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Experiment 5 kirchhoffs rules

Kirchhoff’s Rules 1
EXPERIMENT 5: KIRCHHOFF’S RULES
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Kirchhoff’s Rules 2
Abstract
The arrangement of resistors is fundamental in determining the current that flows through a
circuit. When resistors are connected in parallel, their net effect in reducing the amount of
current in the circuit is lower. Resistors that are in series connection have a net resistance that is
equivalent to the sum of the resistance of all resistors in the circuit. This reduces the current flow
with a big magnitude. Kirchhoff's junction and loop rules aid in the calculation to determine the
effect of resistors in a circuit. In the experiment, errors were experienced from the onset due to
parallax and assumption made on the internal resistance of the batteries. These errors resulted
into non-uniformity of some values such as the resistance of the resistors. For instance, the error
at the first experiment was about 0.3 percent in average. An error of 1 percent was realized in the
100 Ω resistor while others did not show any error at the beginning.
Key words: resistor, connection, resistance.
Kirchhoff’s Rules 3
Experiment 5: Kirchhoff’s Rules
Introduction
Gustav Robert Kirchhoff, who was a German physicist, came up with two circuit laws
that are fundamental in understanding electrical circuits. The two Kirchhoff’s laws deal with
electrical current and the potential difference in circuits. The first law is known as Kirchhoff’s
current law or Kirchhoff’s junction rule while the second law is referred to as Kirchhoff’s
voltage law or Kirchhoff’s loop rule. The junction rule states that the sum of the current getting
into a junction of an electric circuit is equal to the sum of the current leaving the junction
(Makarov, Ludwig, and Bitar 2016, 91). This law clarifies that there is conservation of charge
within an electric circuit. The loop rule, on the other hand, states that the sum of the potential
differences across all elements in a circuit of a closed loop is zero (Makarov, Ludwig, and Bitar
2016, 92). This law proves that energy is neither created nor destroyed but rather be converted
into different forms. Experiment 5 confirms the validity of the two laws generated by Kirchhoff.
Kirchhoff's loop rule and current law depend on the lumped element model that is
compatible with the circuit (Nilsson and Riedel, 2015). In case the model does not conform to
the circuit, the two laws will not be applicable. The junction rule works on the assumption that
electric current flows only in conductors from one end to another. This poses a limitation for the
alternating current (AC) circuits of high frequencies because the lumped element model is never
applicable. Therefore, junction rule is only valid if the total electric charge remains constant
throughout the circuit. The loop rule, on the other hand, works on the assumption that the
magnetic field linking any closed loop is never fluctuating. This is never the case with AC
circuits, which have short wavelengths with varying magnetic fields. The line integral of the
Kirchhoff’s Rules 4
electric field in an AC circuit is never zero, and this contradicts Kirchhoff's voltage law (Nilsson
and Riedel 2015).
Apart from Kirchhoff’s rules, the experiment also investigates the effects of connecting
resistors in series and parallel within a circuit. When resistors are connected one after another (in
series), the resultant resistance will have a greater impact on the current flow than on any single
resistor. The resultant resistance, which is known as the equivalent resistance is the summation
of the resistance of independent resistors in a series connection, that is,
Equivalent resistance R
eq
= R
1
+ R
2
+ R
3
+ R
4
+ …. (Dobson, Grace, and Lovett 2008).
In a series connection, the amount of current flowing through the resistors is same, but the
potential difference across them is different (Nilsson and Riedel 2015).
When resistors are connected in parallel, the voltmeters read the same potential
difference across them. The equivalent resistance of resistors in parallel connection is obtained
by summing up the reciprocal of resistance of the individual resistors, that is,


+
+
+
+……. (Siebel 2016, 10).
Whether resistors are in parallel or series connection, the value of equivalent resistance controls
the amount of current flowing through the circuit for a given potential difference of the energy
source. When the equivalent resistance increases, the current flow decreases. In any circuit, the
current, resistance, and potential difference obey Ohm’s law, which says, “The current flowing
between two points in a conductor is directly proportional to the potential difference across the
two points provided temperature and other physical factors remain constant” (Nilsson and Riedel
2015). Ohm's law relates the three parameters as follows:
I =
Where I is the current, V is the potential difference, and R is the resistance
within the circuit.
Kirchhoff’s Rules 5
The graph of potential difference (V) against current (I) determines whether a conductor
in a circuit is ohmic or non-ohmic. An ohmic conductor will obey Ohm’s law, that is, it will have
a constant resistance that is portrayed by a straight-line graph. A non-ohmic material such as
light dependent resistors (LDR) presents a curve and therefore the resistance varies depending on
other factors such as temperature or sunshine intensity (Zhao, Min, Chen, and Hao 2016). This
report will, therefore, confirm whether the resistors in the circuits obey Ohm’s law.
Experiment 5.1: Obtaining Resistors for the Given Resistors
Procedure
The electrical circuit was set up as shown in figure 1 below. The variable power was
adjusted to 9V using a voltmeter. I took the measurement of current using the ammeter. The
process was repeated for the two remaining resistors.
Figure 1: Obtaining resistance of resistors.
Results
The results of the experiment are clearly shown in Table 1. The table shows the amount
of current that flew through the circuit under different resistors.
Voltage (V)
Measured Current (I)
Calculated Resistance (R)
Denoted Resistance (R)
9V
41Ma
220 Ω
220 Ω
9V
18mA
500 Ω
500 Ω
9V
88mA
102 Ω
100 Ω
Kirchhoff’s Rules 6
Table1: Resistance of resistors from experiment 5.1
Since the resistors showed a consistent current on the multi-meter, they can be said to obey
Ohm’s law. Therefore, the formula
applies, and the calculated resistance were obtained as
follows:
R
1
=




 
R
2
=





R
3
=




 
Analysis
There are minor differences between the calculated resistance in resistor 1 and resistor 2
and their corresponding actual resistance denoted by the manufacturers. These differences show
that there were some errors during the experiment. The errors were calculated as follows:
Error in R
3
=

 and the percentage error =

 
The error in R
1
would be absolved by maintaining the number of decimal points to correlate to
those of the marked resistance. R
2
have a zero error, representing a 100 percent accuracy. The
source of error in measuring R
3
is the limitation of the multi-meter to display zero decimal point
numbers. The absolute error in multi-meter was 0.5; thus, the reading in the ammeter would be
registered as  to cater for the error.
Experiment 5.2: Resistors in Series
Procedure
The circuit was set up as shown in figure 2 below. The power supplies
and
were
calibrated to 5V and 9V respectively using the voltmeter. With resistors R
1
(220 Ω) and R
2
(100
Ω)in place, I recorded the readings on the ammeter. The same procedure was repeated with the
Kirchhoff’s Rules 7
power sources in series connection, that is, the terminals of
were interchanged. The 500 Ω
resistor was replaced with 220 Ω and 100 Ω resistors to note the difference in the readings.
Figure 2: Resistors in series.
Results
The table below (Table 2.1) shows the results of the experiment before interchanging the
terminals of
Ammeter Reading
R
t
From Ohm’s
Law
R
t
from formula of
resistors in series
44mA
73.0 Ω
320 Ω
Table 2.1: Batteries in parallel.
Table 2.2 shows the ammeter readings after interchanging the terminals of
.
Ammeter Reading
R
t
From Ohm’s
Law
R
t
from formula of
resistors in series
6mA
2333 Ω
600 Ω
8mA
1750 Ω
320 Ω
5mA
2800 Ω
720 Ω
5V
9V
Kirchhoff’s Rules 8
Table 2.2: Batteries in series.
Analysis
When the batteries are in parallel connection, the effective potential difference (V
t
) is
determined as follows:


V
t
= 3.214V
Using the recorded current, the effective resistance is calculated as follows using Ohm’s law.



Using the formula for resistors in series, the effective resistance is given by
R
t
= R
1
+ R
2
= 100 + 220 = 320 Ω
When the batteries are in series connection, the resultant potential difference is determined as
follows:
V
t
= V
1
+V
2
=5 +9 = 14V
From Ohm’s law, the effective resistance in the first combination (R
1
= 100 Ω and R
2
=500 Ω) is
given by;



Using the formula for resistors in series, R
t
= R
1
+ R
2
= 100 +500=600 Ω
Other calculated resistances are shown in Tables 2.1 and 2.2 above.
From the two, it is clear that the effective resistance from the formula of resistors in
series does not agree with the ones from calculations using Ohm's law. The errors might have
come up due to parallax when adjusting the voltmeter to the required potential differences. The
same problem might have occurred when reading the ammeter.
Kirchhoff’s Rules 9
Experiment 5.3: Resistors in Parallel
Procedure
The power supply was calibrated to 9V using a voltmeter. The circuit was set up as
shown in figure 3 below and currents read from the three ammeters.
Figure3: Resistors in parallel
Results
Having used 500 Ω resistor and 220 Ω resistor in place of R
1
and R
2
respectively, the
readings on the ammeters were as follows,
A
1
= 35mA, A
2
= 24mA, and A
3
= 10mA
Analysis
The junction rule requires that the sum of the current entering a junction be equal to the
sum leaving. In this case,
A
1
=A
2
+A
3
= 24 + 10 = 34mA
The value of A
1
from the ammeter reading is 35mA. The difference in the two values shows that
Kirchhoff’s junction rule is not satisfied. The percentage error in the reading is given by the
following calculation.
A
1
9V
A
2
A
3
Kirchhoff’s Rules 10
 

 
The 100 Ω resistor is used in the circuit to drop the potential difference from the power source. It
also combines its resistance with the internal resistance of the battery to increase the efficiency in
analyzing the current flow through the resistors R
1
and R
2
.
Using the formula for resistors in parallel, the equivalent resistance between R
1
and R
2
is
calculated as follows.




R
eq
= 152.78 Ω
Combining resistors R
1
, R
2
, and 100 Ω, the total resistance in the circuit is given as follows.
R
t
= 100+152.78 = 252.78 Ω
Applying Ohm’s law, the total resistance in the circuit is given as follows.



The total resistance calculated using the two methods have a minor disparity. The percentage
error, in this case, is


 . The major source of this error is the
assumption made on the internal resistance of the energy to be zero. If the internal resistance of
the battery were not neglected, the two values would be similar.
Experiment 5.4: Complex Circuit
Procedure
The electrical circuit was set as shown in figure 4 below. The power supplies
and
were calibrated to 5V and 9V respectively using a voltmeter. I then took the readings on the
ammeters. I swapped the terminals of battery
and repeated the whole procedure. Readings on
the ammeters were noted.
A
1
A
2
Kirchhoff’s Rules 11
Figure4:Complex circuit.
Results
Having used 100 Ω, 220 Ω, and 500 Ω resistors as R
1
, R
2
, and R
3
respectively, the
following results were obtained before and after changing the terminals of battery
.
Ammeter
A
1
A
2
A
3
Before changing
terminals of
43mA
42mA
74mA
After changing
terminals of
43mA
42mA
47mA
Table 4: Ammeter readings from the complex circuit.
Analysis
The current from the 5V (I
1
) battery branches at the first joint (J1) into I
2
and I
3
which are
measured by ammeters A
2
and A
3
respectively. At J1, the sum of the current entering is 43mA.
The sum of the current leaving the node is    . The amount of the current
leaving the joint is not the same as the one entering; therefore the joint rule is not satisfied. The
percentage difference between the current into and out of the joint is given as follows.
A
3
5V
9V
J2
J1
L2
L1
Kirchhoff’s Rules 12
Percentage error =


 
Assuming R
1
= 100 Ω, the values of R
2
and R
3
can be confirmed using the loop and joint rules as
follows.
The loop rule equation for loop 1(L1):   
 
  

 
R
3
= 125.7 Ω
Equation for loop 2 (L2):
  

    R
2
= 9.3018 Ω
There is a big disparity in the resistance of the resistors when compared to the values from the
first experiment. This disparity arises due to the accumulation of errors from the onset of the
experiment. A repeated error results into a big disparity at the end of any experiment. Failing to
consider the internal resistance of the batteries during the experiment has also led to the
accumulation of errors during calculations.
The ammeter is always connected in series within the circuit because its principal
purpose is to display the amount of current flowing through the circuit, and it has a low
resistance. Ammeter does not inhibit the flow of current; therefore, connecting it in parallel
would lead to a short circuit (Pickover, 2008). The voltmeter, on the other hand, is connected in
parallel because it has a higher resistance. If connected in series, the voltmeter would lower the
potential difference from where it is connected.
The loop and junction rules are based on two fundamental laws of nature. The junction
rule applies the law of conservation of charge, which states that the net charge of an isolated
electric system remains the same (Veltman, Pulle, and De Doncker, 2016, 30). The junction rule
works on the principle of conservation of energy because the voltage that forms its basis is
energy per unit charge. The total amount of energy per unit charge that is gained in a system
Kirchhoff’s Rules 13
must be equal to the amount of energy per unit charge that is lost by the same system. This is
because both energy and charge are always conserved.
Conclusion
The application of the two Kirchhoff's rules provides a deeper understanding of the
circuit systems. They help one to understand the effect of adding resistors of high resistance to a
circuit. The application of the two formulae for resistors in series and parallel also provides
basics in determining the net current flowing through a circuit. To realize a higher current within
an electric circuit, one should consider connecting resistors in parallel because of the low
effective resistance. For one to limit errors in the experimental setup, he or she should consider
factors such as internal resistance and parallax.
Kirchhoff’s Rules 14
References
Dobson, K., Grace, D., and Lovett, D. (2008).Physics. London: Collins.
Makarov, S.N., Ludwig, R. and Bitar, S.J., (2016). Circuit Laws and Networking
Theorems.Practical Electrical Engineering (pp. 89-138). Switzerland: Springer
International Publishing.
Nilsson, J.W. and Riedel, S.A., (2015).Electric circuits.London: Pearson.
Pickover, C. A. (2008). Archimedes to Hawking: Laws of science and the great minds behind
them. Oxford: Oxford University Press.
Siebel, T., (2016).Electricity and Resistance.ATZ worldwide, 118(12): 8-13.
Veltman, A., Pulle, D.W. and De Doncker, R.W., (2016). Simple Electro-Magnetic Circuits.In
Fundamentals of Electrical Drives (pp. 29-45).Switzerland: Springer International
Publishing.
Zhao, T., Min, Y., Chen, Q. and Hao, J.H., (2016).Electrical circuit analogy for analysis and
optimization of absorption energy storage systems.Energy, 104: 171-183.

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