Laboratory #5
Electrical Behavior
of a Simple RC Circuit with Initial Conditions
ENGR225
Engineering
and Computer Science Department
October 16,
2007
David M.
Beams, Lucas P. Niiler
Adapted and edited from work done by
© 2005, David M. Beams and Lucas P. Niiler
I. Project
description
The timedomain behavior of a simple circuit consisting of a series connection of a voltage source, a resistor, and a capacitor is investigated in this laboratory. The voltage across the capacitor is experimentally measured and compared with both the predictions of simple theory and simulation results obtained with circuitsimulation software.
The circuit considered in this laboratory procedure is shown in Fig. 1. It consists of a dc voltage source V_{S}
in series with a switch, a resistance R, and a capacitor C. The voltage across the capacitor is V_{C}(t)
and the current in the circuit is i(t). The switch is presumed to be open for time t
≤ 0 and closes instantaneously at t = 0+. The capacitor voltage at the moment of the
closure of the switch is V_{CO}.
Fig.
1. Series RC circuit.
Kirchhoff’s Current Law requires that the current in the resistor equal the current flowing in the capacitor at all times. This current i(t) will be zero for t ≤ 0 because of the open switch. The following relations apply for t ≥ 0+:
_{} (1a)
_{} (1b)
Eq. (1a) states the current vs. voltage relationship of the resistor R; Eq. (1b) is the corresponding relationship for the capacitor C. Eqs. (1a) and (1b) may be combined to give a firstorder differential equation describing V_{C}(t):
_{} (2)
The solution of Eq. (2) can be obtained by
various methods (e.g., variation of parameters, method of the integrating
factor) as outlined in [1]. The solution
of Eq. (2) takes the form:
_{} (3)
It may be noted that the first term in the expression for V_{C}(t) is an exponential decay toward zero of a capacitor C initially charged to a voltage of V_{CO} in parallel with a resistor R; the second term is the exponential charging of capacitor C (having no initial charge) through resistor R toward the asymptotic limit of V_{S}.
The circuit shown in Fig. 2 is used to for the empirical measurement part of this laboratory procedure.
Fig. 2. Test circuit with independent adjustments for
initial and final capacitor voltages.
Designators CH1 and CH2 indicate connections for oscilloscope channels 1
and 2, respectively.
The operational amplifiers of Fig. 2 serve as unitygain buffer amplifiers. Buffer U1A isolates the resistance R from the 50K potentiometer which adjusts final capacitor voltage V_{S}; otherwise the effective resistance would be greater than R by a value ranging from 0Ω to 25KΩ depending upon the setting of the potentiometer wiper. Buffer U1C prevents loading of the RC network by the input resistance of the oscilloscope. Buffer U1B is not strictly necessary but was included since four identical operational amplifiers are available in a single TL084 package.
The
doublepole, doublethrow (DPDT) switch is placed in position A to establish
the initial capacitor voltage V_{CO}. The value of the 10K resistance is
unimportant; it was chosen to be considerably smaller than R so that the
initial capacitor voltage may be quickly established when the switch is moved
to position A. The capacitor voltage
moves in accordance with Eq. (3) when the switch is moved to position B at
t = 0. Moving the switch to
position B also places +5V across channel 1 of the oscilloscope. The step change from 0V to +5V at CH1 is used
to trigger the sweep of the oscilloscope.
The capacitor voltage after t = 0 is recorded by
channel 2 of the oscilloscope. The
equivalent circuit of Fig. 2 is shown in Fig. 3.
Fig. 3. Equivalent circuit to Fig. 2.
Table 1 lists the laboratory equipment used in this laboratory procedure.
Table 1. Bench laboratory equipment used in this
procedure.
Manufacturer 
Model No. 
Description 
HewlettPackard

E3631A

Tripleoutput
power supply (0– ±25V, 0– +6V)

Tektronix

TDS210

Twochannel
digital oscilloscope

HewlettPackard

HP34401A

Digital
multimeter

The specific apparatus listed in Table 1 is not required to perform this experiment; equivalent apparatus may be substituted. It is important, however, that the oscilloscope be a digital storage oscilloscope.
Simulation of the circuit was performed with the evaluation
(student) version of PSpice 8.0 circuitsimulation software (MicroSim,
Procedures
The experiment was performed in accordance with the
laboratory procedure “Electrical Behavior of a Simple RC Circuit with
Initial Conditions,” written by D. M. Beams, Department of Electrical Engineering,
Values of V_{C}(t) were calculated from Eq. (3) from t = 0 to10s with Excel.
The circuit of Fig. 2 was constructed and connected to the power supply and oscilloscope. The power supply was adjusted to produce outputs of +5V and ±15V as indicated by the display of the power supply. The oscilloscope sweep trigger mode was set for single sweep with triggering on the rising edge of the voltage at channel 1. The oscilloscope time base was set for 1 s per division. The sensitivity of channel 2 was set to 2V per division.
The 50K potentiometers of Fig. 2 were adjusted to produce an initial capacitor voltage V_{CO} of +5V and a final capacitor voltage V_{S} of –5V as measured with the digital multimeter at pins 7 and 1, respectively, of the operational amplifier. The DPDT switch was placed in position “A” for a minimum of five seconds to permit the capacitor to charge to the initial voltage V_{CO}. The switch was then moved to position “B,” triggering the oscilloscope sweep and recording the output voltage of U1C on channel 2 for 10s. Data points were then taken from the oscilloscope screen, using the oscilloscope cursors to assist in reading values of voltage and time from the screen. This process was repeated for the combinations V_{CO} = +2.5V, V_{S} = +10V.
The circuit of Fig.
4 was drawn with MicroSim Schematics 8.0.
A pulsed voltage source with initial (unpulsed) value V_{CO}
and pulsed value V_{S} drives the RC network with the
transition from V_{CO} to V_{S} beginning at t
= 0. A transient analysis was performed
by PSpice 8.0 for 10 s for each of the (V_{CO}, V_{S})
combinations given above; the capacitor voltage was plotted with the
Probe graphical postprocessor. Results
were copied to Excel for graphing.
Fig. 4. Schematic of circuit for simulation with
PSpice 8.0. The marker at “Vc” is a
command to the graphical postprocessor to display the voltage at that node vs.
time automatically when simulation ends.
Graphs were prepared showing simulation results, predicted
values computed from Eq. (3), and measured results for both cases. Figure 5 shows the curves for V_{CO} = +5V
and V_{S} = –5V.
Figure 6 shows curves for the case V_{CO} = +2.5V
and V_{S} = +10V.
Fig. 5. Simulated, computed, and measured values of
capacitor voltage vs. time for the charging of a 1μF capacitor with
initial voltage of +5V through a 1MΩ resistor to a final value of –5V. Computed values were found from Eq. (3).
Fig. 6. Simulated, computed, and measured values of
capacitor voltage vs. time for the charging of a 1μF capacitor with
initial voltage of +2.5V through a 1MΩ resistor to a final value of +10V. Computed values were determined from Eq.
(3).
The apparent
agreement of the measured curves of Figs. 5 and 6 with their counterparts
computed from Eq. (3) substantiates the correctness of the theoretical
development presented above (see section II).
One measured data point in Fig. 6 (for t = 0.5s) deviated
noticeably from the predicted value. It
is surmised that the discrepancy arises from error in positioning the
oscilloscope cursor.
Other possible sources of experimental error are the inputoffset voltage and the inputbias current of operational amplifier U1C of Fig. 2. Figure 7 below shows the equivalent circuit of Fig. 2 including a model of U1C incorporating both the inputbias current (I_{IB}) and inputoffset voltage (V_{IO}). The voltage V_{O}(t) in Fig. 7 is the voltage read at the output of U1C. The input current of the digital sampling oscilloscope (DSO) is represented by I_{inDSO}. Buffer U1C isolates the RC circuit from the input current of the oscilloscope.
Fig. 7. Equivalent circuit of Fig. 2 when the
inputoffset voltage (V_{IO}) and inputbias current (I_{IB})
of operational amplifier U1C and loading of the oscilloscope are included.
Repeating the analysis of
Section II above with the circuit of Fig. 7 gives:
_{} (4)
Manufacturer’s datasheets [2] for the TL084 give a maximal value of 400pA for the inputbias current at an ambient temperature of 25C. The effect of inputbias current is therefore negligible; the term R I_{IB} in Eq. (4) should be no larger than 400μV, which is insignificant compared to the values of V_{S} used in this experiment. Comparison of Eq. (4) with Eq. (3) shows that the effect of inputoffset voltage is to shift the output voltage of U1C relative to capacitor voltage V_{C}(t) by –V_{IO}. Manufacturer’s data for the TL084 give a maximal value of ±15mV for the inputoffset voltage. This error is inconsequential in this experiment. Thus the results of the experiment may be considered valid even if the imperfections of the operational amplifier are ignored.
Another possible source of error arises from the resolution of the readout of cursor position on the oscilloscope screen; the readout of cursor voltage has increments of 80mV when the oscilloscope sensitivity is 2V per division (the resolution for readings beyond ±10V is 100mV). Thus the uncertainty in any voltage measurement is ±40mV (±50mV for readings beyond ±10V). This uncertainty has a negligible effect on the readings taken in this procedure.
Calibration of the readout of the oscilloscope was compared to the digital multimeter at voltages of ±10V and ±5V. Results of this comparison are shown in Table 2. Calibration of the oscilloscope was judged to be valid.
Table 2. Comparison of voltage readings of the Tektronix
TDS210 oscilloscope with the HewlettPackard HP34401A digital multimeter.
DMM
reading, V 
Oscilloscope
reading, V 
+10.0031 
10.1 
+5.0008 
4.96 
–5.0070 
–5.04 
–10.008 
–10.0 
Both simulation and laboratory procedures produced results consistent with expectations. The experiment has thus provided empirical confirmation of the theory of the electrical behavior of RC networks as developed above.
[1] P.D. Ritger and
N.J. Rose. Differential Equations
with Applications.
[2] Texas Instruments, Datasheet for TL081, TL082, and TL084 operational amplifiers, (Rev. E, Feb. 1999). http://wwws.ti.com/sc/ds/tl084.pdf (July 1, 2004).