Resonance and Impedance in AC and LCR Circuits (6/2/2015)

AC RC Circuits

It's hard to tell from the picture above, but what you see is a calculation of Ohm's Law in an AC circuit. Ohm's Law states V = IR, but in an AC circuit, the total resistance is called impedance, Z. The equation for total impedance is Z= sqrt(R^2 + Xc^2), Xc equaling 1/wC. Using these relations we can apply Ohm's Law to a circuit as Vrms = Irms * Z. This is what we will be using for the following experiment.

For this experiment, we had a circuit consisting of a 100uF capacitor, a 100 Ohm resistor, and a function generator (set at 1V and varying frequency of 10Hz/1000Hz). This was all connected to Logger Pro in order to read current and voltage over time. Before gathering readings, we calculated the theoretical impedance that we'd expect from this circuit, which was placed into the table of data shown below. We then took readings (shown above) and used that data to fill in the rest of the table shown below. What observe is that impedance greatly decreases when frequency is increased.

What we did next was calculate the phase shift in our readings. We do this because the total current and the power dissipated in a series RLC circuit depends on the phase shift between the total current and the total voltage. The results can be seen below, which show that as frequency increases, the phase shift decreases. Not shown is the phase shift at 1000Hz, which is 9.03(degrees).

If X_L = X_C, then the total phase shift is zero and we get maximum current and maximum power dissipated in the resistor. This is not the case as seen above, so we are not getting the max power and current dissipation possible in our resistor.

Resonance in a Series LCR Circuit

In this next example, we consider an RLC circuit where R = 10 W, L = 0.2 mH, C = 5 mF and the applied voltage is V_rms = 25 V. We were asked to calculate the resonance frequency, which came out to 5033Hz. We then calculated what the current would be if the frequency were 3000Hz, which turned out to be 2.01A. Then, we calculated the total power dissipation in this 2.01A circuit, which was 42.6 W. These calculations can be seen above. 

Resonance in RLC Circuits

For this final lab, we have a circuit which is once more attached to a function generator, but now has a 470mF capacitor attached to it, along with a 10 Ohm resistor, and a 0.19mH inductor. The purpose here is to calculate the resonance frequency. That is, when impedance Z is minimized. The resulting frequency and calculations can be seen above, along with a diagram of our circuit (and our actual circuit).

Summary:

In summation, we have primarily focused on analyzing resonance and impedance of AC and LCR circuits, which is slightly different than with DC circuits. The main takeaway from this is that we now know how impedance and resonance is affected, and what we can do to achieve desirable levels of each. For example, we see what happens when frequency is increased or decreased, and just how much of a significant change this can have on a circuit.

Resistors, Capacitors, and Inductors in an AC Circuit(5/28/2015)

Day 25, Page 11

Alternating Currents and Voltages

For this experiment, we put together an AC circuit as shown above, which consists of a resistor and function generator (set at 2V and 0.01A). This was all connected to Logger Pro, in order to gather readings on current and voltage flowing through the circuit. We needed this data in order to calculate the V(rms), I(rms), and percent error. The second set of data we are going to use to compare with the first set is the result of readings taken from our circuit using a digital multimeter.

Graphs of Potential V vs Time, Current vs Time, and Current vs Potential Voltage.

After we took readings, we calculated what our expected V(rms) and I(rms) would be based on the graphs shown above. We then tested for our actual V(rms) and I(rms) using a digital multimeter. These values can be seen below, along with our percent errors that we were able to calculate now that we received two sets of data. We received acceptable error percentages, so the experiment went smoothly.

Capacitor Circuit

Depicted above is our calculation and representation of how when a sinusoidal voltage source is connected across a capacitor, then the charge on the capacitor and the current to the capacitor also vary sinusoidally with time.  The voltage and charge are related by v = q/C, so the charge and voltage are in phase.  However, the charge lags the current by 90^o, so it is found that the voltage lags the current by 90^o.

The calculation shown below refers to an example problem, which states that a 0.02 mF capacitor is connected to a 50Vrms AC voltage source which oscillates at 10 kHz. We are asked what the Irms is to the capacitor.

Capacitors in an AC Circuit

As with the first experiment we have a circuit shown above, but instead of a resistor being used we have a capacitor hooked into the circuit along with the function generator. The function generator was set to 2V and 0.16A(rms), so twice as much voltage was used this time and a significantly increased I(rms).

After getting the data shown above, we were able to calculate the expected and actual voltage and current in the system exactly as we did in the first experiment. We used our data and equations for V(rms) and I(rms) to calculate our theoreticals, and then tested the actual voltage and current in the circuit using a multimeter. Unfortunately, this time it seems we ended up with a very high percent error, but this seemed common among the class. Now that we delved into phase shifts, we also calculated phase values for our circuit.

Inductors in Alternating Circuits

For this final lab we used the same circuit as before, except instead of a capacitor or resistor, we now tested out this same system using an inductor. We again ran a current through using a function generator, collected data (shown above), calculated expected values for I(rms) and V(rms) using our data, and then took actual readings using a multimeter. Our values (seen below) are slightly better in regards to the percent error, but the error is still rather high.

Summary:

In summation, we created three circuits which consisted of a function generator and one of three other components, which were a resistor, capacitor, and inductor. What we saw was that the V(rms) value was the highest in the resistor circuit, and lowest in the inductor circuit. The I(rms) was opposite, as it was lowest in the resistor circuit, and highest in the inductor circuit.

Inductors in a Circuit (5/26/2015)

*Disclaimer: I was absent this day due to a severe stomach flu which had me admitted to the hospital. I'll try describing the following as best I can from what I've been able to figure out myself through pictures.*

Introduction to Inductance

For this experiment, we had a 110 turn inductor for which we had to calculate the inductance L of. We did this using the equation L=(u(0)*N^2*A)/l, where u(0) is a constant, N is the number of turns, A is the cross sectional area of a solenoid, and l is the length of the solenoid. The value came out to 760 microHenry, and we will use this expected value to compare with the value we get when we experimentally test the inductor. We also calculated the resistance of the inductor, which came out to be 0.27846 Ohms.

After calculating these values, we created a circuit consisting of the inductor and a function generator connected in series to an oscilloscope. By running a current through, we were able to calculate a time constant for the inductor by looking at the reading shown above, and performing the proper calculations shown below.

After that, we added a resistor to the circuit as shown below, along with black grounding leads:

After running a current using the function generator once again, we received the reading shown below on the oscilloscope:

Using all the data we calculated before, the readings achieved from the latest circuit, and certain assumptions about the system (such as the internal resistance of the function generator), we calculated a half-time of the decay of the induced emf in the solenoid. Afterwards, we used the half-time measurement to calculate the actual inductance of the coil which came out to 7.36 microHenry. This results in a 3.16% error based on our initial theoretical calculations. We then used that inductance to calculate the number of turns N in the solenoid, which came out to 108.3 turns, giving us a 1.55% error. All this work can be see below:

LC Circuit Problem

For this next experiment, we imagine a circuit containing a 35.0 mH inductor in series with a 120 W resistor, both of which are in parallel with a 730 W resistor and a 45.0 V battery. The switch is closed for 170 ms and then opened. Given that, we were asked to answer the follow questions (for simplicity, the answers are placed immediately under the questions): 

(A)	What is the time constant when the switch is first closed? 290ms
(B)	What is the value of the current in both resistors at the moment the switch is opened at 170 ms? 0.061A
(C)	What is the voltage drop across both resistors at 170 ms? 20V
(D)	How long will it take for the voltage drop across the inductor to be equal to 11.0 V after the switch is closed? 105ms
(E)	What is the time constant of the circuit after the switch is open at 170 ms? 41ms
(F)	How much energy is dissipated in the two resistors after the switch is open at 170 ms? 480mJ All the work for the previous questions can be seen in the slides below:

Summary:

In summation, we looked more in depth into inductors and put one to the test by using it in a circuit, from which we measured various experimental values to compare against our theoretical ones. Through this process we had to calculate various things such as time constants, which gave us further insight into what factors affect inductance and how we can use those factors in our calculations. After we performed the inductance lab, we were given a problem to solve, which put our recently gained knowledge to the test.

Induction, Faraday's Law, Maxwell's Equations (5/19/2015)

 ActivPhysics 13.9: Electromagnetic Induction

For this experiment we used the ActivPhysics website to run a simulation, and then answered questions in order to analyze several factors. What we primarily analyzed here was magnetic flux and induced EMF. By answering the questions below we were able to connect the two and see their relationship based on changing values of frequency, magnetic field, and area. More detailed explanations of each analysis can be seen after the questions below.

Magnetism and Electric Current

For this demonstration, we have a large magnet with a metal rod placed between the poles. What we ended up seeing was that when a current is run through the rod, a force is created which pushes the rod outward or pulls it inward (seen below) depending on the direction of the current. This is a physical demonstration of what we have been studying, which is that when you have a current going one direction, and a magnetic field going in another, then you get a resulting force.

ActivPhysics 13.10: Motional EMF

This second ActivPhysics experiment is basically an in-depth analysis of the demonstration which Professor Mason conducted just now. The red bar seen above is our rod, and it moves based on the direction of current and magnetic field. By answering the questions below, we analyzed magnetic flux again in order to see how it changes as the rod moves. This led us to an analysis of the induced EMF in this situation and its relationship with the flux and induced current.

Self Inductance and a Solenoid

Based on the previous experiments today, and information from previous lessons, we used various equations for flux, magnetic field, inductance, and current in order to determine the new equation for inductance, which can be see as L = u(0)*n^2*A/l. This is the physical equation for induction.

Shown above is an example from the lecture in which we tested out the new, derived equation for induction. We were given a 100 turn coil with a length of 4 cm and a radius of 0.1 cm, and had to find the inductance of that coil.

Current Flow Through an Inductor

Shown above is the calculation for the units of induction based on our physical equation for induction. What we ended up with was a kg*m^2/C^2, which is what we now call a Henry.

ActivPhysics 14.1: The RL Circuit

This final ActivPhysics activity had us analyze analyze a simple RL circuit, consisting of a resistor, battery, inductor, and a switch. We had to answer the questions above (answers are seen below), which primarily consisted of analyzing the current through the circuit while changing various factors, such as doubling the inductance and resistance. We also had to find the inductive time constant, which is the time it takes for the current in an RL circuit to reach equilibrium, and is proportional to the inductance, L, and inversely proportional to the resistance, R. The ratio of these two values is defined to be the inductive time constant, τ _L.

Seen below is some more elaboration on how the time constant is related to inductance and circuits, which states what was described above for inductive time constants.

Summary:

In summation, we looked at three different situations related to induction and analyzed every factor related to induction in a circuit. We looked at EMF, flux, magnetism, resistance, and inductors themselves and analyzed and calculated how they're all related and how they affect one another. The calculations done outside of the ActivPhysics demos revolved around the simulations we saw in order to gain deeper insight from them.

Magnetic Fields, Power Lines, Levitation (5/14/2015)

Magnetic Fields

While hard to grasp from the drawing above, what is being depicted is how two magnetic fields around wires affect the field around a neighboring wire. We learned that as current is flowing, the magnetic field is flowing in a circular pattern around a wire. As we see by the circles (dots pointing outward, and x pointing inward), due to two wires near each other, the field is rotating upward on the left and downward on the right.

Now, we had to predict what direction the force is pointing in regards to two power lines with current flowing through. We predicted (incorrectly) that the force is pointing outward. The actual answer is that there is no force. The reason for this is because the current in power lines is alternating back and forth, leading to net forces cancelling out.

The Magnetic Field at the Center of a Current Loop

For this lab, we had to test the strength of the magnetic field around a wire with varying amounts of loops. The results of our data can be seen below:

The table above shows all the values we recovered, and the graph below that shows the values of B and shows that as the loops increase, the strength of B increases.

Michael Faraday's Quest

In the demonstration above, it can be seen that by inserting and removing a magnet into a loop of wire, an electrical force was able to be produced. Below are various ways in which we believe the strength of the electric field can be increased.

All are correct, and we have now established a connection between electricity and magnetism. This connection can be visually seen below:

Magnetic/Electrical Field Demonstration

Shown above is a setup which consists of two coils (with a light bulb attached to one coil), and a magnetic field in between the two (located in the grey ring between the coil and the black pole). As you can see, but simply having the magnetic field next to the coil, the light bulb has been lit. When the coil and bulb are lifted up (toward a weaker point in the magnetic field), the bulb dims due to a weaker electrical field being produced.

By looking at the next pictures shown above, we see that a metal ring is placed at the bottom of the pole. What happened next was that a current was run through the bottom coil, which created an opposing magnetic force that lifted the right upward. 

A magnetic force has been created due to the current, and the metal ring was repelled upward.

Shown above is another example of the magnetic repulsion created by the current flow.

The two depictions above show what is going on inside the demonstration.

In these final two pictures and video, we witness that when dropping two identical, non-magnetic, metal pieces down two tubes (one plastic and one metal), the piece falling through the metal tube falls significantly faster. The reasoning behind this is based on what we have been looking at above. A magnetic field is essentially being created as the metal piece moves downward. A drawing of what is going on can be seen on the top right.

Summary:

In summation, we have looked at how magnetic fields interact with one another, as well as how they can be used to create an electrical field. This ties electrical and magnetic fields, since we saw how the introduction of an electrical field created a magnetic field as well. This is what happened when the metal rings were levitated up and off of the pole during the Faraday demo.