
different settings for f_{0} and W_{b}.

I calculated component values for a switchable equaliser, with a choice of step frequencies and either five (1.5dB steps) or seven (1dB steps) low-frequency gains (I shall probably use seven). These are based on a value for C of 100nF for a step centred at 1kHz. Here are some sample component values:

Step frequency:

Frequency | W_{b} |
C (and practical values) |

250Hz | 46 cm | 400.0nF ~ 100n+100n+100n+100n |

300Hz | 38 cm | 333.3nF ~ 100n+100n+100n+22n+10n |

350Hz | 33 cm | 285.7nF ~ 100n+100n+47n+22n+22n |

400Hz | 29 cm | 250.0nF ~ 100n+100n+47n+2n7 |

450Hz | 26 cm | 222.2nF ~ 100n+100n+22n |

500Hz | 23 cm | 200.0nF ~ 100n+100n |

550Hz | 21 cm | 181.8nF ~ 100n+47n+22n+10n+2n7 |

600Hz | 19 cm | 166.7nF ~ 100n+47n+22n |

650Hz | 18 cm | 153.8nF ~ 100n+47n+4n7 |

700Hz | 16 cm | 142.9nF ~ 100n+22n+22n |

750Hz | 15 cm | 133.3nF ~ 100n+22n+10n |

850Hz | 14 cm | 117.6nF ~ 100n+10n+4n7+2n7 |

1kHz | 12 cm | 100.0nF |

Gain (1dB or 1.5dB steps):

Gain | R_{1} |
R_{2} |

0dB | 13815 ohms | 0 ohms |

1dB | 13815 ohms | 1686 ohms |

1.5dB | 9204 ohms | 1735 ohms |

2dB | 6896 ohms | 1786 ohms |

3dB | 4585 ohms | 1892 ohms |

4dB | 3426 ohms | 2004 ohms |

4.5dB | 3038 ohms | 2062 ohms |

5dB | 2727 ohms | 2122 ohms |

6dB | 2259 ohms | 2248 ohms |

**References**

Olson, H. F. "Direct Radiator Loudspeaker Enclosures", JAES Vol.17, No.1, 1969 October, pp.22-29

Back to DIY page.

Alex Megann, May 2000