Laboratory #5

Electrical Behavior of a Simple RC Circuit with Initial Conditions

ENGR225

Engineering and Computer Science Department

Andrews University

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 time-domain 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 circuit-simulation software.

### II.                Theoretical background

The circuit considered in this laboratory procedure is shown in Fig. 1.  It consists of a dc voltage source VS in series with a switch, a resistance R, and a capacitor C.  The voltage across the capacitor is VC(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 VCO.

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 first-order differential equation describing VC(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 VC(t) is an exponential decay toward zero of a capacitor C initially charged to a voltage of VCO 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 VS.

# Equipment

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 unity-gain buffer amplifiers.  Buffer U1A isolates the resistance R from the 50K potentiometer which adjusts final capacitor voltage VS; 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 double-pole, double-throw (DPDT) switch is placed in position A to establish the initial capacitor voltage VCO.  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.

# 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 circuit-simulation software (MicroSim, Irvine, CA).

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, University of Texas at Tyler, 10 January 2003.

# Computation

Values of VC(t) were calculated from Eq. (3) from t = 0 to10s with Excel.

# Experimental measurements

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 VCO of +5V and a final capacitor voltage VS 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 VCO.  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 VCO = +2.5V, VS = +10V.

The circuit of Fig. 4 was drawn with MicroSim Schematics 8.0.  A pulsed voltage source with initial (unpulsed) value VCO and pulsed value VS drives the RC network with the transition from VCO to VS beginning at t = 0.  A transient analysis was performed by PSpice 8.0 for 10 s for each of the (VCO, VS) combinations given above; the capacitor voltage was plotted with the Probe graphical post-processor.  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 post-processor to display the voltage at that node vs. time automatically when simulation ends.

### IV.              Results

Graphs were prepared showing simulation results, predicted values computed from Eq. (3), and measured results for both cases.   Figure 5 shows the curves for VCO = +5V and VS = –5V.  Figure 6 shows curves for the case VCO = +2.5V and VS = +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).

### V.                 Discussion

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 input-offset voltage and the input-bias 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 input-bias current (IIB) and input-offset voltage (VIO).  The voltage VO(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 IinDSO.  Buffer U1C isolates the RC circuit from the input current of the oscilloscope.

Fig. 7.  Equivalent circuit of Fig. 2 when the input-offset voltage (VIO) and input-bias current (IIB) 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 input-bias current at an ambient temperature of 25C.  The effect of input-bias current is therefore negligible; the term R IIB in Eq. (4) should be no larger than 400μV, which is insignificant compared to the values of VS used in this experiment.  Comparison of Eq. (4) with Eq. (3) shows that the effect of input-offset voltage is to shift the output voltage of U1C relative to capacitor voltage VC(t) by –VIO.  Manufacturer’s data for the TL084 give a maximal value of ±15mV for the input-offset 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 TDS-210 oscilloscope with the Hewlett-Packard 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

### VI.              Conclusions

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.

### VII.           References

[1]  P.D. Ritger and N.J. Rose.  Differential Equations with Applications. New York: McGraw-Hill, 1968.

[2]  Texas Instruments,  Datasheet for TL081, TL082, and TL084 operational amplifiers, (Rev. E, Feb. 1999).  http://www-s.ti.com/sc/ds/tl084.pdf (July 1, 2004).