## Model Validation

To validate the model I made a MDF transmission line with a length equal to one meter and section equal to the diaphragm area of the loudspeaker I used: the Dayton Audio RS100-4, a small wide band with very interesting features. In the table are listed the parameters measured with the Clio Pocket measuring system and added mass technique.

Dayton Audio RS 100-4 Parameters

Nominal Impedance | 4 Ω |

Re | 3,1 Ω |

Fs | 87 Hz |

Sd | 35,3 cm² |

Qms | 2,88 |

Qes | 0,88 |

Qts | 0,68 |

Cms | 0,7822 mm/N |

Mms | 4,2630 g |

Rms | 0,81 Ωm |

Bl | 2,86 Tm |

SPL | 82,11 dB |

VAS | 1,3610 L |

Le (1 kHz) | 0,09 mH |

For practical reasons the line has a rectangular shape, however, the results of the simulation, and the comparison with a similar line obtained from a cardboard tube of circular cross-section, confirm that the shape (remaining within reasonable limits) does not affect the final result.

First of all I tested the validity of the simulation by comparing the graph of the impedance curve of the driver, mounted in the empty line, with the real measurement obtaining the result, in my opinion very promising, visible in **Figure 12**. I then made the same measurements filling the line with Dacron at a density of 16 kg/m^{3}. This is a fibrous material often used as an alternative to the traditional long fiber wool, much loved by TL’s DIYers. The results can be seen in **Figure 13**.

When performing simulations with the damped line it is necessary to take into account the influence of the absorbent on the speed of sound passing through it. To calculate the speed of sound within the damped line I measured the time it takes for the acoustic wave to run through the transmission line by impulse response.

If impedance measurement is important to verify the validity of the model, SPL measurement is fundamental to the design of any loudspeaker system. The intensity of the acoustic pressure of the system measured in axis at a distance *r* is given by:

where *ǀU*_{sys}*ǀ* is the total volume velocity of the system.

The SPL that would measure a microphone at a distance of one meter is:

with *p*_{ref} equal to 2×10^{-5} N/m^{2}.

From the equations above we obtain the expression to be inserted in the LTSpice PROBE graphic post-processor for the measurement of the SPL at one meter:

**SPL=20*log10[59000*frequency*I(VSYS)]**

The circuit set up for SPL measurement is visible in Figure 10.

F1 and F2 are current controlled current sources and are controlled by the current flowing in V4 and V5 respectively. The sum of these currents (which represent the volume velocity of the woofer and the opening) is measured by the VSYS ammeter. The total SPL of the system measured in near field and simulated at 1 m is visible in **Figure 20**.

With the same expression it is obviously possible to simulate individually the SPL of woofer and opening by replacing VSYS with V4 and V5 respectively. The graphs of the SPL measured in near field and simulated at 1 m, for the empty and damped line (with density 7 kg/m^{3 }and 16 kg/m^{3} of damping material), are visible in **Figures 14** to **19**.

In this phase of the research it emerged the need to vary the flow resistance of the absorbent as a function of frequency. The easiest way to create a frequency-dependent resistance in LTSpice is to use the Laplace command. To apply this command to a resistor, you need to think in terms of conductance:

where *g(s) *precisely* *represents conductance.

The LTSpice guide indicates the syntax to implement the command:

**R=1 Laplace= g(s)**

For example a resistance *R=f* can be defined in the frequency domain by entering the following in the value field:

**R=1 Laplace=2*{pi}/abs(s)**

Since the flow resistance can have a consistent value even at low frequency (even in direct current, as the name itself suggests) I placed the variable resistor in series to a fixed resistor. In the absence of data referring to the material, the value of these two resistors can be obtained by means of measurements with the test line, varying it arbitrarily until obtaining a satisfactory correspondence with the simulation. To dimension the fixed resistor I used the measurement of the impedance of the loudspeaker mounted in the line, varying the value until the amplitude of the measured peak is aligned with that of the simulated peak. For the dimensioning of the variable resistor I used the SPL measurement of the opening, increasing the value until the slope of the measured and simulated acoustic low-pass filter was aligned. These simple guidelines allow to obtain an excellent correspondence between measurement and simulation.

In the SPICE guidelines I express these values as unit rates rather than as pure resistances so that they can be used (provided that the filling density remains unchanged) even in case of variation of the line size or in subsequent projects. An example:

**.param RF = (250/SL)*DeltaZ**

where *RF* is the *Ohm* value of the resistor present in the acoustic section of the analog circuit in Figure 9, *250* is the unit value of the flow resistance, *SL* is the line section and DeltaZ is the length of a line segment.

Finally, to simulate the cone excursion measurement I used the circuit visible in Figure 11.

The current (mechanical speed of the cone) flowing in V3 controls the current source with unit gain F1. Since the integral of the speed corresponds to the displacement, the unit value CTEST capacitor, used as an integrator, gives us the value of the excursion in meters. This measure is very useful to verify that the transmission line is not limiting the performance of the loudspeaker when it is driven at its maximum power. **Figure 21** shows an example of a damped line excursion measurement. For convenience I have assigned to CTEST the value of 1 mF; in this way the voltage in volts corresponds to the excursion in mm.

**Andrea Rubino**

*Figure 12. Empty line impedance. a) Measurement. b) Simulation.*

*Figure 13. Damped line impedance. a) Measurement. b) Simulation.*

*Figure 14. SPL woofer (empty line). a) Measurement. b) Simulation.*

*Figure 15. SPL open end (empty line). a) Measurement. b) Simulation.*

*Figure 16. SPL woofer (7 kg/m³ dacron) a) Measurement. b) Simulation.*

*Figure 17. SPL open end (7 kg/m³ dacron) a) Measurement. b) Simulation.*

*Figure 18. SPL woofer (16 kg/m³ dacron) a) Measurement. b) Simulation.*

*Figure 19. SPL open end (16 kg/m³ dacron) a) Measurement. b) Simulation.*

*Figure 20. SPL system (16 kg/m³ dacron) a) Measurement. b) Simulation.*

*Figure 21. Simulated cone excursion (empty line). RMS value (input 2.83 V).*

## References

[1] M. Colloms, “High Performance Loudspeakers”, Pentech Press Ltd., Fourth Edition; 1991

[2] J. Backman, “A Computational Model of Transmission Line Loudspeakers”, 92^{nd} AES Convention, 1992, preprint no. 3326.

[3] G. L. Augspurger, “Loudspeaker on Damped Pipes”, JAES Volume 48 Issue 5 pp. 424-436; May 2000.

[4] B. Olney, “A Method of Eliminating Cavity Resonance, Extending Low Frequency Responce and Increasing Acoustical Damping in Cabinet Type Loudspeakers”, J. Acoust. Soc. Amer. Volume 8; October 1936.

[5] A. R. Bailey, “A Non-resonant Loudspeaker Enclosure”, Wireless World; October 1965.

[6] A. R. Bailey, “The Transmission Line Loudspeaker Enclosure”, Wireless World; May 1972.

[7] L. J. S. Bradbury, “The Use of Fibrous Materials in Loudspeaker Enclosures”, JAES Volume 24 Issue 3 pp.162-170; April 1976.

[8] R. M. Bullock and P. E. Hillman, “A Transmission-line Woofer Model”, 81^{st} AES Convention, 1986, Preprint no. 2384.

[9] B. N. Locanthi, “Application of Electric Circuit Analogies to Loudspeaker Design Problems”, JAES Volume 19 Issue 9 pp. 778-785; October 1971.

[10] G. L. Augspurger, “Transmission Lines Updated, Part 1”, Speaker Builder; 2/00.

[11] G. L. Augspurger, “Transmission Lines Updated, Part 2”, Speaker Builder; 3/00.

[12] G. L. Augspurger, “Transmission Lines Updated, Part 3”, Speaker Builder; 4/00.

[13] L. L. Beranek, “Acoustics”, New York: McGraw-Hill; 1954.

[14] W. M. Leach, Jr., “Computer-Aided Electroacustic Design with SPICE”, JAES Volume 39 Issue 12; December 1991.

[15] V. Landi, “La linea di trasmissione”, AUDIOreview, n.104-106-112-132-133; 1991-1993.