AC RC Circuit & Resonance in RLC Circuits

In AC RC circuits, the impedance can be found using the formula below. We also found the current in terms of voltage, resistance, frequency and capacitance. Using the derived formula, we were able to find the relationship between frequency and current. If the frequency increases, the current increases as well. The capacitance reactance formula includes both frequency and capacitance.

Here we connected the function generator across the resistor and capacitor with a current probe setup to register the current through the circuit. We connected the voltage probe across the resistor and capacitor.

                                     

Using Logger Pro we recorded the max voltage and max current by interpreting the graphs. We then calculated for the impedance. The capacitance was given and assumed to be 100uF. The resistor used was the 10 ohm resistor. We repeated these steps at 2 different frequencies, 10Hz and 1000Hz. 

For the first run, we set the frequency at 10 Hz.

These are the graphs used to interpret the max voltage and current. We took values from the graphs peaks. 

Once we increased the frequency, the graph produced looked like this. We then had to change the time/lapse so that we could read the graphs peaks easier. 

Now we calculated for our experimental values of Irms and Vrms so that we could calculate impedance. We then calculated for our percent difference between our experimental values for impedance and our theoretical values. The percentage difference was under 3%. The experiment was a success. 

Here is our calculated experimental values for the phase angle when the frequency is 10Hz and when it is 1000Hz. The formulas used are shown.

Since we did not use an inductor for this experiment, the inductance reactance was zero. 

Our theoretical values for the phase angles are shown with calculations. Our experimental angles were slightly off. This is possibly because of our methods when reading the graphs from logger pro. We should have autoscaled or zoomed in more to find the exact Vmax and Imax. Then our phase change might be closer in value. 

Below is an example of how to solve for resonance frequency in RLC Circuits when given inductance and capacitance. Inductors and capacitors do not dissipate power; only resistors do. We also solved a problem for finding power in an RLC Circuit when given Vrms and the frequency. 

Using our previously found value for the inductance of our 110 turn inductor and the given capacitance, we found a theoretical value for our resonance frequency. 

In order to find the experimental resonance frequency, we connected our RLC circuit with the 110 turn inductor and connected it to our function generator. We then set our our voltage to 2.0V. Next we used a multimeter to find the current through the circuit by setting it in series with the circuit components. We increased the frequency until the current reached its peak at 76.6mA. When at its maximum current, the frequency was 280.0Hz.

Next we used a multimeter to find the current through the circuit. We increased the frequency until the current reached its peak at 76.6mA. When at its maximum current, the frequency was 280.0Hz. We knew that the current was at its maximum because it decreased when the frequency was increased again. 

Summary:

• Resonance frequency is reached when the current it at its maximum. 
• Using the formulas for impedance and reactance, we can solve for Vrms, Irms, and frequency. 
•  Inductors and capacitors do not dissipate power; only resistors do.
• If the frequency increases, the current increases as well.
• This experiment still follows that AC circuits follow Ohm's Law with a different equation for impedance, or total resistance. 

AC Circuits & Voltage

The actviphysics activity showed us that once current reaches a certain value, the emf is no longer induced. The current left is produced by the battery used. We also solved for current using the formula I=V/R and solved for the time constant. 

Below are graphs of voltage and current over time. The derivations for Vrms, Irms,  Imax, and Vmax are shown too. 

Here we set up our circuit board with a resistor known resistance. We connected the voltage probe across the source and set the frequency to 10Hz on the function generator connected. 

Our graph on logger pro was as shown. We used the peaks on graph to find the Vmax and Imax. 

The voltage vs current ended up being a linear graph. 

The linear fit graph is shown.

We then compared our experimental values with our theoretical values and found the percent error. 

Here is a derivation for current as a function of capacitance. Current is directly proportional to the frequency. 

The difference between AC circuits and DC circuits in regards to Ohm's Law is the value of Z, where Z represents impedance. The proportionality of current and voltage can be checked by testing for linearity in the current vs voltage graph.

Next we set up our function generator with a current probe in series with the capacitor. 

Our graphs for current vs time, voltage vs time, and voltage vs current are as shown. The peaks of voltage and current are different. We later used the difference in the peaks to determine the phase change. 

Below are calculations for Vrms, Irms, frequency, percent difference and phase change. 

Here are derivations for current and voltage in terms of inductance, omega, time, and theta.

Finally we experimented on the inductor, and connected our function generator across the inductor with a current probe in series with the inductor. 

We viewed the graphs using logger pro.

We performed the calculations for Vrms, Irms, impedance and percent difference. Our values were very different from the theoretical value. We noticed that lab groups around us had the same large percentage difference. This large difference could be due to miscalculations or the inductor used.  

Summary:

• An AC circuit is a circuit in which voltage and current vary sinusoidally with time.
• the average current and voltage on any AC circuit is zero because of the negative and positive values due to frequency
• In each cycle, voltage and current hit certain peaks that can be used to find the Vmax, Imax, Vrms and Irms. 
• In AC circuits obey Ohm's Law but have one important difference. The impedance is calculated differently. 

Inductance

An inductor is a coil of wire which stores energy in a magnetic field when it carries current. The inductance of an inductor is defined as the magnetic flux through the inductor per unit current. Any change in the inductor leads to a change in the magnetic field it produces. This causes a change in magnetic flux through the inductor and produces an induced emf in the inductor.

The graphs for current vs time, voltage vs time and current vs voltage are shown below. The current cannot change instantaneously. The formula for inductance is shown below.

This problem required us to solve for inductance when given turns and dimensions. Since we knew the charge density of copper, we were able to find the resistance.

Here we solved for the time constant when given the inductance and resistance.

The next 3 images are in reference to an online activphysics activity we completed. 

Below you can see that the direction of the magnetic field is in the same direction as the velocity of the charged particle. 

Here we solved for the magnetic flux using the initial magnetic field, area and angle between the area vector and magnetic field vector. 

We found that as L decreases, the magnetic flux decreases because the area gets smaller. Therefore, when the area increases, the magnetic flux will increase.

The magnetic flux is also proportional to the area.

The area is proportional to the velocity because the velocity determines the rate of change of the area.

Derivations for induced emf are shown at the bottom.

The induced current depends on the direction of the magnetic flux. Induced current is proportional to the velocity, magnetic field, size of the loop and change in area. When the max current is induced, the max emf is also induced and the resistance is at its minimum.

The current and voltage relationship in terms of capacitance is exponential. The derivations are as follows:

The derivation of inductance and voltage in terms of inductance are shown. The length of a coil is directly proportional to its permeability. 

Here we solved an inductance problem.

The units for inductance are as follows:

In this problem we showed how after a long period of time, an inductor has no effect. A conductor however still acts as a resistor and drops the current. The slope of the current vs time graph is the voltage. Inductors resist rapid change in current. 

In this activity we saw how the frequency changes the graphs for emf and magnetic flux over time.

When the area decreases, the magnetic flux decreases. The flux is proportionally dependent on the magnetic field, area and rotational angle. 

An induced emf still occurs even if the frequency is zero. The induced emf is at its maximum when the plane is parallel to the magnetic field. When the magnetic field is perpendicular to the plane, the induced emf is zero. When the frequency increases, the amplitude of the emf graph increases. 

Below Mason showed the class how a charged metal rod reacts when placed into a magnetic field. Initially the rod moved away from the magnetic. Then Mason changed the direction of the magnetic field and the metal rod moved toward the magnet. 

These  2 graphs depicts a square function produced by a function generator onto an oscilloscope.

For this experiment we used a 110 turn inductor. 

We connected the inductor to an oscilloscope and a function generator in order to produce a graph of voltage vs time. 

Using the graph above and the formula from, we found the time constant and calculated the half time.

Using the half time, total resistance and the time constant, we solved for the resistance of the inductor an the number of turns within the inductor. Our calculations were very similar to the actual number of turns which was 110.

Summary:

• An inductor is a coil of wire which stores energy in a magnetic field when it carries current. 
• The inductance of an inductor is defined as the magnetic flux through the inductor per unit current. 
• Any change in the inductor leads to a change in the magnetic field it produces. 
• This causes a change in magnetic flux through the inductor and produces an induced emf in the inductor.
• We found that as L decreases, the magnetic flux decreases because the area gets smaller. 
• Therefore, when the area increases, the magnetic flux will increase.
• The magnetic flux is also proportional to the area.
• The area is proportional to the velocity because the velocity determines the rate of change of the area.
• The induced current depends on the direction of the magnetic flux. 
• Induced current is proportional to the velocity, magnetic field, size of the loop and change in area. 
• When the max current is induced, the max emf is also induced and the resistance is at its minimum.
• An induced emf still occurs even if the frequency is zero. 
• The induced emf is at its maximum when the plane is parallel to the magnetic field. 
• When the magnetic field is perpendicular to the plane, the induced emf is zero. 
• When the frequency increases, the amplitude of the emf graph increases. 
• The length of a coil is directly proportional to its permeability. 
• When the area decreases, the magnetic flux decreases. 
• The flux is proportionally dependent on the magnetic field, area and rotational angle.