Exp 20 - Kirchhoff's Laws for Circuits

Experiment 20
Kirchhoff's Laws for Circuits
Advanced Reading:
(Serway & Jewett 8th Edition)
Chapter 28, section 28-3.
Equipment:
1 Circuit board
2 D cell batteries with
holders & leads
1 Kelvin DMM with leads
1 10 Ω resistor
1 12 Ω resistor
1 15 Ω resistor
1 18 Ω resistor
1 22 Ω resistor
R1
a
i1
f
Objective:
The object of this experiment is to apply
Kirchhoff's rules for circuits to a two loop circuit
in order to determine the currents and voltage
drops in each loop.
Theory:
The two basic laws of electricity that are most
useful in analyzing circuits are Kirchhoff's laws
for current and voltage. Kirchhoff's Current
Law (KCL) states that at any junction of a
circuit, the sum of all the currents entering the
junction equals the sum of the currents leaving
the junction. In other words, electric charge is
conserved. Kirchhoff's Voltage Law (KVL)
states that around any closed loop or path in a
circuit, the algebraic sum of all the voltage drops
must equal zero. In other words potential has to
return to the original value.
There are three generally accepted ways to solve
multiple loop circuit problems, the branch
method, the nodal method, and the loop method.
The loop method will be used in this experiment.
In this method a current loop is drawn for each
closed loop of the circuit. To avoid confusion, it
is good to arbitrarily have all the currents going
clockwise. The currents in a loop always flow
through a junction, so the KCL is satisfied. We
need only to worry about satisfying the voltage
requirements. To do this the following rules
need to be followed:
(1)
If a current traverses a resistor in the
direction of the current (loop) flow, the change
1.5v
R2
1.5v
A
c
b
e
A
R3
R4
i2
R5
d
A
figure 20-1
in potential is –iR. If a 2nd current traverses the
same resistor (e.g., R2 in figure 20-1 above) in
the opposite direction the change in potential is
+iR.
(2) If a seat of emf (voltage source) is traversed
in the direction of the emf (from - to + on the
terminals), the change in potential is +ε; if it is
traversed in the opposite the emf (from + to -),
the change in potential is -ε.
For example, the equation for the loop one in
figure 20-1 would be:
ε − i1 R1 − i1 R2 + i2 R2 − i1 R3 = 0
or:
ε − i1 ( R1 + R2 + R3 ) + i2 R2 = 0
A similar equation can be written for the other
loop and by solving the two equations
simultaneously, the values for i1 and i2 can be
obtained.
Procedure:
1. Measure and record the resistance of each of
the five resistors on the lab table with the
ohmmeter.
2. Construct a circuit with the five resistors and
the two batteries on the circuit board as shown in
figure 20-1. Make sure the battery polarities are
correct.
3. Measure the potential difference across each
of the batteries while the circuit is complete.
When this is done, disconnect the batteries.
4. Calculate the currents i1 and i2 using the
measured resistor values. To do this, write the
equation for each of the loops as given in the
theory section. Move the emf term (ε) to the
right-hand side of the equation. (WATCH
YOUR ± SIGNS! This is one of the most
common causes of incorrect answers.) The
coefficients for the equations are the values of
the resistances. Solve for i1 and i2. (If either
current has a negative value, do not be alarmed.
This means that the real current flow is opposite
to the arbitrary current direction chosen.)
5. Place the DMM between R3 and f on the
circuit board. (See figure 20-1 for these
locations.) Place the batteries back into the
circuit and measure the current flow in the loop.
You will have to remove one prong of the
resistor plug and then complete circuit using the
ammeter. After measuring the current flow,
return the resistor to its original configuration.
6. Repeat for the other ammeter positions. Try
to do these measurements as quickly as possible.
If the batteries are nearly dead, then the potential
difference across the battery could change rather
quickly, causing the currents to be different. Be
sure to check all of your connections. If one
is loose, the probability of erroneous
measurements is high.
7. Compare the experimental values of i1 and i2
with the calculated values obtained in part 2. If
they are not the same, check your calculations
from part 2 and retest your circuit (or see
warning above.)
8. Measure the voltage drops and emf's around
each loop using the DMM. Do the loops obey
Kirchoff's voltage law?
Part 2: Can you trust a measuring
instrument all the time?
15. Put the 5 resistors back in the bag.
Next, construct a series circuit using one battery,
two 10M (10 x 106 ) ohm resistors and a jumper.
Measure the potential differences across the
battery and each of the resistors of the circuit.
Record these values. Disassemble circuit when
finished.
Do the potential differences (of the resistors
from each circuit) add to the potential difference
across the power supply? (See question 4
below).
Questions/Conclusions:
1. Explain what effect the DMM might have on
the circuit when inserted to measure the current
(i.e., comment on whether the current into a
junction is equal to the current out of a junction
when uncertainty is considered.) Refer to what
you observed in the Series & Parallel lab when
you measured the resistance of the ammeter.
2. Would disconnecting the battery on the lefthand side of the circuit board affect the current
i2? Calculate what i2 is for this case.
3. Using Kirchhoff’s current and voltage laws
explicitly (i.e., use sum of voltage around loops
and sum of current in and out of junctions)
derive the current equations to the circuit used in
this experiment.
Do not solve. Simply set up equations and put in
a form that is conducive to computer solution.
4. Discuss results of part 2 (Can you trust a
measuring instrument all the time?) in the
context Kirchhoff’s voltage law and the effect of
DMM on the circuit.
You will need to calculate the equivalent
resistance of the 10 M resistor circuit assuming
that the voltmeter (i.e., DMM) is a
10 × 10 6 Ohm resistor. Does the voltmeter
affect the circuit? Would this affect your
answer?
Download PDF