### Charging Characteristics of a Series RC Circuit

In the circuit in Figure 1, a voltage source (V_{S}) is initially isolated from a resistor (R) and capacitor (C) connected in series by an open switch. When the switch is closed, the capacitor will begin to charge, and the voltage across its terminals (V_{C}) will increase exponentially.

**Figure 1.** A series RC network charging circuit

**Figure 1.**A series RC network charging circuit

The charging and discharging rate of a series RC networks are characterized by its RC time constant, $$tau$$, which is calculated by the equation:

$$tau = R · C$$

Where:

- $$tau$$ is the time constant in s
- R
- C is the capacitance in F

The voltage across the capacitor as it charges over time is given by the equation:

$$V_C = V_S · (1 – e^{(frac{-t}{tau})})$$

Where:

- V
_{C}is the voltage across the capacitor in V - V
_{S}is the voltage of the source in V - t is the time since the closing of the switch in s
- $$tau$$ is the RC time constant in s

Using that equation, we can construct the following table to see how the voltage across the capacitor changes with time.

**Table 1. **Voltage charging values for an RC series network.

**Table 1.**Voltage charging values for an RC series network.

Ratio (t/RC) | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

V_{c} (% of V_{s}) |
63.21 | 86.47 | 95.02 | 98.17 | 99.33 |

After one time constant, the capacitor has charged to 63.21% of what will be its final, fully charged value. After a time period equal to five time constants, the capacitor should be charged to over 99%. We can see how the capacitor voltage increases with time in Figure 2.

**Figure 2. **Capacitor voltage charging over time in a series RC network circuit

**Figure 2.**Capacitor voltage charging over time in a series RC network circuit

As a general guideline, three time constants (95%) are often considered charged. However, for high precision circuits like analog-to-digital converters (ADC) with many bits of resolution, the charging period must increase to eliminate this “settling time” charging error from affecting the accuracy.

### Discharging Characteristics of a Series RC Circuit

Now let’s consider what happens when we discharge a capacitor through a series resistor as in the circuit in Figure 3.

**Figure 3. **A series RC network discharging circuit

**Figure 3.**A series RC network discharging circuit

When discharging this series RC network, the time constant is the same, but the exponential discharge equation is given by:

$$V_C = V_{C0} · (e^{(frac{-t}{tau})})$$

Where:

$$V_{C0}$$ is the initial voltage of the capacitor at time = 0 before the switch is closed.

The following table provides the discharge percentages over time for the series RC circuit.

**Table 2.** Voltage discharging values for an RC series network.

**Table 2.**Voltage discharging values for an RC series network.