An RC circuit experiment measures the time-dependent changes in voltage and current as a capacitor charges and discharges in a circuit where a resistor and capacitor are connected in series, confirming transient behavior and the circuit time constant.
Immediately after connection to a DC power supply, the capacitor voltage does not change instantaneously but approaches the supply voltage over time. Conversely, when a charged capacitor is discharged through a resistor, the capacitor voltage approaches 0 with time.
This change is not linear but exponential, and its rate is determined by the time constant τ, the product of resistance R and capacitance C.
This article assumes a charging and discharging experiment using a DC power supply, resistor, capacitor, switch, and voltmeter or oscilloscope, and presents the introduction, precautions, items to record, sample experimental data, calculations, sources of error, example results, and example discussion in that order.
Note:
This article is a study reference for discussing results obtained in physics experiments at schools, universities, and similar institutions. The experimental conditions and measured values shown here are illustrative examples and are not actual measurement results.
Electrolytic capacitors have polarity. Connecting one in reverse or using it above its rated voltage can cause heating, leakage, or damage. Turn off the power before changing wiring and confirm that no charge remains on the capacitor. For actual procedures, follow your school’s laboratory manual and the instructions of your instructor or TA.
- What Is an RC Circuit?
- Main Items to Record in the Results
- Sample Experimental Conditions
- Sample Experimental Data
- Calculation Methods
- Calculating the Theoretical Time Constant
- Voltage One Second After Charging Begins
- Voltage One Second After Discharging Begins
- Obtaining the Experimental Time Constant from 63.2%
- Obtaining the Experimental Time Constant from 36.8%
- Linearizing the Discharge Curve
- Calculating Capacitance from the Experimental Time Constant
- Relative Error of the Time Constant
- Initial Current
- Main Sources of Error
- Example of How to Write the Results
- Points to Connect to the Discussion
- Example Discussion
- Example Discussion When the Experimental Time Constant Is Larger Than the Theoretical Value
- Example Discussion When the Experimental Time Constant Is Smaller Than the Theoretical Value
- Example Discussion When the Initial Value of the Charging Curve Is Not Zero
- Example Discussion When the Semilogarithmic Graph Is Not Linear
- Summary
What Is an RC Circuit?
An RC circuit contains a resistor and a capacitor. Immediately after connection to a DC power supply or switching, transient behavior occurs in which voltage and current change with time.
Time Constant
The quantity that represents the rate of change in an RC circuit is called the time constant and is calculated as follows.
τ = RC
When resistance is expressed in Ω and capacitance in F, the unit of the time constant is seconds.
Charging a Capacitor
When an initially uncharged capacitor is connected through a resistor to a DC supply of voltage V0, the capacitor voltage is given by:
VC(t) = V0{1 − exp(−t/RC)}
The charging current decreases exponentially with time.
I(t) = (V0/R)exp(−t/RC)
Discharging a Capacitor
When a capacitor charged to an initial voltage V0 is connected to a resistor and discharged, the capacitor voltage is given by:
VC(t) = V0exp(−t/RC)
Voltage After One Time Constant
One time constant after charging begins, the capacitor voltage reaches about 63.2% of its final value.
VC(τ) = V0(1 − e−1) ≈ 0.632V0
One time constant after discharging begins, it falls to about 36.8% of its initial voltage.
VC(τ) = V0e−1 ≈ 0.368V0
After about five time constants, charging reaches about 99.3% of the final value and discharging falls to about 0.67% of the initial value, so the change can be regarded as practically complete.
Main Items to Record in the Results
- Nominal and measured resistance of the resistor
- Nominal capacitance and tolerance of the capacitor
- Rated voltage and polarity of the capacitor
- Supply voltage
- Theoretical time constant RC
- Elapsed time and capacitor voltage during charging
- Elapsed time and capacitor voltage during discharging
- Voltage across the resistor
- Circuit current
- Voltage after one time constant
- Experimental time constant obtained from 63.2% or 36.8%
- Time constant obtained from the slope of a semilogarithmic graph
- Relative error between theoretical and experimental values
- Shapes of the charging and discharging curves
- Comparison when resistance or capacitance is changed
- Oscilloscope time scale, voltage scale, and probe attenuation
Sample Experimental Conditions
| Item | Sample Condition |
|---|---|
| Resistor R | 10.0 kΩ |
| Capacitor C | 100 μF |
| Supply voltage V0 | 5.00 V |
| Theoretical time constant τ | 1.00 s |
| Measurement method | Digital voltmeter or oscilloscope |
| Measurement time | 0–5.0 s |
| Measurement interval | 0.5 s |
| Room temperature | 23 ℃ |
Sample Experimental Data
Capacitor Voltage During Charging
| Time t (s) | Theoretical Voltage (V) | Measured Voltage (V) | Percentage of Final Value |
|---|---|---|---|
| 0.0 | 0.000 | 0.02 | 0.4% |
| 0.5 | 1.967 | 1.95 | 39.0% |
| 1.0 | 3.161 | 3.14 | 62.8% |
| 1.5 | 3.884 | 3.86 | 77.2% |
| 2.0 | 4.323 | 4.30 | 86.0% |
| 2.5 | 4.590 | 4.56 | 91.2% |
| 3.0 | 4.751 | 4.73 | 94.6% |
| 4.0 | 4.908 | 4.89 | 97.8% |
| 5.0 | 4.966 | 4.95 | 99.0% |
Capacitor Voltage During Discharging
| Time t (s) | Theoretical Voltage (V) | Measured Voltage (V) | Percentage of Initial Value |
|---|---|---|---|
| 0.0 | 5.000 | 4.98 | 99.6% |
| 0.5 | 3.033 | 3.05 | 61.2% |
| 1.0 | 1.839 | 1.86 | 37.3% |
| 1.5 | 1.116 | 1.14 | 22.9% |
| 2.0 | 0.677 | 0.70 | 14.1% |
| 2.5 | 0.410 | 0.43 | 8.6% |
| 3.0 | 0.249 | 0.27 | 5.4% |
| 4.0 | 0.092 | 0.10 | 2.0% |
| 5.0 | 0.034 | 0.04 | 0.8% |
Time Constant When Resistance Is Changed
| Resistance R | Capacitance C | Theoretical Time Constant RC | Experimental Time Constant |
|---|---|---|---|
| 5.0 kΩ | 100 μF | 0.50 s | 0.49 s |
| 10.0 kΩ | 100 μF | 1.00 s | 0.99 s |
| 20.0 kΩ | 100 μF | 2.00 s | 1.96 s |
Time Constant When Capacitance Is Changed
| Resistance R | Capacitance C | Theoretical Time Constant RC | Experimental Time Constant |
|---|---|---|---|
| 10.0 kΩ | 47 μF | 0.47 s | 0.46 s |
| 10.0 kΩ | 100 μF | 1.00 s | 0.99 s |
| 10.0 kΩ | 220 μF | 2.20 s | 2.15 s |
Calculation Methods
Calculating the Theoretical Time Constant
For a resistance of 10.0 kΩ and capacitance of 100 μF, convert the units to Ω and F before calculating.
R = 10.0 kΩ = 1.00 × 104 Ω
C = 100 μF = 1.00 × 10−4 F
τ = RC
= (1.00 × 104)(1.00 × 10−4)
= 1.00 s
Voltage One Second After Charging Begins
VC(t) = V0{1 − exp(−t/RC)}
= 5.00{1 − exp(−1.00/1.00)}
= 5.00(1 − 0.3679)
= 3.16 V
Voltage One Second After Discharging Begins
VC(t) = V0exp(−t/RC)
= 5.00exp(−1.00/1.00)
= 1.84 V
Obtaining the Experimental Time Constant from 63.2%
For a supply voltage of 5.00 V, the voltage corresponding to 63.2% during charging is:
0.632 × 5.00 = 3.16 V
Read from the graph the time at which the charging curve reaches 3.16 V. If the measured curve reaches 3.16 V at 0.99 s, the experimental time constant is 0.99 s.
Obtaining the Experimental Time Constant from 36.8%
For an initial voltage of 4.98 V, the voltage corresponding to 36.8% during discharging is:
0.368 × 4.98 = 1.83 V
Read the time at which the discharge curve falls to 1.83 V and use it as the time constant.
Linearizing the Discharge Curve
Taking the natural logarithm of both sides of the discharge equation gives:
lnVC = lnV0 − t/RC
A graph with time t on the horizontal axis and lnVC on the vertical axis is a straight line with slope −1/RC.
Slope a = −1/τ
τ = −1/a
For example, if the slope of the best-fit line is −1.01 s−1:
τ = −1/(−1.01)
= 0.990 s
Calculating Capacitance from the Experimental Time Constant
For an experimental time constant of 0.990 s and measured resistance of 10.1 kΩ:
C = τ/R
= 0.990 / (10.1 × 103)
= 9.80 × 10−5 F
= 98.0 μF
Relative Error of the Time Constant
For a theoretical value of 1.00 s and experimental value of 0.99 s:
Relative error (%) = |Experimental value − Theoretical value| / Theoretical value × 100
= |0.99 − 1.00| / 1.00 × 100
= 1.0%
Initial Current
The theoretical current immediately after charging begins can be found by treating the capacitor as a short circuit.
I(0) = V0/R
= 5.00 / 10.0 × 103
= 5.00 × 10−4 A
= 0.500 mA
Main Sources of Error
| Source of Error | Effect on Measurement | Likely Trend in Results | Improvement |
|---|---|---|---|
| Resistor tolerance | Actual R differs from nominal value | Time constant differs from theoretical value | Use measured resistance in calculations |
| Capacitor tolerance | Actual C differs from nominal value | Overall charging/discharging time becomes longer or shorter | Determine actual capacitance from experimental data and compare |
| Capacitor leakage current | Charge is lost internally | Final charging voltage may be lower and discharge may be faster | Use a capacitor with low leakage |
| Equivalent series resistance of capacitor | Nonideal internal resistance is added | Initial current and short-time behavior deviate from theory | Use a component with low ESR |
| Internal resistance of power supply | Effective circuit resistance increases | Charging time constant increases | Account for output resistance of the power supply |
| Output resistance of signal generator | Typically about 50 Ω is added in series | Error becomes significant in experiments using small resistances | Use total resistance including output resistance |
| Input resistance of voltmeter/oscilloscope | A parallel resistance is added across the capacitor | Discharge time constant decreases | Use a high-input-resistance instrument or 10× probe |
| Probe input capacitance | Added to the capacitance being measured | Time constant increases in low-capacitance circuits | Use a low-capacitance probe and correct for it |
| Switching time | Delay occurs in determining time zero | Initial portion of curve is shifted | Use the oscilloscope trigger function |
| Reaction time in manual measurement | Time and voltage records are not synchronized | Time constant varies | Use automatic recording or a sufficiently long time constant |
| Initial charge on capacitor | Voltage is not 0 at start of charging | Charging curve deviates from theoretical equation | Fully discharge before measurement |
| Insufficient charging | Voltage at start of discharge has not reached supply voltage | Assuming an initial value of 5 V introduces error | Charge for at least 5τ before discharging |
| Supply-voltage fluctuation | Final or initial voltage changes | Curves from different trials do not coincide | Use actual initial and final values |
| Data sampling interval | Too few points near the time constant | Reading the 63.2% position becomes coarse | Measure more finely around 0–2τ |
| Noise in low-voltage region | Relative error becomes large near end of discharge | End of semilog graph deviates from a straight line | Exclude points near the noise floor from the fit |
Why Is the Current Maximum Immediately After Charging Begins?
Before charging, the capacitor voltage is 0, and immediately after connection the capacitor temporarily behaves nearly like a short circuit. Therefore, almost the entire supply voltage is applied across the resistor, and the current reaches its maximum value of V0/R.
As charging proceeds, the capacitor voltage rises and the voltage across the resistor decreases, so the current gradually decreases.
Why Does the Capacitor Voltage Not Change Instantaneously?
The charge on a capacitor is expressed as Q = CV. Changing the voltage instantaneously would require charge to move instantaneously, which would require infinite current.
Because an actual circuit has finite resistance, capacitor voltage changes continuously.
Why Does a Larger Resistance Slow the Change?
With a larger resistance, less current flows for the same voltage. Because less charge moves to or from the capacitor per unit time, charging and discharging require more time.
Why Does a Larger Capacitance Slow the Change?
A capacitor with larger capacitance requires more charge to reach the same voltage. Therefore, when charged or discharged through the same resistance, more time is required for the change.
Why Is the Process Considered Complete After About Five Time Constants?
The exponential function theoretically never reaches the final value during charging or exactly 0 during discharging. However, after five time constants the remaining difference is only about 0.67%.
Because this difference is comparable to or smaller than the measurement error in many experiments, charging or discharging is regarded as practically complete.
Example of How to Write the Results
A series RC circuit was constructed using a 10.0 kΩ resistor, a 100 μF capacitor, and a 5.00 V power supply, and the capacitor voltage during charging and discharging was measured as a function of time.
During charging, the capacitor voltage increased from 0.02 V immediately after the start to 3.14 V at 1.0 s and 4.95 V at 5.0 s. During discharging, it decreased from an initial voltage of 4.98 V to 1.86 V at 1.0 s and 0.04 V at 5.0 s.
The time constant obtained from the time at which the charging curve reached 63.2% of the final value was 0.99 s, corresponding to a relative error of 1.0% from the theoretical value of 1.00 s.
Points to Connect to the Discussion
- Did the charging and discharging curves have exponential shapes?
- Was the voltage after one time constant close to 63.2% or 36.8%?
- Did the experimental time constant agree with the theoretical value RC?
- Did the semilogarithmic graph become a straight line?
- Did the time constant increase proportionally when resistance was increased?
- Did the time constant increase proportionally when capacitance was increased?
- Was the change nearly complete after about five time constants?
- How did current change immediately after charging began and after sufficient time had passed?
- How did capacitor tolerance affect the results?
- How did instrument input resistance and input capacitance affect the time constant?
- Could the determination of time zero or delay from manual operation be ignored?
- Why did measurements near the end of discharge deviate from the theoretical curve?
Example Discussion
In this experiment, a series RC circuit with a 10.0 kΩ resistor, 100 μF capacitor, and 5.00 V supply was used to measure the time-dependent capacitor voltage during charging and discharging.
During charging, the capacitor voltage was nearly 0 V immediately after the start, then increased rapidly and approached 5.00 V with progressively smaller increases over time. During discharging, the voltage fell rapidly from its initial value and then approached 0 V with progressively smaller decreases.
These curves were not linear and agreed with the exponential changes described by VC = V0{1 − exp(−t/RC)} for charging and VC = V0exp(−t/RC) for discharging.
The theoretical time constant calculated from τ = RC was 1.00 s. From the charging curve, the time at which the voltage reached 3.16 V, corresponding to 63.2% of the final voltage of 5.00 V, was approximately 0.99 s.
In the discharge curve, the time at which the voltage fell to approximately 1.83 V, corresponding to 36.8% of the initial voltage of 4.98 V, was also about 1.0 s, so nearly the same time constant was obtained from both charging and discharging.
The experimental time constant of 0.99 s was close to the theoretical value of 1.00 s, with a relative error of 1.0%. This confirms that the rate of change in an RC circuit is determined by the product of resistance and capacitance.
When the resistance was changed to 5.0 kΩ, 10.0 kΩ, and 20.0 kΩ, the experimental time constants were approximately 0.49 s, 0.99 s, and 1.96 s, respectively. Doubling the resistance approximately doubled the time constant, showing that the time constant is proportional to resistance.
When capacitance was changed, larger capacitance also required more time for charging and discharging. This is because a capacitor with greater capacitance requires more charge to reach the same voltage.
Immediately after charging begins, the capacitor voltage is close to 0, so almost the entire supply voltage is applied across the resistor and the current is maximum. As charging proceeds, capacitor voltage rises and resistor voltage falls, causing the current to decrease exponentially.
After sufficient time has passed in DC steady state, no current flows through an ideal capacitor, which behaves as an open circuit. In contrast, immediately after charging begins, the capacitor behaves nearly like a short circuit. This difference produces the transient behavior of the RC circuit.
Possible causes of the difference between theoretical and experimental values include tolerances in the nominal resistor and capacitor values, capacitor leakage current and equivalent series resistance, internal resistance of the power supply or signal generator, and input resistance of the measuring instrument.
In particular, the input resistance of an oscilloscope or voltmeter is connected in parallel with the capacitor being measured. If this input resistance is comparable to the circuit resistance, the effective resistance during discharge decreases and the actual time constant also becomes smaller.
An oscilloscope probe also has input capacitance, which is added to the circuit capacitance. Its effect is small for a large capacitance such as 100 μF, but it cannot be ignored when measuring small capacitances in the nF or pF range.
When a switch is operated manually and time is measured with a stopwatch, reaction time can shift time zero. The shorter the circuit time constant, the larger this timing difference becomes as a relative error, so an oscilloscope trigger function is more suitable for short time constants.
Near the end of discharge, the voltage becomes small, so instrument resolution and electrical noise constitute a larger fraction of the measured value. Therefore, low-voltage points tend to deviate from a straight line on a semilogarithmic graph, and an appropriate fitting range must be selected.
Measurement accuracy can be improved by measuring the resistance and using that value in calculations, fully discharging or charging the capacitor before measurement, taking finer data near the time constant, and accounting for instrument input resistance and capacitance.
Thus, the experiment confirmed that charging and discharging voltages in an RC circuit change exponentially and that the rate of change is determined by the time constant τ = RC.
Example Discussion When the Experimental Time Constant Is Larger Than the Theoretical Value
Possible reasons why the experimental time constant was larger than the theoretical value include the actual resistance or capacitance being larger than the nominal value, the output resistance of the power supply or signal generator being added in series, and the oscilloscope probe capacitance being added to the circuit capacitance.
Example Discussion When the Experimental Time Constant Is Smaller Than the Theoretical Value
Possible reasons why the experimental time constant was smaller than the theoretical value include the actual capacitance being smaller than the nominal value, the input resistance of the voltmeter or oscilloscope being connected in parallel with the capacitor and reducing the effective resistance during discharge, and capacitor leakage current.
Example Discussion When the Initial Value of the Charging Curve Is Not Zero
A nonzero capacitor voltage at the start of charging may be caused by charge remaining from the previous measurement. When an initial voltage is present, the charging equation that assumes a start from 0 V cannot be used directly, so the capacitor should be fully discharged through a resistor before measurement.
Example Discussion When the Semilogarithmic Graph Is Not Linear
Possible reasons why the semilogarithmic graph of discharge voltage was not linear include the increasing influence of noise and instrument resolution in the low-voltage region, capacitor leakage current causing deviation from a simple exponential function, and a mismatch between the switch transition time and measurement start time.
Summary
During charging and discharging in an RC circuit, capacitor voltage and circuit current change exponentially with time.
The rate of change is determined by the time constant τ = RC. After one time constant, the charging voltage reaches about 63.2% of its final value, while the discharging voltage falls to about 36.8% of its initial value.
In the discussion, it is important to connect the comparison between theoretical and experimental time constants with the exponential curves, semilogarithmic graph, effects of resistance and capacitance, component tolerances, instrument input resistance and capacitance, and error in determining time zero.
