Phase
Data

The TL's system response
(1 meter measurement) is the vector summation of the near field woofers response
and the near field Terminus response. Since vector summation requires magnitude and
phase, the following data sample is used to illustrate this relationship. 
The woofer magnitude and
phase can be modeled as a small signal response for Q_{tc}=0.5 close box
response. Mathematically this can be described by the Beranek equations of a Plane
Circular Piston in Infinite Baffle
_{(1)}. This implies a linear phase response. While in a general case there
are serious problems with this model, this is close to what is measured for the TL36,
Fig. 1.
Note that for F_{r}=60 Hz the phase angle = 80° and magnitude = 8 dB. It should
be noted where the phase transition occurs, vs that of the terminus response in Fig.2. 
Fig. 1 TL Terminus
response for a line D_{t}=0.6, Dacron HoloFill II fiber
click image for larger
charts

Corresponding to the near
field woofer response is the near field Terminus response, Fig. 2. The Terminus response
for F_{r} = 60 Hz the phase angle = +19.3° and magnitude = 3.8 dB. The
measured magnitude has a 20 dB attenuation, thus the reference magnitude is 203.8
= +16.2 dB in reference to the near field woofer measurement.
The D phase ~99° is very close to the TL system
requirement of 90° for maximum gain @ F_{r}.
From this we can infer that the D_{t} is close to optimum for the fiber type
used.
To relate the phase of Fig.2 vs Fig.1, you have to compare the front of the woofer's
response to that of the back of the woofer's response, ie the interior of the TL
line, shown in Fig.5.
For the Terminus the phase transition does not occur at the line resonance frequency
F_{r} related to the 1/4l, but at the 2^{nd}
harmonic frequency. This is important in analyzing the nulls in the TL system response
and the shift with frequency with stuffing density change. 
Fig. 2 TL Terminus
Response, Magnitude and Phase
click image for larger
charts


______________________________________________________________________________________
^{(1)} Acoustics, L. Beranek, Section 5. pg 118. 

Unstuffed
Line Response

To gain an understanding
of the TL Terminus change with D_{t }it is useful to examine the Terminus
response for the unstuffed line since it is the limiting case and one gains an understanding
of the 1/4l TL line resonance definition. 
Fig.3 TL 36 TL Line
Harmonics, Terminus Response, D_{tr} = 1.0
click image for larger
chart

The TL line length
is characterized by F_{r}, the line 1/4 l resonance frequency and the associated
harmonics.
Two interesting effects
of the Terminus response for the unstuffed line are apparent in the plot of Fig.
3:
 The low frequency
attenuation slope, 18 dB/octave.
 The asymptotic attenuation
of the harmonics.
From the asymptotic
attenuation of the harmonics we can infer that even for the unstuffed line, D_{tr}=1,
air at a density of 1.18kg./m^{3}, the attenuation function is nonlinear
with frequency.


Thus we can infer that
with fiber having a greater density than air, a more pronounced attenuation function
as well as a change in air velocity in the fiber mass will occur 
Table 01 TL 0.91 meters, D_{tr}
= 1


Calculated

Measured

Shift

Magnitude



Type 
(Hz)

(Hz)

(Hz)

(dB)

Phase

Transitions 

*F_{r} 
94.2

104.3

+12.1
+8.6%

+0dB

56.5°


1^{st} Harmonic 
188.3

180.8

7.5
4.0%

7.6dB

159°


2^{nd} Harmonic 
282.5

273.2

9.3
3.3%

+6.8dB

+80.9°

233.5

3^{rd} Harmonic 
376.7

352.8

23.9
6.3%

7.3dB

10.3°


4^{th} Harmonic 
470.9

446.6

24.3
5.2%

+5.7dB

134°

464.6

5^{th} Harmonic 
565.0

533.1

31.9
5.7%

8.1dB

+151°


6^{th} Harmonic 
659.2

599.8

59.4
9.0%

+6.7dB

+66.5°


7^{th} Harmonic 
753.4

688.3

65.1
8.6%

7.6dB

34.5°


8^{th} Harmonic 
847.6

774.5

73.1
8.6%

+3.7dB

116°

789.9

9^{th} Harmonic 
941.7

871.4

70.3
7.5%

6.4dB

+145°


10^{th} Harmonic 
1035.9

924.4

111.5
10.8%

+1.0dB

+56.7°



Note the shift in frequency
of the measured peak values from the calculated. This is not an instrumentation problem
but the byproduct of speed of sound shift. 

* For TL_{B} the
F_{r} for the unstuffed line @ 104 Hz moves to ~ 65 Hz= as shown in Fig.5.2, TL_{B} system response
when stuffed. This is one of the differences from the generic TL line where the F_{r}
shift due to fiber density would be much smaller. 

Stuffed
Line Response 

Fig. 4 TL 36 Terminus
Response D_{tr} = 6.8/ D_{tr} = 9.5
click image for larger
chart

Fig. 4 shows the progressive
change in the bandwidth of the Terminus with the change of D_{tr}. Note that
the low frequency slope hinge point at 1/4l does not change, this is a
function of line length. However the bandwidth from the 1/4l frequency changes: the high
end 3 dB point moves to a higher frequency and the odd harmonic nulls are attenuated.
These factors are a function of the fiber attenuation characteristics defined by
Bradbury's
equations.
It is interesting to compare the low frequency response of the interior of the TL
line, ie the back of the woofer, Fig. 5 to that of Fig. 4. In summary the high frequency
slope point is set by line length, however the low frequency slope frequency changes
with stuffing density. As the density is increased it approaches the fundamental
of a pressure response. 

The
TL's Interior response, ie back of the woofer's spectral response, as shown in Fig.5
documents that contrary to the popular conception that it is a mirror image of the
front of the woofer, it is fundamentally changed by the TL's line loading and is
essentially a pressure phenomena. This phenomena is complex and not discussed further
in this document.
Note that the phase from DC to the 1/4l frequency is linear. This sets
the phase response of the Terminus, though for the Terminus the additional factor
of frequency change vs. fiber density and type has to be added. 
Fig. 5. TL Line, Interior Response
click image for larger chart


[ Back to the Transmission
Line Speaker Page
 The TL_{B}  Appendix TOC ]
