To calculate e (Euler's number, ~2.71828) on a scientific calculator, press [SHIFT] then [ln] on Casio models, or [2nd] then [LN] on Texas Instruments models. In electrical engineering, e is the base of the natural logarithm and is primarily used to model exponential decay and growth. The most common application is the RC transient discharge formula: V(t) = V0 · e-t/RC. This guide breaks down how to input e, derive the formula, track units, and avoid the syntax errors that plague bench calculations.
The Core Formula: Exponential Decay in RC Circuits
When a charged capacitor is disconnected from a DC source and allowed to discharge through a resistor, the voltage does not drop linearly. It decays exponentially. The rate of this decay is governed by Euler's number (e), which naturally describes continuous compounding processes in physics.
The standard discharge equation is:
V(t) = V0 · e-t / (R · C)
| Symbol | Name | Standard Unit | Description & Assumptions |
|---|---|---|---|
| V(t) | Voltage at time t | Volts (V) | The remaining voltage across the capacitor plates at a specific moment. |
| V0 | Initial Voltage | Volts (V) | The voltage on the capacitor at t = 0 (the exact moment discharge begins). |
| e | Euler's Number | Dimensionless | Mathematical constant (~2.71828). Represents the base of the natural logarithm. |
| t | Time elapsed | Seconds (s) | Duration since the discharge path was completed. |
| R | Resistance | Ohms (Ω) | Total series resistance. Assumes ideal resistor (no parasitic inductance). |
| C | Capacitance | Farads (F) | Capacitor value. Assumes ideal capacitor (no equivalent series resistance/ESR). |
When this formula applies: This equation assumes a simple first-order RC network with a constant resistance and an ideal capacitor. It applies to DC transient analysis where the power source has been removed or switched to 0V. For AC circuits or circuits with inductors (RLC), the math shifts to differential equations involving complex impedance and damping ratios.
Rearranged Forms: Solving for Every Variable
On the bench, you rarely just solve for V(t). More often, you need to size a bleeder resistor to meet a safety discharge time, or calculate how long a backup capacitor will keep a microcontroller alive during a brownout. To do this, we isolate the variables using the natural logarithm (ln), which is the inverse function of ex.
- Solve for Initial Voltage (V0):
V0 = V(t) · et / (R · C) - Solve for Time (t):
t = -R · C · ln( V(t) / V0 ) - Solve for Resistance (R):
R = -t / [ C · ln( V(t) / V0 ) ] - Solve for Capacitance (C):
C = -t / [ R · ln( V(t) / V0 ) ]
Note: The product of R · C is known as the time constant, denoted by the Greek letter tau (τ). One time constant (τ = 1) represents the time it takes for the voltage to decay to ~36.8% of its initial value (since e-1 ≈ 0.3678).
Worked Examples with Unit Tracking
The most common reason engineers and hobbyists get wrong answers on their calculators is unit mismatch. The formula strictly requires base SI units: Ohms, Farads, and Seconds. Let's walk through two real-world scenarios.
Problem 1: Finding Voltage at a Specific Time
Scenario: You have a 24V DC relay circuit. When power is cut, a 470 μF capacitor discharges through a 10 kΩ bleeder resistor. What is the voltage across the capacitor 2 seconds after power-off?
- Convert to base SI units:
R = 10 kΩ = 10,000 Ω
C = 470 μF = 0.000470 F
t = 2 s
V0 = 24 V - Calculate the time constant (τ):
τ = R · C = 10,000 · 0.000470 = 4.7 seconds - Calculate the exponent:
-t / τ = -2 / 4.7 = -0.42553 - Calculate the decay factor (ex):
On your calculator, typee(via SHIFT+ln), then^, then-0.42553.
e-0.42553 = 0.6534 - Multiply by initial voltage:
V(t) = 24 · 0.6534 = 15.68 V
Result: After 2 seconds, the capacitor still holds 15.68V.
Problem 2: Finding Time to a Microcontroller Brownout Threshold
Scenario: An ESP32 dev board is powered by a 5V supercapacitor backup. The ESP32 brownout detector triggers a reset at 3.3V. The supercap is 100 μF and the equivalent load resistance is 4.7 kΩ. How many milliseconds do you have to save data to flash memory before the brownout hits?
- Convert to base SI units:
R = 4,700 Ω
C = 0.0001 F
V0 = 5 V
V(t) = 3.3 V - Calculate the time constant (τ):
τ = 4,700 · 0.0001 = 0.47 seconds - Apply the rearranged time formula:
t = -τ · ln( V(t) / V0 )
t = -0.47 · ln( 3.3 / 5 ) - Calculate the natural log:
ln(0.66) = -0.4155 - Final multiplication:
t = -0.47 · -0.4155 = 0.195 seconds (or 195 ms)
Result: You have exactly 195 milliseconds to execute your EEPROM write routine before the ESP32 resets. For reference, the Espressif ESP-IDF documentation notes that deep sleep entry can take several milliseconds, making this margin critical.
Common Unit Mistakes and Realistic Magnitudes
If your calculator output looks completely wrong, you likely fell victim to one of these bench-calculation traps:
| The Mistake | What Happens | The Fix |
|---|---|---|
| Leaving Capacitance in μF | Your time constant (τ) will be off by a factor of 1,000,000. The decay will appear instantaneous. | Always multiply μF by 10-6 (e.g., 100 μF = 0.0001 F). |
| Mixing ms and Seconds | If t is in milliseconds but τ is in seconds, your exponent will be 1000x too large. | Convert all time variables to base Seconds before calculating. |
| Dropping the Negative Sign | e+t/RC yields a number greater than 1, implying the capacitor is magically generating voltage. | Decay exponents must be negative. If V(t) > V0, check your sign. |
Realistic Answer Magnitudes: The term e-t/RC is a dimensionless scaling factor. For any positive time t, this factor must be a decimal between 0 and 1. If your calculator displays a scaling factor of 2.4, you dropped the negative sign in the exponent. Conversely, when using the rearranged formula to solve for time, the natural log of a fraction (V(t)/V0) will always yield a negative number. Multiplying that negative log by the negative sign in the formula yields a positive time. If your calculated time is negative, you swapped V(t) and V0 in the logarithm.
Frequently Asked Questions
How do I type 'e' on a Casio fx-991EX or TI-84 calculator?
On a Casio fx-991EX (or similar ClassWiz models), press the yellow [SHIFT] button, then press the [ln] button (which has a small 'e' printed above it). This inputs Euler's number. On a TI-84 Plus, press the green [2nd] button, then press the [LN] button (which has 'e' printed above it), or press [2nd] then the [e^x] dedicated key depending on your exact OS version. Always follow it with the exponentiation key ^ and wrap negative exponents in parentheses: e^(-2/4.7).
What is the difference between 'e' and 'E' on my calculator display?
This is a massive source of confusion. Lowercase e (accessed via the SHIFT+ln method) is Euler's number (2.71828...). Uppercase E (often displayed automatically by the calculator) denotes scientific notation (base-10 exponent). If your calculator screen reads 1.5E3, it means 1.5 × 103 (or 1500). It does not mean 1.5 · e3. When manually entering scientific notation, use the dedicated [EXP] or [EE] button, not the Euler e button.
Why does my calculator give a domain error when calculating e-t/RC?
A 'Domain Error' or 'Math Error' usually happens when you are using the rearranged formula to solve for time or resistance, specifically during the natural log (ln) step. The natural logarithm is only defined for positive numbers. If you accidentally input ln(V0 / V(t)) instead of ln(V(t) / V0) while forgetting to adjust the negative signs, or if you input a target voltage that is higher than the initial voltage (e.g., trying to find when a 5V cap discharges to 12V), the calculator will throw a domain error because the physics are impossible.
How many time constants until a capacitor is considered fully discharged?
In electrical engineering practice, we use the 5τ (five time constant) rule. After 5 time constants, the exponent becomes -5. Calculating e-5 yields approximately 0.0067, meaning the capacitor has discharged to 0.67% of its initial voltage. For a 24V circuit, 5τ leaves only 0.16V, which is safely considered 0V for almost all practical logic and power applications. According to Georgia State University's HyperPhysics reference, this 1% threshold is the standard benchmark for transient settling times in DC networks.
