6/2 RC and LRC Circuits

Lab Day: 6/2 RC and LRC Circuits

Purpose: Today we will look at AC,RC and LRC circuits where we will analyze how resistors and capacitors, and inductors when arranged together affect the current and voltage relationships. 

We began class by using a function generator connected to a circuit board across a resistor and a capacitor where we derived equations using the impedence factor. The impedence is a measure of the resistances across an inductor, capacitor and resistor and this is important when looking at a whole circuit. It works well even when one factor is not connected. We calculated the Irms as we double the frequency.

Experiment AC: RC Circuit Analysis (AC)

We have a circuit board showing a connection of a resistor and a capacitor(AC)

We began the experiment setup seen using an alternating current and connected a resistor and a capacitor to the circuit. We connected the current meter to logger pro and the results showed a sinusoidal wave form of current and voltage.

We initially setup the function generator at 10Hz and then we looked at the graph using logger pro for the same experiment with 1000 Hz and found a relationship between frequency and current.

We measured the current and voltage seen in the picture and found that the current graph had many bumps which was due to the sampling rate at high frequencies. After adjusting the sampling rate, we fit the graph and used the values for our calculations of theoretical/experimental values.

Using the new equations derived, we eliminate the inductor in our impedence calculations and found the vrms values and Irms values. We recorded our experimental results in the chart displayed on the whiteboard. 

We then calculated the phase angle when it is at 10 Hz and 1000Hz and found that as frequency increases the phase angle becomes a lot smaller. If we were to calculate the power dissipated in our circuit we would use P=I^2R and only look at the resistor.

RLC Circuits

The experiment was repeated with an RLC circuit by connecting a resistor, inductor, and a capacitor to the circuit. The results in the white board on the right shows the results of experiment using the circuit board and multimeter to compare to theoretical calculations. In fact in an RLC circuit, we find resonance when the capacitive inductance of the inductor is equal to the capacitive reactance of resistor. We get that W=1/sq(LC) which occur only in correlation to maximum power.

Conclusion: We looked at different types of circuits in alternating current where we found experimentally with resistors and capacitors (RC circuit) connected in series that the current and voltage is different by a phase angle. When we looked at RLC circuits which is a whole circuit, we found that there was a special case of resonance where the angular frequency is inversely proportional to the inductor and capacitor. 

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. 

5/26(RL circuits/ Oscilloscope)

Lab Day: 5/26 RL circuits (Oscilloscopes, Inductors)

Purpose: We look at RL circuits and use oscilloscopes to analyze the relationship. 

We began class by looking at inductors and applying Faraday's law where a changing current will cause an induced emf in the inductor.  The graph below shows that as time goes on, the current will increase exponentially.  Initially when the switch in the circuit is closed, no current is flowing and the voltage across the inductor is the same as the voltage source. Then as the current increases it starts plateauing and reaches a steady state consistent with our equation.

Next, we used the equation for inductance using a given resistance and found that the inductor had an inductance of 760 mH. we found the resistance of the copper wire in the given inductor based on calculations below after find the inductance of the given inductor. The resistance calculated is quite small.

Introduction to the Oscilloscope: Measuring Inductance

In this experiment, we used the oscilloscope shown in the picture connected to a function generator. The display shows a square wave form initially shown in the picture. We adjusted the knobs and played around with the controls to get the voltage to be displayed in this shape.

We then attached an inductor in parallel to the oscilloscope and adjusted the positions again and it showed an altered wave form from our original display.

The results showed expected behavior as the induced emf was shown across the inductor. We then calculated the inductance using the wave form measured from our oscilloscope and found experimentally that the inductance across the inductor is 703 mHenrys. The percent error is 7.5% from our theoretical calculations of our inductor used which is pretty good. 

LR Circuit Problem

We went further to look at an LR circuit where we have an inductor and two resistors connected in our circuit. Using the equation I=Imax(1-e^-t/tao) we found the current flowing through. Below shows calculations using kirchoff's rule and ohms law to solve for time constant, energy, and current.

Conclusion: The voltage and current is seen to behave differently when an inductor is added to the circuit and the current is seen to increase rapidly and reaches a steady state as seen in the experiments of our oscilloscope and RL Circuit problems.

5/19 Electromagnetic Induction, Inductors

Purpose: The purpose of today's class is to look at inductance and electromagnetic inductors using activphysics and to look at how inductors work.

Electromagnetic Inductors

We started class by using activphysics to learn about the magnetic flux and induced emf. The change in magnetic flux caused an induced current or emf which is consistent with faraday's law. When the magnetic flux changes the most, we get the highest induced emf and when the angle between the magnetic field and the area is 90 degrees we get that there is no induced emf. Also the negative emf indicates a positive change in magnetic flux. 

Copper Rod Experiment

In this experiment, Professor Mason setup a rod that would move when connected to a supply of current. As a result when we gave the rod a current, the rod moved and when we changed the current direction, we found that the rod also moved in the opposite direction.

The whiteboard shows our predictions and the experiment was consistent as the rod reversed its direction when we changed the current direction by going inward. This is due to the fact that the force on the stick will be inverted as we invert the current direction. Using the right hand rule, we can see that as the magnetic field is pointing upward, we have a current going along the rod, we will get a force pushing out.

In the next activity, we look at motional EMF and answered the questions on the whiteboard and we see that the velocity is reversed as we reverse the current as seen in the previous experiment.

Inductors

Inductors are important because they serve as an energy storage in a magnetic field when there is a current in a wire. We define inductance as the negative emf produced by the inductor over the change in current.  We can then plug in the equation given the number of turens to calculate the magnetic flux. Therefore we can find the voltage gain and drop across an inductor using inductance 

We derived the emf using the lengh, area, and number of turns in regards to the changing current using these factors to identify the inductance.

Next, we looked at the inductance of a Solenoid and found with given values using the equation derived previously to solve for inductance knowing the number of coils, area, and length of solenoid.

We were able to identify the units of inductors using our equation and found that it is equal to one Henry.

After a long time has passed, we find that the current will eventually plateau and remain constant and goes toward infinity. The graph on the right of the board shows this relationship as voltage is the inductance times the changing current.

Finally we looked at activphysics again regarding RL circuits where we have resistors and inductors. We found that when an inductor is connected, we have an induced emf that is opposite of the emf from the battery supplying the current. As time increases, the inductance increases and as resistance increases the inductance decreases. The relationship is shown in the questions answered below. If the inductance is large, we get that the current will reach equilibrium slower than if we increase the resistance which will allow for the current to reach equilibrium faster. The relationship is confirmed in our time constant relationship with resistance.

Conclusion: By looking at activphysics, we learned that any current supplied will cause an induced back emf going the opposite direction as the emf supplied by the battery. Therefore, we can identify how forces in related to velocity and movement of objects in a magnetic field. We also found that by changing magnetic flux, we get an induced emf to oppose the changing flux. The second part of class, we learned about inductors and identified that inductors are useful in magnetic fields in order to store energy when there is a current supplied. We learned using the number of coils, area, and length the idea of self-inductance and solenoids. When inductors are added to the circuit seen in RL circuits, we find that it causes a change in magnetic flux which in turn produces a induced emf.

5/14 (Lenz's Law, Faraday)

Lab Day 5/14/15: Magnetic flux, Lenz Law, Faraday

Purpose: To apply Lenz's law and Faraday's Law to magnetic fields and explore the idea of induction. 

At the beginning of class, we were given two lines that have a current, a magnetic field, and a magnetic force on each. The current was predicted to go upward when the magnetic field points out of the board. When the magnetic field goes into the board on the right line, the current also goes upward using the right hand rule. Once again, force is equal to the current, length of the wire times the magnetic field. The magnetic dipole moment is used to find the magnetic field of an infinitely long wire.

Professor mason then performed the experiment by switching on the power supply causing an alternating current in the two parallel lines. There are in fact no net forces acting as there due to the alternating current. The graph created by logger pro below shows magnetic field in respect to time where two cycles of B is observed in which we can interpret the magnetic field oscillations caused by the north and south poles by applying Faraday's law.

Magnetic Field at the Center of a current loop

We found the proportionality of loops and magnetic fields by measuring the magnetic field when applying different number of loops N to it. In this experiment, we initially turned on the power which supplied a current to measure the magnetic field change due to one loop. Next, we measured the magnetic field as we increased the coiled number of loops of the insulated wire. The ammeter was used to measure the current through the wire which displayed measurements in the graph below connected to logger pro.

As a result, the magnetic field is proportional to the number of loops and the current. Therefore, the more loops, the larger the magnetic field when B is proportional to NI. 

Given the surface area ab, we can find the magnetic flux through a surface. It is seen that when the magnetic field is parallel to the surface area, the magnetic flux is zero. When the magnetic field is perpendicular to the area, we see that magnetic flux=BA=Bab.

Next we take a look at how to create a current in a coil of a wire using magnetic fields. In the experiment above, Prof. Mason used a bar magnet and galvanometer and allowed the magnet to go back and forth causing movement (changing magnetic fields) which stimulates a current seen in the movement of the meter through induction. As he stops in the center of the coil there is no change in the meter. 

There are four ways to maximize the current. If we want to maximize the current on an induced EMF we can add more loops on the coil, have a bigger loop by increasing the area, use a bigger magnet, and also move the magnet faster. 

Major points:

  (1)currents can be induced in a conductor(coiled wire) in the presence of a changing magnetic field 

Lenz's law is explored in this experiment.

In the left picture, when the metal is placed on top of the magnet, the light bulb lights up. The picture to the right shows a thin silver loop of copper of aluminum that is placed on top of the metal in which it floats when he turns on the electricity. However when this metal is cut, it would not suspend anymore.

When the north pole of the magnet is going toward a loop, the flux increases and an upward secondary magnetic field is created causing a counterclockwise current. The changing flux created by the magnetic field created by the induced emf causes the the loop to float due to the force upward.

In this part, when we place two metals inside with one made of plastic and one made of aluminum at the same time. As a result the aluminum fell much slower than the plastic.

The reason for the magnet moving slower is because there exists a magnetic field that generates a magnetic force upward. The induced current generates a magnetic field which then creates a force that opposes the induced EMF causing it to move slower.

Mathematical Representation of Faraday's Law

The picture above derives the emf and flux. We get B=B0sinwt where we can see the coil of wire with N loops and radius R, and when placed perpendicular to B a uniform magnetic field, can be identified with factors of angular and time sinusoidally. The electromotive force in the pickup coil is derived. 

Then we look at the electromotive force in respect to time in comparison to the B vs time graph. There is a phase shift seen from the original graph of magnetic field.

Conclusion: Today we applied Lenz's Law and Faraday's Law to see how we could induce a current using magnetic fields. We were able to come up with a mathematical derivations of these concepts to find that an EMF and flux explains the relationship between the two.