21.6 DC Circuits Containing Resistors and Capacitors

DC Circuits with Resistors and Capacitors

An RC circuit combines a resistor and a capacitor in the same circuit path. Because the capacitor charges and discharges over time rather than instantly, RC circuits produce voltages and currents that change with time. Understanding this time-dependent behavior is central to analyzing real DC circuits.

Time Constant Calculation for RC Circuits

The time constant ($\tau$) tells you how quickly a capacitor charges or discharges. It's calculated by multiplying the resistance and capacitance in the circuit:

$\tau = RC$

The result is in seconds. A larger $\tau$ means the capacitor charges and discharges more slowly; a smaller $\tau$ means it responds faster.

What does one time constant actually represent? After a time $t = \tau$, the capacitor voltage reaches about 63.2% of its final value during charging. The math behind that number:

$V_C = V_f(1 - e^{-1}) \approx 0.632\,V_f$

After five time constants ($5\tau$), the capacitor has reached 99.3% of its final value and is considered fully charged (or fully discharged, depending on the process).

Example calculations:

• A 100 kΩ resistor with a 10 μF capacitor: $\tau = (100 \times 10^3\,\Omega)(10 \times 10^{-6}\,\text{F}) = 1\text{ s}$
• A 220 kΩ resistor with a 47 nF capacitor: $\tau = (220 \times 10^3\,\Omega)(47 \times 10^{-9}\,\text{F}) \approx 10.3\text{ ms}$

Notice how changing either $R$ or $C$ directly scales the time constant. Doubling the resistance doubles $\tau$; halving the capacitance cuts $\tau$ in half.

Voltage Changes During Capacitor Charge/Discharge

The voltage across a capacitor doesn't jump instantly. It follows an exponential curve, and the two key equations describe what happens during charging and discharging.

Charging (capacitor starts at 0 V, source voltage is $V_f$):

$V_C = V_f(1 - e^{-t/\tau})$

The voltage rises quickly at first, then gradually levels off as it approaches $V_f$.

Discharging (capacitor starts at $V_0$, no source driving it):

$V_C = V_0\,e^{-t/\tau}$

The voltage drops quickly at first, then tapers toward zero.

Both processes are governed by the same exponential behavior, just in opposite directions.

How $\tau$ affects the rate of change:

• Smaller $\tau$ → faster charging/discharging (the curve is steep and settles quickly)
• Larger $\tau$ → slower charging/discharging (the curve is gradual)

Worked examples:

• Charging: If $\tau = 1$ s and $V_f = 5$ V, then after 1 s: $V_C = 5(1 - e^{-1}) \approx 3.16\text{ V}$ That's 63.2% of 5 V, exactly as the time constant predicts.

• Discharging: If $\tau = 10$ ms and $V_0 = 3.3$ V, then after 20 ms (which is $2\tau$): $V_C = 3.3\,e^{-20/10} = 3.3\,e^{-2} \approx 0.45\text{ V}$ The capacitor has lost most of its stored voltage after just two time constants.

RC Circuit Applications in Technology

RC circuits show up in many real devices because they provide controllable time delays and frequency-dependent filtering.

Timing and pulse applications:

• Camera flashes: The flash unit charges a capacitor through a resistor, then discharges it rapidly through the flash tube. The RC time constant controls how long the flash lasts.
• Pacemakers: An RC circuit generates regularly timed electrical pulses that stimulate heart contractions. The time constant sets the pacing rate.
• Defibrillators: A large capacitor is charged to high voltage through a resistor, then discharged through the patient's chest to restore normal heart rhythm. The RC values control both the charging time and the energy delivered.

Signal conditioning:

• Touchscreens: Your finger changes the local capacitance on the screen. The circuit detects this change in $C$ (which changes $\tau$) to determine where you touched.
• Switch debouncing: Mechanical switches produce rapid, noisy voltage spikes when pressed. An RC circuit smooths these out so the circuit registers a single clean transition.
• Power supply filtering: RC low-pass filters smooth out voltage ripple in DC power supplies, producing a steadier output.
• Audio equalizers: RC high-pass and low-pass filters selectively pass or block certain frequency ranges, letting you adjust bass and treble levels.

Transient and Steady-State Behavior in RC Circuits

Every RC circuit goes through two distinct phases after a voltage change occurs.

Transient response is the initial adjustment period. Voltage and current are changing rapidly as the capacitor charges or discharges. This phase lasts roughly $5\tau$. During this time, the exponential equations from above describe exactly what's happening.

Steady state is what the circuit settles into after the transient dies out. In a DC circuit at steady state, no current flows through the capacitor because it's fully charged. At that point, the capacitor behaves like an open circuit (a break in the wire). All the source voltage appears across the capacitor, and none drops across the resistor.

The role of the electric field: Energy in a capacitor is stored in the electric field between its plates. During charging, the field builds as charge accumulates. During discharging, the field weakens as charge flows back out. The strength of this electric field is directly proportional to the charge on the plates, which is why the voltage curves are exponential rather than linear.