Experiment 2
Thévenin equivalence
We verify the validity of the Thévenin’s theorem experimentally.
Any battery or power supply composed of passive components can be represented by an equivalent ideal
voltage source, VTh , in series with an equivalent internal resistance, RTh . Knowledge of VTh and RTh
permits the determination of the terminal voltage and output current produced by the supply for any load.
2.1
Preparation
In preparation for the proper understanding of the experimental procedures, be sure to review Thevenin’s
Voltage and Current laws; if the material has not yet been covered during the lectures, there is a wealth
of information on the topic available on the Web.
☛✟
For the voltage divider circuit shown in Figure 2.2 determine the equations for the Thévenin voltage
!✠
✡
VTh and resistance RTh .
2.2
Circuit assembly techniques
You will be using a breadboard to assemble and test your hands-on circuits. The breadboard provides a
convenient and organized way of implementing circuits, making quick component changes and providing
trouble-free circuit connections. On the board you have access to five pairs of banana jacks, red and black.
Also, there are five BNC coaxial connectors. A cable supplies the board with ±15, +5 and 0 Volts DC.
The black banana jacks and the outer ring of the BNC connectors have a common connection to ground,
or 0V. These signals are all available to the protoboard, the matrix that you will build your circuits on, as
shown in Figure 2.1.
To make the experience of assembling a working circuit more enjoyable:
1. Be sure that the power is OFF;
2. verify by direct measurement the values of all the circuit components;
3. verify that the jumper wires used are not broken (test for 0 Ω resistance);
4. assemble the circuit in a systematic and organized fashion;
5. check off each component as you add it to your circuit;
6. verify that your assembled circuit connections correspond to those of the schematic diagram;
7. turn ON the power and test the circuit for proper opearation.
9
10
EXPERIMENT 2. THÉVENIN EQUIVALENCE
Figure 2.1: Electronics lab breadboard connection matrix
If the circuit does not behave as expected, you will need to do some troubleshooting. Use the schematic
diagram as a guide to determine the voltage levels that should be present at various points of the circuit,
then use a voltmeter to measure these nodes. Develop a systematic approach to assembly and verification
of the circuit that you are building. As the circuits get more complicated, you will find it advantageous to
construct the circuit in stages, verifying the proper operation of the circuit after each progressive step.
2.3
Load line of a simple voltage divider
Equipment and components required
•
•
•
•
breadboard workstation, with +5V and ±15V power rails
digital multi-meter
various resistors
jumper wires
☛✟
Design and construct the voltage divider circuit on a protoboard using the +15V available on the
!✠
✡
breadboard and a pair of fixed resistors R1 , R2 with their values chosen arbitrarily somewhere in the
range of 1 to 10kΩ. Record the nominal and measured values of these resistors; you will need them
later.
☛✟
To measure voltages, you will use a precision digital multimeter set to measure DC Volts. This is the
!✠
✡
default setting when the multimeter is turned on. Connect the multimeter to your circuit by first
connecting a wire from the V-Ω terminal to a red banana jack, then connect the ground/common
terminal to the corresponding black banana jack.
Complete the connections with jumper wires from the appropriate protoboard pins to your circuit.
Do not connect the multimeter directly to your circuit; this will bypass the short-circuit protection
built into the breadboard (Review and follow the scheme shown in Appendix A).
2.3. LOAD LINE OF A SIMPLE VOLTAGE DIVIDER
11
Figure 2.2: Voltage divider circuit for demonstrating Thévenin’s equivalence
☛✟
Check that the digital multi-meter is set to measure DC Volts, then obtain a precise voltage for the
!✠
✡
+15V supply.
☛✟
When measuring the voltage V across the load resistor RL , the current through the resistor is easily
!✠
✡
calculated from the Ohm’s law:
V
RL
Start at RL = ∞, i.e. have no load resistor connected. In this way, you are measuring the “opencircuit” voltage, or Voc .
I=
☛✟
Select, measure and record a series of six RL values, in the approximate range 0.2R2 ≤ RL ≤ 2R2 .
!✠
✡
Load the circuit and measure the voltage across each RL .
☛✟
Tabulate your data, and then plot V as a function of I. You can use the command-line physica or
!✠
✡
the more convenient physicalab (the web-based version is at www.physics.brocku.ca/physica).
Perform a fit of your data.
? What fitting equation is appropriate? Explain. What is the physical significance of the V -intercept?
What does the slope of the line represent?
? Calculate the I-intercept of the line you fitted through your data points. What is its physical
significance?
☛✟
Reconfigure your multimeter to measure direct current by removing the wire from the V-Ω terminal
!✠
✡
and inserting it into the I terminal, then pressing the shift key followed by the IDC key.
☛✟
Remove RL , and connect the meter leads between the same two points in the circuit. Record your
!✠
✡
measurement and compare this to the previously determined I-intercept value.
? What is the purpose of above step? What assumption are you making regarding the instrument
connected across the load points?
? Last week, there was a point made about never doing this; why is it OK for this circuit?
☛✟
Reconfigure the multimeter wiring and setting to measure DC Volts.
!✠
✡
12
EXPERIMENT 2. THÉVENIN EQUIVALENCE
2.4
Determination of Thévenin equivalents
Figure 2.3: Thévenin equivalent circuit
You will recall that the Thévenin’s equivalent voltage VTh is the effective voltage across the load points.
The Thévenin’s equivalent resistance RTh of the circuit is determined by shorting all voltage sources in the
circuit and calculating the effective resistance across the load points.
☛✟
Use Paint or a similar program to sketch the VT h and RT h Thévenin equivalent circuits for inclusion
!✠
✡
in your lab report. ALternately, you can create these diagrams with EWB components. Be sure to
also include the mathematical derivations for these quantities.
☛✟
Calculate from your derived equations the Thévenin’s equivalent voltage VTh and resistance RTh of
!✠
✡
the circuit in the previous section. Include these results and derivations as part of your Discussion.
☛✟
On the protoboard, construct the equivalent circuit using the HP variable-voltage power supply to
!✠
✡
set VTh . A variable resistor, or potentiometer, could be used to set RTh . Can you think of a better,
more accurate way to obtain the resistance equivalent to RTh ?
☛✟
Measure the terminal voltage VL for the same values of RL that you used in the previous section,
!✠
✡
then perform the previous analytical steps.
? Plot and compare with the plot from the previous section. Did you build a good equivalent circuit?
The answer should be based on an error analysis of the results.
Discussion
Complete this lab by summarizing the results and discussing the Thévenin’s equivalence and its use in
simplifying electrical circuits. You should tabulate three sets of results for VT h , RT h and Imax , the maximum
circuit current; the results from the two fits and the theoretical results calculated using V , R1 and R2 and
Thévenin’s equations. Include also the two experimentally measured values of Imax .
Include a thorough error analysis as part of the critical evaluation of your results.