Physics UIUC 199POM/Physics 498POM Physics of Music/Musical Instruments PC-Controlled Sonic Mechanical Resonance DAQ System
Use of A PC-Based Data Acquisition System for Measurement of Complex Mechanical Resonances/Vibrations of Musical Instruments For the UIUC Physics 199POM/Physics 498POM courses, we have developed a PC-based data acquisition (DAQ) system for the purpose of measuring the so-called complex (i.e. phase-sensitive) mechanical vibrations of various kinds of musical instruments as a function of frequency. As shown below in the block diagram of this experimental setup, the PC is used to set the amplitude of, and step the frequency of an Agilent sine-wave function generator over a user-defined frequency range f min f f max , usually in the audio band. In order to achieve a high signal-tonoise ratio, the AC sine-wave signal output from the Agilent function generator (which has a maximum amplitude of 5.0 Volts) is input to a custom-built high-voltage amplifier that has a voltage gain of 10x. The 10x-amplified AC sine-wave signal output from the high-voltage amplifier is then used to drive a wafer-thin 1 diameter piezo-electric transducer, which due to the nature of the piezo-electric material converts the electrical energy into mechanical vibrational energy at the driving frequency f of the function generator. Thus, the transmitter piezo-electric transducer acts like a miniature loudspeaker, injecting mechanical energy into a musical instrument, e.g. a guitar, placed in proximity to the bridge of the guitar in order to mimic as closely as possible the natural mechanical vibrations of the bridge when the guitar is played normally.
Transmitter Piezo-Electric Transducer
Personal Computer
National Instruments Lab PC+ DAQ Card ADC5: Im(Vin) ADC4: Re(Vin)
Agilent Function Generator
10x Voltage Amplifier
Sine-Wave Voltage Signal
~
5x Preamp SRS 830 DSP Lock-In Amplifier
Receiver Piezo-Electric Transducer
Reference Signal Input, Vref(t)
X Output: Re(Vin)
Signal Input, Vin(t)
Y Output: Im(Vin)
Figure 1: Setup for Sonic PC-Based DAQ Experiment A second wafer-thin receiver piezo-electric transducer, used as a microphone, is placed at any other interesting/ convenient location on the musical instrument in order to detect the mechanical vibrations present at that location caused by the mechanical vibrations from the driving force(s) input from the transmitter piezo-electric transducer e.g. located at the bridge of a guitar. The AC signal output from the receiver piezo-electric transducer is amplified 5x via a custombuilt low-noise preamplifier instrumentation op-amp circuit, which has high input impedance and low output impedance, which is input to a Stanford Research Systems SRS-830 DSP-based lock-in amplifier. The AC sine-wave signal output from the Agilent function generator VFG ( t ) = VoFG sin t is also sent to the SRS-830 lock-in amplifier, and is used as a reference signal. The lock-in
1 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, IL 2000-2008.
UIUC Physics 199POM/Physics 498POM Physics of Music/Musical Instruments PC-Controlled Sonic Mechanical Resonance DAQ System
amplifier compares this AC sine-wave reference signal VFG ( t ) = VoFG sin t to the AC sine-wave
where ( ) is a {frequency-dependent} phase shift; = 2 f (radians/second) is the so-called angular frequency and f = the frequency of oscillation (Hz, or cycles per second). An example of these two waveforms is shown in Figure 2 below for numerical values of VoFG = 1.4 Volts ,
Rcvr signal output from the receiver piezo-electric transducer VPzo ( t ) = VoRcvr Pzo sin t + ( ) ,
VoRcvr Pzo = 1.0 Volts , = 10 radians second and ( ) = 0.57 radians .
Figure 2: Waveforms associated with a.) the Agilent sine-wave function generator (input stimulus to the musical instrument) and b.) the receiver piezo-electric transducer (output response from the musical instrument). Note that in the above figure, the ( ) > 0 ( ( ) < 0 ) waveform of the receiver piezoelectric transducer is respectively shifted earlier (later) in time relative to the sine-wave function generator waveform in this situation we say that the receiver piezo-electric transducer signal leads (lags) the sine-wave function generator signal by the amount ( ) > 0 ( ( ) < 0 ) radians. Using the trigonometric identity sin [ A + B ] = sin A cos B + cos A sin B we see that we can write:
Rcvr VPzo ( t ) = VoRcvr Pzo sin t + ( ) = VoRcvr Pzo sin t cos ( ) + cos t sin ( )
2 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, IL 2000-2008.
UIUC Physics 199POM/Physics 498POM Physics of Music/Musical Instruments PC-Controlled Sonic Mechanical Resonance DAQ System
This relation can also be rewritten in the following form as:
Rcvr VPzo ( t ) = VoRcvr Pzo cos ( ) sin t + VoRcvr Pzo sin ( ) cos t
The first term in the above expression can be seen to be in-phase with the driving signal from the Agilent function generator VFG ( t ) = VoFG sin t whereas the second term is 90o out-of-phase with the driving signal from the Agilent function generator. The SRS-830 DSP lock-in amplifier is a truly miraculous, highly sophisticated and extremely versatile electronic device. Since the lock-in amplifier uses the sine-wave signal output from the Agilent function generator as its reference signal, it knows precisely what frequency f to be looking for in the input signal, even if the input signal is noisy (n.b. the noise can be at any frequency) - and if much of the noise is not at the reference frequency f, it is almost completely ignored by the lock-in amplifier! The digital signal processor (DSP) inside the guts of the lock-in amplifier carries out some fancy mathematical calculations to figure out how much of the input signal from the receiver piezo-electric transducer is in-phase with the reference signal VoRcvr Pzo cos ( ) and how much of the input signal is 90o out-of-phase with the reference signal, the DSP then outputs two DC voltages that are proportional to the in-phase and 90o out-of-phase components of the input signal from the receiver piezo-electric transducer! We can graphically represent the in-phase and 90o out-of-phase DC voltage components output from the lock-in amplifier that are associated with the AC signal from the receiver piezo-electric transducer as x- and y-components in a two-dimensional graph known as the complex plane, with the horizontal/x-axis representing the in-phase component VoRcvr Pzo cos ( ) and the vertical/y
axis representing the 90o out-of-phase component VoRcvr Pzo sin ( ) , as shown in Figure 3 below:
y-axis imaginary axis
VoRcvr VoRcvr
Pzo
Pzo
cos ( ) VoRcvr
Pzo
sin ( )
VoRcvr
Pzo
sin ( )
( )
V
Rcvr Pzo o
cos ( )
x-axis real axis
Figure 3: The complex plane
3 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, IL 2000-2008.
UIUC Physics 199POM/Physics 498POM Physics of Music/Musical Instruments PC-Controlled Sonic Mechanical Resonance DAQ System
From the right triangle in the above figure and use of Pythagoras theorem hyp = adj 2 + opp 2 (i.e. r = x 2 + y 2 ) we see that the magnitude (i.e. the length) of the hypotenuse is given by:
VoRcvr Pzo = VoRcvr Pzo cos ( ) + VoRcvr Pzo sin ( ) = VoRcvr Pzo cos 2 ( ) + sin 2 ( ) = VoRcvr Pzo
2 2
We also see from the above figure that:
( ) = tan 1 (VoRcvr Pzo sin ( ) VoRcvr Pzo cos ( ) ) = tan 1 ( sin ( ) cos ( ) ) = tan 1 ( tan ( ) ) = ( )
In the jargon of mathematicians and physicists, the in-phase component of an arbitrary complex signal V ( t ) is known as the real part of V ( t ) and is denoted as e V ( t ) ; the 90o
{
}
out-of-phase component of the complex signal V ( t ) is known as the imaginary part of V ( t ) and is denoted as m V ( t ) . An arbitrary complex signal can thus be in represented, general as: V ( t ) e V ( t ) + i m V ( t ) where i 1 . The i is there to remind us that this component is 90 out of phase with the reference signal. The magnitude of complex V ( t ) is denoted as
o
{
{
}
}
{
}
{ } where V ( t ) is the so-called complex conjugate of V ( t ) and is defined as V ( t ) e {V ( t )} i m {V ( t )} and i = 1 {thus, i i = 1 1 = ( 1) = +1 }. The phase angle is defined as tan ( m {V ( t )} e {V ( t )} ) .
V ( t ) = V ( t ) V ( t ) = e V ( t ) + m V ( t )
2
{
}
2
1
Note also that in the above figure, if the real, or in-phase component VoRcvr Pzo cos ( ) < 0 , the minus sign physically means that this component of the signal is 180o out-of-phase with the reference signal. Similarly, if the imaginary, or 90o out-of phase component VoRcvr Pzo sin ( ) < 0 the minus sign here physically means that this component of the signal is 90o out-of-phase with the reference signal, as opposed to being +90o out-of-phase, if VoRcvr Pzo sin ( ) > 0 . The DC voltages output from the lock-in amplifier that are proportional to the in-phase component VoRcvr Pzo cos ( ) and 90o out-of-phase component VoRcvr Pzo sin ( ) of the AC signal receiver piezo-electric transducer are then each input to an ADC on a National Instruments LabPC+ DAQ card. What is an ADC? An ADC is an acronym for Analog-to-Digital Converter. An ADC is another very useful circuit that converts a DC voltage into a digital number. The National Instruments LabPC+ DAQ card has 8 ADC channels, however for the Sonic Mechanical Resonance DAQ setup, we only need to use two of the eight ADC channels, in order to digitize the two DC voltages output from the lock-in amplifier. The ADCs on the National Instruments LabPC+ DAQ card have 12 bits of digital resolution spanning an input voltage range of 5.0 Volts . This means that any DC voltage input to a 12-bit ADC within this 10.0 Volt span can be represented by a binary number (i.e. 0s and 1s) ranging from a lowest value of 000000000000b (= 0 decimal) to its highest value of 111111111111b (= 4095 decimal).
4 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, IL 2000-2008.
UIUC Physics 199POM/Physics 498POM Physics of Music/Musical Instruments PC-Controlled Sonic Mechanical Resonance DAQ System
For 12-bit ADCs, there are 212 = 4096 (decimal) possible distinct numbers (0,1,2,3,4, .4093,4094,4095). Thus, a 12-bit ADC quantizes a continuum of DC voltages between 5.0 Volts to 4096 values between these limits, with 10V/4096 = 0.00244 Volt least-significantbit resolution. For a high-quality/well-designed ADC, the relationship between DC voltage input vs. digital number output is nearly perfectly linear, as shown in Figure 4 below:
Digital # Output from ADC
4095 n.b. y = mx+b straight line, y-intercept b = 2047, slope of straight line m = 4096/10.0 = 4.096 ADC counts per Volt
5.0 Volts
2047 0.0 Volts +5.0 Volts
DC Voltage input to ADC
0
Figure 4. The linear relationship between ADC value vs. DC voltage. Note that we also employ the use of signal-averaging techniques in the Sonic DAQ program in order to minimize the effects of (relatively small amounts of) electrical noise fluctuations and electromagnetic noise pickup associated with the ADC measurements of the DC voltages output from the lock-in amplifier. For each frequency setting of the Agilent function generator f , we (rapidly) take a total of 5000 ADC measurements for each ADC and then compute the average, or mean ADC voltage value <V_ADC> for each of the two ADC channels. It turns out that when we have no signal present from the Agilent function generator, we find that the <V_ADCs> are NOT precisely equal to 0.000 volts, but something very slightly different from zero, typically on the order of a fraction of a milli-Volt, tiny in comparison to 5.0 Volts. These slight DC voltage offsets associated with each of the ADC channels occur for a number of reasons the electronic circuitry in each of the ADC channels isnt perfect internally, each ADC circuit often has a slight DC voltage offset. Also, for no signal present at the input of the lock-in amplifier, the two DC voltages output from the lock-in amplifier arent perfectly equal to 0.0000 volts either. Furthermore, there can exist very small contact potential differences between different metals e.g. chrome-plated BNC connectors and gold-plated pins, copper pc board traces, solder, etc. on the pc boards inside the lock-in amplifier, etc. Periodically, we take no-signal Sonic DAQ runs in order to explicitly measure these no-signal <V_ADC> DC voltage offsets. We call these no-signal DC
5 Professor Steven Errede, Department of Physics, Univers...