In order to test non-linearity, the effects of different transfer functions of an AD633 multiplier in a given electrical circuit were investigated and compared with the theoretical expectations. First of all, the phenomenon of frequency doubling was found to occur when squaring the input voltage. Secondly, the multiplier was reconfigured to give a square-root response. This allowed us to vary the degree of non-linearity by choosing the parameters of input voltage and DC offset such that we could determine which terms in the Taylor expansion of the transfer function were relevant and hence to what degree the circuit behaved non-linearly. For a small, sinusoidal variation about a large DC level, the system was found to be weakly non-linear. For high amplitude and a low DV offset we observed strong non-linearity. Compared to weak non-linearity, we were able to detect the third harmonic as well as the first and the second one. The existence of harmonics was investigated on the PicoScope screen and verified by plotting output amplitude (dBV) versus input amplitude (dBV) and finding the gradient of the slope corresponding to the respective harmonic. Finally, frequency mixing was explored in its broader context by investigating amplitude modulation and demodulation on the same circuit board.
Table of Contents
1. Introduction
2. Theoretical Background
2.1. Non-linear circuits in constrast to linear circuits.
2.2. Frequency doubling.
2.3. Varying the degree of non-linearity.
2.4. Amplitude modulation and demodulation.
3. Methods and Results
3.1. Frequency doubling.
3.2. Square-root circuit.
3.3. Weak Non-Linearity.
3.4. Strong Non-Linearity.
3.5. Amplitude Modulation.
3.6. Amplitude Demodulation.
4. Discussion
4.1. Common occurence of outliers.
4.2. Potential Errors.
4.3. Trendline fitting of logarithmic plots.
5. Conclusion
Research Objectives and Key Topics
This document investigates the behavior of non-linearity within electrical circuits, specifically utilizing an AD633 multiplier to demonstrate and analyze non-linear phenomena such as frequency doubling, amplitude modulation, and demodulation.
- Investigation of non-linear circuit transfer functions compared to theoretical models.
- Experimental verification of frequency doubling (Second Harmonic Generation).
- Analysis of non-linearity degrees by varying input voltage and DC offset.
- Exploration of amplitude modulation and demodulation principles through frequency mixing.
Excerpt from the Book
2.2. Frequency doubling.
In the given experimental setup, frequency doubling can be explained by looking at the effect of the multiplier. For an input voltage Vin = A cos ωt, the output of the AD633 multiplier is
(2) W(t) = X(t)2 = BA2(cos ωt)2 = BA2 (1 + cos 2ωt) / 2
Hence, there are two different frequencies in the output: a DC-offset and twice the frequency of the input voltage. The transfer function of a general non-linear system, Vout = f(Vin) can be approximated by a Taylor series, which with sinusoidal input A cos ωt, gives
(3) Vout = f(Vin) = ∑n anVn in = ∑n anAn(cos ωt)n
Using power-reduction formula, the (cos ωt)n terms can be written as sums of terms with frequencies that are multiples of ωt. Thus, a non-linear system generates an output with frequency components of ω, 2ω, 3ω, 4ω,etc.
Summary of Chapters
1. Introduction: The chapter defines the motivation for studying non-linear behavior in physical systems and identifies frequency doubling as a primary consequence of non-linearity.
2. Theoretical Background: This section provides the mathematical foundations, including transfer functions, Taylor series approximations for non-linear systems, and the principles of amplitude modulation.
3. Methods and Results: This chapter details the experimental procedure using the AD633 multiplier circuit and presents the measured data across various operational configurations.
4. Discussion: The discussion addresses the occurrence of outliers in the data, evaluates potential measurement errors, and critiques the validity of trendline fitting for logarithmic plots.
5. Conclusion: The concluding section summarizes the successful verification of the initial theoretical predictions regarding the AD633 multiplier's non-linear characteristics.
Keywords
Non-linearity, AD633 multiplier, Frequency doubling, Second Harmonic Generation, Taylor expansion, Amplitude modulation, Demodulation, Electrical circuits, Signal processing, Harmonic coefficients, Signal-to-Noise Ratio, Logarithmic plots, Power-law, Transfer function, Sine wave.
Frequently Asked Questions
What is the core subject of this research?
The research focuses on the behavior of non-linearity in electrical circuits using the AD633 multiplier as the primary component.
What are the primary thematic fields addressed?
The study covers electrical circuit design, non-linear system theory, signal modulation/demodulation techniques, and experimental data analysis.
What is the main objective or research question?
The goal is to experimentally investigate and verify theoretical models of non-linearity, specifically frequency doubling and amplitude modulation, using electrical circuit components.
Which scientific methodology is applied?
The study uses experimental measurement, comparative analysis between empirical data and Taylor series predictions, and graphical data plotting to evaluate circuit behavior.
What topics are covered in the main section?
The main section covers the setup and testing of frequency doubling, square-root circuits, weak and strong non-linearity, and the modulation/demodulation of signals.
What are the characterizing keywords of this work?
Key terms include non-linearity, AD633 multiplier, frequency doubling, harmonic generation, amplitude modulation, and Taylor expansion.
How does the DC-offset influence the non-linearity of the circuit?
The research demonstrates that varying the DC-offset and input amplitude changes the number of relevant terms in the Taylor series, effectively shifting the circuit between weak and strong non-linear regimes.
Why was the third harmonic observed during the strong non-linearity experiment?
The third harmonic was observable because the strong non-linear regime requires an additional term of cos(ωt)^3 from the Taylor expansion to accurately describe the system's output.
How was the amplitude modulated signal recovered?
The signal was recovered by performing a further multiplication of the modulated waveform with the carrier frequency, followed by the use of a low-pass filter to remove the high-frequency carrier.
- Quote paper
- Laura Imperatori (Author), 2013, Non-Linearity. Frequency-doubling, degree variation, amplitudemodulation and demodulation, Munich, GRIN Verlag, https://www.grin.com/document/266748
