11.8 Resistor-Capacitor (RC) Circuits

Resistor-capacitor (RC) circuits combine resistors and capacitors, creating unique electrical behaviors. These circuits are essential in electronics, controlling how quickly capacitors charge and discharge. Understanding RC circuits helps explain timing in various electronic devices.

The time constant of an RC circuit, calculated as the product of resistance and capacitance, determines charge and discharge rates. This concept is crucial for analyzing capacitor behavior, including voltage changes and current flow over time in different circuit configurations.

Equivalent Capacitance

When multiple capacitors are connected in a circuit, they can be analyzed as a single equivalent capacitance ($C_{eq}$). This simplifies circuit analysis by treating multiple components as one.

For capacitors connected in series:

• The equivalent capacitance ($C_{eq,s}$) is calculated using the inverse sum formula: $\frac{1}{C_{\text{eq}, \text{s}}}=\sum_{i} \frac{1}{C_{i}}$
• The equivalent capacitance is always less than the smallest individual capacitance in the series
• For example, with two capacitors: $\frac{1}{C_{eq,s}} = \frac{1}{C_1} + \frac{1}{C_2}$

For capacitors connected in parallel:

• The equivalent capacitance ($C_{eq,p}$) is simply the sum of all individual capacitances: $C_{\text {eq.p }}=\sum_{i} C_{i}$
• For example, with two capacitors: $C_{eq,p} = C_1 + C_2$

RC Circuit Behavior

RC circuits exhibit distinctive time-dependent behavior that makes them useful in timing applications, filters, and signal processing.

Time Constant

The time constant ($\tau$) of an RC circuit measures how quickly a capacitor charges or discharges:

• Calculated as the product of equivalent resistance and capacitance: $\tau=R_{\text{eq}} C_{\text{eq}}$
• Measured in seconds when resistance is in ohms and capacitance in farads
• Represents a characteristic time scale for the circuit's response

For a charging capacitor:

• After one time constant, the charge reaches approximately 63% of its final value
• After five time constants, the capacitor is considered practically fully charged (>99%)

For a discharging capacitor:

• After one time constant, the charge decreases to approximately 37% of its initial value
• After five time constants, the capacitor is considered practically fully discharged (<1%)

Charging and Discharging Processes

When a capacitor charges in an RC circuit:

1. Initially, the uncharged capacitor acts like a wire, allowing easy charge flow to the plates
2. As charge accumulates, the potential difference across the capacitor increases
3. This increasing potential difference opposes further current flow
4. The branch current gradually decreases as the capacitor charges
5. Eventually, the capacitor approaches its fully charged state with maximum potential difference and zero branch current

When a capacitor discharges:

1. The process begins with the capacitor at maximum charge and potential difference
2. Current flows in the opposite direction compared to charging
3. The plate charge, potential difference, and branch current all decrease exponentially
4. Eventually, all three quantities approach zero as the capacitor fully discharges

After a time much longer than the time constant (typically 5τ or more), the capacitor and its branch can be modeled using steady-state conditions.

Practice Problem 1: Equivalent Capacitance

Solution:

(a) For capacitors in series, we use the formula: $\frac{1}{C_{\text{eq}, \text{s}}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}$

Substituting the values: $\frac{1}{C_{\text{eq}, \text{s}}}=\frac{1}{3 \text{ μF}}+\frac{1}{6 \text{ μF}}+\frac{1}{12 \text{ μF}}$ $\frac{1}{C_{\text{eq}, \text{s}}}=\frac{4}{12 \text{ μF}}+\frac{2}{12 \text{ μF}}+\frac{1}{12 \text{ μF}}=\frac{7}{12 \text{ μF}}$ $C_{\text{eq}, \text{s}}=\frac{12 \text{ μF}}{7}=1.71 \text{ μF}$

(b) For capacitors in parallel, we use the formula: $C_{\text{eq}, \text{p}}=C_1+C_2+C_3$

Substituting the values: $C_{\text{eq}, \text{p}}=3 \text{ μF}+6 \text{ μF}+12 \text{ μF}=21 \text{ μF}$

Practice Problem 2: RC Circuit Time Constant

Solution:

The time constant of an RC circuit is calculated using: $\tau = R \times C$

Substituting the values: $\tau = 220 \text{ Ω} \times 470 \times 10^{-6} \text{ F}$ $\tau = 103.4 \times 10^{-3} \text{ s}$ $\tau = 0.1034 \text{ s}$

By definition, after one time constant (0.1034 seconds), the capacitor will be charged to approximately 63% of its maximum value.