Introduction During This Experiment, The Class Continued

Introduction during this experiment, the class continued to learn about

During this experiment, the class continued to learn about how voltage varies with frequency. This experiment explored passive filters, also known as band pass filters. The procedure involved working with two series RLC circuits and two parallel circuits containing an inductor, a capacitor, and resistors. Initial calculations included deriving the transfer function H(jω) for each circuit, calculating the magnitude |H(jω)|, identifying the resonant frequency (ωr), and determining the bandwidth. Specifically, for the series RLC circuit, H(jω) was found to be 1 / [1 + j((ωL / R) - (1 / R C ω))], with a magnitude expression of 1 / sqrt[1 + ((1/R)*(ωL - 1 / (ωC)))^2]. The resonant frequency was calculated to be approximately 6.1 kHz, and the bandwidth was about 11.9 kHz. Similar calculations were performed for the parallel circuit, with H(jω) = 1 / [1 + (R1 / R2) + j(ωC - (1 / ωL))], with corresponding magnitude and bandwidth calculations.

The main procedure involved constructing the circuits with specific component values, such as R = 5.1 kΩ, L = 68 mH, C = 0.01 μF, and varying the frequency from 10 Hz to 100 kHz while measuring V₂ using an oscilloscope. The experimental magnitude of V₂ versus frequency was plotted, revealing a resonant frequency around 5.12 kHz, slightly different from the theoretical 6.1 kHz, which was attributed to insufficient data points near the peak. Similar procedures were followed for circuits with different resistor values (2 kΩ and 20 kΩ), leading to consistent observations. The plots obtained using PSPICE matched experimental results, confirming the reliability of the measurements. The experiments were repeated with different resistor configurations to observe the effects on resonance and bandwidth.

In the discussion, it was emphasized that prior knowledge of mathematical models such as H(jω), |H(jω)|, ωr, and bandwidth was essential for analyzing the circuits efficiently. The discrepancies between theoretical and experimental values, approximately 16%, were largely due to limited data near the resonant peak. The importance of taking sufficient data points was highlighted, as this improved the accuracy of the experimental verification of theoretical predictions. Graphical analysis proved useful but relied heavily on adequate data sampling.

The conclusion summarized the key outcome: students observed how output voltage varies with frequency in band pass filters and developed skills in measurement and mathematical modeling. The experiment also underscored the significance of thorough data collection to accurately match experimental results with theoretical calculations, which is vital in designing and analyzing electronic filters.

Paper For Above instruction

Understanding the frequency-dependent behavior of electrical circuits is fundamental in electronics engineering, especially when designing filters that manipulate signals for various applications. Passive band pass filters, comprising inductors, capacitors, and resistors, are widely used to allow signals within a specific frequency range to pass while attenuating signals outside this band. This experiment provided valuable hands-on experience in analyzing these filters, integrating theoretical calculations with practical measurements, and validating results through simulation.

The core concept explored was the transfer function H(jω), which describes how voltage is transferred through the circuit as a function of frequency. For the series RLC circuit, the transfer function was derived as H(jω) = 1 / [1 + j((ωL / R) - (1 / R C ω))], indicating the frequency dependence of the circuit's response. The resulting magnitude |H(jω)| reflects how the output voltage amplitude varies with frequency. Critical parameters such as the resonant frequency (ωr ≈ 6.1 kHz) — where the circuit's response peaks — and the bandwidth (~11.9 kHz) were calculated from these expressions, providing benchmarks for experimental comparison.

In practical application, the experiment involved constructing the circuits, applying a sinusoidal input, and measuring the output voltage V₂ across the circuit components. Using an oscilloscope, students plotted the magnitude of V₂ against a range of frequencies, observing a peak near the predicted resonant frequency. The resulting experimental resonant frequency (~5.12 kHz) deviated slightly from the theoretical value, which was attributed to insufficient data points around the peak. This highlighted a significant lesson: capturing enough data points near resonance is crucial for accurate experimental validation of theoretical models.

The use of PSPICE, a circuit simulation tool, enhanced the understanding by providing precise plots that closely matched the experimental results. This cross-validation underscored the importance of simulation in verifying analytical and experimental findings, especially when experimental limitations are present. Repeating the measurements with different resistor values demonstrated how component variations influence resonant behavior and bandwidth, reinforcing the concepts of circuit design considerations.

The experiment also involved analyzing circuits with different resistor configurations: R = 2 kΩ, 5.1 kΩ, and 20 kΩ, each affecting the quality factor (Q) and the selectivity of the filter. The theoretical calculations predicted no change in the resonant frequency due to identical inductor and capacitor values; however, the bandwidths varied with R, illustrating the influence of resistance on filter sharpness.

A key insight from the discussion was the importance of data sampling density. Limited data points lead to inaccuracies in determining resonant frequency and bandwidth from graphs. Accurate measurement requires dense sampling near the peak and cutoff frequencies to precisely characterize the filter’s frequency response. This has practical implications in designing filters for communication systems, audio processing, and other electronic applications, where precision is vital.

The experiment fostered essential skills: applying mathematical models to real circuits, using measurement instruments effectively, and interpreting graphical data. It also provided a foundation for understanding how passive components work together to shape signal behavior. The importance of thorough experimentation and simulation validation was emphasized, especially when working with complex frequency responses where small measurement errors can significantly impact performance predictions.

In conclusion, the experiment demonstrated the critical relationship between theoretical analysis, simulation, and practical measurement in electronic filter design. The ability to predict and verify filter behavior is indispensable for engineers working on signal processing, telecommunications, and electronic instrumentation. Moving forward, more extensive data collection and refined measurement techniques will improve the accuracy of experimental validations, ultimately leading to better-designed electronic systems that meet specific frequency requirements.

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