Lab #18
Prof. Susanne M. Lee
Parallel/Series Resistors
page 1
University at Albany, SUNY
Parallel Resistance, Series/Parallel Circuit Combinations,
and Power Consumption
Reading: Giambatista, Richardson, and Richardson – Chapter 18 (18.1-18.9, 18.11).
Summary: This week’s lab is the second half of the lab you started last week and you will
begin the lab by quickly and accurately measuring the resistance of the resistors at your lab
bench. After constructing a parallel resistor circuit with these resistors, you will measure the
current and voltage through and across each resistor and experimentally determine the
equivalent circuit resistance. Lastly you will construct a combination series and parallel
resistor circuit and again measure the current and voltage drops in the circuit. You will then
use your data to determine how much power is consumed during operation of these circuits.
Note1: if you have difficulty answering the Pre-Lab questions, a detailed appendix at the end of the lab
contains explanations that may help you.
Note2: if you find yourself spending more than 2 hours on this Pre-Lab, please see the lab web site for
tutoring times.
Pre-Lab Analysis
The Appendix at the end of this write-up gives a detailed explanation of the physics for
parallel circuits. Reading this appendix before doing Problem 1 will make this question
considerably easier.
1.) In this problem you will derive from basic principles the equivalent resistance formula
for parallel circuits. (See Lab Appendix)
a.) Write an equation relating the power supply voltage (Vps) to the voltage drop (V1)
across R1 and V2 across R2 in Figure 1. [3!pts]1
b.) Write the relation between the total current (I) in
R1
I1
the circuit to the current I1 in resistor R1 and the
C
I A
current I2 in resistor R2. [3!pts]2
+
c.) Use Ohm’s law to relate each current to the voltage
B
D
and resistance that produced it. For example, Vps
I
2
R2
–
I1=V/R1. Do the same for I2, R2 and I, Requivalent.
[6!pts]3
d.) Combine these current/resistance relations with Figure 1: Parallel resistor circuit.
your answers to (a) and (b), to show that [5!pts]4
1
1
1
=
+
.
R equivalent R 1 R 2
e.) Combining all the above information for parallel resistors, show that the power
dissipated in the whole parallel circuit is P=P1!+!P2, where P1=the power dissipated in
R1 and P2=the power dissipated in R2. (Hint: use the formula for power that involves voltage
in Pre-Lab 11, Question 4e.) [8 pts]5
Parallel/Series Resistors
Lab #18
Prof. Susanne M. Lee
page 2
University at Albany, SUNY
R1
R3
2.) For the series/parallel resistor combination
shown in Figure 2, determine the equivalent
resistance of the circuit by:
a.) first finding the equivalent resistance of R3
Vps
R4
R2
and R4. Call this R1,equivalent, [3 pts]6
b.) second finding the equivalent resistance of
R5
R2 and R1,equivalent.
Call this R2,equivalent,
[3!pts]7
Figure 2. Parallel and series circuit.
c.) lastly finding the equivalent resistance of R1,
R2,equivalent, and R5. This is the total equivalent resistance of the whole circuit. [3!pts]8
d.) If R1=100Ω, R2=200Ω, R3=150Ω, R4=450Ω, and R5=750Ω and the power supply is set to
1V, what is the equivalent resistance of the circuit and how much current will flow
through the 750Ω resistor? [8!pts]9
3.) Outline the lab following the format of “Outline Format” posted on ERes. [20 pts]10
Equipment to be used in this lab:
m 2DVMs - 1 ammeter & 1 voltmeter
m 1 power supply
m 1 breadboard
m 5 resistors between 100Ω and 5kΩ – the sum of two of these should be approximately the same
as one of the others
m connecting wires (~ 0!Ω)
1.) Determining Resistor Values
A. From Color Codes
Black Brown
0
1
Red
2
Orange
3
Yellow Green
4
5
Blue
6
Violet Gray
7
8
White
9
Table 1: Resistor color codings.
r Figure 3 is a reminder of how to read resistor colors:
• Gold = 5% tolerance
• Silver = 10%
Tolerance
Value Exponent
r In a data table in your lab notebook, record the colors
and values of each resistor (3 total) at your lab bench. Figure 3: Reading
Don’t forget to include units where appropriate and
resistor codes.
title your table. [16 pts]11
B. From DVM Measurements
r As you did last week, use the DVM to measure the resistance of each resistor
directly. Connect one end of one resistor to the “VΩ” receptacle and the other end
of the resistor to the “COM” receptacle. Set the DVM to the Ω range that is closest
to the nominal value determined from the resistor’s colors. Read the resistance
value and record it in your notebook. [16!pts]12
Lab #18
Prof. Susanne M. Lee
Parallel/Series Resistors
page 3
University at Albany, SUNY
r Make sure all of your resistance values are between 100Ω and 5kΩ. If any are
outside this range, please notify your TA and he or she will get you a smaller
valued resistor replacement.
2.) Equivalent Parallel Resistance & Power Consumption
r Turn the power supply voltage down to 0V and turn it off.
r Using three resistors that are within 1000Ω of
the maximum and minimum resistance
Power Supply
values, build the parallel circuit shown in
+
–
Figure 4 and connect the voltmeter and
ammeter as shown in this figure.
• The ammeter arrangement is the same as that
shown in black in Figure 5 and allows you to
DVM2
measure the total current in the circuit.
R1
R2
R3
Voltmeter
• The voltmeter arrangement is the same as that
shown in green in Figure 5 and allows you to
+ DVM1 –
measure the voltage drop across R1.
Ammeter
r Sketch the resistor arrangement in your
notebook, labeling the measured value of Figure 4: 3 resistor parallel circuit.
each resistor used in the circuit. [7!pts]13
Ammeter
A. Current & Voltage in Parallel Circuits
+–
r Turn on the power supply and
+–
+–
+–
increase the power supply voltage
until about 30-35mA of total circuit Vbattery
V1 R2
V2 R3
V3
R1
current appears on the ammeter.
r Record the precise value of the
total circuit current and the voltage
Figure 5. Schematic for measuring parallel
drop V1 across R1 in a new data
resistance.
table in your notebook.
Don’t
forget to label this title to distinguish from all your other data tables in this lab.
[10 pts]14
r Now connect the ammeter to read the current through R1, green circle with + and –
inside it in Figure 5. Record the current in your data table. Do the same for the
currents through R2 and R3 and record the corresponding currents in your table.
[6!pts]15
r Connect the voltmeter across each of the other two resistors and record the
voltages in your notebook. [4!pts]16
r Turn the power supply voltage to zero and turn off the power supply.
r Make a sketch in your notebook as to where each current and voltage was
measured. [7 pts]17
r Are all three currents the same or different (a complete sentence is required for this
answer)? If different, what is the relation between them? Explain why your
answer is correct in terms of where the + charges (current) go in the circuit (see
Lab Appendix). [5 pts]18
Lab #18
Prof. Susanne M. Lee
Parallel/Series Resistors
page 4
University at Albany, SUNY
r Are all three voltages the same or different? Why should or shouldn’t they be
different? (Again, see Lab Appendix) [5 pts]19
• If your data does not make sense in terms of the model, do NOT go on. Figure out what
is wrong with your setup and make it work correctly.
B. Equivalent Parallel Resistance Calculation
r Using the appropriate voltage and total current measurements you just made,
directly calculate the equivalent resistance of the whole circuit (don’t forget units).
[4 pt]20
r How does this experimentally determined equivalent resistance relate to the R1,
R2, and R3 values you determined in Section 1? How does this formula compare to
that you found in Pre-lab Question 1? (A full sentence is expected for all such
questions.) [7 pt]21
C. Power Consumption
r Calculate the power consumption for each resistor and for the equivalent circuit
resistance. Enter your results in a new column of your previous data table.
(Don’t forget to show how you did your power calculation and include units
wherever appropriate … including in the data table.) [12!pts]22
r Write the relation between the total power consumption in the circuit and that in
each resistor (give an equation). Derive this relation from the parallel equivalent
resistance formula (Pre-lab Question 1). [12!pts]23
3.) Equivalent Resistance for a Combination Series & Parallel Circuit:
r Choose R3 and R4 in Figure 6 such that R3!+!R4 is about the same value (to within a
factor of 3) as one of your other resistors, which you will make R2. Sketch this
circuit in your notebook and label all resistors with their measured values.
R1
[13!pts]24
R3
r Turn on the power supply and increase
its voltage until the total current in the
circuit reads roughly 10mA.
R4
R2
r By measuring the appropriate current Vbattery
and voltage, find the equivalent
R5
resistance
of
this
circuit
(experimentally). Sketch the circuit and
indicate on the sketch at which points
Figure 6. Parallel and series circuit.
you connected the two DVMs in order
to measure the current and voltage you used to determine Requivalent. Record all
measured and calculated values in your lab notebook and explain how you found
Requivalent. [19!pts]25
r Calculate the “theoretical” equivalent resistance of this circuit. See Pre-lab Question
2. [13!pts]26
4.) Extra Credit
r Calculate the total power consumption in the circuit. Don’t forget to include units
and to show all work. [8!pts]27
Lab #18
Prof. Susanne M. Lee
Parallel/Series Resistors
page 5
University at Albany, SUNY
Lab Appendix:
A. Physics of Parallel Resistors
R1
I1
In the parallel resistor circuit of Figure 7, +!charges
C
I A
leave the “+” power supply terminal and move through
the essentially resistanceless wire until they reach the
+
intersection at AB. Some charges then decide to go up
B
D
Vps
through R1, while the rest go down through R2. Thus the
I
2
R2
–
current I splits into I1 and I2 and since all the charges go
either up or down, the sum of the two split currents must
be the same as the total current I, i.e. I=I1+I2.
Figure 7: Parallel resistor circuit.
How many charges go which way is determined by
which path offers the least resistance. The more resistance, the slower the rate at which
charges will flow through that resistor and the less current will go there.
To determine quantitatively how much current flows through R1, notice that the “+”
power supply terminal is directly connected to AB (with low resistance wires) and the “–”
power supply terminal is directly connected to CD (with low resistance wires). Similarly a
low resistance wire connects A to one side of R1 while a low resistance wire connects C to the
other side of R1. This means that the power supply voltage (Vps) is really connected directly
across R1. Following similar logic, the power supply voltage is also directly connected
across R2. Thus the voltage drops across the two resistors in parallel are the same and are
also the same as the voltage supplied by the battery.
To come back to our original question – how much of the total current goes through R1
and how much through R2? We are now ready to answer that since we know the voltage
drop across each resistor and we know each resistance. Ohm’s law gives each current:
I1!=!Vps/R1 and I2!=!Vps/R2. Knowing these currents we can now find an equivalent resistance
(Requivalent) for the whole circuit similar to series circuit equivalent resistance. As a reminder,
the equivalent parallel resistance replaces R1 and R2 with a single resistor whose value
produces the same total current (I) in the circuit. Thus the voltage drop across Requivalent is Vps
and the current through it is I. Since we know I1 andI2 in terms of Vps and R1 and R2, and we
1
1
1
know I in terms of I1 and I2, we can relate Requivlent to R1 and R2 as
=
+
. For
R equivalent R 1 R 2
the details of deriving this last equation, see Pre-lab Question #1.