RL Circuit Analysis

Build a hybrid VI to analyze the RL circuit shown in Figure 8 using Fourier series.

Figure 8: RL Series Circuit with Periodic Input Voltage

The input voltage for the circuit is to be either a square wave or a triangular wave with a period T=2 seconds.

Compute and display the following:

Hints:

RMS Value

The RMS value of a periodic function v(t)v(t) size 12{v \( t \) } {}with period TT size 12{T} {}is given by

The RMS value of a waveform consisting of sinusoids with different frequencies is equal to the square root of the sum of the squares of the RMS value of each sinusoid. If a waveform is represented by the following Fourier series

then, the RMS value VRMSVRMS size 12{V rSub { size 8{ ital "RMS"} } } {}is given by

Average power

The average power of the Fourier series can be expressed as

Doppler Effect

The Doppler effect denotes the change in frequency and wavelength of a wave as perceived by an observer moving relative to the wave source. The Doppler effect can be demonstrated via time scaling of Fourier series. The observer hears the siren of an approaching emergency vehicle with different amplitudes and frequencies as compared to the original signal. As the vehicle passes by, the observer hears another amplitude and frequency. The reason for the amplitude change (increased loudness) is because of the proximity of the vehicle. The closer it is, the louder it gets. The reason for frequency (pitch) change is due to the Doppler effect. As the vehicle approaches, each successive compression of the air caused by the siren occurs a little closer than the last one, and the opposite happens when the vehicle passes by. The result is the scaling of the original signal in the time domain, which changes its frequency. When the vehicle approaches, the scaling factor is greater than 1, resulting in a higher frequency, and, when it passes by, the scaling factor is less than 1, resulting in a lower frequency. More theoretical aspects of this phenomenon are covered in reference (Reference).

Define the original siren signal as x(t)x(t) size 12{x \( t \) } {}. When the vehicle approaches, one can describe the signal by

where B1(t)B1(t) size 12{B rSub { size 8{1} } \( t \) } {}is an increasing function of time (assuming a linear increment with time) and aa size 12{a} {}is the scaling factor having a value greater than 1. When the vehicle passes by, one can describe the signal by

where B2(t)B2(t) size 12{B rSub { size 8{2} } \( t \) } {}is a decreasing function of time (assuming a linear decrement with time) and bb size 12{b} {}is the scaling factor having a value less than 1.

First, generate a signal and create an upscale and a downscale version of it. Observe the Fourier series for all the signals. Set the amplitude and frequency of the original signal and the scaling factors as controls. In addition, play the sounds using the LabVIEW Play Waveform function. Figure 9 shows a possible front panel for this type of system.

Figure 9: Front Panel of a Doppler Effect System

Synthesis of Electronic Music

In electronic music instruments, sound generation is implemented via synthesis. Different types of synthesis techniques such as additive synthesis, subtractive synthesis and frequency modulation (FM) synthesis are used to create audio waveforms. The simplest type of synthesis is additive synthesis, where a composite waveform is created by summing sine wave components, which is basically the inverse Fourier series operation. However, in practice, to create a music sound with rich harmonics requires adding a large number of sine waves, which makes the approach inefficient computationally. To avoid adding a large number of sine waves, modulation with addition is used. This exercise involves the design of algorithms used in the Yamaha DX7 music synthesizer, which debuted in 1983 as the first commercially available digital synthesizer.

The primary functional circuit in DX7 consists of a digital sine wave oscillator plus a digital envelope generator. Let us use additive synthesis and frequency modulation to achieve synthesis with six configurable operators. When one adds together the output of some operators, an additive synthesis occurs, and when one connects the output of one operator to the input of another operator, a modulation occurs.

In terms of block diagrams, the additive synthesis of a waveform with four operators is illustrated in Figure 10.

Figure 10: Additive Synthesis

The output for the combination shown in Figure 10 can be written as

Figure 11 shows the FM synthesis of a waveform with two operators.

Figure 11: FM Synthesis

The output for the combination shown in this figure can be written as

Other than addition and frequency modulation, one can use feedback or self-modulation in DX7, which involves wrapping back and using the output of an operator to modulate the input of the same operator as shown in Figure 12.

Figure 12: Self-Modulation

The corresponding equation is

Different arrangements of operators create different algorithms. Figure 13 displays the diagram of an algorithm.

Figure 13: Diagram of an Algorithm

And the output for this algorithm can be written as

With DX7, one can choose from 32 different algorithms. As one moves from algorithm No. 32 to algorithm No. 1, the harmonics complexity increases. In algorithm No. 32, all six operators are combined using additive synthesis with a self modulator generating the smallest number of harmonics. Figure 14 shows the diagram for all 32 combinations of operators. More details on music synthesis and the Yamaha DX7 synthesizer can be found in the (Reference)-(Reference).

Next, explore designing a system with six operators and set their amplitude and frequency as controls. By combining these operators, construct any three algorithms, one from the lower side (for example, algorithm No. 3), one from the middle side (for example, algorithm No. 17) and the final one from the upper side (for example, algorithm No. 30). Observe the output waves in the time and frequency domains (find the corresponding Fourier series).

Figure 14: 32 Algorithms in the Yamaha DX7