
Appendix L Hframe and Uframe openbaffle woofersThe electrical circuit diagram below models the essential acoustic parameters of the Hframe. Reducing the length of the left transmission line to zero (T1 => 0) and increasing the length of the right transmission line to d2 = L, such that T2 = L/c, leads to the Uframe. The front and rear voltages are shown below for the case that L = 0.6 m, (24"), the transmission line or waveguide impedance Z_{w} = 1000, and the impedance seen out the open end is R_{r} = 400. Addition of the front and rear outputs for distant observation points at different angles a yields magnitude and phase of the radiated signal. Added to the graphs below are magnitude and phase of the Hframe at zero degree for comparison purpose. At low frequencies the Uframe analyzed has 3.6 dB higher output onaxis than the Hframe. At 180^{0} offaxis the output is about 9 dB down. The phase shift of the Uframe radiation is 90^{0} onaxis and +90^{0} degrees towards the rear of the frame, relative to the source. Magnitude and phase are quite frequency dependent as the resonance frequency is approached. The polar diagram (3) has been sketched for different frequencies using the magnitude graph above. It indicates somewhat unidirectional radiation, whereas the Hframe is bidirectional (1). The introduction of losses in the Uframe opening in the form of an acoustic flow resistor R_{F} tends to give the polar diagram a more uniform appearance (4). With R_{F} + R_{r} = Z_{w} = 1000 the waveguide is terminated and exhibits no resonance. The rear output voltage, V_{R}, has almost the same magnitude as V_{F}. Magnitude and phase of the radiated signal for this case are shown below. Graph (4) is drawn from these data. I have seen before an oversimplified analysis of the Uframe and Hframe. While it works for the Hframe due to the complete symmetry of the structure and the resulting cancellation of effects from both waveguide openings, it leads to incorrect answers for the Uframe. The implied assumption in this analysis is that there is no change in impedances. It is as if the waveguide continued in front and rear all the way to each observation point and only the length of the front and the rear waveguide changes as a function of the angle a. In the electrical circuit model this corresponds to setting R_{r} = Z_{w} = 1000. There is no resonance anymore, V_{F} = V_{R}, the two voltages only differing by 180^{0} in phase. The resulting "radiated" magnitude and phase for different angle a are shown below and the corresponding Hframe onaxis response for comparison. The polar diagram in this flawed analysis is a cardioid (5) with 6 dB higher output than the Hframe on axis. If on the other hand the radiation resistance is assumed to be even less than the value 400 of the previous examples and set to R_{r} = 100, then the polar diagram (6) of the Uframe tends more towards that of the dipole at low frequencies. The lower value for Rr would be indicated by a 14 dB resonance peak in the rear opening of the Uframe. None of this leads to a cardioid polar response, though it would be nice if a simple Uframe could provide it. If the objective were to build a unidirectional woofer, then a different approach is necessary. One way is to use two closed box woofers backtoback, driven 180^{0} out of phase (A). The electrical signal to the rear woofer is delayed by a time T corresponding to the time it takes the acoustic output from the front woofer to travel the distance l to the rear. A commercial example of such cardioid woofer can be seen at Meyer Sound. Rather than using an electrical delay T it is possible to derive the necessary delay from lumped acoustic elements as sketched in (B). The compliance of the air volume inside the box corresponds to an acoustic capacitor. In combination with the flow resistor Z_{F} it forms an RC lowpass filter. Below its cutoff frequency the lowpass has nearly constant delay. The difficulty in this approach lies in finding a practical flow resistor of the necessary value, linearity and flat frequency response. It is difficult to obtain a resistor that can handle the large volume flows required for a woofer and (A) becomes the best solution. For higher frequencies variations of the concept in (B) can lead to cardioids with very wide bandwidth. By proper dimensioning of various parameters the rolloff of the rear radiation, due to the lowpass filter, can be made to coincide with the decrease in rearward radiation from the front side of the box, due to front baffle size and driver diameter becoming acoustically large. When combined with a coaxial tweeter, that takes advantage of the horn like nature of the midrange cone, it becomes possible to extend the unidirectional frequency range even further. A commercial example is the Revolution loudspeaker by Gradient Ltd. . To go deeper into unidirectional speaker design you may
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