Power in dc circuits lab report

Physics Lab Report

I need some one do my physics lab reports. Do not copy from other lap report, please.

Each student should write his/her own laboratory report.

Duplicating reports will result in an "E" in your final grade.

•Lab Manuals (contained within each week)

•KET simulations: http://virtuallabs.ket.org/physics/. Students will receive an e-mail from the KET Virtual Physics Labs with an invitation to enroll into the class.

•

•PhET Interactive simulations: http://phet.colorado.edu/en/simulations/category/physics.

Expression of the experimental results is an integral part of science. The lab report should have the following format:

• Cover page (10 points) - course name (PHY 132), title of the experiment, your name (prominent), section number, TA’s name, date of experiment, an abstract. An abstract (two paragraphs long) is the place where you briefly summarize the experiment and cite your main experimental results along with any associated errors and units. Write the abstract after all the other sections are completed.

The main body of the report will contain the following sections, each of which must be clearly labeled:

•Objectives (5 points) - in one or two sentences describe the purpose of the lab. What physical quantities are you measuring? What physical principles/laws are you investigating?

•Procedure (5 points) - this section should contain a brief description of the main steps and the significant details of the experiment.

•Experimental data (15 points) - your data should be tabulated neatly in this section. Your tables should have clear headings and contain units. All the clearly labeled plots (Figure 1, etc.) produced during lab must be attached to the report. The scales on the figures should be chosen appropriately so that the data to be presented will cover most part of the graph paper.

•Results (20 points) – you are required to show sample calculation of the quantities you are looking for including formulas and all derived equations used in your calculations. Provide all intermediate quantities. Show the calculation of the uncertainties using the rules of the error propagation. You may choose to type these calculations, but neatly hand write will be acceptable. Please label this page Sample Calculations and box your results. Your data sheets that contain measurements generated during the lab are not the results of the lab.

•Discussion and analysis (25 points) - here you analyze the data, briefly summarize the basic idea of the experiment, and describe the measurements you made. State the key results with uncertainties and units. Interpret your graphs and discuss what trends were observed, what was the relationship of the variables in your experiment. An important part of any experimental result is a quantification of error in the result. Describe what you learned from your results. The answers to any questions posed to you in the lab packet should be answered here.

•Conclusion (5 points) - Did you meet the stated objective of the lab? You will need to supply reasoning in your answers to these questions.

Overall, the lab report should to be about 5 pages long.

Each student should write his/her own laboratory report.

Duplicating reports will result in an "E" in your final grade.

All data sheets and computer printouts generated during the lab have to be labeled Fig.1, Fig. 2, and included at the end of the lab report.

Lab report without attached data sheets and/or graphs generated in the lab will automatically get a zero score.

EQUIPMENT:

Computer with Internet access.

INTRODUCTION AND THEORY:

I. Kirchhoff’s Rules.

For complex circuits analysis commonly used are two Kirchhoff’s rules known as the junction rule and the loop rule.

The first Kirchhoff’s rule, the junction or current rule has been derived from the conservation of charge principle and states that the algebraic sum of currents into any junction is zero. In other words: the sum of currents entering any junction must be equal the sum of currents leaving that junction. A junction (or a node) is the technical term for a point in electrical circuit where two or more branches are joined together.

Since current is the flow of electrons through a conductor, it cannot build up or disappear at any point of the circuit: what comes in must come out. By convention currents directed towards a junction are regarded as positive, while currents going away from the junction are given a negative sign.

The second Kirchhoff’s rule, the loop or voltage rule, based on the conservation of energy, states that the algebraic sum of the potential differences around any closed loop is zero. It holds because the electrostatic field within an electric circuit is a conservative force field. As you go around a loop, when you arrive back at the starting point it has the same potential as it did when you began. Any increases or decreases along the loop have to cancel out for a total change of 0. If it did not, then the potential at the start/end point would have two different values. Voltage is the energy per unit charge and conservation of energy demands that energy is neither created nor destroyed.

Page 2 of 9

The accepted sign convention for use with the second Kirchhoff’s rule is as follows:

- when traveling from positive to negative (+ to -) in an emf source the voltage drops, so the potential difference is negative; when going from negative to positive (- to +) in an emf source the voltage goes up, so the value of the potential difference is positive,

- when crossing a resistor, the voltage drop (aka potential difference, V = IR) is determined by the direction of the current; crossing in the same direction as the current means the voltage goes down so the value of the potential difference is negative. When crossing a resistor in the direction opposite the current, the voltage is increasing and the potential difference is positive.

To better understand this sign convention let’s apply Kirchhoff’s loop rule to the circuit presented in Fig.2.

Fig. 1. Examples with applications of the Kirchhoff’s Current Rule.

Page 3 of 9

For loop # 1: ∑ " = VB1 – I1R1 + I3R4 = 0

For loop # 2: ∑ " = - VB2 + I2R2 + I2R3 – I3R4 = 0

From the two Kirchhoff’s rules and Ohm’s Law one can derive a relationship for a resistance equivalent to any combination of resistors. When resistors are connected along a single path, so the same current flows through all of them, we call it a series connection. In a series circuit the current through each component is the same and the voltage across the circuit is the sum of the voltages across each element. The equivalent (total) resistance is equal to the sum of all their individual resistances.

(Eqn. 1)

If the same voltage is connected to all components we call the circuit parallel. In a parallel circuit voltage across each of the components is the same, and the total current is the sum of the currents through each element. To find the total resistance of all components, we have to add the reciprocals of the resistances Ri of each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance.

Fig. 2. Example of application of voltage polarity convention in Kirchhoff’s Voltage Rule.

Fig. 3. Resistors connected in series.

Page 4 of 9

(Eqn. 2)

For only two resistors in parallel the solution for an equivalent resistance is reasonably simple:

Rtotal = #$#%

#$&#% .

II. Electric Power.

The electric power associated with a complete circuit or a circuit component represents the rate at which energy is converted from the electrical energy of the moving charges to some other form, like heat or mechanical energy, or energy of electric field or magnetic field.

In case of a resistor the electrical energy supplied to it appears in a form of thermal energy. For a resistor R the power P dissipated in it is given by the product of applied voltage V (potential drop across the resistor) and the electric current I flowing through it:

P = V I (Eqn. 3)

By using Ohm’s Law this relationship can be also conveniently expressed as:

P = ' %

# = I 2R (Eqn. 4)

When I is in amperes, V is in Volts, and R is in ohms, the SI unit of power is the watt, W.

Fig. 5.

Fig. 4. Resistors connected in parallel.

Page 5 of 9

If we consider a simple circuit consisting of voltage source VS with a finite internal resistance RS connected to a load resistance RL as pictured in Fig. 5, it can be shown that to obtain the maximum external power from the source, the resistance of the load must equal the resistance of the source. This is called the maximum power transfer theorem or the resistance matching principle.

For the circuit in Fig. 6 the power dissipated in the load resistor RL is given by

PL = ( ')#)& #+, 2 RL (Eqn. 5)

The value of RL for which this expression is a maximum could be calculated by differentiating it in

respect to RL. The solution is -.+ -#+

= 0 / RL = RS.

The condition for maximum power transfer does not result in maximum system efficiency. If by definition the efficiency η is the ratio of power dissipated in the load PL= I

2RL to power delivered by the source PS = I

2(RL+RS) , it can be calculated from Fig.5 and presented as follows:

η = #+

#+ & #)

When RL = RS , then η = 0.5. When RL = ∞ or RS = 0, then η = 1, When RL = 0 , then η = 0.

The efficiency is only 50% when maximum power transfer is achieved but approaches 100% as the load resistance approaches infinity (note that the total power tends in this case toward zero). Efficiency also approaches 100% when the source resistance goes to zero, and 0% is the load resistance approaches zero, like in a short circuit. In the latter case all the power is consumed inside the source.

It should be emphasized once more that the maximum power transfer must not be confused with maximum efficiency. If the resistance of the load is larger the resistance of the source, a higher percentage of the source power is transferred to the load but the magnitude of the power dissipated in the load is lower since the total circuit resistance goes up. On the other hand if the load resistance is made smaller than the source resistance, then most of the power ends up being dissipated in the source.

A real-life example of this phenomenon is matching loudspeakers to an audio amplifier. So to get the loudest boom-box connection in your car, you should connect an "8 ohm resistance" loud-speaker to an 8 ohm resistance connection on the amplifier. A mechanical analog is matching a propeller (load resistor) to a motorboat (power source) – the boat speed is optimized for proper pitch.

Page 6 of 9

PROCEDURE

Part 1. Testing the Kirchhoff’s rules.

Using Firefox login to VirtualPhysicsLabs: http://virtuallabs.ket.org/physics/ . Click the Labs tab and select lab 16 -DC Circuits. Before running the simulation read the full description and detailed information.

From the” Pick a Circuit” options select “Diagonal” and adjust the values of the circuit elements as shown below. R1=10 Ohms; R2=10 Ohms; R3=20 Ohms; V1=30 V; V2=5V.

Your circuit has two junctions, A and B, and two small loops – see Fig. 6.

Fig. 6. Schematics representation of the diagonal circuit used in Part 2.

Connect virtual ammeters (in series with circuit components) and voltmeters (in parallel with circuit components) as necessary to test the Kirchhoff’s current rule for junction B and the Kirchhoff’s voltage rule for both loops. When doing the calculations be aware of the polarity of measured currents and voltages! Use the accepted sign convention.

To document your experiment capture the simulation screen of the circuit with all the instruments connected and displaying the measured quantity and paste it into MS Word – you will have to attach it to your lab report. It is recommended to use the landscape orientation for the page layout in MS Word.

Page 7 of 9

It should look something like an example in Fig. 7. Note that the values of the circuit elements in the example are different from your experimental numbers.

Fig. 7. A sample connection of ammeters and voltmeters used to test Kirchhoff’s Rules in diagonal circuit. Did the currents for junction B add up to zero? Did the voltages around loop 1 and loop 2 add up to zero? Part 2. Electric power in DC Circuits. Start a new simulation of the DC Circuits lab in the VirtualPhysicsLabs environment. Set up a circuit shown below. Rs=15 Ohms; V=10 V; RL=0.5 Ohms

Page 8 of 9

In order to test the maximum power transfer theorem we need a measurable internal resistance of the voltage source. Since the virtual batteries are ideal voltage sources with zero resistance, we have created a fictional power supply that better resembles a real-life device by adding a 15 Ω resistor in series with the 10 V battery. Collecting the experimental data will include the measurements of voltage and current through the variable load resistance set initially as 0.5 Ω. Open LoggerPro and prepare 3 columns for recording RLOAD (in Ω), I (in A) and VLOAD (in V). To add the third column, go to Data → New Manual Column. Next, create a new calculated column for the power dissipated in the load resistance, P = “I”× """, where I and V should be chosen from the pull down variables selection. Configure the graph to display the power on the vertical axis – left click on the vertical axis label and select the “Power” from the list of variables. Now you are ready to collect the data as described in steps 1 through 3.

• Step 1 - In the simulation circuit close the switch (double click) and record the current and voltage displayed by the measuring instruments.

• Step 2 – Open the switch (double click).

• Step 3 - With the left click select the load resistor and change its value in the “Resistance” window to a new value.

Repeat steps 1 – 3 while changing the load resistance by 2 ohms in the range of 2 Ω to 18 Ω, and then from 20 Ω up to 60 Ω changing the load resistance by 5 Ω. You should end up with 19 data rows in the data table. Double click the graph window and deselect the “Connect Points” option in the graph option menu. To find the load resistance maximizing the power transferred to the load, you will use the curve fit feature of LoggerPro. In the upper tool bar menu click the Analyze tab and choose Curve Fit. In the Curve Fit window click the Define Function button. This will open a new User Defined Function window where in the “f(x) =” field you need to type in function of the same format as derived in eqn. 5. For example:

As shown above for simplification you can use parameter A to represent VS

2 and parameter B to represent the internal resistance of the voltage source, RS . After clicking the OK button and then the Try Fit button, the software will automatically find the values for the parameters which assure the best fit to your data. When the fitted line nicely matches your data accept the fit by clicking OK. To add the uncertainties in the fit parameters, double click the small Auto Fit window and choose the Show Standard Error (or Show Uncertainty) option. Save the LoggerPro file for future reference. After making sure the LoggerPro file shows all the data points in the table and the graph with the curve fit, capture the screen and paste it into MS Word – you will have to attach it to your lab report. It is recommended to use the landscape orientation for the page layout in MS Word.

Page 9 of 9

Fig. 8. A sample file in LoggerPro with the power transfer data. It can be proven analytically that the pick of the power curve happens when x = B (or RL= B). From your fit what value of RL maximizes the power in the load? How does it compare to our simulated internal resistance of the battery set to 15 Ω? * Include answers to all questions in lab report