An RC circuit contains a resistor and a capacitor, often connected to a battery or voltage source. It is one of the simplest circuits that changes with time instead of reaching its final voltage instantly. The key idea is that the capacitor stores charge while the resistor controls how fast charge can flow.
RC circuits matter because they appear in timers, filters, camera flashes, sensors, and many electronic control systems.
The speed of charging or discharging is set by the time constant, tau = RC. During charging, the capacitor voltage rises toward the source voltage V0 in an exponential curve, while the current starts large and decreases. During discharging, the capacitor voltage and current both decrease exponentially toward zero.
After one time constant, the capacitor has completed about 63 percent of its total voltage change, and after about five time constants it is usually considered nearly fully charged or discharged.
Key Facts
- Time constant: tau = RC
- Charging capacitor voltage: VC(t) = V0(1 - e^(-t/RC))
- Discharging capacitor voltage: VC(t) = Vinitial e^(-t/RC)
- Charging current: I(t) = (V0/R)e^(-t/RC)
- Capacitor charge: Q = CVC
- After t = tau, a charging capacitor reaches about 0.632V0 and a discharging capacitor falls to about 0.368Vinitial
Vocabulary
- RC circuit
- An RC circuit is an electric circuit that contains a resistor and a capacitor connected so that voltage and current change over time.
- Capacitance
- Capacitance is the ability of a capacitor to store electric charge per volt, measured in farads.
- Time constant
- The time constant tau is the product RC and tells how quickly an RC circuit charges or discharges.
- Exponential decay
- Exponential decay is a decrease in which a quantity falls by the same fraction over equal time intervals.
- Steady state
- Steady state is the long-time condition when circuit voltages and currents no longer change significantly.
Common Mistakes to Avoid
- Treating the capacitor voltage as changing linearly is wrong because RC charging and discharging follow exponential curves, not straight lines.
- Forgetting that tau = RC must use ohms and farads is wrong because using kilo-ohms or microfarads without conversion can give a time constant off by factors of 1000 or more.
- Assuming the capacitor is fully charged after one time constant is wrong because it has reached only about 63 percent of the way to its final voltage.
- Using the charging equation for discharging is wrong because charging approaches V0 while discharging falls from an initial voltage toward zero.
Practice Questions
- 1 A 10 kΩ resistor is connected in series with a 100 μF capacitor. Calculate the time constant tau in seconds.
- 2 A capacitor charges from a 12 V source through a resistor. If tau = 2.0 s, what is the capacitor voltage after 2.0 s? Use VC = V0(1 - e^(-t/tau)) and e^(-1) = 0.368.
- 3 Explain why adding a larger resistor to an RC timing circuit makes an LED stay on longer after the switch is opened.