RC charging time constant lab report

1. Title and aim

RC charging and the capacitor time constant. Determine the charging time constant of an RC circuit from capacitor-voltage observations and a logarithmic graph.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of RC charging time constant of 0.9995 s, differing by 0.046% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• 5.000 V DC step source: Provide a constant applied voltage E from t = 0.
• 10.0 kΩ resistor: Limit charging current and set the resistance in the model time constant.
• 100 μF capacitor: Store charge; preserve polarity and voltage rating when using a physical polarized capacitor.
• Switch and voltage-time recorder: Start charging from an initially discharged state and sample capacitor voltage with adequate time resolution.
• Supply E = 5.000 V; resistance R = 10.00 kΩ.
• Capacitance C = 100.0 μF; model time constant τ = RC = 1.000 s.
• Initially uncharged ideal capacitor; voltage rounded to 0.001 V.

4. Theory and graph choice

Watch a capacitor approach its final voltage and turn an exponential curve into a straight line. Record capacitor voltage at several elapsed times, transform the readings and estimate the time constant. Every virtual observation starts from an initially discharged capacitor; these are simulated transients.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units s⁻¹.

5. Method

1. With a physical source disconnected, discharge through a suitable resistor and verify near-zero capacitor voltage. Check component polarity and ratings before wiring.
2. Use E = 5.000 V and identify t = 0 as the start of the applied voltage step. Choose a recording method suited to a roughly one-second transient.
3. Choose an elapsed time in the 0.2–3.0 s range and request the capacitor-voltage observation. Record Vc in volts and time in seconds.
4. Collect at least five distinct times across the range. Each model observation restarts the same initially discharged transient; real repeat runs require a verified discharge first.
5. Check that Vc is below E. Calculate E − Vc, then the dimensionless ratio E/(E − Vc), then its natural logarithm. Fit this logarithm against time.
6. Use τ = 1/m in seconds. Compare with RC = 1.000 s, inspect the intercept and residuals, and discuss voltage-dependent transformed uncertainty and timing calibration.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0459%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 3.9745e-09; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.8892e-05. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.8526e-05.

Propagating only the relevant fit coefficient gives a fit-only standard error of 1.8875e-05 s. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Source voltage, timebase, voltage calibration, capacitor tolerance, leakage and probe loading contribute uncertainty. Logarithmic transformation amplifies voltage error near the supply voltage. Equal-weight fitting in transformed coordinates is a teaching approximation; consider a direct nonlinear fit and an uncertainty model for real data.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Check capacitor polarity and voltage rating; discharge through an appropriate resistor before reconnecting physical components.
• Measure actual source voltage in physical work and use the same E in every transformation.
• Use natural logarithms of dimensionless ratios; exclude Vc ≥ E from this analysis.
• Use synchronized fast recording and reset to a discharged state before each physical repeat; label virtual data simulated.

The model uses E = 5.000 V, R = 10,000 Ω and C = 100 μF, with zero initial voltage, instantaneous source switching, exact elapsed times and no leakage, equivalent series resistance, dielectric absorption or meter loading. Vc is rounded to 0.001 V. Each requested observation starts an identical transient. The unweighted logarithmic fit treats times as exact and does not model the voltage-dependent uncertainty introduced by transformation. The generator calculates Vc = 5[1 − exp(−t/1.000)] V and rounds to 0.001 V. Analysis uses x = t and y = ln[5/(5 − Vc)], estimating τ = 1/gradient. The reference 1.000 s is the product of the fixed model R and C, not a measurement of actual components.

12. Conclusion

The worked example illustrates how to obtain RC charging time constant from measurements, graph variables and a fitted coefficient. The estimate is 0.9995 s. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.