Experiment 2K: Kirchhoff's Rulest



Experiment 22: DC Circuits - Kirchoff’s Laws

Purpose:

To study the principles of DC networks and the use of Kirchoff’s Rules.

Apparatus

1) a pre-wired circuit board with batteries, DC millimeter

2) a DC voltmeter electrical connectors

Theory: Definitions

1) A junction is a point where two or more leads connect in an electrical circuit.

2) A branch connects two neighboring junctions. Each branch carries its own branch current.

3) A loop is a closed path, which starts and ends at the same junction.

Theory

A) Kirchoff’s Current Law: The algebraic sum of all branch currents at a junction equals zero.

Σ Ii = 0 (1)

[Sign Convention(: all currents entering the junction are assumed positive (+I), while all those leaving it are assumed negative (-I).]

So, e.g., at a junction with three currents, I1 enters, while I2 and I3 leave, would be written, using Kirchoff's current law as: +I1 - I2 - I3 = 0. From which one may deduce that: I1 = I2 + I3. This makes sense since I1 is supplying current to the junction while I2 and I3 are removing current from the junction, or, ΣIin = ΣIout. (See Fig. 1.)

[pic]

B) Kirchoff’s Voltage Law: The algebraic sum of all potential differences in a loop is equal to zero.

Σ Vi = 0 (2)

[Sign Convention: The side of a resistor, capacitor, or inductor (passive circuit component) where the current enters is positive (+V) since the current loses energy as it passes through the component and the entry side is at a higher potential. The other side is negative (-V) since it is at a lower potential. The side of a battery or power supply (active circuit component) where the current enters is negative and the other side is positive (meaning that the current gains energy as it passes through the component.]

There are two types of potential differences in any circuit: 1) a voltage difference, (designated by a "V") and 2) an electromotive force (emf or ε, aka batteries or power supplies) designated by “+/- ε”.

To set up an equation like (2), trace a path around a closed loop. Then, when you pass through a battery write +ε if your path gains energy. Passing through a resistor, capacitor, or inductor, write the voltage as "-V" if you are losing energy (same direction as the current). Use labels appropriately.

For example, look at the simple series circuit with three voltages: ε, V1, and V2 shown in Figure 2. The current direction is clockwise. The sign convention above gives us the positive and negative signs for the 2 resistors and the battery in the circuit as shown. Going around the loop in the clockwise direction, there is a voltage gain through the battery and voltage losses through the resistors. So in this direction, Kirchoff's voltage law gives us + ε - V1 - V2 = 0. From which one may deduce that ε = V1 + V2. The two resistors (V1, and V2) use up all of the voltage supplied by the battery (ε). (See Fig. 2.)

C) Ohm's law: This is the basic form of all voltage drops. The current, I, passes through the resistance, R, which requires a potential difference of V to push the electrons through it. The equation is one of the simplest in all of physics:

V = I · R (3)

Preliminary Procedure

a) Your network board is shown in Fig. 3. Familiarize yourself with it and make sure that the batteries are mounted with polarities conforming to Fig. 3.(

b) Check the zero reading of your ammeter with the terminals disconnected, and the zero reading of your voltmeter with the terminals shorted. If they do not exactly read zero, adjust them with the small thumbscrew in the center (under the needle indicator) of the meter.

c) Record the values of resistances R1, R2, R3, and R4. Measure and record the values of ε1 and ε2 . Note: ε1 should be 3 volts and ε2 1.5 volts if the batteries are fresh.

Procedure Part I: Kirchoff's Current Law

d) When the gaps a-b, c-d, and e-f are closed with wires (called jumpers), there are three different currents in the network: I1in the left loop, I2 in the right loop, and I3 down the center. We assign them directions given in Fig. 4.

e) To measure I1, use the 500mA scale on your ammeter, connecting it across the gap a-b as shown. Be careful to connect the + and – terminals of the ammeter properly. This will allow for either a positive or negative reading.

Close the other two gaps (c-d, and e-f) with jumpers. Your reading determines the true direction of I1. (See Fig. 4.) If your reading is positive, the current is going from the positive side to the negative side. Your ammeter dial should read a positive current; if not, make sure that your batteries are correctly oriented and that your ammeter is correctly connected. If in doubt, call your instructor.

f) If everything is correct, then switch the ammeter to the 50ma scale and record the current I1, estimating it to 3 significant figures. As soon as you finish this, disconnect the ammeter and reconnect gap a-b.

g) To measure I2, connect the ammeter across the e-f gap instead of the jumper, observing the same procedure as in (e) and (f). As soon as you finish this, disconnect the ammeter and reconnect gap e-f.

h) Repeat all this for I3 using gap c-d. Check your original assumed direction of I3 with the direction you found for I3 as it appears when read by the ammeter in the gap c-d. As soon as you finish this, disconnect the ammeter and reconnect gap c-d.

i) According to your measurements, is the equation (1) satisfied or not? If you are not certain, check with your instructor before proceeding.

Procedure Part II: Kirchoff's Voltage Law

j) On your data-sheet, prepare a table as shown. Set your voltmeter to the 5-volt full scale (Hint: attach a red connector to the plus (+) terminal of the voltmeter, and a black connector to the – terminal.) Remember voltage is measured by attaching the voltmeter to the appropriate points without disrupting the circuit.

|Kirchoff's Voltage Law |

|POTENTIAL |SIGN |MAGNITUDE |

|DIFFERENCE |(+ OR -) |(VOLTS) |

|(VOLTS) | | |

|Va - Vk | | |

|Vk - Vj | | |

|Vj - Vc | | |

|Vc - Va | | |

|ALGEBRAIC | |

|SUM: | |

k) Read carefully the sign conventions in Section B. Label resistor + and – in Fig 4 using the true directions of currents I1 , and I3.

Connect the voltmeter properly as you go around the loop. Remember that the voltmeter will show you only the magnitudes of the potential difference. The signs (plus and minus) must be determined by you. Measure and record all potential difference magnitudes and signs in the table. AS SOON AS YOU FINISH THIS, DISCONNECT ALL GAPS.

l) According to your measurements, is the equation (2) satisfied or not? If you are not certain, check with your instructor.

Procedure Part III: The Modified Network

m) With all gaps open, connect a wire across the resistor R4. This effectively removes it from the circuit. Finally, reverse the polarity of emf, ε2 .

n) When the gaps are closed, there will be 3 new values for I1, I2, and I3. (See Fig. 5.) Measure and record the currents in the Modified Network by the same procedure you used in Part I, to wit: start with the 500 ma scale as in (d); then proceed as in (e) and (f).

BEFORE YOU LEAVE THE LAB:

a) Place ε2 into its original position.

b) Disconnect all jumpers from the circuit board.

c) Make sure you recorded the values of your resistances R1, R2, R3, R4 (they are accurate to within 0.1%)

d) Make sure you recorded the values of ε1 and ε2, from the voltmeter readings (accurate to within 1.0%).

Lab Report

Part I: Verification of Kirchoff’s Laws

1) Using your measured values of currents in Basic Network, display the actual value (in mA) of the left-hand-side of equation (1), when applied to junction g.

2) Using your measured magnitudes and signs of potential differences in Basic Network, display the actual value of the left-hand-side of equation (2) when applied to loop a-k-j-g-c-a (counter-clockwise about the left side loop).

Part II: Basic Network

3) Set up 3 equations with 3 unknowns (the currents) for the Basic Network: one for junction g, and the other two for loops a-k-j-g-c-a and i-g-c-f-i.

4) Solve these equations by a method of your choice, but show clearly your method and include most of the steps in your calculation.

WARNING: You are not allowed to use your measured values of currents in any of your calculations. You may use only the recorded values of resistances and the emf's in order to determine the theoretical values of the currents.

NOTE: To achieve high grades, solve your equations algebraically. I.e. do not insert any values for the resistances and emf's. Display a formula for I1, in the following form:

However, once I1 is determined from the algebraic solution, then the remaining currents, I2 and I3 may be determined by substitution of this value for I1.

5) Display your results in Table 1 as shown. Note that the discrepancies should be "absolute," in mA, and not in %.

|TABLE 1: BASIC CIRCUIT |

|CURRENT |CALCULATED |MEASURED |ABSOLUTE |

| |(mA) |(mA) |DISCREPANCY |

| | | |(mA) |

|I1 | | | |

|I2 | | | |

|I3 | | | |

Question #1: Why are the internal resistances of the batteries not considered in this experiment?

Part III: Modified Network

6) Set up 3 equations with 3 unknowns, for the modified network, as in (3) above. Display these modified equations without solving them.

Hint: You should now appreciate the importance of solving the equations algebraically, as in (3), because by reversing the sign of ε2 and setting R4 = 0 in (3) you automatically have the solution for the Modified Network.

7) Solve the modified equations. (Note: if you have done (3), just follow the hint above.)

8) Display your results for the modified network in a table similar to Table 1 above.

( It makes no difference whether this is the correct direction or not. You will find the correct direction after you solve the equations because solutions with the negative sign imply that the actual direction of the current is opposite to the assumed direction.

( A preceding lab class may have left the batteries in the reversed position

-----------------------

V2

-

Fig. 2 Kirchoff's Voltage Law

I

eð

I

I

V1

I

+

-

+

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a

c

e

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d

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k

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1

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Fig. 3 Basic Network (Open)

Fig. 4 Basic Network (Closed)

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3

R

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2

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Fig. 3 Basic Network (Open)

Fig. 4 Basic Network (Closed)

2

R

4

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3

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1

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1

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2

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Ammeter

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Fig. 5 Modified Network (Closed)

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Terms with ε's and R's only

I1 =

Terms with R's only

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Note: R4 is no longer part of the circuit.

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