5/28 AC Circuits: Resistance, Capacitance, Inductors

Lab Day: 5/28 Alternating Current: Resitor, capacitor, Inductors

Purpose: We look at alternating current circuits where we consider the relationship between resistors, capacitors, and inductors by analyzing sinusoidal current and voltage through an AC source. 

When we look at an alternating current we get an oscillating voltage and we can consider what will happen to the current. We began class by looking at the oscillating voltage of an AC power supply. We calculated the Vrms in relationship to Vmax in the calculations using integration shown on the right. Since we are using an alternating current we need to find the root mean square values because the

average current and voltage is zero, therefore converting it to positive values.

Next, using a circuit board we preformed three experiments using an AC power supply across a resistor, capacitor, and inductor where we used logger pro to analyze the current and voltage.

Alternating Current: Resistor

In this experiment, we first looked at the current and voltage through an AC power supply when attached to a resistor. The results show a sinuisoidal wave form where we identified the Vmax and Imax from the peaks(amplitude, (A) value on our sine equation) directly from the graphs. The setup was connected using a function generator(power supply) connected to a circuit board that has resistors and also a current meter in order to measure voltage and current on logger pro 

We took the slope of the current and found that the average current is zero.

The graph shows that as voltage increases, current also increases as seen in our sinusoidal graph and fit. Most importantly, the graph shows that the voltage and current is in phase with each other.

We found our theoretical values of Vrms and Irms which was calculated using the Imax and Vmax from our graph above. We compared this value to our measured value by taking the vrms and Irms directly using a multimeter. It was shown that the percentage error was 24% for our voltage and 7.1% for the current. The voltage error was quite small while the error was higher in our current measurements. 

Resistor

Alternating Current: Capacitors

In this experiment, we used a function generator as our power supply connected to a capacitor (450 microfarad) and a current meter that allowed logger pro to measure the current. 

By considering a capacitor in AC circuits we take into account the resistance where we know that a capacitor has a capactive reactance (Xc) which is in the same units as a resistors through ohms law where we can use Vmax=ImaxXc

We derived another set of equations with capacitance and found that there is a phase shift. Since we know that the angular frequency is 2pif. We can calculate the capacative reactance with the given frequency and capacitance then apply ohms law to find the Irms using the Vrms value. Therefore we can calculate the capacitive reactance to find the Irms shown in the whiteboard.

The graph shows the current and voltage with an capacitor with results indicating that the voltage graph is lagging by 90 degrees meaning that they are out of phase. This is consistent with our derivations. When there is a large voltage across a capacitor, that means the capacitor is fully charged Q=CV. When Q is max, we get that the capacitor will impede the flow of charge. Therefore the current will be zero when voltage is at maximum and minimum seen in the graph. When the voltage goes negative, it will induce a current causing the current to be at a maximum. 

Below we have a circular graph of current vs voltage where it represents a current going in a circular motion which is consistent with the fact that it is 90 degrees out of phase. Therefore, the power cannot be dissipated in a capacitor. 

In this part we summarized the results derived from our graph of Vmax and Imax similar to the previous experiment with resistors. However, since capacitors exhibit a resistance, we call it the capacitive reactance which is inversely proportional to the capacitance. We once again compared the measured values using a multimeter to the theoretical values calculated using the graphs and found that there is a percentage of 79% error which is quite large and this due to the fact that we did not account for the resistance found in the power supply causing us to get higher resistance in experimental capacitance reactance. 

Alternating current: Inductors

The last experiment we did was an AC circuit connected to an inductor. 

We derived the inductive reactance by taking the integral of the voltage in the white board below. 

We solved for the Irms and inductive reactance given the voltage, inductance, and frequency.

The results showed an interesting elliptical shape for the current vs voltage graph and this is consistent as the phase shift is under 90 degrees in comparison to the capacitor.

We summarized the results of inductors in AC circuits and we found that there was a percentage error of 25% which is pretty good and could be accounted for by the resistance that we did not account for from our power supply and the uncertainty

Conclusion: We learned that in AC circuits, we get a sinusoidal wave form in our voltage and current where the average for both is zero. Therefore by relating Vmax and Imax values, we can find our Vrms and Irms in order to analyze our results. We further preformed three experiments of resistors, capacitors, and inductors in AC circuits and found the relationship between each of them within our uncertainty. This is important because when we analyze RC circuits and RLC circuits, we are taking into account all two or three of these components and by combining our results there will be overlapp.