Charging Discharging Capacitor Equations: Formulas, Worked Examples & Sizing

The fundamental behavior of any resistor-capacitor (RC) network is governed by the time constant, tau (τ = R × C). Whether you are designing a soft-start delay for a power supply enable pin or calculating the bulk capacitance needed to ride through a 20ms AC brownout, the charging discharging capacitor equations dictate exactly how voltage and current evolve over time. Below is the complete mathematical framework, rearranged for practical bench use, followed by worked examples and a chemistry selection guide.

The Core Charging Discharging Capacitor Equations

These equations assume a simple series RC circuit connected to a DC voltage step. The charging equation models the voltage across the capacitor as it accumulates charge, while the discharging equation models the voltage decay when the source is removed and the capacitor drains through the resistor.

Symbol	Parameter	Standard Unit	Description
V(t)	Instantaneous Voltage	Volts (V)	Voltage across the capacitor at time t.
Vs	Source Voltage	Volts (V)	The DC supply voltage charging the capacitor.
V0	Initial Voltage	Volts (V)	The starting voltage on the capacitor before discharge begins.
R	Resistance	Ohms (Ω)	Total series resistance (include ESR and source impedance in real circuits).
C	Capacitance	Farads (F)	Capacitor value. Must be converted from µF to F for calculations.
t	Time	Seconds (s)	Elapsed time since the voltage step or discharge initiation.
e	Euler's Number	Dimensionless	Mathematical constant ≈ 2.71828.
τ (tau)	Time Constant	Seconds (s)	τ = R × C. Time to reach 63.2% of final voltage (charge) or drop to 36.8% (discharge).

Charging Equation:
V(t) = V_s * (1 - e^(-t / RC))

Discharging Equation:
V(t) = V_0 * e^(-t / RC)

Rearranged Forms: Solving for Time, Resistance, and Capacitance

On the bench, you rarely need to find V(t); you already know the voltage threshold you need to hit (like a microcontroller's brownout detection or a MOSFET gate threshold). You need to solve for t, R, or C. Here are the algebraically rearranged forms using the natural logarithm (ln).

For Charging Circuits:

• Solve for t: t = -R * C * ln(1 - (V(t) / V_s))
• Solve for R: R = -t / (C * ln(1 - (V(t) / V_s)))
• Solve for C: C = -t / (R * ln(1 - (V(t) / V_s)))

For Discharging Circuits:

• Solve for t: t = -R * C * ln(V(t) / V_0)
• Solve for R: R = -t / (C * ln(V(t) / V_0))
• Solve for C: C = -t / (R * ln(V(t) / V_0))

Assumptions, Unit Traps, and Realistic Magnitudes

These equations are derived from ideal first-order differential equations. If your physical circuit violates the underlying assumptions, your math will fail. According to foundational circuit theory documented by sources like All About Circuits, the ideal RC model assumes a pure resistance and pure capacitance.

Unit Mistakes That Break the Math:
The most common bench error is plugging microfarads (µF) or milliseconds (ms) directly into the formula without converting to base SI units (Farads and Seconds). If you type 100 into the C variable instead of 0.0001 (for 100µF), your calculated time will be off by a factor of one million. Always convert to Farads, Ohms, and Seconds before calculating, then convert back.

Realistic Answer Magnitudes:
For timing circuits (delays, debouncing), τ typically ranges from 1ms to 5 seconds, utilizing resistors in the 10kΩ to 1MΩ range and capacitors from 10nF to 10µF. For power supply hold-up (bulk filtering), τ is usually 10ms to 100ms, utilizing low resistances (milliohms to single-digit ohms representing the load) and massive capacitances (1,000µF to 10,000µF).

Worked Example 1: Sizing a Soft-Start RC Delay

Scenario: You are designing an enable pin delay for a buck converter. The IC's EN pin threshold is exactly 2.5V. Your system logic rail (V_s) is 5.0V. You want the converter to turn on exactly 100ms after the 5V rail stabilizes. You have chosen a 100kΩ pull-up resistor to minimize quiescent current. What capacitor value do you need?

Knowns:

• V(t) = 2.5V
• V_s = 5.0V
• t = 0.100 seconds
• R = 100,000 Ω

Step-by-Step Solution:

1. Select the rearranged charging equation for C:
   C = -t / (R * ln(1 - (V(t) / V_s)))
2. Substitute the known values:
   C = -0.100 / (100,000 * ln(1 - (2.5 / 5.0)))
3. Simplify the inner fraction:
   2.5 / 5.0 = 0.5
4. Calculate the natural log:
   ln(1 - 0.5) = ln(0.5) ≈ -0.693147
5. Multiply R by the ln result:
   100,000 * -0.693147 = -69,314.7
6. Divide -t by the result:
   C = -0.100 / -69,314.7 ≈ 0.0000014427 Farads
7. Convert to microfarads:
   C = 1.44 µF

Concrete Pick: The closest standard E12 capacitor value is 1.5 µF. To guarantee the delay is at least 100ms (accounting for typical +20% capacitor tolerance), select a 2.2 µF X7R MLCC.

Worked Example 2: Calculating Power Supply Hold-Up Time

Scenario: A 24V DC motor controller requires a minimum of 18V to maintain logic state during a power dropout. The total steady-state load of the controller draws 200mA, which we can model as a constant equivalent resistance. You need the system to ride through a 50ms brownout. What bulk capacitor is required?

Knowns:

• V_0 = 24V (Initial charged voltage)
• V(t) = 18V (Minimum operational threshold)
• t = 0.050 seconds
• R = V / I = 24V / 0.2A = 120 Ω

Step-by-Step Solution:

1. Select the rearranged discharging equation for C:
   C = -t / (R * ln(V(t) / V_0))
2. Substitute the known values:
   C = -0.050 / (120 * ln(18 / 24))
3. Simplify the inner fraction:
   18 / 24 = 0.75
4. Calculate the natural log:
   ln(0.75) ≈ -0.28768
5. Multiply R by the ln result:
   120 * -0.28768 = -34.5216
6. Divide -t by the result:
   C = -0.050 / -34.5216 ≈ 0.001448 Farads
7. Convert to microfarads:
   C = 1448 µF

Concrete Pick: A standard 1500 µF or 2200 µF Aluminum Electrolytic capacitor rated for at least 35V (24V nominal + 20% derating margin). As noted in Electronics Tutorials, always account for the capacitor's ESR, which will cause an immediate voltage step-drop when the discharge current initiates, effectively reducing your starting V_0.

Decision Tree: Picking the Right Capacitor Chemistry for RC Circuits

Calculating the value is only half the battle. If you use the wrong dielectric, parasitic effects will invalidate the math. Use this decision path to select the exact component type.

Application Scenario	Required Characteristic	Chemistry / Dielectric	Concrete Part Recommendation
Precision Timing (t < 100ms, tight tolerance, e.g., EN pin delays, oscillator networks)	Zero DC bias derating, low leakage, tight temperature coefficient.	C0G / NP0 MLCC	Kemet C0805C105J5GACTU (1.0µF, 50V, C0G). Do not use X7R here; X7R loses up to 50% capacitance at rated voltage.
Long-Duration Timing (t > 1 second, e.g., reset delays, watchdog timers)	High capacitance in small footprint, low leakage current.	Film (Polyester/Polypropylene) or Tantalum	KEMET T491 Series Tantalum (e.g., T491C476K016AT for 47µF). Warning: Add a series resistor to limit surge current if hot-plugging.
Bulk Hold-Up / Decoupling (t > 10ms, high current discharge, e.g., brownout ride-through)	Massive capacitance, high ripple current handling, acceptable high ESR.	Aluminum Electrolytic	Nichicon UWT1V222MNL1GS (2200µF, 35V). Check datasheet for leakage current; it can be several milliamps, which ruins high-impedance timing circuits.
High-Temperature / Automotive (Under-hood, >125°C ambient)	Stable capacitance at extreme heat, no liquid electrolyte to dry out.	MLCC (X8R) or Solid Polymer Aluminum	Panasonic OS-CON Solid Polymer (e.g., 20SVF100M). Offers extremely low ESR compared to standard wet electrolytics.

Final Rule of Thumb: If your calculated RC time constant relies on a resistor greater than 100kΩ, you must use C0G/NP0 MLCC or Film capacitors. The leakage current of standard X7R ceramics and electrolytics (often modeled as a parallel high-value resistor) will bleed off the charge, causing the capacitor voltage to stall before it ever reaches your calculated V(t) threshold.