, 2004
Engineering 214
Circuits Analysis I
Lab Section 1
Partner: Kevin LaBeau
Instructor: Dr. Blauch Abstract

The relationship between voltage and current for capacitance will be verified and applied to a
specific integrating amplifier circuit. The analysis will be broken into hand calculations,
verification using PSpice Schematics Software and on an oscilloscope, verified through the
actual building and measuring of the circuit. The results of this experiment will either verify
or disprove the voltage-current relationship for capacitance in a circuit with an integrating
amplifier.
1.0
Introduction

The elements or components of any circuit are related through various proportions. Some
components, such as resistors, are just multiplied by the value of the current flowing through
them to result in a voltage across the resistor. This laboratory will examine the relationship
between voltages and currents when dealing with the component of a capacitor. The
capacitor, identified as C, will be connected in an integrator circuit that resembles the one
shown in Figure 1.

Figure 1: Integrating Amplifier Circuit to be analyzed
2.0
The Voltage-Current Relationship for Capacitors

A capacitor is a device that temporarily stores charge in a circuit. It consists of two metallic
plates separated and insulated from each other by a dielectric. The amount of charge that is
stored is proportional to the voltage across the capacitor. The relationship between the
voltage and charge of the capacitor is separated by a constant known as the capacitance. In an
equation, the relationship can be expressed as,
C
v
C
q
×
=

(1)
with
q
representing the charge in coulombs,
C
being the relationship constant known as
capacitance, and
C
v
being the voltage going across the capacitor in the circuit. Current is the
change in charge over a change in time, which can be expressed as,

dt
dq
i =

(2)
By substituting the charge in equation (1) into the charge of equation (2), the relationship
becomes,

dt
dC
v
dt
dv
C
i
C
C
×
+
×
=

(3)
However, the capacitor in this laboratory will have a constant capacitance value, so equation
(3) becomes,

dt
dv
C
i
C
×
=

(4)
In the circuit shown in Figure 1, the 100k resistor provides DC feedback to stabilize the
output voltage of the op-amp. So it is possible to just ignore that resistor and use Kirchoffs
Current Law at the negative input node of the op-amp. The application of KCL there looks
like,

0
=
dt
dv
C
R
v
O
S







+ =
k
dt
RC
v
v
S
O







(5)
If the function generator displays the input signal as a square wave such as the one displayed
in Figure 2, then
O
v is a triangular wave representing the integral of the
S
v square wave, shown
in Figure 3.

Figure 2: v
S
Square Wave

Figure 3: v
o
Triangular Wave
Since the input signal has an average value of zero, the op-amp output signal also has to have
an average value of zero. By selecting an appropriate integration constant, k, the average
value of zero can be accounted for. Equation (5) can then be stated as,
k
t
RC
V
v
O
+ =
for
2
0
T
t




(6)
T/2 T t
v
S


V



0

-V
T/2 T t
-Vo
Vo



0 Equation (6) shows that the straight line slope is negative. This means that when time is zero,
the output signal must start from positive V
o
, so
o
V
k =
. When the time is at T/2, the output
voltage signal is stopped at negative V
o
. If the frequency, f, represents the inverse of time, T,
V
o
can be calculated to be the equation displayed in (7).


o
o
V
T
RC
V
V
+ = 2



fRC
V
V
o
4
=








(7)
3.0

Verification of Input and Output Voltage Relationship


3.1
Analysis and Design

It was necessary to decide on a frequency around 1 kHz to calculate the ideal
resistance and capacitance from equation (7). The resistance that was chosen was 3
k and the capacitance chosen had a value of 100 nF. The actual value calculated for
V
was calculated to be 6 volts from equation (7) when the output voltage was chosen
to be 5 volts. This ideal calculation is displayed below.


(
)(
)
(
)
( )
volts
6
5
10
100
3000
1000
4
4
9
=
×
=
= V
V
fRCV
V
o




(8)
3.2
Simulation and Design Verification

The resulting circuit on PSpice is depicted below in Figure 2. VPULSE was used as
the voltage source and a transient analysis configured to display at least 50ms of the
output was used.
Figure 4: Integrating Amplifier Circuit with Design Element Values

Before simulation in PSpice, the actual resistance values were measured on the DMM
so that they could be put into the circuit drawn, not the nominal values. Table 1 shows
those values.
Table 1: Resistance Values
Resistor
Nominal Value
Measured Value
R
3 k
2.981 k
R
1

100 k
99.8 k

Once the actual resistance values were put into PSpice, the circuit was simulated. The
resulting element values of the simulation can be found in Table 2 with the actual
measured values from the building and measuring procedure of the laboratory. Also,
the simulation of the circuit output voltage graph can be found in the Appendix.

The circuit in Figure 4 was built on the CADET circuit wiring board. The square
wave of the voltage source was controlled by the function generator. All of the
measurements made were made on the SCOPE. There are two available channels,
channel 1 which is yellow on the screen and channel 2, which is blue on the screen. The scope probes were connected to Ch1 and Ch2 inputs on the SCOPE. The Ch1
scope was connected between the function generator and the datum. The Ch2 probe
was connected to pin 6 of the op-amp. The "measurements feature on the scope was
used to make the peak voltage, average voltage, and the measurements for frequency.
The values of the measurements are all on Table 2.
4.0
Laboratory Measurements

4.1
Laboratory Equipment Used

Without the necessary tools, this lab could not have been performed. The materials
used to build and measure the circuit were an Oscilloscope Tektronix Model TDS
3012 (SCOPE), a digital multimeter (DMM), a C.A.D.E.T. Trainer Circuit Wiring
Board, an Op-Amp 741, a 100k ¼ W resistor, a ¼ W resistor of a chosen resistance
value, a non-polarized polyester capacitor of chosen value, and various leads and
connectors.
4.2
Description of Measurements

Before the circuit was built, resistors were measured for their actual resistance values,
recorded previously as Table 1. The capacitor could not be measured for its true
value, so the assumption that it was ideal was made. The circuit was built on a
breadboard that was connected to a voltage of 12.0 volts. The voltages across an
element were measured in parallel with the DMM.
5.0
Comparison

5.1
Measured and Calculated Values
The values that could be measured were recorded and are displayed below in Table 2.
These values can be compared to the chosen design values and the values simulated on
PSpice. The capacitance value was not measured.
Table 2: Element Values
Element
Design Value Simulated Value Measured Value
V
6 V
6 V
6 V
V
o

5 V
5.0577 V
5.12 V
f
1000 Hz
-
1006 Hz

5.2
Error Analysis
The percent discrepancies between each of the measured, simulated, and calculated
voltage values were calculated using the formula,
100


%
× =
Voltage
Calculated
Voltage
Calculated
Voltage
y
discrepanc
x

(9)
where Voltage
x
is representative of either the measured voltage or the simulated
voltage. The percent errors were computed and organized into Table 3.
Table 3: Voltage Discrepancy
Voltage
Simulation Error
Measured Error
V
o
1.22%
2.34%

It was assumed that there was no voltage going across resistor, R
1
, when there actually
is. Another source of error could be due to measurement error. Digital multimeters
only display readings to a certain amount of decimal places. The resistor value for R
was read to be 2.981 k when the actual resistance could have been 2.98149987 k .
This applies to everything that was measured. The more components in a design
experiment, the more places of error there can be.
6.0
Conclusion

This laboratory experiment was designed to verify the relationship between current and
voltage relationship for capacitance in an integrating amplifier circuit. The discrepancies were above 1%, but they were still small. The relationship between V and V
o
in the circuit in
Figure 2 was verified.










Appendix