Physics 182 – Summer 2013 – Experiment #5
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Experiment #5, Series and Parallel Circuits, Kirchhoff’s Laws
1 Purpose
Our purpose is to explore and validate Kirchhoff’s laws as a way to better understanding simple DC
circuits
2 Introduction
As we learned from the Ohm’s law experiment, an electrical circuit is any continuous path or array of
paths along which current may flow. A circuit usually contains a battery or other sources of EMF
(electromotive force) to create the current. Without a source of energy to drive the circuit no current
(electric charge) will flow. Between the terminals of our power source can be any combination of
elements through which the electrons may pass; anything from a single wire to a complicated collection
of wires, diodes, transistors, and other circuit elements as resistors and capacitors.
Whatever the elements that make up the circuit there are some simple rules that must be obeyed. Two of
these rules are Kirchhoff’s laws regarding current (flow of electric charge) and voltage (electrical
potential difference).
Since, as far as we know, charge can be neither created nor destroyed, if we pick a single point in a circuit
all of the charge that flows into that point must also flow out of it in a steady state, or unchanging
situation. Put in terms of currents Kirchhoff’s law derived from the conservation of charge states that at a
given node (point in the circuit) the sum of the currents flowing into and out of that node will be zero.
This is shown schematically in figure 1 where the sum of the three current flowing into and out of the
node, signified by the black dot, adds to zero: I1  I2  I3  0 . Note in this case two of the currents are
negative, that is to say, they flow out of the node, where the node is the low potential point. The negative
(-) black node dot is the low connection reference point and the three positive (+) dots are the high
connection reference points for the voltage measurements in part 3.2.3 of this lab. It should be noted that
this node is our arbitrary reference point for making measurements.
Figure 1.
Kirchhoff’s law for voltages in a circuit is a consequence of the fact that the electric field exerts a force
which is conservative. Thought of in terms of potential energies, the potential energy of a charge at a
particular point in space does not depend on the path the charge took to get to that point. Since the
potential energy of a charge in a circuit is its charge, Q, times the voltage at the point in the circuit, V, the
sum of the voltages measured around any loop in a circuit must add to zero. Otherwise, if the charge went
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around that loop its energy would be changed when it returned to its starting point. This is shown
schematically in Figure 2. Here the sum of the voltages will be zero, Va-b + Vb-c + Vc-d + Vd-a = 0.
Assuming that the bottom left hand node “a” is at the higher potential V1, V2 and V3 would be positive
and Vd-a would be negative as the voltages are measured counterclockwise. In this part of the lab, Vemf =
10.00V will be the power supply that is driving the circuit. This yields:
V1 + V2 + V3 – 10.00V = 0, or V1 + V2 + V3 = 10.00V = Vemf
IR1 + IR2 + IR3 = 10.00V = Vemf (Current I is a constant.)
Figure 2
Kirchhoff’s laws are important because, if we take all the loops and all the nodes in a complicated circuit
and apply Kirchhoff’s laws to them then we end up with a series of algebraic equations which can be
solved to uniquely determine all of the voltages and currents in our problem.
3 Experimental Apparatus and Procedure
3.1 Apparatus
1. DC power supply.
2. Agilent 34405A digital volt-ohmmeter (to be used for voltage and resistance measurements).
3. Fluke 75 digital multi-meter (to be used for all current measurements).
4. Carbon resistors and connecting cables.
If you are curious, read the manuals concerning the operation of the power supply and the digital meters
or read the short description in the appendix.
Direct Resistance Measurements with a Digital Ohmmeter.
3.2.1 Resistance in Parallel and Series. Measurements go in the report data section.
1. Measure and record in lab report data section the resistance of the carbon resistors R1, R2, R3 directly
using the digital voltmeter (Agilent 34405A) set to measure resistance in ohms (Ohms function switch
setting). The correct range will determine the amount of digits after the decimal point.
2. Connect the resistors in parallel and series and measure and record the equivalent parallel and series
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resistance of these networks. The correct range will determine the amount of digits after the decimal
point. Measurements go into the data section of the lab report. Compare your measured values with
the theoretical values calculated from the following formulas:
a. Equivalent parallel resistance:
1/Rp = 1/R1 + 1/R2 + 1/R3
b. Equivalent series resistance:
Rs = R1 + R2 + R3
Calculations of Rp and Rs go into the calculations and
analysis section of the lab report. Show both formulas
and your calculations.
3.2.2 Kirchhoff's Voltage Law
Assemble the series circuit using three carbon resistors as
shown in the figure to the right. Connect the power supply
across this network with the milliammeter (Fluke 75) in
series. The milliammeter (current meter) measures the
current (1 mA = 0.001 A, where A = ampere). The current
is a constant in this series circuit. Set the power supply to
Vemf = 10.00 V. The correct range on the voltmeter will
determine the amount of digits after the decimal point.
Record the current Iexp in the circuit and measure the
voltage drop V across each resistor using the voltmeter
(Agilent 34405A). Results go into the data section of the report.
.
3.2.3 Kirchhoff's Current Law
In this section we will be studying Kirchhoff’s current law.
The circuit we will be examining is shown to the right. The
node we are going to explore is the point where the three
resistors connect. Because we need to insert the current
meter in series with each leg of the circuit of interest there
will be a series of measurements with the ammeter in
different place in the circuit. All measurement will be made
with the same voltage, 5.00 volts, applied to the circuit.
First, record the voltage drop across each of the three resistors. All voltage measurements should be made
with the node as the low side of the potential. As R2 and R3 form a parallel connection, V2 and V3 should
be the same negative voltage. V1 is a positive voltage. Next, insert the ammeter between the end of the
first resistor and the low side which is the node. To do so, cables will have to be removed when the
ammeter is connected. Record the current running through the first resistor. Next, insert the ammeter
between the second resistor and the node. Record the current running through the second resistor. Finally,
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insert the ammeter between the third resistor and the node. Record the current running through the third
resistor. Both I2 and I3 should have a negative reading. I1 is a positive current. Results go into the data
section of the report.
If I2 and I3 were measured relative to the direction of the current flow, these two values would have been
positive. They are negative as we used the node as common reference point. Same applies to V2 and V3.
4 Calculations and Analysis for Lab Report. Show all formulas and calculations.
4.1 Analysis of Resistance in Parallel and Series
1. Calculate Rp. Use percent error to compare the experimentally measured and theoretically calculated
value of Rp. When doing the % error, substitute the theoretical value for the accepted value.
2. Calculate Rs. Use percent error to compare the experimentally measured and theoretically calculated
value of Rs.
4.2 Analysis of a simple series circuit and Kirchhoff’s Voltage Law
1. Use percent error to compare the experimentally measured value of the current I in the circuit with the
theoretical value calculated using the equation I = V/Rs where Rs is the equivalent series resistance of
the resistors as directly measured experimentally (not calculated).
2. Use percent error to compare each experimentally measured value of the voltage drop across each
resistor with the theoretical value calculated using the formula V = IR (use the experimentally
measured value for I and the measured resistance of each resistor for V = IR).
3. Verify Kirchhoff's voltage law, that both the experimental and theoretical algebraic sum of the voltage
drops around the circuit in a complete loop is zero. Note that the power supply is a source of EMF so
that the current flows through the power supply from negative to positive terminal. Thus, denoting the
power supply EMF in volts by Vemf, Kirchhoff's voltage law can be written as:
Experimental Value: Vemf,exp = V1,exp + V2,exp + V3,exp
Theoretical Value: Vemf,th = IexpR1 + IexpR2 + IexpR3
Where the V's refer to corresponding measured voltage drops across each of the three resistors in the
circuit as indicated in Figure 2. Use R measured for the theoretical value. Calculate % error.
4.3 Analysis of a simple circuit and the Kirchhoff Current Law.
1. Use percent error to compare the experimentally measured value of the current, Iexp, in the circuit with
the theoretical value calculated using the equation I1,th = V1/R1 whereV1 is the measured voltage drop
and R1 is the measured resistance of the first resistor. Do the same for the second and third resistor.
2. Verify Kirchhoff’s current law, that the algebraic sum of currents in a node is zero. Note that one
current flows into the node (positive) and two flow out (negative). Show that:
I1,exp + I2,exp + I3,exp = 0 and I1,th + I2,th + I3,th = 0
5 Questions
1. Did you verify the Kirchhoff’s voltage law for the simple series circuit which you constructed?
Explain how this was shown. If the verification was not exact, how large was the deviation?
2. Did you verify the Kirchhoff’s current law for the circuit which you constructed? Explain how this
was shown. If the verification was not exact, how large was the deviation?
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3. In the circuit used to verify Kirchhoff’s current law, R2 and R3 are connected in parallel.
a.
Calculate the equivalent parallel resistance Rp using the measured values of R2 and R3.
b.
Your value of Rp, calculated above in part 3.a, is in series with R1. Calculate Rs using the
measured value of R1 and your calculated value of Rp in part 3.a.
c.
Using the value of Rs from part 3.b above, and 5.00 volts, calculate the current. Call this
current Ical.
d.
Use percent error to show the difference between the calculated current in 3.c and the positive
sum of I2,exp and I3,exp. (For this calculation, simply change both negative values to positive
values. Nothing complicated.) Example. If I2,exp = - 0.45 mA and I3,exp = - 0.93 mA, than their
sum for this question is | I2+3,exp | = 1.38 mA. Simple. Took the absolute value. Use the 3.c
value as the experimental current value when doing percent error.
e.
Why should the positive sum of I2,exp and I3,exp equal the current in 3.c?
6 Conclusion
This section should have a clear statement of the results of the experiment and the extent to which the
results are in agreement with the theory being tested. Include percent error for this comparison.
To make this comparison meaningful, you should include the impact of the experimental error on your
results. In addition, please include a statement of what you have learned, a critique of the experiment, and
any suggestions you have which you think could improve the experiment or the lab handout.
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Instrumentation
1. The DC Voltmeter measures the voltage difference between two points
to which its terminals are connected. The voltmeter is always connected
in parallel to the part of the circuit across which the potential
difference is to be measured (Figure A1). In this experiment the Agilent
34405A multi-meter (multimenter) is used to measure voltage
differences and the resistance of the carbon resistor.
2. The DC Milliammeter measures the electric current between any
two points to which its terminals are connected. To measure the
current in any part of the circuit, the circuit must be broken at that
point and the ammeter must be inserted in the gap with loose ends
connected to its terminals, i.e., the meter is connected in series
with that part of the circuit where the current is to be measured
(Figure A2). The Fluke 75 battery powered multi-meter is used to
measure the current in milliamperes in this experiment.
3. Connecting wire leads. Assumed to be made of a good conducting material, e.g., a metal such as copper,
and of a large enough cross-section to have negligible resistance and voltage drop across them.
4. The DC Power supply is a device used to supply a
constant source of EMF between its output terminals.
The electric current flows internally in the power
supply from the minus to the plus terminal. In the
external circuit it flows from the plus to the minus
terminal (Figure A3). The voltmeter on the actual
power supply is not accurate and should not be used to
determine the output voltage. The Agilent 34405A
multi-meter is used to measure voltages on the power
supply.
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Data Sheet. All measurements go into the data section of the lab report.
1
Individual Resistors. The correct range will determine the amount of digits after the decimal point.
Measured using Agilent 34405A
Measured value R1 = ______________ Put in correct units for all measurements.
Measured value R2 = ______________
Measured value R3 = ______________
2
Resistor Combinations. The correct range will determine the amount of digits after the decimal
point. All calculations and % errors go into the calculation section of the lab report.
Parallel Combination.
Experimental: Rp,exp = ______________
Theoretical:
Rp,th = [1/R1 + 1/R2 + 1/R3] -1 = ______________ (use measured values of R)
% error = ______________
Series Combination
Experimental: Rs,exp = ______________
Theoretical:
Rs,th = R1 + R2 + R3 = ______________ (use measured values of R)
% error = ______________
3
Study of Simple Circuits: Kirchhoff's Voltage Law
A. Applied Voltage Vemf = ______________ (10 volts) (Enter as a positive value.)
B. Current I Measured using the Fluke 75.
Experimental: Iexp = ______________ (1 mA = 1 milliAmpheres = 0.001 A = 1x10-3 A)
Theoretical: Ith = Vemf /Rs,exp = ______________ (Use experimentally measured Rs,exp)
% error = ______________
C. Voltage drops across the individual resistor. All values should be positive.
The correct range will determine the amount of digits after the decimal point.
Measured (Agilent 34405A) Theoretical V = IR (Use Experimental Iexp & R Measured)
V1,exp = ____________
V2,exp = ____________
V3,exp = ____________
V1,th = ____________
V2,th = ____________
V3,th = ____________
% error = ______________
% error = ______________
% error = ______________
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D. Verification of Kirchhoff's Law. Sum should equal Vemf = 10 Volts
Vemf,exp = V1,exp + V2,exp + V3,exp
Vemf,th = V1,th + V2,th + V3,th
Using experimental values from part C: Vemf,exp = ______________
Using theoretical values from part C: Vemf,th = ______________
% error = ______________
4 Study of Simple Circuits: Kirchhoff's Current Law
A. Applied Voltage = ______________ (5 volts)
B. Individual measured experimental voltages and currents. (Results will be both positive and
negative.)
V1 = _______________
I1,exp = ________________
V2 = _______________
I2,exp = ________________
V3 = _______________
I3,exp = ________________
C. Predicted theoretical currents (use voltages just measured in circuit and resistances that were
measured) and the % error with the experimentally measured currents.
V1/R1 = I1,th = ________________
% error = ________________
V2/R2 = I2,th = ________________
% error = ________________
V3/R3 = I3,th = ________________
% error = ________________
D. Verification of Current Law. Sum should equal zero.
Using Measured values; I1,exp + I2,exp + I3,exp = ________________
Using Predicted Values I1,th + I2,th + I3,th = ________________
For lab report, all measured values must go into the data section. All calculations, including %
errors, goes into the calculations and analysis section of the lab report. Keep these separate. Keep
things neat and well organized.
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