First, a quick terminology correction that will save you embarrassment in a lab or on a jobsite: voltage does not go through a capacitor; current goes through, while voltage is measured across the terminals. However, because 'voltage through a capacitor formula' is a common search phrase, we will address the exact math you need to calculate the potential difference across a capacitor's plates as it charges or discharges in a DC circuit.
The direct answer for the voltage across a charging capacitor in a series RC (Resistor-Capacitor) circuit at any given time is: V(t) = Vs(1 - e-(t/RC)). For a discharging capacitor, the formula is V(t) = V0(e-(t/RC)). Below, we break down every variable, rearrange the math for practical bench work, and solve real-world problems with strict unit tracking.
The Core RC Charging Formula and Symbol Definitions
The fundamental equation governing the transient response of a capacitor charging through a resistor from a constant DC source is derived from Kirchhoff's Voltage Law and the defining integral of capacitance (I = C * dV/dt). This formula applies strictly to ideal lumped-element circuits with a constant DC voltage source, a single series resistor, and a single capacitor.
| Symbol | Parameter | Standard Unit | Typical Bench Magnitude |
|---|---|---|---|
| V(t) | Voltage across the capacitor at time t | Volts (V) | 3.3V, 5V, 12V, 24V |
| Vs | Source voltage (Supply voltage) | Volts (V) | 5V to 48V DC |
| e | Euler's number (mathematical constant) | Dimensionless | ~2.71828 |
| t | Time elapsed since charging began | Seconds (s) | Milliseconds to seconds |
| R | Series resistance | Ohms (Ω) | 1kΩ to 10MΩ |
| C | Capacitance | Farads (F) | 100pF to 10,000μF |
| τ (tau) | Time constant (R × C) | Seconds (s) | μs to minutes |
Rearranged Forms: Solving for R, C, and t
On the workbench, you rarely need to find V(t) blindly. Usually, you have a target voltage threshold (like the 2/3 Vcc trigger point of a NE555 timer) and need to select a resistor or capacitor, or calculate the exact delay time. Here are the algebraically rearranged forms of the charging equation, solved for each variable:
- Solve for Time (t):
t = -R × C × ln(1 - (V(t) / Vs))
Use when: Designing a delay circuit and you need to know how long until a microcontroller GPIO or comparator trips. - Solve for Resistance (R):
R = -t / (C × ln(1 - (V(t) / Vs)))
Use when: You have a fixed capacitor in your BOM and need to calculate the current-limiting or timing resistor. - Solve for Capacitance (C):
C = -t / (R × ln(1 - (V(t) / Vs)))
Use when: Sizing a soft-start capacitor for a power supply or a debounce capacitor for a mechanical switch. - Solve for Target Voltage (V(t)):
V(t) = Vs × (1 - e-(t / (R × C)))
Use when: Debugging an existing board to verify if the measured voltage at a specific millisecond matches the design.
Worked Examples with Strict Unit Tracking
The most common reason hobbyists and students get 'impossible' answers (like a capacitor reaching 500V on a 12V battery) is unit mismanagement. Let's solve two problems with explicit unit tracking. The Golden Shortcut: If you multiply kilo-ohms (kΩ) by microfarads (μF), the result is directly in milliseconds (ms). If you multiply Mega-ohms (MΩ) by microfarads (μF), the result is in seconds (s). We will use base SI units (Ohms, Farads, Seconds) below to prove the math, but keep the shortcut in mind.
Problem 1: Finding Voltage at a Specific Time
Scenario: A 12V DC source charges a 100μF capacitor through a 47kΩ resistor. What is the voltage across the capacitor exactly 2.0 seconds after the switch closes?
- Convert to base SI units:
R = 47,000 Ω
C = 100 × 10-6 F = 0.0001 F
Vs = 12 V, t = 2.0 s - Calculate the time constant (τ = RC):
τ = 47,000 × 0.0001 = 4.7 seconds. - Apply the charging formula:
V(2) = 12 × (1 - e-(2.0 / 4.7)) - Compute the exponent:
-2.0 / 4.7 = -0.42553
e-0.42553 ≈ 0.6534 - Final Calculation:
V(2) = 12 × (1 - 0.6534) = 12 × 0.3466 = 4.16 Volts
Sanity Check: At 1 time constant (4.7s), the capacitor reaches ~63.2% of 12V (7.58V). At 2.0s, it hasn't reached one time constant yet, so 4.16V is a highly realistic magnitude.
Problem 2: Finding Time to Reach a Threshold Voltage
Scenario: You are designing a soft-start circuit. You need a 5V logic enable pin to trigger when an RC network reaches 3.3V. Your supply is 5V, R is 10kΩ, and C is 470μF. How long does the delay take?
- Convert to base SI units:
R = 10,000 Ω
C = 470 × 10-6 F = 0.00047 F
Vs = 5 V, V(t) = 3.3 V - Calculate τ:
τ = 10,000 × 0.00047 = 4.7 seconds. - Apply the rearranged time formula:
t = -4.7 × ln(1 - (3.3 / 5.0)) - Compute the logarithm argument:
3.3 / 5.0 = 0.66
1 - 0.66 = 0.34 - Final Calculation:
ln(0.34) ≈ -1.0788
t = -4.7 × -1.0788 = 5.07 seconds
Assumptions, Limits, and Real-World Parasitics
The formulas above assume an ideal capacitor and an ideal voltage source. In physical reality, every component has parasitics that alter the curve, especially as you push toward the edges of the component's specifications. According to standard circuit theory models referenced by Electronics Tutorials, the lumped element model holds true until the physical dimensions of the circuit approach the wavelength of the signal, which is never an issue for DC timing circuits.
However, you must account for these three non-ideal behaviors when your calculated math doesn't match your oscilloscope trace:
- Equivalent Series Resistance (ESR): Real capacitors have internal resistance. An aluminum electrolytic capacitor might have an ESR of 0.5Ω to 2Ω. This adds to your external R, slightly altering the time constant and creating an immediate voltage step (V = I × ESR) the millisecond the switch closes.
- Dielectric Absorption: If you rapidly discharge a large electrolytic capacitor and leave it open-circuited, the voltage will 'creep' back up. This is caused by the slow relaxation of polarized molecules in the dielectric. It ruins precision sample-and-hold circuits but is negligible in standard RC timing.
- Tolerance and Drift: A standard Y5V ceramic capacitor can lose up to 80% of its capacitance when DC bias voltage is applied. If your formula calls for 10μF, a 10μF Y5V part at 5V might actually act like a 2μF part, drastically shortening your calculated time delay.
Component Selection Decision Tree
Calculating the math is only half the job; picking the right physical capacitor to execute that math is where designs succeed or fail. Use this decision path to select the exact component type and part number for your RC network.
| Design Requirement | Capacitance Range | Dielectric / Type | Concrete Part Recommendation |
|---|---|---|---|
| Precision Timing (<1% drift, tight tolerances, stable over temp) | 10pF to 1μF | C0G / NP0 Ceramic | KEMET C0805C104J5GACTU (0.1μF, 50V, 5% tol, C0G) |
| Low-Frequency Timing (Delays of 1s to 10s, moderate precision) | 10μF to 100μF | Solid Polymer / Tantalum | KEMET T491D476M016AT (47μF, 16V, 20% tol, MnO2 Tantalum) |
| Bulk Energy / Smoothing (Power supply filtering, NOT for precision timing) | 100μF to 10,000μF | Aluminum Electrolytic | Panasonic EEU-FR1E101 (100μF, 25V, Low ESR, FR Series) |
| High-Voltage Snubber (Across MOSFETs/Relays, >50V DC) | 1nF to 1μF | Polypropylene Film | WIMA MKP1O121004B00KSSD (0.01μF, 1000V DC, 10% tol, PP Film) |
The Final Verdict: If you are building a 555-timer astable multibrator or a microcontroller reset delay and your calculated capacitance is under 1μF, always default to a C0G/NP0 ceramic capacitor (like the KEMET C0805 series listed above). Avoid X7R or Y5V dielectrics for timing networks, as their capacitance drops non-linearly with applied DC voltage, rendering your perfectly calculated 'voltage across the capacitor' formula entirely useless in the real world.
For deeper reading on transient response and RC network behaviors, the All About Circuits textbook chapter on DC RC time constants provides excellent foundational theory, while application notes from silicon vendors like Texas Instruments detail how parasitic inductance affects high-speed switching edges.
