Alternating Current

Purpose:  We take a look at how a circuit behaves when the current switches directions periodically. We do this by examining the current and the voltage on Logger Pro and comparing the behavior of the two.

Derivations Of The Root Mean Square And Average Power

We do some integration to derive the values of rms voltage and current. The reason we determine these values is so that we are able to get a relationship between the maximum values and the rms. The relationship proves useful as the values detected by the equipiment used are root mean square values because in an alternating current, the average of the values proves to be zero so we use the rms, which makes all values positive to get an alternative average. We also come up with a definition for average power.

The Alternating Current Experiment

We put together a circuit using a circuit board. The circuit board consists of 3 resistors of different values, 3 capacitors of different values, and an inductor. We are only connecting resistors in the circuit in this case. The function generator serves as a power supply and we connect the currentometer and the voltometer to the circuit to measure the current and the voltage in the circuit. The measured values will show in Logger Pro. In this experiment, we use 100 Ohms.

When we record the current and voltage vs. time, we get a sinusoidal graph. Both graphs appear to be symmetrical. When we graph the current vs. voltage, the graph produced is linear. This is consistent, as the sinusoidal graphs show us that when voltage increases, current increases, and vise versa. We fit the lines to get a measurable function so that we may us the values of the function to compare to theoretical values. 

We summarize the experiment by comparing the values of theoretical rms voltage and current with the experimental voltage and current and find that the values were relatively close with percentages of 9.7%

We solve a problem with given values to solve for the maximum current. We used the definition of impedance to solve for the rms current and the rms current to solve for the maximum current.

The next experiment involves only a capacitor in a circuit. The resistor is not a part of the circuit. We see by the diagram on the top right corner shows a circular motion which say that the sine graph produced by the voltage graph is roughly 90 degrees from the cosine graph produced in the current graph, which makes sense that one is a sine and one is a cosine graph.

Our whiteboard shows the comparison of the experimental values to the graph values and the theoretical values. The large difference were by the comparisons between the experimental and graph values. This is due to large uncertainties in the data.

We do some derivations to find a function for the current. We do this by taking the integral of the voltage with respect to time. We find through our experiments that the voltage with respect to time is a sine function multiplied by the maximum voltage. When we integrate, we get that the current is the cosine times the max voltage and the negative inductive reactance. The orange writing shows more arithmetic for finding the rms current using the given values. The integral confirms what we saw in our experiments, that the phase difference is just 90 degrees.

The final experiment has us looking at just and inductor in a circuit. We take a look at the behavior of the inductor the same way we looked at the capacitor and the resistor. We find that the current vs. voltage graph produces an elliptical graph. This shows us that the phase difference is not at 90 degrees but somewhere in between 0 and 90.

We, again, compare the theoretical values to the experimental values. Shockingly, there was no difference in the values found. We, then, calculate the phase difference by examining the time differences in a period.

Conclusion:  We distinguish the behaviors of current and voltage of three different circuits with alternating currents and found that the graphs produce sinusoidal graphs which express that the current and the voltage is oscillating back and forth. The differences we seen is the phase difference in each. For resistors, the phase difference is 0 between the current and the voltage graphs. For the capacitors, the phase difference is 90 degrees. And, for the inductor, the phase difference was somewhere in between 0 and 90, which we found by examining the time difference within the period.