The Limit of Nearfield Measurement for Loudspeaker

The Limit of Nearfield Measurement
Near-field measurement means that we put the microphone very near to the loudspeaker and then do measurement. This kind of measurement gives highest signal to noise ratio since the microphone is very near to the source and the reflected sound and noise can be ignored.
However, loudspeaker is not a point source which radiates sound on a surface. This causes inaccuracy in frequency response measurement at high frequency when Near-field technique is used. Assume a woofer with diameter 12 inch is reproducing a 1 kHz sound. The sound radiated from the middle and the sound radiated from the edge of woofer will travel to the microphone and summed up (see figure 1). As distance d2 is longer than d1, there will be a phase difference between 2 signals. Thus, the amplitude is different from 2 sound wave radiate from a point source. Since different frequency will have different phase shift, the measured frequency response will be distorted.
Figure 1. Sound radiation.
In actual case, there are more than 2 radiating wave and the calculation is more complicated. Here, I would like to derive the equation on the accuracy of Near-field measurement:
- Assume the loudspeaker is a flat, circular non-vibrating plane.
- Assume the radiation loss of all sound waves are the same
- Take the sound wave radiated at the middle as reference signal.
- Assume the microphone is 2 cm in front of the middle of loudspeaker.
time delay = t d = (d2– d1) / v s (1)
d2 = √(r2+d12) (2)
phase shift = p = (2πt d)(frequency of sound) (3)
total received signal power= s1 = 2π∫sin(wt+p)dr (4)
where
p is the phase shift
t d is the time delay
v s is the velocity of sound
w is the frequency of radiated sound
for r, d1 and d2, please see Figure 1.
In ideal case, there should be no phase shift in all radiated sound:
total received signal power in ideal case = s2= 2π∫sin(wt)dr (5)
Then,
Percentage error = (s2 – s1)/ s2 (6)
Where
s1 is the total received signal energy
s2 is the total received signal energy in ideal case
In order to find out the maximum frequency available for Near-field measurement, a MATLAB program was written to calculate the percentage error in measured amplitude at different frequency:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [frequency error] = Nearfield(diameter,distance)
clc;
Rmax = diameter/2/100; %radius of loudspeaker
d1 = distance/100; % distance between the mic and loudspeaker in meter
vs = 340; % velocity of sound
frequency = logspace(1,5,300);
error=0;
for k=1:300
t = 0:1/frequency(k)/100:1/frequency(k); %100 sampling point in each frequency
w = 2*pi*frequency(k); %angular frequency of the signal
signal_2 = sin(w*t); %the ideal signal without phase shift
s1=0;s2=0; %initialize s1 and s2
for r=0:0.001:Rmax
td = (sqrt(r^2 + d1^2)-d1)/vs; %time delay
phase = 2*pi*td*frequency(k); %phase shift
signal_1 = sin(w*t - phase); %shifted signal
s1 = s1 + 2*pi*r*signal_1;
s2 = s2 + 2*pi*r*signal_2;
end
error(k) = 100*((rms(s2)-rms(s1))/rms(s2));
end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Results with different driver’s diameter have been simulated as follow:
Figure 2. Percentage error of sound power with 2 cm driver.
Figure 3. Percentage error of sound power with 4 cm driver.
Figure 4. Percentage error of sound power with 10 cm driver.
Figure 5. Percentage error of sound power with 15 cm driver.
Figure 6. Percentage error of sound power with 20 cm driver.
Figure 7. Percentage error of sound power with 25 cm driver.
Figure 8. Percentage error of sound power with 30 cm driver.
By comparing these figures, we can see that Nearfield measurement introduce more error with increasing driver diameter. As discussed in the first example, if a driver with 12 inch (30 cm) diameter is measured by a microphone which is 2 cm apart from the middle of driver, there will be approximately 11% error in measured power at 1k Hz.
The above figures and MATLAB program can be used to estimate the limit of Nearfield measurement under different situations and requirement.
Author: Alvis Chan
Email: alvischan900@。