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The aim of this experiment is to gain an introduction into the tools used for AC circuits, which will help understand the representation of the AC sine wave ^{[1]}. Furthermore, it will aim to investigate the relationship between peak to peak, peak, and RMS measurements and allow the understanding of the use of oscilloscopes and digital multimeters to make these measurements in an AC circuit ^{[1]}.

An AC wave with a DC offset can be represented mathematically in the form:

D + A sin(2πft + θ) ^{[1]}

Where D is the offset caused by the addition of a DC voltage, A is the amplitude of the wave, f is the frequency in Hz, and θ is the phase of the wave.

The magnitude of a sine wave can be expressed in three forms: the peak (the max value of the waveform measured from the mean value of the wave), the peak-to-peak (the sum of the magnitudes of the positive and negative peak), and RMS (the root mean squared value of an AC wave, which is the effective value or DC equivalent) ^{[3]}.

The relationship between these values can be expressed as:

V_{pp} = 2V_{pk} = 2√2V_{rms} ^{[1]}

When measuring the values of an AC wave, oscilloscopes are most suitable for peak to peak, while digital multimeters output in RMS values only. When using AC voltage in an ideal environment, the same principles of circuits such as Ohm's law apply ^{[3]}.

For this section, the amplitude, frequency, period, DC offset, and phase were calculated from the sinusoidal waveform:

1 + 3sin(2π50t) V

Using the function generator, an AC Voltage was created by setting the amplitude of the wave to 3 or 6pp, adjusting the frequency to 50Hz, and changing the DC offset to 1V.

The function generator was then connected to the oscilloscope where the AC and DC measurements of the wave were compared.

A visual measurement of the wave's period and amplitude was made by counting the number of divisions in the wave's amplitude and the number of time divisions of one cycle. This was then converted into actual values for the wave.

Using the oscilloscope's 'measure' function, the wave's max, frequency, period, and pk-pk were recorded. These values were then compared to the ones calculated previously.

In this section of the experiment, we first used measured resistor values to calculate the expected RMS current flow of an AC Circuit and the voltage drop across the 3.3k resistor. We then constructed the circuit using a function generator and measured the RMS current flow and peak-to-peak voltage drop across the 3.3k resistor using a digital multimeter. These measurements were then taken again by using the oscilloscope instead of the multimeter.

With the same circuit configuration, we then calculated and measured the same values when the frequency of the source was changed to 250kHz, 500kHz, and 1000kHz.

Parameter | Value |
---|---|

Amplitude | 3 V |

Frequency | 50Hz |

Period | 0.02 sec |

DC Offset | 1 V |

V (Peak to Peak) | 6 V |

Phase | 0 |

**Visual Measurements:**

Period: 4 divisions (at 5ms/div) = 20 milliseconds

Amplitude: 3 divisions (at 1V/div) = 3 volts

Max: 3.03

Frequency: 50 hertz

Period: 20 milliseconds

Pk-Pk: 6.06

**Section 1 Comments:**

In this section, the measured values from the oscilloscope replicated those made via calculations. Although the visual calculation of the wave isn't the most accurate method, it produced the same numbers calculated. This means that this would be a valid method to quickly measure/estimate the values that are inputted into the oscilloscope. Using the oscilloscope's automated measuring produced a measurement slightly above the calculated values, however, within reasonable accuracy. This would suggest that there may be some discrepancies either from the production of the AC voltage from the function generator or the outside environment may affect the circuit.

When comparing the DC equivalent to the AC waveform, the DC wave was of the same phase; however, slightly lower in peak values. This DC value would represent the effective/RMS value of the AC wave.

Frequency | Calculated Current (RMS) | Multi-meter Measurement (Current RMS) | Oscilloscope Measurement (Current RMS) | Calculated Resistor Voltage | Multi-meter Resistor Voltage | Oscilloscope Resistor Voltage |
---|---|---|---|---|---|---|

100 Hz | 1.01 mA | 0.99 mA | 1.01 mA | 9.33 V | 9.15 V | 9.33 V |

250 Hz | 1.01 mA | 0.89 mA | 0.92 mA | 9.33 V | 8.23 V | 8.51 V |

500 Hz | 1.01 mA | 0.7 mA | 0.76 mA | 9.33 V | 6.47 V | 7.02 V |

1000 Hz | 1 mA | 0.48 mA | 0.50 mA | 9.33 V | 4.43 V | 4.62 V |

**Part 2 Comments:**

Looking at the results of this section, it can be seen that for the theory holds true for the 100Hz circuit as we see that the calculated and both measurements are fairly similar. However, as the frequency increased, the current across the resistor substantially decreased. This may have been caused by a capacitance or impedance in the circuit or discrepancies in the production/measurement of the AC voltage. While the measured values didn't meet the calculated, the two measured values from the multimeter and oscilloscope remained consistent. Which would suggest that the relationship between the peak to peak and RMS values is still constant. Judging by the results found, it would suggest that the frequency is dependent.

In the experiment, skills were gained from using the function generator and oscilloscope, which showed that the components of an AC sine wave could be extracted from the mathematical expression and that these results would hold true using both visual and automatic measurements.

It was also observed in the experiment that the relationship between the RMS and peak values is relatively constant across a circuit for lower frequencies and consistent in higher frequencies for measured values. The understanding of the measurements at higher frequencies is inconclusive and would require further research.

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