Free Pearson Edexcel International A Level Physics lesson

Log capacitor curves and calibrate a thermostat

A measuring circuit must be understood as part of the experiment.

What you’ll learn

  • Plan and evaluate core practical 11.
  • Calibrate a thermistor potential divider for core practical 12.
  • Quantify sensitivity and diagnose loading or thermal lag.

Core practical 11: use a logger to measure charging and discharge

Choose a resistor and capacitor giving a decay long enough to resolve with the logger but short enough to record several time constants. Measure the resistor or include its tolerance. Connect the voltage input across the capacitor, check polarity and voltage rating, and use a switch arrangement that selects charging from a low-voltage source or discharge through the known resistor.

Record before switching so that the initial value and switch time are visible. Use a stable timebase and sufficient voltage resolution across the full useful range. A charging curve approaches the supply asymptotically; a discharge curve approaches zero. Repeat with the same initial voltage and inspect whether switching delays or incomplete discharge change the starting conditions.

Determine RC from the 1/e point or a straight log-voltage plot during discharge. Include input resistance, leakage and offset in the evaluation. A high-resistance discharge resistor can make voltmeter loading significant, and the low-voltage tail is vulnerable to zero errors. A realistic improvement changes one of these identified limitations rather than merely collecting more points from the same biased circuit.

Vdischarge = V₀e^(−t/RC); Vcharge = Vs(1 − e^(−t/RC)); gradient ln(V/Vref)–t = −1/(RC)

Symbols and units

  • V is capacitor voltage in V
  • V₀ and Vs are initial and supply voltages in V
  • R is effective resistance in Ω
  • C is capacitance in F
  • t is time in s.

How the equation explains the idea

  1. Discharge voltage falls as V₀ exp(−t/RC). Charging voltage rises towards Vs as Vs[1 − exp(−t/RC)]. Both have the same time constant for the same effective circuit.
  2. A natural-log plot of positive discharge voltage divided by a fixed reference has slope −1/(RC). Late-time noise or offsets can distort the logarithm, so choose a justified fitting range.

Worked example

A capacitance from logged values

  1. Voltage drops from 8.0 to 8.0/e = 2.94 V in 3.0 s.
  2. Thus RC = 3.0 s.
  3. If R = 150 kΩ and loading is negligible, C = 3.0/150000 = 20 μF.

The 1/e reading must refer to the actual initial discharge voltage.

Keep this idea. Measure the starting value, resolve the early curve and account for the logger’s electrical loading.

Core practical 12: turn resistance change into a temperature threshold

A negative-temperature-coefficient thermistor decreases in resistance as temperature rises. Put it in a potential divider with a fixed resistor and use the divider output as a temperature-dependent signal. State which component the output is measured across: warming lowers the output across the thermistor but raises the output across the fixed resistor.

Place the thermistor and a reference thermometer close together in a stirred water bath, with electrical connections protected from water as appropriate. Use a low sensing current to limit self-heating. Change the bath temperature through a safe useful range, wait for stable equilibrium and record the output voltage and reference temperature at each setting. Warming and cooling points can reveal thermal lag or hysteresis in the system.

Plot output against temperature without assuming a straight line over a wide range. Use interpolation to select a switching voltage near the intended threshold, then test it with repeated approaches from both directions. The calibration belongs to the whole divider and supply configuration; changing the fixed resistor or supply voltage changes the voltage–temperature relation even if the thermistor is unchanged.

Vthermistor = Vs RT/(Rfixed + RT); Vfixed = Vs Rfixed/(Rfixed + RT)

Symbols and units

  • Vs is fixed supply voltage in V
  • RT is thermistor resistance in Ω
  • Rfixed is the fixed resistance in Ω
  • Vthermistor and Vfixed are voltages across those components.

How the equation explains the idea

  1. With negligible output loading, the current is Vs/(Rfixed + RT). Multiply by each resistance to obtain its voltage fraction.
  2. For an NTC thermistor, warming lowers RT. The voltage across it falls while the voltage across the fixed resistor rises. The two voltages sum to Vs in the ideal unloaded series model.

Worked example

Choose the output arm

  1. Vs = 6.0 V, Rfixed = 10 kΩ and thermistor resistance falls from 20 to 5 kΩ.
  2. Across the thermistor, output changes from 6 × 20/30 = 4.0 V to 6 × 5/15 = 2.0 V.
  3. Across the fixed resistor, the complementary output rises from 2.0 to 4.0 V.

The voltage trend depends on where the output is measured.

Keep this idea. Calibrate the complete divider, controlling thermal contact, equilibrium and self-heating.

Use a calibration curve locally and test its limitations

A calibration graph relates a known input to sensor output under specified conditions. Its local gradient gives sensitivity, which may change across a nonlinear curve. A large output change per degree is useful only if noise, resolution and drift remain sufficiently small. Convert voltage uncertainty into temperature uncertainty using the local slope rather than a single global average when the curve is strongly curved.

Interpolate inside the measured range instead of extending a polynomial or straight segment far beyond it. Collect more points near a threshold where the decision matters. Record the supply voltage and any reference-sensor correction, because a stable-looking output can still be biased by the calibration reference. Recalibrate if the circuit or measurement range changes.

A thermostat is a control system, so a static calibration is only part of its behaviour. Thermal lag can make the controlled temperature overshoot a threshold, while a narrow switching band can cause rapid repeated switching. Describe these dynamic limitations and test heating and cooling under safe conditions. Do not mistake a high sensitivity for guaranteed control accuracy.

Local sensitivity S = ΔV/Δθ; approximate temperature uncertainty δθ ≈ δV/|S|

Symbols and units

  • S is local voltage sensitivity in V °C⁻¹ when θ is temperature in °C
  • δV is voltage uncertainty in V
  • δθ is temperature uncertainty in °C.

How the equation explains the idea

  1. Near an operating point, a small change obeys ΔV ≈ SΔθ. Divide by sensitivity magnitude to estimate δθ ≈ δV/|S|.
  2. If |S| = 0.020 V °C⁻¹ and δV = 0.010 V, this contribution is about 0.5 °C. A flatter portion gives larger temperature uncertainty; reference thermometer uncertainty and hysteresis need separate consideration.
Temperature uncertainty grows where calibration is flatter

Conditions. Illustrative monotonic calibration V = 1 + 0.040θ V, where numerical θ is in °C, compared with a lower-sensitivity V = 1 + 0.020θ V sensor. Both share a 1 V offset and constant ideal slopes.

Temperature / °C; Sensor output / V.

A voltage uncertainty of 0.010 V implies 0.25 °C at sensitivity 0.040 V °C⁻¹, versus 0.50 °C at 0.020 V °C⁻¹. This treats only voltage-reading uncertainty, not calibration bias or hysteresis.

Open the interactive lesson to view the plotted curves and their coordinate tables.

Worked example

Turn voltage resolution into temperature resolution

  1. Near the threshold, output changes by 0.24 V over 4.0 °C.
  2. Local sensitivity magnitude is 0.060 V °C⁻¹.
  3. A voltage uncertainty of 0.012 V contributes about 0.012/0.060 = 0.20 °C.

Reference and thermal uncertainties must still be considered separately.

Keep this idea. Use local sensitivity to interpret uncertainty and verify the threshold under realistic conditions.

Open the interactive lesson for free practice and reviewed three-question samples when available. New full question sets and timed unit checks are in preparation. Existing complete Physics revision packs retain paid access; existing free teaching PDFs remain free.

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