Download Experiment 2: Measurements on DC circuits

Experiment 2: Measurements on DC circuits
1
Objectives
The objective of this experiment is to analyze simple resistive circuits in DC. The circuits considered here
are: resistors in series, resistors in parallel, series-parallel combination, voltage divider, current divider and
the Wheatstone bridge. This experiment will allow the experimental verification of the theoretical analysis.
2
Introduction and Test Circuits
The theoretical analysis of the circuits under study is based on Ohm’s and Kirchhoff’s laws. The main
equations relating the electrical parameters of each circuit are presented next.
1. Ohm’s law. The voltage V (in volts, V) across a resistor is directly proportional to the current I (in
amperes, A) flowing through it. The constant of proportionality is the resistance R (in ohms, Ω).
V = RI
(2-1)
2. Resistors in series. (See Figure 2-1).
• The current through N elements in series is the same for all of them.
Is = I1 = I2 = · · · = IN
(2-2)
• The voltage across the ith element is Ri Ii . The sum of the voltages across each element is equal
to the voltage applied to the entire series combination.
Vs = V1 + V2 + · · · + VN =
N
X
Vi
(2-3)
i=1
Equation 2-3 is formulated from Kirchhoff’s voltage law.
• The equivalent resistance of the series combination is the sum of the individual resistances.
Req = R1 + R2 + · · · + RN =
N
X
Ri
(2-4)
i=1
3. Resistors in parallel. (See Figure 2-2).
• The voltage across N elements in parallel is the same for all of them.
Vs = V1 = V 2 = · · · = VN
(2-5)
• The current through the ith element is Vi /Ri . The sum of the currents through each element is
equal to the current provided to the entire parallel combination.
Is = I1 + I2 + · · · + IN =
Equation 2-6 is formulated from Kichhoff’s current law.
1
N
X
i=1
Ii
(2-6)
• The reciprocal of the equivalent resistance of the parallel combination is the sum of the reciprocal
of the individual resistances.
N
X 1
1
1
1
1
=
+
+ ··· +
=
Req
R1
R2
RN
Ri
i=1
(2-7)
4. Series-parallel combination. An example of a series-parallel combination circuit is shown in Figure 2-3.
The analysis of this type of circuit is accomplished by substituting the series (or parallel) combinations
by their equivalent resistances, such that the circuit is transformed into a pure parallel (or series)
circuit. Once the electrical parameters (voltage and/or current) have been determined for the equivalent
resistances, the voltages and/or currents for the individual resistors in the series or parallel combinations
can be obtained by using these parameters as Vs and Is for the corresponding combination.
5. Voltage divider. From Equation 2-3, a series circuit with two resistors will divide the applied voltage
Vs into two voltages V1 and Vo across each resistor. Notice that Vo is the output of the voltage divider
(see Figure 2-4), as it is referenced to ground. The proportion in which the input voltage Vs is divided
is given by
R2 Vs
Vo =
(2-8)
R1 + R2
In order for this circuit to operate as a voltage divider, the output current Io must be zero or very
small compared with the current through R2 .
6. Current divider. (See Figure 2-5). From Equation 2-6, a parallel circuit with two resistors will divide
the applied current Is into two currents I1 and IL through each resistor. The proportion in which the
input current Is is divided is given by
R 1 Is
(2-9)
IL =
R1 + RL
7. Wheatstone bridge. Figure 2-6 shows the Wheatstone bridge circuit. The output of this circuit is the
difference of potential across nodes a and b. Notice that it is composed of a pair of voltage dividers,
therefore, no significant current can be drawn from the output. The output voltage is given by
’
“
R2
R4
Vab =
−
(2-10)
Vs
R1 + R2
R3 + R4
3
Preparation
In preparation for the lab compute the required quantities.
1. Resistors in series. Consider the circuit in Figure 2-1 when N =3 and the component values shown in
Table 2-1. Complete the entries corresponding to the theoretical values.
2. Resistors in parallel. Consider the circuit in Figure 2-2 when N =3 and the component values shown
in Table 2-2. Complete the entries corresponding to the theoretical values.
3. Series-parallel combination. Consider the circuit in Figure 2-3 and the component values shown in
Table 2-3. Complete the entries corresponding to the theoretical values.
4. Voltage divider. Consider the circuit in Figure 2-4 and the component values shown in Table 2-4. Find
an expression for R2 in terms of Vs , R1 and Vo . Complete the entries corresponding to the theoretical
values.
5. Current divider. Consider the circuit in Figure 2-5 and the component values shown in Table 2-5. Find
an expression for RL in terms of Is , R1 and IL . Complete the entries corresponding to the theoretical
values.
2
6. Wheatstone bridge. Consider the circuit in Figure 2-6 and the component values shown in Table 2-6.
Assume R1 =R2 =R3 =R and find an expression for R4 in terms of Vs , R and Vab . Complete the entries
corresponding to the theoretical values.
4
Procedure
This part of the experiment requires assembling the resistive circuits presented in the previous section
and measuring data from all of them. Refer to the appendices regarding the use of the equipment and
breadboard. These activities allow for a continuing familiarization with the use of the equipment. Set the
current compliance value of the voltage source to 15 mA.
1. Resistors in series. Assemble the circuit in Figure 2-1 with N =3 and the component values shown in
Table 2-1. Take measurements to complete the entries corresponding to the experimental values.
2. Resistors in parallel. Assemble the circuit in Figure 2-2 with N =3 and the component values shown
in Table 2-2. Take measurements to complete the entries corresponding to the experimental values.
3. Series-parallel combination. Assemble the circuit in Figure 2-3 with the component values shown in
Table 2-3. Use Vs =10 V. Take measurements to complete the entries corresponding to the experimental
values. Notice that the resistor experimental values can be taken from the previous measurements in
Table 2-1 and Table 2-2. Measure the voltage across each resistor and use Ohm’s law (Equation 2-1)
and the resistor experimental values to determine the experimental values of IRi , i=1,2,. . . ,4.
4. Voltage divider. Assemble the circuit in Figure 2-4 with the component values shown in Table 2-4.
Take measurements to complete the entries corresponding to the experimental values.
5. Current divider. Assemble the circuit in Figure 2-5 with the component values shown in Table 2-5.
Set the voltage compliance value of the current source to 15 V. Take measurements to complete the
entries corresponding to the experimental values.
6. Wheatstone bridge. Assemble the circuit in Figure 2-6 with the component values shown in Table 2-6.
Take measurements to complete the entries corresponding to the experimental values.
5
Analysis
This section is intended for the analysis and comparison of the experimental and theoretical results. Answer
all the questions.
1. Calculate the error percentage between the measured and theoretical data and complete all the corresponding entries in Tables 2-1 through 2-6. The error percentage is given by
%error =
dth − dm
× 100
dth
where dth and dm are the theoretical and measured data respectively.
2. From the above results comment on the three parameters with the highest error percentages and the
three with the lowest error percentages. Discuss the possible causes for error and comment on the
overall agreement between the measured and theoretical data.
3
Parameter
Units
Theoretical
Experimental
%Error
R1
1
R2
R3
kΩ
2.2 5.6
Req
Vs
VR1
VR 2
V
VR3
Is
mA
10
…
…
Table 2-1: Resistors in series.
Parameter
Units
Theoretical
Experimental
%Error
R1
1.5
R2
R3
kΩ
6.8 8.2
Req
Vs
V
10
…
…
IR 1
IR2 IR3
mA
Is
Table 2-2: Resistors in parallel.
3. From the results of the series-parallel combination circuit select the true statements: (a) Is = IR1 +
IR2 + IR3 + IR4 , (b) Vab = Vs − Vbc , (c) Is = IR1 + IR2 = IR3 + IR4 .
4. A voltage divider can provide a voltage at its output, but it cannot provide current. If a device is
connected to the output of a voltage divider, what value of resistance should it have?
5. In a Wheatstone bridge, assume R1 =2R2 and R3 =2R4 . What is the value of Vab ?
Param
R1
R2
R3
R4
Ra−c
Vab
Unit
kΩ
Theor
1.0
2.2
5.6
8.2
Exper
%Error
Param
IR 1
IR 2
IR 3
IR 4
Is
Vbc
V
Unit
Theor
mA
V
Table 2-3: Resistors in series-parallel combination.
4
Exper
%Error
Parameter
Units
Theoretical
Experimental
%Error
R1
R2
kΩ
2.2
Vs
Vo
V
10
…
…
4.5
Table 2-4: Voltage divider.
Parameter
Units
Theoretical
Experimental
%Error
R1
RL
kΩ
2.2
Is
IL
mA
10 5.5
…
…
Table 2-5: Current divider.
Parameter
Units
Theoretical
Experimental
%Error
R1
R2
R3
2.4
kΩ
2.4 2.4
R4
Vs
Vab
V
10
…
…
-2
Table 2-6: Wheatstone bridge.
Vs
+
−
R1
R2
RN
Is
Figure 2-1: Resistors in series.
Is
Vs
+
−
R1
R2
Figure 2-2: Resistors in parallel.
5
RN
Is
Vs
+
−
a
R1
b
R3
R2
R4
c
Figure 2-3: Series-parallel combination.
Io
Vs
+
R1
+
−
R2
Vo
-
Figure 2-4: Voltage divider.
R1
Is
IL
RL
Figure 2-5: Current divider.
R1
Vs
+
−
R3
+ -
a
R2
Figure 2-6: Wheatstone bridge.
6
b
Vab
R4