Download Lab #2 Voltage and Current Division

ECE 170
Lab #2 Voltage and Current Division
Lab #2 Voltage and Current Division
In this experiment, we will be investigating the concepts of voltage and current division.
Voltage and current division is an application of Kirchoff’s Laws.
Kirchoff’s Voltage Law
Kirchoff’s Voltage Law (KVL) states that the sum of all voltage drops around any loop in
any circuit sum to zero. In mathematical form,
n
v
i
0
(2.1)
i 1
Where the vi in Equation (2.1) are the voltages across the individual components in any
circuit. As an example of how to use Kirchoff’s Voltage Law to solve a circuit, consider
the circuit shown in Figure 2.1.
+ V1 R1
Vs
+
+
R2 V
2
R3 V
3
-
-
Figure 2.1: Application of Kirchoff’s Voltage Law.
If we apply Equation (2.1) to the loop contained in the left half of Figure (2.1), and then
the right half of the circuit, while traversing each loop in a clockwise direction, we obtain
the two equations
0  V s  V1  V 2
0  V 2  V3
This approach produces two equations for which there are three unknowns. We could
use the equation for the outer loop Figure 2.1 to obtain a system of three equations and
three unknowns. This would work, but there is an easier way. Let us do away with all the
voltages shown in Figure 2.1 except the source and define currents flowing in the two
main loops. This is shown in Figure 2.2. Since we are defining the currents, we are free
to choose their directions. Call the currents I1 and I2.
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ECE 170
Lab #2 Voltage and Current Division
R1
Vs
R2
I1
I2
R3
Figure 2.2: KVL Example.
Let us now write equations by applying Kirchoff’s Voltage Law for the left and right
loops as before, but using our defined currents. Each loop is traversed in the clockwise
direction.
0  Vs  R1 I1  R2 ( I1  I 2 )
0  R2 ( I 2  I1 )  R3 I 2
We now have two equations with two unknowns that can easily be solved. To obtain the
voltages as shown in Figure 2.1, just use Ohm’s Law. These voltages are given as
V1  R1 I 1
V2  R2 ( I 1  I 2 )
V3  R3 I 2
Solving a circuit by defining your own currents allows you to configure the currents that
would provide an easy solution. Once these are obtained, any quantity desired can be
solved for. Also, by defining your own currents, you will make fewer mistakes, as it is
easier to keep track of polarities of components in your equations.
Kirchoff’s Current Law
Kirchoff’s Current Law (KCL) is similar to his voltage law. It states that all currents
entering or leaving any circuit node (connection point) sum to zero. Mathematically, this
is
n
i
i
0
(2.2)
i 1
To solve a circuit with Kirchoff’s Current Law, consider the circuit shown in Figure 2.3
where we have arbitrarily chosen the currents as I1, I2, and I3.
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ECE 170
Lab #2 Voltage and Current Division
R1
Vx
I1
Vs
R2
I2
I3
R3
Figure 2.3: KCL Example.
Applying Equation (2.2) at the node symbolized by Vx,
I1  I 2  I 3
We can then substitute for the currents using Ohm’s Law. This yields,
Vs  Vx Vx Vx


R1
R2 R3
Thus we now have one equation with one unknown (Vx). This allows us to easily solve
the circuit and obtain any circuit voltage or current desired.
Equivalent Resistance
Consider the circuit shown in Figure 2.4.
Req
R1
+
R2
I
V eq
R3
-
Figure 2.4: Resistors in Series.
Since the resistors have the same current flowing through them, by Ohm’s Law
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Lab #2 Voltage and Current Division
V eq  I R1  I R 2  I R3
V eq  I ( R1  R 2  R3 )
V eq  I Req
Where
Req  R1  R2  R3 ...
Thus, we can conclude the equivalent resistance of any number of resistors in series is the
sum of the resistances.
For resistors in parallel, consider the circuit shown in Figure 2.5.
Ieq
+
Req
V
I1
R1
I2
R2
I3
R3
-
Figure 2.5 Resistors in Parallel.
By Kirchoff’s Current Law,
I eq  I 1  I 2  I 3
I eq 
V
V
V


R1 R2 R3
 1
1
1
I eq  V  
 
 R1 R2 R3 
Req 
V
I eq
Req 
1
1
1
1


R1 R2 R3
The formula for the equivalent resistance of resistors in parallel that is given above can
be extended to any number of resistors by adding another term to the denominator. If
two resistors are in parallel, this formula can be reduced to a much more convenient
form. Simplifying, the equivalent resistance of two resistors in parallel is given by
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Lab #2 Voltage and Current Division
Req (2 resistors in parallel ) 
R1 R2
R1  R2
Instructional Objectives
2.1
2.2
2.3
2.4
2.5
Take voltage readings at various points in a circuit.
Take current readings at various points in a circuit.
Explain the operation and function of a potentiometer.
Identify and verify voltage and current division configurations in a circuit.
Verify Kirchoff’s Laws.
Procedure
1.
Adjust the DC power supply to output 10V. Obtain two 10k resistors and one
27k resistor as shown in Figure 2.6. Measure the actual values of these resistors
and the power supplies voltage. Record your data in Table 2.1. Be sure to keep
track of resistors and not mix them up since you have measured their values.
Vs ______________________
+
V1 10k 
+
27k  V2
10V
10k 
+
-
V3 -
Figure 2.6: Circuit to Verify Kirchoff’s Voltage Law.
2.
Measure the voltages as shown in Figure 2.6 and record their values in Table 2.1.
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ECE 170
Lab #2 Voltage and Current Division
Component/
Value
Nominal R
Value
(k)
Measured
R Value
(k)
Measured
Voltage
(V)
Calculated
Voltage
(Pre-Lab)
(V)
R1, V1
R2, V2
R3, V3
Table 2.1: Measured Data for Figure 2.6.
3.
Using your measured values for R1, R2, R3, and Vs, calculate the voltage drops for
V1, V2, and V3. Place these in the “Actual Calculated Volts” column in Table 2.2.
Calculate the % error of your measured voltages to the pre-lab voltages (data in
Table 2.1) using the pre-lab values as the accepted values. Also calculate the %
error of your “actual calculated values” to the measured values (from Table 2.1).
Use the measured values as the accepted values in your calculations. Place these
results in Table 2.2.
Component/
Value
Actual
Calculated
Volts
(V)
% Error
Calculated to
Measured
% Error
Actual Calculated
to
Measured
R1, V1
R2, V2
R3, V3
Table 2.2: Calculated Data and Errors for Figure 2.6.
4.
The errors you obtained in Table 2.2 should be less than 5%. If they are not, try
to find the reason why.
5.
Construct the circuit shown in Figure 2.7. Measure the actual value of the supply
voltage and the values of the resistors you use.
Vs ______________________
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ECE 170
Lab #2 Voltage and Current Division
10k 
I1
10V
27k 
I2
I3
10k 
Figure 2.7: Circuit to Verify Kirchoff’s Current Law.
6.
Repeat steps 2 and 3 for the currents I1, I2, and I3. Record your data in Tables 2.3
and 2.4. Always turn off power when you are making changes in the circuit,
such as moving the ammeter to measure a different current.
Component/
Value
Nominal R
Value
(k)
Measured
R Value
(k)
Measured
Current
(mA)
Calculated
Current
(Pre-Lab)
(mA)
R1, I1
R2, I2
R3, I3
Table 2.3: Measured Data for Figure 2.7.
Component/
Value
Actual
Calculated
Amps
(mA)
% Error
Calculated to
Measured
% Error
Actual Calculated
to
Measured
R1, I1
R2, I2
R3, I3
Table 2.4: Calculated Data and Errors for Figure 2.7.
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ECE 170
7.
Lab #2 Voltage and Current Division
In this part of the experiment, we will be investigating the properties of the
potentiometer.
The potentiometer is a variable resistor.
A functional
representation of a potentiometer is shown in Figure 2.8. Measure the resistance
between the upper and lower terminals of the potentiometer. Record this value.
Also record what happens to the measured resistance as you rotate the shaft of the
device. Record what happens to the resistance between the wiper and the upper
terminal and the wiper and lower terminal as the shaft of the pot is rotated.
upper terminal
center or "wiper" terminal
lower terminal
Figure 2.8: Potentiometer.
8.
Construct the circuit shown in Figure 2.9. Adjust the potentiometer until Vout is
1V. Record the value of Vs and Vout.
Vs ______________________
Vout ______________________
Vs=6V
+
Vout
-
Figure 2.9: Potentiometer Circuit.
9.
Shut off the power and disconnect the power supply from the circuit. Carefully,
without moving the arm of the potentiometer, measure the resistance between the
upper and center terminal and then between the lower and center terminal.
Record these values.
Rupper-center ___________________ Rlower-center ___________________
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ECE 170
10.
Lab #2 Voltage and Current Division
This is a design problem. Your task is to design a circuit that will deliver 1.5V 
5% across a load resistor of 10k. Figure 2.10 demonstrates the problem. The
resistors available for use are listed in Figure 2.10. As with most engineering
design problems, there are constraints. In this case, your design must cost less
than 20 cents. Each resistor costs 6 cents. Assume your time is free. Design and
document a circuit that will meet the design criteria. Record your proposed
solutions and brainstorm until you have found a solution. Construct your
proposed circuit. Verify that it meets the design criteria by measuring Vin and
Vout. When your circuit is working, demonstrate the design to the lab instructor.
You may use more than one resistor of a particular value if you wish.
Your Circuit Design
Available Resistors
1k
6.2k 
10k 
15k 
27k 
Vin = 5V
+
Rload Vout = 1.5V
1.5V 
+/-5%
5%
-
Figure 2.10: Circuit Design Schematic for Step 10.
Vin ______________________
Vout ______________________
Post Lab Questions
2.1.
Draw a schematic diagram similar to Figure 2.6 except label the elements with the
measured resistance and computed voltage. Compare the voltages you calculated
to the voltages you measured for the circuit in Figure 2.6. Explain why they may
not be exactly equal. Verify that Kirchoff’s Voltage Law applies to this circuit
using your measured values.
2.2
Compare the currents you calculated to the currents you measured for Figure 2.7.
Explain why they may not be exactly equal. Verify that Kirchoff’s Current Law
applies to this circuit.
2.3
For the potentiometer circuit of step 8 and 9, draw the equivalent voltage divider
circuit and label the resistances with their actual measured values. Use the
measured values and the voltage divider relationship to show that the
potentiometer functions as a voltage divider.
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ECE 170
Lab #2 Voltage and Current Division
2.4
Draw the circuit you designed in step 10. Explain the reasoning you used to get
to your final solution and discuss how you verified that the circuit met the design
criteria.
2.5
Assume that the supply voltage was exactly 5V and the resistors were ideal in step
10. For what range of Rload will your circuit deliver 1.5V  5%? In other words,
what is the maximum and minimum resistance of Rload for which the circuit will
still operate?
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ECE 170
Lab #2 Voltage and Current Division
Name: ____________________
Section: ____________________
Pre-Lab #3: Linearity, Proportionality, and Superposition
1.
For the circuit shown in Figure 3.0a, calculate the proportionality constant that
relates the output voltage to the input voltage, k=Vout/Vin.
10k 
10k 
+
10k 
Vin
10k 
10k 
Vout
-
Figure 3.0a: Circuit for Problem 1.
2.
Calculate the voltages V5V and V15V as shown in Figures 3.0b and 3.0c
respectively.
k
k
k
+
+
5V
V5V
k
6.2k
V15V
6.2k
15V
-
-
Figure 3.0c: Circuit 2 for Problem 2.
Figure 3.0b: Circuit 1 for Problem 2.
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