EEE117L Lab 9

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California State University, Sacramento *

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117L

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Electrical Engineering

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Dec 6, 2023

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ECE117L Lab 10 Operational Amplifier Oscillators and Fourier Series Objective 1. Derive the Fourier series for common periodic signals; sine, square, triangle and saw- tooth waveforms 2. Design and simulate oscillators for the various waveforms and use the Fast Fourier Transform function in PSPICE to display the Fourier Series. Section 1: Fourier Series Calculations According to Fourier analysis, a periodic function can be represented by an infinite sum of sine and cosine functions that are harmonically related. For instance, a square wave may be considered to be a superposition of an infinite number of odd harmonic frequencies whose amplitudes decrease inversely with frequency. The fundamental, or lowest frequency, is the frequency of the square wave. Figure 1 depicts how the harmonics add together to form the square wave. In this diagram, 2L is the period of the wave form and the frequency is 1/2L Figure 1, Odd harmonics add together to form a square wave. Table 1, Normalized Fourier Coefficients Wave Form Fourier Series Equation Harmonics 1 st (1Hz) 2 nd (2Hz) 3 rd (3Hz) 4 th (4Hz) 5 th (5Hz) 6 th (6Hz) 7 th (7Hz) Square Wave 1.273 0 0.423 0 0.254 0 0.182 Triangle Wave 0.811 0 -0.090 0 0.034 0 -0.016 Note: π L = 2 π 2 L For a 2V square wave with 100Hz frequency, the Fourier’s series for the first 7 harmonics is:
ECE117L Lab 10 Operational Amplifier Oscillators and Fourier Series x(t) = 2.546 sin ( 2 π 100 t ) +0.846 sin ( 2 π 300 t ) +0.508 sin ( 2 π 500 t )+ ¿ 364 sin ( 2 π 7 00 t ) In our lab, we will first design and simulate a Square Wave Oscillator and then a Triangle Wave Oscillator . We will simulate the Transient Response ( time domain ) and then use the Fast Fourier Transform ( FFT ) function in PSPICE to converter and display the signal in the frequency domain . Before simulating the circuits, let’s first calculate the Fourier coefficients and harmonic frequencies for a square wave and then a triangle wave. Scale the parameters in table 1 to fill-in the table below. Table 2: Calculated Fourier coefficients and harmonic frequencies. Wave Form Parameter Harmonics 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th Square Wave 10V, 1 kHz Coefficient (amplitude) 0 0 0 Harmonic Frequency 1kHz 0 0 0 Triangle Wave 5V 500 Hz Coefficient (amplitude) 0 0 0 Harmonic Frequency 500Hz 0 0 0 Section 2: Square Wave Oscillator Operational Amplifiers make excellent low frequency square wave generators by wiring them as relaxation oscillators. Figure 2 shows a common circuit for a basic relaxation oscillator. Examination of this circuit shows that it contains two voltage dividers, each driven by the output of the op-amp and each with its output going to one of the op-amp input terminals. One of the voltage dividers is resistive and feeds back to the non-inverting input . The other voltage divider comprises of and RC circuit, R1 and C1, and generate a timing waveform to the inverting input of the op-amp. The op-amp behaves as a voltage comparator (switch) which is activated by the relative voltage levels at the two input signals. To understand the operation, assume that C1 is initially fully discharged. The output is driven to positive saturation level by the voltage divider at the op amp’s positive input (+). This applies a positive voltage to both of the voltage dividers. Under this condition, half of the positive saturation voltage is applied to the non-inverting input of the op-amp via R2-R3 resistive divider and a rising positive voltage applied to the inverting input through R1 and C1. As C1 charges up exponentially through R1 and the positive output of the op amp, the voltage at the inverting input eventually exceeds the potential at the non- inverting input. At that point, the op-amp comes out of saturation and its output swings negative. Under this condition, the voltage at the non-inverting input of the op-amp swings negative through R2 and R3, but the inverting input tends to be held steady by the charge on C1. The result is that the op-amp output abruptly switches to negative saturation. C1 begins to discharge in the negative direction via R1 and
ECE117L Lab 10 Operational Amplifier Oscillators and Fourier Series applies raising negative voltage to the inverting input. Eventually, the negative potential at the inverting terminal exceeds the negative voltage at the non-inverting terminal, causing the output to swing back into positive saturation. The sequence continues to repeat, creating a series of square waves. The period of the square wave depends on the time constant of the R1-C1 combination and on the voltage dividing ratio or R2 and R3. The operating frequency can be changed by altering any one of these variables. Note that we want to switch half way through the period of the square wave. So the switching frequency is set by using the equation, ¿ 1 2 R 1 C 1 . Figure 2, Square Wave Oscillation Amplifier 1. Calculate the RC time constant to produce a 1k Hz oscillation frequency and calculate values for R1 and C1. With PSPICE, run a transient analysis of the square wave oscillator to capture the voltage waveform at the op-amp output. Explain the relationship between the RC time constant and the frequency. In your transient response, display 5 periods of the waveform. 2. To run a Fast Fourier Transform, and select the FFT icon the PSPICE menu bar. In your lab report, show the PSPICE FFT response and compare it to the calculated Fourier series coefficients that were listed in table 2. Note: To convert the transient response to a Fourier Series, select he FFT icon on the display task bar. Also, when PSPICE converts from the transient display to the frequency domain, it will provide a very wide bandwidth. Adjust the x-axis of the display to provide better f = 1 2 R 1 C 1
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