Source: Physics 212 Lab Activity v1.1, University of Illinois at Urbana-Champaign
Tags: RC circuit, time constant, capacitance, capacitor, resistor, exponential decay, charging, discharging, tau, manufacturing tolerance, 22µF, IOLab, DAC, breadboard
Difficulty: Intermediate | Prerequisites: Basic circuit concepts (voltage, current, resistance), Ohm's law, series circuits, Prelab 6 material.
This lab sits at the point in the course where you move from ideal circuit theory to real measurement. You already know what capacitance is on paper; now you are measuring it on an actual component and discovering that the number printed on the side is only approximate. The broader lesson is about manufacturing tolerance: every real component deviates from its labelled value, and quantifying that deviation is a core experimental skill in any electronics or physics lab. You should be comfortable with series RC circuits and the idea that voltage across a capacitor changes exponentially with time.
You charge or discharge a capacitor through a known resistor, measure the RC time constant from the resulting voltage curve, then calculate C = τ / R. The result will differ slightly from the labelled 22 µF because real components have manufacturing tolerances, and your job is to quantify both the value and its uncertainty.
Capacitance (C)
The ability of a component to store electric charge per unit voltage. Measured in farads (F). For this lab, you are working with capacitors labelled 22 µF (microfarads), where 1 µF = 10⁻⁶ F. In simple terms, it tells you how much charge the capacitor can hold for a given voltage across it.
RC Time Constant (τ)
The product of resistance and capacitance in a series RC circuit: τ = RC. It has units of seconds. After one time constant, the voltage across a charging capacitor reaches about 63.2% of its final value (or drops to about 36.8% during discharge). Think of it as the "speed dial" for how quickly the capacitor charges or discharges.
Manufacturing Tolerance
The range within which a component's actual value may differ from its labelled (nominal) value. Expressed as a percentage. A resistor with a gold stripe has ±5% tolerance; a 10 kΩ resistor could be anywhere from 9.5 kΩ to 10.5 kΩ. In simple terms, it is the manufacturer saying "we got it close, but not exact."
DAC (Digital-to-Analogue Converter)
The IOLab's output that produces a set voltage. In this experiment, it provides the step voltage that begins the charging process. When using the A1 and A2 analogue inputs, the DAC must be set to 3 V or below.
Nominal Value
The value printed on the component. For your capacitors, this is 22 µF. Your measured value will likely differ.
Exponential Charging / Discharging
The voltage across the capacitor follows V(t) = V₀(1 − e^(−t/τ)) during charging and V(t) = V₀ · e^(−t/τ) during discharging, where V₀ is the supply voltage and τ = RC. Think of it as: the capacitor fills quickly at first, then the rate slows down as it approaches full charge, like pouring water into a funnel.
A 10 kΩ resistor is placed in series with one 22 µF capacitor.
The DAC output connects to the resistor; the GND connects to the negative side of the capacitor.
Voltage is monitored at two points:
V₁: between the DAC and the resistor
V₂: between the resistor and the capacitor
The IOLab's A1 and A2 analogue inputs measure V₁ and V₂.
A breadboard and four pin-to-pin wires complete the physical connections.
Switching the DAC on applies a step voltage to the RC series circuit.
The capacitor charges through the resistor; the voltage across the capacitor rises exponentially toward the DAC voltage.
By recording how the voltage changes over time, you extract the time constant τ from the shape of the curve.
Once you know τ and R, you calculate C = τ / R.
From the definition: Find the time at which the capacitor voltage reaches 63.2% of its final value (during charging) or drops to 36.8% (during discharging). That elapsed time equals one time constant.
From a curve fit: Fit the data to an exponential function. The fit parameter in the exponent gives τ directly. If using the natural log method, plot ln(V) against time; the slope of the linear region is −1/τ.
With τ measured in seconds and R measured in ohms, capacitance is:
C = τ / R
Report the result in µF for easy comparison with the labelled 22 µF.
The labelled value (10 kΩ) carries a ±5% tolerance, so using it introduces up to ±500 Ω of uncertainty.
A multimeter reduces this uncertainty considerably. On the 20 kΩ range, the accuracy rating is ±(0.8% + 2 counts), where each "count" is 10 Ω (the resolution of that range).
Always use the measured resistance, not the nominal value, in your capacitance calculation.
Quantity | Formula |
|---|---|
Time constant | τ = RC |
Capacitance from τ | C = τ / R |
Charging voltage | V(t) = V₀(1 − e^(−t/τ)) |
Discharging voltage | V(t) = V₀ · e^(−t/τ) |
Voltage at t = τ (charging) | V(τ) = 0.632 × V₀ |
Voltage at t = τ (discharging) | V(τ) = 0.368 × V₀ |
Manufacturing tolerance is not just a textbook idea. Every resistor, capacitor, and inductor in consumer electronics has a tolerance band, and engineers must design circuits that function correctly across the full range. Timing circuits (like the 555 timer) rely on RC time constants, so a 10% shift in capacitance directly shifts the timing. This is why precision components cost more: tighter tolerance means tighter quality control.
"The labelled value is the true value." It is not. The label gives the nominal value; the actual value sits somewhere within the tolerance band. Your measurement is likely closer to the truth than the label.
"One time constant means the capacitor is fully charged." At t = τ, the capacitor is only about 63% charged. It takes roughly 5τ to reach 99% of the final voltage.
"The DAC voltage does not matter." While the final capacitance result should not depend on the DAC voltage, using A1 and A2 inputs requires the DAC to be set at or below 3 V. Exceeding this can damage the analogue inputs or give incorrect readings.
"You only need to measure one capacitor." The lab asks you to measure as many as possible so you can comment on the spread of values and estimate the manufacturing tolerance from real data.
⚠️ Be able to identify τ from a voltage-vs-time graph, both for charging and discharging.
⚠️ Know how to rearrange τ = RC to solve for any one of the three quantities.
⚠️ Understand why 63.2% (charging) and 36.8% (discharging) are the signature fractions for one time constant.
⚠️ Expect questions that give you a voltage curve and ask you to extract τ, then calculate C or R.
⚠️ Manufacturing tolerance may appear in conceptual questions about why measured values differ from expected ones.
True or false: After two time constants, a charging capacitor has reached about 86% of its final voltage.
Fill in the blank: The time constant of a series RC circuit is defined as τ = ______.
True or false: If you double the resistance in an RC circuit while keeping C the same, the capacitor charges twice as fast.
Fill in the blank: On a discharging curve, the voltage has dropped to approximately ______% of its initial value after one time constant.
True or false: A gold stripe on a resistor means the tolerance is ±10%.
Q: You measure the time constant of an RC circuit to be 0.215 s. The resistance is 9.87 kΩ. What is the capacitance?
A: C = τ / R = 0.215 / 9870 ≈ 2.18 × 10⁻⁵ F ≈ 21.8 µF.
Q: A 22 µF capacitor is charged through a 10 kΩ resistor with a 3 V supply. What voltage do you expect across the capacitor at t = τ?
A: V(τ) = 0.632 × 3 V = 1.896 V, approximately 1.90 V.
Q: Why might your measured capacitance differ from 22 µF even if your experiment is performed correctly?
A: Manufacturing tolerance means the actual capacitance is not exactly the labelled value. The component could be anywhere within its tolerance band (often ±10% or ±20% for electrolytic capacitors).
Q: You want to find the time constant from a charging curve. Describe one method.
A: Identify the final (steady-state) voltage V₀ from the plateau of the curve. Calculate 0.632 × V₀. Find the time on the curve where the voltage first reaches that value. The elapsed time from the start of charging to that point is τ.
Q: If you use the nominal 10 kΩ resistance instead of a multimeter-measured value, how does this affect your reported capacitance?
A: The nominal value has ±5% uncertainty, which propagates directly into your capacitance calculation. Using the multimeter-measured value (with its smaller ±(0.8% + 2 counts) uncertainty) gives a more precise and likely more accurate result.
This material connects directly to AC circuits later in the course, where the interplay between resistance, capacitance, and frequency determines impedance and phase. The exponential charging behaviour is also the basis for understanding RL circuits (inductors replace capacitors, but the maths is structurally identical). Uncertainty propagation, covered in the companion notes, is a skill you will use in every remaining lab.
RC circuit, time constant, tau, capacitance measurement, exponential decay, exponential charging, series RC, 22 microfarad, 22µF, 10 kilohm, IOLab, DAC voltage, breadboard circuit, manufacturing tolerance, component tolerance, gold stripe resistor, multimeter resistance, PHYS 212, Physics 212, University of Illinois, E&M kit, capacitor lab, Lab 6