Download Experiment 2K: Kirchhoff`s Rulest

Revised 4/14
Experiment 22: DC Circuits - Kirchhoff’s Laws
Purpose:
To study the principles of DC networks and the use of Kirchhoff’s Rules.
Apparatus
1) a pre-wired circuit board with batteries, DC multimeter
2) a DC voltmeter electrical connectors
Theory: Definitions
1) A junction is a point where two or more wires connect.
2) A branch is a wire between two neighboring junctions. Each branch carries its own current.
3) A loop is a closed path, which starts and ends at the same junction.
Theory
The two Kirchhoff Laws which we will apply to our circuit are Kirchhoff’s Current Law and
Kirchhoff’s Voltage Law.
A) Kirchhoff’s Current Law: The sum of all branch currents at a junction equals zero:
Σ Ii = 0
(1)
Currents have no intrinsic sign. The sign convention we use is that currents entering the
junction are positive (+I), while all those leaving it are negative (-I). 
Example: Fig. 1 shows a junction where three branches (wires) meet. Each branch has its own
current. At this junction, I1 enters and I2 and I3 leave: Σ Ii = +I1 - I2 - I3 = 0. This makes sense
since I1 is supplying current to the junction while I2 and I3 are removing current from the
junction.

In calculations you can start with either sign. It makes no difference whether this is the correct
direction or not since you will find the correct direction after you solve the equations. Solutions
with the negative sign imply that the actual direction of the current is opposite to your assumed
direction.
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Experiment 22
B) Kirchhoff’s Voltage Law: The sum of all potential changes in a loop must be equal to zero.
This tells us that the energy remains the same at the starting point no matter what path you take:
Σ Vi = 0
(2)
There are two types of potential changes in any circuit: 1) voltage differences designated
by “V”; and 2) electromotive forces (emf) given by batteries or power supplies designated by
“”.
When you apply the Voltage Law Equation (2) to a circuit, you trace a path around any
closed loop. When you pass through a battery, write + if your loop direction is from negative to
positive because there is an energy gain. Write -V if you go through a resistor, capacitor, or
inductor in the direction of the current since there is a loss of energy going through a resistor.
Example: Take a clockwise loop through the series circuit below (Fig. 2). The current
direction is also clockwise. There are three current elements , R1 and R2 that give voltages ,
V1, and V2. Going in the clockwise direction, there is a voltage gain through the battery and
voltage losses through the resistors. So, in this direction, Kirchhoff's voltage law gives us
+ - V1 - V2 = 0. We can rearrange this to:  = V1 + V2 which shows that the two resistors (V1
and V2) use up all of the voltage supplied by the battery ().
C) Ohm's law: This is the basic form of all voltage drops. The current, I, passes through the
resistance, R, which causes a potential loss of V as electrons are pushed through it. The equation
is one of the simplest in all of physics:
V=I·R
(3)
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Experiment 22
Preliminary Procedure
a) Start by checking 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 read exactly zero, adjust
them with the small screw in the center of the meter (under the needle indicator).
b) Your network board (shown with gaps closed by jumper wires) is shown in Fig. 3. Make sure
that the batteries are mounted with polarities as shown in Fig. 3. Copy the circuit on your data
sheet.
c) Record the values of resistances R1, R2, R3, and R4 on your data sheet. Measure and record the
values of ε1 and ε2. (Note: ε1 is 3 volts and ε2 1.5 volts if the batteries are fresh.)
Procedure Part I: Kirchhoff's Current Law
d) When the gaps a-b, c-d, and e-f are closed with short 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.
e) To measure the direction of I1, use the 500mA scale on your ammeter, connecting it to the gap
connectors a and b as shown in Fig. 4. below. This puts the ammeter in series with the resistors.
Be careful to connect the + and – terminals of the ammeter properly. Make sure the other two
gaps (c-d, and e-f) are closed with jumpers. If your reading is negative, switch the + and – wires.
Once you have a positive reading, the current is going from the positive side to the negative side.
Draw an arrow on your circuit diagram to record the direction of your current. If you’re not sure,
call your instructor.
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Experiment 22
f) If everything is correct and the current is reasonable, switch the ammeter to the 50ma scale in
order to get greater precision in your reading. Record the current I1 to 3 significant figures.
Then disconnect the ammeter and reconnect gap a-b with a jumper.
g) To measure I2, remove the e-f jumper and connect the ammeter across the e-f gap, observing
the same procedure as in (e) and (f) to find the magnitude and direction of the current. Then
disconnect the ammeter and reconnect gap e-f with a jumper.
h) Repeat all this for I3 using gap c-d. Then disconnect the ammeter and reconnect gap c-d with
a jumper.
i) Choose a junction (either c or g). Write the Kirchhoff’s Law Equation for this junction taking
into account your measured directions of the currents. According to your measurements, is the
equation (1) satisfied or not at this junction? If you are not certain, check with your instructor.
Procedure Part II: Kirchhoff's Voltage Law
j) On your data-sheet, prepare a Voltage Law table as shown. Attach a red connector to the plus
(+) terminal of the voltmeter, and a black connector to the – terminal. Set your voltmeter to the
5-volt full scale. Remember voltage is measured by attaching the voltmeter to the appropriate
connectors without disconnecting any wires in the circuit.
k) Connect the voltmeter to points in the circuit as
indicated in this table as you go around the loop in the
counter-clockwise direction. Remember the voltmeter
will show you only the magnitude of the potential
difference. The signs (plus and minus) must be
determined by you by reading the sign conventions
below carefully.
[Note: If you don’t get a reading, you must switch
the positive and negative wires.] Measure and record
all potential difference magnitudes and signs in the
table.
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Kirchhoff's Voltage Law
VOLTAGE
BETWEEN
POINTS:
SIGN
(+ OR -)
a-k
k-j
j-c
c-a
SUM
MAGNITUDE
(VOLTS)
Experiment 22
VOLTAGE LAW SIGN CONVENTIONS
Resistor, Capacitor, Or Inductor
(Passive Circuit Component)
Loop in the same
direction as current
Current loses energy as
it passes through the
Loop in the
opposite direction
+ component.
from current
Battery Or Power Supply
( Active Circuit Component)
from
- to +
+
from
+ to -
-
Current gains energy as it passes
through the component from
negative to positive potential.
l) According to your measurements, is the Kirchhoff Voltage Equation (2) satisfied or not? If you
are not certain, check with your instructor.
AS SOON AS YOU FINISH THIS SECTION, DISCONNECT ALL GAP JUMPERS SO
THAT THE BATTERY DOES NOT RUN DOWN.
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. Then reverse the polarity of emf, ε2 as shown in Fig. 5 .
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
starting with the 500 ma scale as in (d); then proceed with (e) and (f).
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Experiment 22
BEFORE YOU LEAVE THE LAB:
a) Place ε2 into its original position.
b) Disconnect all jumpers from the circuit board.
c) Make sure you have recorded the values of your resistances R1, R2, R3, R4. (They are accurate
to within 0.1%)
d) Make sure you have recorded the values of ε1 and ε2, from the voltmeter readings. (They are
accurate to within 1.0%)
Lab Report
Part I: Verification of Kirchhoff’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 (counterclockwise 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. You should include two loops equations and
the current equation in terms of ’s, I’s and R’s.
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: Don’t use your measured values of currents in any of your calculations. You will
use the recorded values of resistances and the emf's in order to determine the theoretical values of
the currents.
NOTE: It is useful to solve your equations algebraically without inserting values for the
resistances and emf's. Then it will be easier to calculate the currents for Part III: the Modified
Network. For example, display the formula for I1, in the following form:
Terms with 's and R's only
I1 =
Terms with R's only
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Experiment 22
5) Display your results in Table 1 as shown. The discrepancies should be stated as the difference
between the calculated and measured values, not in the % difference.
TABLE 1: BASIC CIRCUIT
CURRENT
CALCULATED
(mA)
MEASURED
(mA)
ABSOLUTE
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.
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) Show your results for the modified network in a table similar to Table 1 above.
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