Objective:
The objective of this experiment is to verify Kirchhoff’s rules applied to a two-loop circuit.
Equipment:
A few ceramic resistors (200 to 500 ohms), two dc power sources (0 to 20 volts), 1 to 5 multi-meters, a calculator, and a few connecting wires with alligator clips
Theory:
Ohm’s law alone is not sufficient to solve for unknown currents in multi-loop circuits. In such circuits, Kirchhoff’s rules are used to solve for the unknowns. There are two rules:
a. Kirchhoff’s loop rule (KLR)
b. Kirchhoff’s junction rule (KJR)
KLR:
KLR simply states that the algebraic sum of voltage jumps and drops across the elements of a closed loop is zero. To apply this law to a selected closed loop, a point must be selected, and then moving from that point in one direction, voltage ups and downs must be written with a plus sign for a jump and minus sign for a drop, until that point is arrived at again. The sum then must be set equal to zero. The reason is that if both terminals of a voltmeter are placed at the same point, the voltmeter will show a potential difference (voltage) of zero. There are a few points that should be considered when applying KLR.
1) Assume a direction for the current in each branch.
2) If moving with the current, then the voltage will drop when the current passes through a resistor (for example, if the assumed current is I_{1}, and the resistance is R_{1}, then there will be a voltage drop of (-V_{1} = -R_{1} I_{1})).
If moving against the assumed current, then the voltage will increase as the current goes through a resistor, and the voltage jump will be (+V_{1} = +R_{1} I_{1}).
3) For a battery (inside of it), going from (-) to (+) is associated with a voltage jump, and from (+) to (-) with a voltage drop. In Fig. 1, arbitrary directions for currents I_{1}, I_{2} , and I_{3} are assumed and shown.
KLR for loop efabe is written as follows:
- V_{1} - I_{1}R_{2} - I_{1}R_{1} + I_{3}R_{3 } = 0. (1)
KLR for loop ebcde may be written as:
- I_{2}R_{3 }+ I_{1}R_{3} + V_{2} – I_{2}R_{4} = 0. (2)
KJR:
A junction is a point in a circuit where more than two wires are connected.
Points b and e are the two junctions in Fig. 1.
KJR states that the algebraic sum of currents going toward and away from a junction is zero. If the currents going to the junction are taken to be positive (+), then the currents going away from the junction will be negative (-).
The current in branch be is assumed to be I_{3}. KJR at junctions e and b may be written as
At junction e: I_{2} – I_{1} – I_{3} = 0 or I_{3} = I_{2} – I_{1}
At junction b: I_{3} + I_{1} – I_{2} = 0 or I_{3} = I_{2} – I_{1}
Fig. 1
A two-loop circuit has three branches. For each branch, the current must be determined; therefore, there are three unknowns. Three equations are needed to solve for three unknowns. Two KLRs and one KJR will provide the three equations.
Procedure:
Click on the following link: http://www.physics.uoguelph.ca/applets/Intro_physics/kisalev/java/kirch1/index.html . A 2-loop circuit containing 3 batteries and 3 resistors appears. If you keep clicking on the negative pole (blue) of any of the batteries, its voltage decreases until it becomes 1.0V. Another click changes its polarity. Clicking on its positive pole (red) increases its voltage to a maximum of 10.0 volts. Experiment this for verification. Also clicking on the top-end of any of the resistors, increases its resistance and clicking on the bottom end of it decreases its resistance.
As shown below, name the resistors R_{1}, R_{2}, and R_{3} from left to right, respectively. Each branch of the circuit in the applet has a battery in it. Name the voltages of the batteries V_{1}, V_{2}, and V_{3} from left to right, respectively, as well.. On the applet, ignore the voltmeters that show the voltages across the resistors. They are rectangle-shaped. The oval-shaped ammeters are important for this experiment. Name the current in the left branch, I_{1}, in the middle branch, I_{2}, and in the right branch, I_{3}. Of course the ammeters, A1, A_{2}, and A_{3} measure the currents I_{1}, I_{2}, and I_{3}.
Set the applet at the data given in the first row of Table 1.
Double-check the values for battery voltages and resistances.
Read and record the currents I_{1}, I_{2}, and I_{3 }as measured by the oval ammeters of the applet. These are your measured values. DO NOT substitute these measured values in the equations you will write in the following steps. Also, ignore the the +/- signs of these values. Just write their absolute values in Table 1.
Procedure for Calculations:
Note: I_{1}, I_{2}, and I_{3} must appear in the following equations as unknowns.
If you do not know how to use your calculator to solve simultaneous equations, click here. Instructions are given for using TI 83, TI 85, and TI 86 and more.
Assume one arbitrary direction for each of the currents I_{1}, I_{2}, and I_{3} in the left, middle, and right branches and show each by an arrow. You need to draw the 2-loop circuit completely on paper and show your assumed direction for the current in each branch. One possible assumption is shown in Fig. 2. At the end of your calculations, you may get positive or negative numbers for the currents. A positive value indicates that your assumed direction was correct. A negative value shows that your assumed direction was wrong and that the real direction for that current is the opposite.
Apply Kirchhoff's Loop Rule (KLR) once to the left loop and once to the right loop to obtain 2 equations involving the 3 unknowns I_{1}, I_{2}, and I_{3}.
Apply Kirchhoff's Junction Rule (KJR) to one junction (either Junction B, or Junction E). This gives the 3rd equation you need to solve for the three accepted values of I_{1}, I_{2}, and I_{3}.
Calculate a %error for each current.
Repeat the experiment for the remaining rows of Table 1.
Fig. 2
Data: Given and Measured:
Case | V_{1} _{Volts} |
V_{2} _{Volts} |
V_{3} _{Volts} |
R_{1} _{Ω} |
R_{2} _{Ω} |
R_{3} _{Ω} |
_{Currents:} | I_{1} _{Amps} |
I_{2} _{Amps} |
I_{3} _{Amps} |
% Error | |||
1 | + | 5.0 | + | 4.0 | - | 3.0 | 4.0 | 5.0 | 7.0 | Measured | ||||
- | - | + | Accepted | |||||||||||
2 | + | 3.0 | - | 6.0 | - | 8.0 | 5.0 | 4.0 | 9.0 | Measured | ||||
- | + | + | Accepted | |||||||||||
3 | - | 6.0 | - | 6.0 | + | 6.0 | 5.0 | 3.0 | 7.0 | Measured | ||||
+ | + | - | Accepted |
Calculations:
With the given values of V_{1}, V_{2}, V_{3} , R_{1}, R_{2}, and R_{3}, apply KLR and KJR to solve for I_{1}, I_{2}, and I_{3, }and use these calculated values as accepted values.
Comparison of the Results:
Corresponding to every measured value, there is an accepted value. Calculate a percent error on each current.
Conclusion: To be explained by students
Discussion: To be explained by students