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THE BUILDING BLOCKS OF CIRCUIT ANALYSIS: THE RESISTOR, OHM’S LAW, AND CONSERVATION OF POWER
Kristine McCarthy Joshua Pratti
Alexis Rodriguez-Carlson October 11, 2006
2
Objectives:
• To learn the color labeling system of resistors
• To measure resistors and determine if they are within their given tolerance
• To measure current, voltage, and resistance using digital and analog multimeters
• To verify Ohm’s Law and to utilize it to calculate values of resistance, current, and
voltage
• To show that the conductance value of a resistor is equal to the inverse of the resistance
value
• To determine the voltage drop across an element by two methods
• To find power absorbed or dissipated by an element by using one of three equations, and
to show that the value is equal regardless of which equation is used
• To observe the change in current through a resistor when the applied voltage is changed
• To use Microsoft Excel to create graphs displaying current according to voltage values,
and to determine equations that model that data
• To determine the resistance and conductance of a circuit by finding the slope of those
equations
• To verify the mathematical relationship of two or more resistor’s equivalent value of
resistance depending on whether the resistors are in parallel or series
• To build a variety of circuits, including those that include multiple power sources
• To observe the conservation of energy principle
3
Discussion of Theory:
A resistor is one of the most basic elements of a circuit and is defined as any element that
dissipates power in the form of heat when current flows through it.
Figure 1: A photograph of a resistor
Resistance is measured in ohms (Ω), and the higher a material’s value the harder it is for
current to flow through the material. The opposite of resistance is conductance (G), which is
measured in Siemens (S). Conductance describes how easily current can move through a
material, and is defined as being the inverse of resistance.
G = 1 Equation 1 R
Resistors are manufactured with different values of resistance. To help discern the value of a
resistor, every resistor has a color code on it. On every resistor, there are four stripes; each
colored stripe stands for a number. To determine which number the color represents, a color
code chart is used.
Color Abbreviation Significant Digit Multiplier Black BL 0 100 Brown BR 1 101 Red RD 2 102 Orange OR 3 103 Yellow Y 4 104 Green GR 5 105 Blue B 6 106 Violet V 7 107 Gray GY 8 108 White W 9 109
Table 1: Resistor Color Code Chart
4
The first and second stripes represent the first two numbers in the resistor’s value. The third
stripe is a multiplier, it indicates by what power of ten the first two numbers should be
multiplied. The fourth and final stripe indicates the tolerance that the resistor is supposed to be
within of its given value.
Figure 2: Close up view of resistors
To determine the value of the resistors in Figure 2, we start by matching each stripe to a
number according to Table 1.
Stripe 1 (first digit) – Green – 5
Stripe 2 (second digit) – Blue – 6
Stripe 3 (multiplier) – Red – 102
Stripe 4 (tolerance) – Gold – +/- 5%
Therefore, the value of this resistor is 5,600 Ω, with a tolerance of +/- 5%.
Of course, finding the theoretical values of each resistor is insufficient. In calculation,
theoretical values may cause incorrect results. A resistor can be within its tolerance and still be
far from its theoretical value, especially when working with large resistances. To find out how
dependable the color codes are it was we found the actual values of several resistors and
compared them to their theoretical values. This can be done with a digital or analog multimeter,
5
though we found that the digital is more accurate and easier to read. The value that the
multimeter measures will give you accurate values to use in calculations and will also tell you if
the resistor is within its tolerance. Table 2 shows the measured and theoretical resistances of the
seven resistors that we tested.
Measured Band A Band B Band C Band D Theoretical Value (Ω)
Tolerance Range (Ω) Analog
(Ω)
Measured Digital
(Ω)
BR BL BL G 10 5% 9.5-10.5 10 10.03 B GY BL G 68 5% 64.6-71.4 55 66.13 B GY BR S 680 10% 612-748 650 681 B GY RD G 6,800 5% 6,460-
7,140 5,500 6,753
OR OR OR G 33,000 5% 31,350-34,650
33,000 33,400
RD OR Y G 330,000 5% 313,500-346,500
290,000 330,000
RD RD GR G 2,200,000 5% 2,090,000-2,310,00
1,400,000 2,130,000
Table 2: Theoretical and Measured Resistances of 7 Resistors
As Table 2 shows, all of the resistors we measured were within their tolerances. This means
that they are within the manufacturer’s stated range of resistance. A resistor’s tolerance is given
by the last of the colored bands on the casing of the resistor. There are gold (5%), silver (10%),
and no color (20%) bands. In order to determine whether or not the resistor is in fact within
tolerance range, you add and subtract the appropriate percentage from the ideal value of the
resistor and these numbers are the extremes of the tolerance as illustrated in the “Range” column
of Table 2. After taking the measurement of the actual resistance, you can determine if it meets
the tolerance requirements and know the exact value to use in any calculations.
6
When combining two or more resistors in a circuit their physical connections will determine
how much total resistance there is in that circuit. One of the goals of this project was to show
how the total resistance in a circuit containing two or more resistors changes if the resistors are
in series or parallel with each other. From our research, we expected to find the relationship for
two resistors in series to be
RTotal = R1 + R2 Equation 2
and
1 = 1 + 1 Equation 3 RTotal R1 R2
for two resistors in parallel.
R1
R2
Figure 3: Two resistors in series
R1 R2
Figure 4: Two resistors in parallel
7
Ohm's law relates resistance to two other electrical values, current and voltage. In an
electrical circuit, the current (I) passing through an element is directly proportional to the voltage
(V) drop across it. The mathematical model for this relationship is
V = I / R Equation 4
The SI units for voltage, current, and resistance are volts (V), amperes (A), and ohms (Ω),
respectively. If any two of the values are known, the third can be found by using Equation 4 and
solving for the unknown value. Rearranging Equation 4 to solve for current yields
I = V/R Equation 5
which indicates that current is equal to the voltage divided by the resistance. If the resistance is
constant, then the voltage will rise at the same rate that the current decreases. This can be
verified by measuring all the values for V, I, and R for a given resistor and comparing them to
the values that can be derived from Equation 4. Since the values of current and voltage always
differ by a factor of resistance, if an equation of this relationship were created it would be in the
form of
y = mx +b Equation 6
where “y” is the voltage, “x” is the current, and “m” is the resistance. Or, if it is more
convenient to speak in terms of conductance, “y” would be current, “x” would be voltage, and
“m” would be conductance.
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Ohm’s law also describes the relationship between power, measured in watts (W), and
current, voltage, and resistance. There are three equations describing these relationships, the first
of which is
P = VI Equation 7
which indicates that increasing either voltage or current will increase the power absorbed or
released by the element. By combining Equations 4 and 7 relationships can be derived to solve
for power given any two of the values for resistance, current, or voltage. P = VI combined with
V = IR yields
P = I2R Equation 8
while P = VI in combination with I = V/R results in
P = V2/I Equation 9
If all of the values are known and power is computed using all of the equations the value
should be equal, so it does not matter which equation is used. Likewise, if the value for power is
known, given any one of the other values one can determine the other two. This is a powerful
tool in circuit analysis.
9
Part 1: Verifying Ohm’s Law:
Procedure:
To better understand resistors we began with three different circuits. The first circuit
included only one resistor, the second circuit included two resistors in series, and the final circuit
included the same two resistors in parallel with each other. Each circuit also included a variable
DC voltage supply, an ammeter, and a digital multimeter. For the most accurate calcuations, the
actual resistance of each resistor was measured before the circuits were built.
R1 R2 224.3 Ω 217.84 Ω
Table 3: Measured Value for Each Resistor
Figure 5: Circuit 1, with only 1 resistor
10
Figure 6: Circuit 2, with 2 resistors in series
Figure 7: Circuit 3, with two resistors in parallel
11
For each of the above circuits, the following steps were followed:
1 - Starting the power supply at 0 V, step up in increments of one volt up to 10 V, measuring
and recording in the current through and the voltage across the resistors at each supplied voltage.
2 - Reverse the conections from the ammeter and voltage meter and repeat step one. This
gave readings in negative voltage.
Data and Analysis: Circuit 1:
The data for Circuit 1 was as follows:
VoltageSupplied VoltageDMM (V) Current (A) RMeasured (Ω) -10 -9.832 -0.0445 224.3 -9 -8.858 -0.0405 224.3 -8 -7.874 -0.0355 224.3 -7 -6.89 -0.031 224.3 -6 -5.905 -0.0265 224.3 -5 -4.921 -0.022 224.3 -4 -3.937 -0.0175 224.3 -3 -2.953 -0.013 224.3 -2 -1.785 -0.0081 224.3 -1 -0.892 -0.0041 224.3 0 0 0 224.3 1 0.8772 0.0042 224.3 2 1.77 0.0082 224.3 3 2.937 0.0135 224.3 4 3.921 0.018 224.3 5 4.906 0.022 224.3 6 5.89 0.026 224.3 7 6.873 0.031 224.3 8 7.857 0.036 224.3 9 8.841 0.04 224.3 10 9.825 0.0445 224.3
Table 4: Data collected from Figure 5: Circuit 1
From this data two graphs were created. The first graph plotted the current on the x-axis with
the supplied voltage on the y-axis. Using Excel’s trendline feature, an equation was created to
12
model the data. In theory, the slope of that equation should be equal to the total resistance in
each circuit.
Circuit 1: Voltage vs Current
y = 225.25x - 0.0075
-15
-10
-5
0
5
10
15
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
Current (A)
Volta
geSu
pplie
d (V)
Graph 1: Graph of Voltage vs. Current for Figure 5: Circuit 1
The next step was to compare the measured value for total resistance from Circuit 1 to the
value of the equation’s slope. The results were as follows:
RMeasured = 224.3 Ω
RSlope = 225.25 Ω
%ERROR = RMeasured - RSlope * 100 = |224.3 Ω - 225.25 Ω| * 100 = 0.0418 % RMeasured 225.25 Ω Equation 10
These results indicate that the resistance as calculated by finding the slope of the equation
was within .0418% of the measured value.
13
After finding that the calculated resistance was close to the measured value, Equation 1
was used to determine the expected value of conductance.
GExpected = 1 = 1 = 4.439 * 10-3 S Equation 11 RMeasured 225.25 Ω
We expected this value to be equal or close to the value of the slope of the line describing
the data when plotted with supplied voltage on the x-axis and current on the y-axis.
Circuit 1: Current vs Voltage y = 0.0045x + 7E-05
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
-15 -10 -5 0 5 10 15
VoltageSupplied (V)
Cur
rent
(A)
Graph 2: Graph of Current vs. Voltage for Figure 5: Circuit 1
GExpected = 4.439 * 10-3 S = 0.004439 S
14
GSlope = 0.0045 S
%ERROR = GExpected - GSlope * 100 = |0.004439 S -0 .0045 S| * 100 = 1.3742 % GExpected 0.004439 S Equation 12
Equation 12 shows that the conductance value calculated from the slope is less than 1.5%
different than the expected value, veryifying this relationship.
To verify Ohm’s law (Equation 4), the chart below was created, showing the values of
voltage measured across the circuit and the value of voltage calculated by using Equation 4.
VoltageDMM (V) Current (A) RMeasured (Ω) Voltage (I * R) % Difference -9.832 -0.0445 224.3 -9.98135 1.519019528 -8.858 -0.0405 224.3 -9.08415 2.553059381 -7.874 -0.0355 224.3 -7.96265 1.125857252 -6.89 -0.031 224.3 -6.9533 0.918722787 -5.905 -0.0265 224.3 -5.94395 0.6596105 -4.921 -0.022 224.3 -4.9346 0.276366592 -3.937 -0.0175 224.3 -3.92525 0.298450597 -2.953 -0.013 224.3 -2.9159 1.256349475 -1.785 -0.0081 224.3 -1.81683 1.783193277 -0.892 -0.0041 224.3 -0.91963 3.097533632
0 0 224.3 0 0 0.8772 0.0042 224.3 0.94206 7.393980848 1.77 0.0082 224.3 1.83926 3.91299435
2.937 0.0135 224.3 3.02805 3.100102145 3.921 0.018 224.3 4.0374 2.968630451 4.906 0.022 224.3 4.9346 0.582959641 5.89 0.026 224.3 5.8318 0.98811545
6.873 0.031 224.3 6.9533 1.168339881 7.857 0.036 224.3 8.0748 2.772050401 8.841 0.04 224.3 8.972 1.481732836 9.825 0.0445 224.3 9.98135 1.591348601
Table 5: Comparing Measured and Calculated Voltage from Circuit 1
15
With the percent difference being calculated according to the following formula:
|VDMM - VCalculated| * 100 Equation 13 VDMM
With most of the values for the measured and calculated voltage within 3.11% of each
other, we felt confident that Ohm’s law had been verified and looked to the data from Circuits 2
and 3 to confirm this.
The very last step with Circuit 1 was to verify that Equations 7, 8, and 9 would all yield
the same value for power. To this end, the chart below was created with all the calculated values
for power at VSupplied = 5V:
Voltage Supplied
VoltageDMM (V)
Current (A)
RMeasured (Ω)
P (W) = V * I
P (W) = I^2 * R
P (W) = V^2 / R
5 4.906 0.022 224.3 0.1079 0.1086 0.1073
Table 6: Calculated Values for Power at VSupplied = 5V for Circuit 1
Since all of the values for power are within .0013 W of each other, we can assume that all three
equations are equally accurate in calculating the power dissipated by the resistor.
Data and Analysis: Circuit 2:
The same proceses were performed for Circuit 2.
VoltageSupplied (V) VoltageDMM (V) Current (A) RMeasured (Ω) -10 -9.912 -0.0225 442.14 -9 -8.928 -0.0205 442.14 -8 -7.937 -0.018 442.14 -7 -6.944 -0.016 442.14 -6 -5.952 -0.013 442.14 -5 -4.96 -0.011 442.14 -4 -3.77 -0.00875 442.14
16
-3 -2.828 -0.0066 442.14 -2 -1.885 -0.0044 442.14 -1 -0.942 -0.0022 442.14 0 0 0 442.14 1 0.953 0.0022 442.14 2 1.896 0.0044 442.14 3 2.838 0.0068 442.14 4 3.781 0.0088 442.14 5 4.971 0.011 442.14 6 5.963 0.014 442.14 7 6.954 0.016 442.14 8 7.946 0.018 442.14 9 8.938 0.0205 442.14 10 9.93 0.0225 442.14
Table 7: Data collected from Figure 6: Circuit 2
Circuit 2: Voltage vs Currenty = 443.59x - 0.0264
-15
-10
-5
0
5
10
15
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025
Current (A)
Volta
geSu
pplie
d (V
)
Graph 3: Graph of Voltage vs. Current for Figure 6: Circuit 2
RMeasured = 442.14 Ω
17
RSlope = 443.59 Ω
%ERROR = |442.14 Ω - 443.59 Ω | * 100 = 0.3280 % Equation 14 442.14 Ω
Circuit 2: Current vs Voltagey = 0.0023x + 6E-05
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
-15 -10 -5 0 5 10 15
VoltageSupplied (V)
Cur
rent
(A)
Graph 4: Graph of Current vs. Voltage for Figure 6: Circuit 2
GExpected = 1 = 1 = 2.2617 * 10-3 S = 0.0022617 S Equation 15 RMeasured 442.14 Ω
GSlope = 0.0023 S
%ERROR = |0.0022617 S - 0.0023 S | * 100 = 1.6922 % Equation 16 0.0022617 S
VoltageDMM (V) Current (A) RMeasured (Ω) Voltage (I * R) % Difference -9.912 -0.0225 442.14 -9.94815 0.364709443 -8.928 -0.0205 442.14 -9.06387 1.521841398 -7.937 -0.018 442.14 -7.95852 0.27113519 -6.944 -0.016 442.14 -7.07424 1.875576037 -5.952 -0.013 442.14 -5.74782 3.430443548 -4.96 -0.011 442.14 -4.86354 1.944758065 -3.77 -0.00875 442.14 -3.868725 2.618700265
18
-2.828 -0.0066 442.14 -2.918124 3.186845827 -1.885 -0.0044 442.14 -1.945416 3.205092838 -0.942 -0.0022 442.14 -0.972708 3.259872611
0 0 442.14 0 0 0.953 0.0022 442.14 0.972708 2.067995803 1.896 0.0044 442.14 1.945416 2.606329114 2.838 0.0068 442.14 3.006552 5.939112051 3.781 0.0088 442.14 3.890832 2.904839989 4.971 0.011 442.14 4.86354 2.161738081 5.963 0.014 442.14 6.18996 3.80613785 6.954 0.016 442.14 7.07424 1.72907679 7.946 0.018 442.14 7.95852 0.157563554 8.938 0.0205 442.14 9.06387 1.408256881 9.93 0.0225 442.14 9.94815 0.182779456
Table 8: Comparing Measured and Calculated Voltage from Circuit 2
With Circuit 2, we again found that most of the values for the measured and calculated
voltage were within 3% of one another. This is close enough to suggest that Ohm’s law does
hold true, and the differences may be attributed to the rounding of values.
In addition to verifying Ohm’s Law, we wanted to find a mathematic relationship
between the total resistance value of the circuit and the values of the two resistors in it. In
Circuit 2 the resistors were in series with each other and the values were:
R1 = 224.3 Ω
R2 = 217.84 Ω
RTotal = Rmeasured = 422.14 Ω
If the total resistance is equal to the sum of the two resistors, using Equation 2, then:
RTotal Expected = R1 + R2 = 224.3 Ω + 217.84 Ω = 442.14 Ω Equation 17
19
RTotal Expected is equal to the measured amount RTotal, which confirms Equation 2 from our
research.
Data and Analysis: Circuit 3:
The same data was gathered for Circuit 3, with the results below:
VoltageSupplied (V) VoltageDMM (V) Current (A) RMeasured (Ω) -10 -9.653 -0.089 110.52 -9 -8.683 -0.08 110.52 -8 -7.72 -0.071 110.52 -7 -6.748 -0.062 110.52 -6 -5.778 -0.053 110.52 -5 -4.807 -0.044 110.52 -4 -3.838 -0.035 110.52 -3 -2.874 -0.026 110.52 -2 -1.914 -0.017 110.52 -1 -0.796 -0.00735 110.52 0 0 0 110.52 1 0.8143 0.0075 110.52 2 1.952 0.018 110.52 3 2.923 0.0265 110.52 4 3.894 0.0335 110.52 5 4.865 0.0445 110.52 6 5.836 0.0535 110.52 7 6.806 0.0625 110.52 8 7.777 0.0715 110.52 9 8.748 0.0805 110.52 10 9.719 0.089 110.52
Table 9: Data collected from Figure 7: Circuit 3
20
Circuit 3: Voltage vs Currenty = 112.66x - 0.0142
-15
-10
-5
0
5
10
15
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
Current (A)
Volta
geSu
pplie
d (V
)
Graph 5: Graph of Voltage vs. Current for Figure 7: Circuit 3
RMeasured = 110.52 Ω
RSlope = 112.66 Ω
%ERROR = |110.52 Ω – 112.66 Ω | * 100 = 1.9363 % Equation 18 110.52 Ω
21
Circuit 3: Current vs Voltage y = 0.0089x + 0.0001
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
-15 -10 -5 0 5 10 15
VoltageSupplied (V)
Cur
rent
(A)
Graph 6: Graph of Current vs. Voltage for Figure 7: Circuit 3
GExpected = 1 = 1 = 9.0481 * 10-3 S = 0.009048 S Equation 19 RMeasured 110.52 Ω
GSlope = 0.0089 S
%ERROR = |0.009048 S - 0.0089 S | * 100 = 1.6372 % Equation 20 0.009048 S
In Circuit 2 the resistors were in parallel with each other and the values were:
R1 = 224.3 Ω
R2 = 217.84 Ω
RTotal = RMeasured = 110.52 Ω
22
Using Equation 4 to calculate the expected total resistance:
1 = 1 + 1 = 0.0090 Equation 21 RTotal 224.3 Ω 217.84 Ω RTotal = 110.52 Ω
This value is the same as the measured total resistance, confirming Equation 4.
VoltageDMM (V) Current (A) RMeasured (Ω) Voltage (I * R) % Difference -9.653 -0.089 110.52 -9.83628 1.898684347 -8.683 -0.08 110.52 -8.8416 1.826557641 -7.72 -0.071 110.52 -7.84692 1.644041451 -6.748 -0.062 110.52 -6.85224 1.544754001 -5.778 -0.053 110.52 -5.85756 1.37694704 -4.807 -0.044 110.52 -4.86288 1.162471396 -3.838 -0.035 110.52 -3.8682 0.786868161 -2.874 -0.026 110.52 -2.87352 0.016701461 -1.914 -0.017 110.52 -1.87884 1.836990596 -0.796 -0.00735 110.52 -0.812322 2.050502513
0 0 110.52 0 0 0.8143 0.0075 110.52 0.8289 1.792951001 1.952 0.018 110.52 1.98936 1.913934426 2.923 0.0265 110.52 2.92878 0.197742046 3.894 0.0335 110.52 3.70242 4.919876733 4.865 0.0445 110.52 4.91814 1.092291881 5.836 0.0535 110.52 5.91282 1.316312543 6.806 0.0625 110.52 6.9075 1.491331178 7.777 0.0715 110.52 7.90218 1.609618105 8.748 0.0805 110.52 8.89686 1.701646091 9.719 0.089 110.52 9.83628 1.206708509
Table 10: Comparing Measured and Calculated Voltages from Circuit 3
Circuit 3 yielded the best results for the comparison of the measured and calculated
voltage, with most values being within 2% of each other.
23
Part 2: Observing the Conservation of Energy Principle with Multi-Source Circuits:
Procedure:
To observe the conservation of energy principle, we built a circuit consisting of six
resistors and two power supplies. To have the most accurate information for our circuit possible,
we first measured the values of all the resistors.
ResistanceSpecification (kΩ) 8.1 10 12 15 22 18
ResistanceMeasured (kΩ) 8.103 9.78 12.45 14.6 21.48 17.7
Table 11: Actual Resistance Measurements for Circuit 4 Next, we built the circuit on a breadboard. This breadboard had an on-board power supply,
which we used to provide 15.04V DC to the circuit. We also added an ammeter, a bench power
source (providing 9.02 V DC), and an analog multimeter (or Simpson meter). The purpose of
the Simpson meter was to measure the current provided by the breadboard power supply. We
read the current from the bench power supply directly from the supply.
24
Figure 8: Circuit 4
Our next step was to measure the voltage across each of the components of the circuit and
to place them in a chart along with the resistance values for each resistor (see Table 11). We
could then calculate the current through each component using Equation 5. Knowing the current,
the power absorbed by each resistor could be calculated by using Equation 8. By the principle of
conservation of energy, the power absorbed by all of the resistors should be equal to the power
supplied by the two power supplies.
To prove that the value of the voltage drop across a resistor is equal to the voltage with
reference to ground in the node before the resistor and the voltage with reference to ground in the
node after the circuit, we also measured each of those values and found the voltage drop from
that data.
25
Data and Analysis: Circuit 4:
To start the calculations and analysis of the circuit, we first measured the voltages across
each element in the circuit. The voltages were read by putting the DMM’s leads on either side of
the resistor. Once these values were read, we could calculate the current through each resistor
using Equation 5 and the known resistance values.
Figure 9: Measuring the voltage across the 6.693 kΩ resistor
ELEMENT 15 V SOURCE
9 V SOURCE 8.1 kΩ 10 kΩ 12 kΩ 15 kΩ 22 kΩ 18 kΩ
Resistance (kΩ) 6.6934 9.8250 12.6780 14.6340 21.7040 17.8360 Resistance (Ω) 6693 9825 12678 14634 21704 17836 Voltage (V) -15.040 -9.020 14.960 5.940 6.050 9.020 0.198 8.700 Current (A) 0.0032 0.000 0.002 0.001 0.000 0.001 0.000 0.000 Power (W) (V2/R) 0.033 0.004 0.003 0.006 0.000 0.004 Power (W) (I2*R) 0.033 0.004 0.003 0.006 0.000 0.004 Power (W) (V*I) -0.048 0.033 0.004 0.003 0.006 0.000 0.004
Table 12: Voltage and calculated current and power values from Figure 8: Circuit 4
The fact that all of the power computations were equal again confirmed our findings that
all the power equations work equally well. The next step was to find if the principle of
26
conservation of power held true. To find this we added the values of the power dissipated by all
the resistors to see if it would equal the power supplied to the circuit, which was –0.048 W.
P6.66934 kΩ = 0.0330 W
P9.825 kΩ = 0.0004 W
P12.678 kΩ = 0.0003 W
P14.634 kΩ = 0.0060 W
P21.704 kΩ = 0.0000 W
P17.8360 kΩ = 0 0040 W
PTotal = 0.0500 W
The value calculated for total dissipated power was 0.002 W more than the power input into the
circuit. We felt that was erroneous and suspected that we had made a mistake in our
measurements. However, when the voltage drop across each resistor was calculated by
subtracting the voltage with respect to ground of the lower node from the larger one, our results
were the same. To explain these findings, we created a drawing of the circuit with all of the
current and voltage values in place. Figure 10 shows Circuit 4 with the values for current,
voltage, and nodes added
27
Figure 10: Circuit 4 with voltage and current values
Looking at Figure 10 brought to our attention that there was a voltage drop across the
ammeter. Using PSpice, we were able to model the circuit including the resistance of the
ammeter, as seen in Figure 11.
28
Figure 11: PSpice Schematic for Circuit 4
29
With the help of this model, we were able to ascertain that the resistance value of the
ammeter was about 23 Ω. Unfortunately, this did not help with the issue that the circuit was
dissipating more power than it was receiving, since the added resistance from the ammeter meant
that a negligible amount more power was being dissipated over all. To discover the margin of
error in our power calculation we found the percent difference between the supplied power and
the dissipated power.
PSupplied = 0.048 W
PDissipated = 0.050 W
%Difference = |PSupplied – PDissipated | * 100 = |0.048 – 0.050| * 100 = 4.167% Equation 22 PSupplied 0.048
The difference of 4.167% was small enough that we determined that it could be attributed
to rounding errors. Therefore, the principle of conservation of power had been fulfilled. To
have a clearer illustration of this principle, we decided to set up a theoretical circuit.
Figure 12: Circuit 5
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The following values were obtained for Circuit 5 by using loop and node analysis:
Figure 13: Circuit 5 with Voltage and Current Values
Element Voltage (V) Current (A) Power (Delivered or Absorbed) (W) 6 V Source -6 8 -48 10 V Source -10 8 -80 18 V Source -18 3 -54 3 A Source -12 3 -36 5 A Source 0 5 0 2 Ω Resistor 16 8 128 4 Ω Resistor 12 3 36 6 Ω Resistor 18 3 54
Table 13: Voltage, Current, and Power Values for Circuit 5
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By the same method used with Table 12, the total power was found.
P6 V Source = -48 W
P10 V Source = -80 W
P18 V Source = -54 W
P3 A Source = -36 W
P5 A Source = 0 W
P2 Ω = 128 W
P4 Ω = 36 W
P6 Ω = 54W
PTotal = 0 W
The theoretical Circuit 5 shows that in an ideal situation the sum of all the powers will
equal zero, confirming the conservation of energy principle.
Conclusion:
In this paper, we have shown evidence of several things. First, the value of a resistor can
be well within its tolerance and still be far from its theoretical value (Table 2). It is important to
measure the actual resistance of any resistor used so that you will have accurate numbers to use
in any calculations that may involve the resistance. To find the resistance, it can be measured
directly with a multimeter or an equation describing the relationship of current to voltage may be
derived and the slope used as the resistance. This method works for one or more resistors,
regardless of whether they are in series or parallel. With a known value of resistance,
calculations using Ohm’s Law (Equation 4) can be performed. With accurate data for all of the
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known quantities, Ohm’s law can be used to calculate any missing values for resistance, current,
or voltage. The voltage across an element may be accurately measured in two ways, either
directly across the element or by subtracting the voltages of the surrounding nodes from one
another. Also, if the power absorbed or supplied by an element is known along with either the
resistance, current, or voltage, the other two values can be found. Any of the Equations 7 - 9 can
be used for analysis when the power is known, they are all equally accurate. One way to check
to see if all of the values are correct in a circuit analysis is to compute the power supplied to the
circuit and see if it equals the power dissipated.
Because accurate values are so important, it is a good idea to measure the value of every
resistor before a circuit it built. Resistance values may vary with the resistor’s age, flexing of the
wire, quality, or the operating temperature. In this paper we discussed two ways to get this
information. The first is to measure the resistance value directly with a multimeter. The second
can be used if you do not have access to either the resistor or the multimeter but you do have the
current and voltage information available over a range of voltages. Then, a graph may be plotted
(see Graphs 1 and 2) showing the current on the x-axis and voltage on the y-axis. The
relationship of voltage to current will be linear, and the value of the slope of that line (the “m”
value in Equation 6) will be equal to the resistance. Once the value of resistance is known,
Equation 1 can be used to find the value of conductance.
Both of these methods can also be used to find the combined resistance of two or more
resistors wired in parallel or series with one another (see Figures 3 and 4). However, when
calculating the value of the combined resistances there are two equations that can be used. The
first, Equation 2, is to be used for resistors in series, and the second, Equation 3, should be used
for resistors in parallel.
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Armed with the values for resistance and at least one known value for current or voltage
in a circuit, the other values can be found by using Ohm’s Law. Evidence of Ohm’s Law can be
found in Tables 5, 8, and 10 where the values calculated for voltage across an element are all
within 7.5% of the actual measured voltage. The voltage across an element can be measured by
either placing the leads to a DMM directly around the element, or by measuring the values of the
nodes on either side of the element and subtracting the smaller from the greater (see Figure 10).
Either way, the same value will be obtained.
Another important value in circuit analysis is power. To solve for power, any of
Equations 7, 8, or 9 can be used, depending on what information is already known. Table 6
shows that each of these equations gives the same value for power dissipated or supplied by an
element. This is a powerful tool, because if the power is known you need only one other known
quantity to solve for all of the values for an element. This is helpful in verifying the data of a
complete circuit, since the sum of all of the powers supplied and dissipated by a circuit will
always be zero, as displayed in Table 13. This law, known as the conservation of energy
principle, will always be true. Combined with Ohm’s law, the conservation of energy is a
powerful tool for extracting information from or about a variety of circuits.
