RC Circuit

Learning objectives

By the end of this module, the learner should be able to:

• explain why a capacitor does not charge instantly;
• define the RC time constant;
• interpret charging and discharging curves;
• calculate capacitor voltage during transient response;
• explain how resistance and capacitance affect circuit response speed;
• describe practical engineering uses of RC circuits.

First-principles explanation

An RC circuit contains a resistor and a capacitor. When a DC supply is connected, the capacitor stores charge, but the resistor limits the rate at which current can flow. This causes the capacitor voltage to rise gradually rather than instantly.

When the charged capacitor is disconnected from the supply and allowed to discharge through the resistor, its voltage falls exponentially. The same time constant controls both charging and discharging behaviour.

Equations and units

• R is resistance, measured in ohms (Ω).
• C is capacitance, measured in farads (F).
• τ is the time constant, measured in seconds (s).
• VC is capacitor voltage, measured in volts (V).
• VS is supply voltage, measured in volts (V).
• I is current, measured in amperes (A).
• E is stored energy, measured in joules (J).

Worked example

Problem: A 1000 Ω resistor is connected in series with a 1 mF capacitor and a 5 V supply. Find the time constant and the capacitor voltage after one time constant.

Known values:

• R = 1000 Ω
• C = 1 mF = 0.001 F
• VS = 5 V

Calculation:

τ = RC = 1000 × 0.001 = 1.0 s

VC(τ) = 5(1 − e−1) = 3.16 V

Answer: The time constant is 1.0 s, and after one time constant the capacitor has charged to approximately 3.16 V, or 63.2% of the supply voltage.

Interactive simulator

The RC Circuit simulator lets the learner adjust resistance, capacitance, and supply voltage, then observe the effect on charging voltage, discharging voltage, current decay, stored energy, and time constant.

Local launch command:

python -m streamlit run simulations/rc_circuit/streamlit_web_app.py --server.port 8507

Engineering meaning

RC circuits appear in timing circuits, filters, sensor interfaces, power-supply smoothing, microcontroller reset circuits, and transient response control. The same first-order behaviour also prepares learners for later control-system and signal-processing topics.

Increasing resistance or capacitance increases the time constant, making the circuit respond more slowly. Reducing resistance or capacitance decreases the time constant, making the circuit respond more quickly.

Assumptions and limits

• The resistor and capacitor are ideal.
• The supply voltage is constant DC.
• Leakage current is ignored.
• Parasitic resistance and inductance are ignored.
• The circuit is treated as a first-order system.

Challenge questions

1. What happens to the time constant if resistance is doubled?
2. What happens to the charging curve if capacitance is increased?
3. Why is the charging current largest at the start?
4. Why does the capacitor approach the supply voltage but not reach it instantly?
5. Give one engineering use of an RC time delay.