7.1 RC circuits: charging and discharging

Components

Capacitor and Resistor in an RC Circuit

A capacitor stores electrical charge and opposes changes in voltage across its terminals. A resistor opposes the flow of electric current and dissipates energy as heat.

In an RC circuit, these two components are connected in series. The resistor limits how much current can flow at any moment, which controls how quickly the capacitor charges or discharges. The current through the resistor follows Ohm's law:

$I = \frac{V_R}{R}$

Voltage and Current Relationships

Kirchhoff's Voltage Law (KVL) applies around the loop at every instant:

$V_{applied} = V_C + V_R$

This single equation drives all the behavior in an RC circuit. Here's why: as the capacitor charges, $V_C$ rises, so $V_R$ must shrink (since their sum is fixed). A smaller $V_R$ means less current through the resistor, which means the capacitor charges more slowly. That's the reason the charging curve isn't linear; it's exponential.

• Early in charging: $V_C$ is small, so $V_R$ is large and current is high. The capacitor charges quickly.
• Late in charging: $V_C$ is close to $V_{applied}$, so $V_R$ is tiny and current is nearly zero. Charging slows to a crawl.
• Fully charged: $V_C = V_{applied}$, $V_R = 0$, and no current flows. The capacitor acts as an open circuit.

Charging and Discharging Behavior

Exponential Charging and Discharging

The voltage across the capacitor follows these equations:

• Charging (capacitor starts at 0 V, source voltage $V_S$):

$V_C(t) = V_S\left(1 - e^{-t/\tau}\right)$

• Discharging (capacitor starts at $V_S$, no source):

$V_C(t) = V_S \, e^{-t/\tau}$

The time constant $\tau$ is defined as:

$\tau = RC$

where $R$ is in ohms and $C$ is in farads, giving $\tau$ in seconds.

After one time constant ($t = \tau$):

• A charging capacitor reaches 63.2% of $V_S$
• A discharging capacitor drops to 36.8% of its initial voltage

These percentages come directly from the math: $1 - e^{-1} \approx 0.632$ and $e^{-1} \approx 0.368$.

Transient and Steady-State Response

The transient response is the time-varying part of the circuit's behavior while the capacitor is charging or discharging. The steady-state is what the circuit settles into after enough time has passed.

A common rule of thumb: after 5 time constants ($5\tau$), the capacitor has reached about 99.3% of its final value. For most practical purposes, that counts as "done."

For example, if $R = 10 \, \text{k}\Omega$ and $C = 100 \, \mu\text{F}$, then $\tau = 1 \, \text{s}$, and the circuit effectively reaches steady-state in about 5 seconds.

Capacitor Characteristics

Charge Storage and Time Constant

A capacitor's ability to store charge is measured by its capacitance ($C$), in farads (F). The charge stored relates to voltage by:

$Q = CV$

This means doubling the voltage across a capacitor doubles the stored charge.

The time constant $\tau = RC$ tells you the "speed" of the circuit. A larger $\tau$ means slower charging and discharging. You can get a larger $\tau$ by increasing either $R$ or $C$ (or both).

Voltage and Steady-State Behavior

The steady-state behavior depends on whether the capacitor is fully charged or fully discharged:

• Fully charged: $V_C = V_{applied}$, no current flows, and the capacitor behaves like an open circuit (a break in the wire).
• Fully discharged: $V_C = 0$, and the capacitor initially behaves like a short circuit (a plain wire), allowing maximum current to flow.

These two extremes are useful shortcuts for circuit analysis. When you see a DC circuit that has been sitting for a long time, you can replace every capacitor with an open circuit and analyze what's left. When a voltage source is first connected, you can treat an uncharged capacitor as a short circuit to find the initial current.