Finding e on a scientific calculator is the first step to moving beyond basic Ohm's law and into time-domain circuit analysis. Euler's number ($e \approx 2.71828$) and its inverse, the natural logarithm ($\ln$), are the mathematical engines behind capacitor charging curves, inductor current ramps, and diode voltage drops. If you are designing a soft-start circuit, a 555 timer delay, or an RC snubber, you need to know exactly which buttons to press and how to track your units so you don't accidentally order a 10,000µF capacitor when you needed a 10nF.
The Core Exponential Formula in Electronics
The most common application of the exponential function on the bench is the RC charging curve. When a DC voltage is applied to a series resistor-capacitor network, the voltage across the capacitor does not rise linearly; it follows an exponential asymptote.
SHIFT or 2nd followed by the ln button. The natural log ($\ln$) is the primary ln button. You will need both for RC calculations.
The standard charging formula is:
$V(t) = V_s(1 - e^{-\frac{t}{R \cdot C}})$
| Symbol | Definition | Standard SI Unit | Realistic Bench Magnitude |
|---|---|---|---|
| $V(t)$ | Voltage across the capacitor at time $t$ | Volts (V) | 1.8V to 48V |
| $V_s$ | Source (supply) voltage | Volts (V) | 3.3V, 5V, 12V, 24V |
| $e$ | Euler's number (base of natural log) | Dimensionless | $\approx 2.71828$ |
| $t$ | Elapsed time since voltage was applied | Seconds (s) | 1ms to 10s |
| $R$ | Series resistance | Ohms ($\Omega$) | $1k\Omega$ to $10M\Omega$ |
| $C$ | Capacitance | Farads (F) | 1nF to 1000µF |
| $\tau$ | Time constant ($R \times C$) | Seconds (s) | 1ms to 5s |
Assumptions & Limits: This formula assumes an ideal capacitor with zero Equivalent Series Resistance (ESR) and zero leakage current, a perfectly linear resistor, and a step-function input (the voltage goes from 0V to $V_s$ instantaneously). In reality, electrolytic capacitors leak, and power supplies have finite current limits. For timing circuits requiring >5% accuracy, use C0G/NP0 ceramic or film capacitors instead of aluminum electrolytics.
Rearranged Forms for Bench Work
You rarely need to solve for $V(t)$ on the bench. Usually, you know the target threshold voltage (like a MOSFET gate threshold or a logic IC trip point) and the desired delay time, and you need to find $R$ or $C$. Here are the algebraically rearranged forms. Note that the $e^x$ button is replaced by the $\ln$ (natural log) button when the variable is in the exponent.
- Solving for time ($t$): $t = -R \cdot C \cdot \ln\left(1 - \frac{V(t)}{V_s}\right)$
- Solving for resistance ($R$): $R = \frac{-t}{C \cdot \ln\left(1 - \frac{V(t)}{V_s}\right)}$
- Solving for capacitance ($C$): $C = \frac{-t}{R \cdot \ln\left(1 - \frac{V(t)}{V_s}\right)}$
- Solving for source voltage ($V_s$): $V_s = \frac{V(t)}{1 - e^{-\frac{t}{R \cdot C}}}$
Unit Mistakes That Break the Math
The number one reason a calculated RC delay fails on the breadboard is a unit conversion error. The exponential function $e^x$ is dimensionless; it demands that the exponent $(-t/RC)$ be a pure number. If your units don't cancel out to seconds, the math breaks.
10 for a 10µF capacitor, the calculator assumes 10 Farads (a massive supercapacitor). You must enter 10 \times 10^{-6} or 0.00001. Similarly, if your time is 500ms, you must enter 0.5 seconds. Always convert to base SI units (Volts, Ohms, Farads, Seconds) before hitting the equals button.
Worked Problems with Unit Tracking
Problem 1: Finding Voltage at a Specific Time
Scenario: You have a 12V DC source charging a 10µF capacitor through a 100kΩ resistor. What is the capacitor voltage exactly 0.5 seconds after the circuit is energized?
- Convert to SI units: $V_s = 12V$, $R = 100,000\Omega$, $C = 0.00001F$, $t = 0.5s$.
- Calculate the exponent denominator ($RC$): $100,000 \times 0.00001 = 1.0$ second.
- Calculate the full exponent: $-0.5 / 1.0 = -0.5$.
- Apply $e^x$: On your calculator, press
SHIFT+ln, then-0.5. Result: $e^{-0.5} \approx 0.60653$. - Subtract from 1: $1 - 0.60653 = 0.39347$.
- Multiply by $V_s$: $12 \times 0.39347 = \mathbf{4.72V}$.
Problem 2: Sizing a Resistor for a Target Delay
Scenario: You need a 5V logic line to cross the 3.3V threshold exactly 500ms after power-on. You have a 10µF capacitor in your bin. What resistor do you need?
- Identify knowns (SI units): $V(t) = 3.3V$, $V_s = 5V$, $t = 0.5s$, $C = 0.00001F$.
- Select the rearranged formula: $R = \frac{-t}{C \cdot \ln(1 - \frac{V(t)}{V_s})}$.
- Calculate the fraction: $3.3 / 5 = 0.66$.
- Subtract from 1: $1 - 0.66 = 0.34$.
- Apply natural log ($\ln$): Press
lnthen0.34. Result: $\ln(0.34) \approx -1.0788$. - Multiply by C: $0.00001 \times -1.0788 = -0.000010788$.
- Divide $-t$ by the result: $-0.5 / -0.000010788 = \mathbf{46,347\Omega}$.
Result: You need a 46.3kΩ resistor. The nearest standard E24 value is 47kΩ, which will yield a slightly longer delay (~507ms).
Decision Path: Sizing an RC Delay for a MOSFET Gate
When designing a soft-start or delayed turn-on for a power MOSFET (like the ubiquitous IRFZ44N), you use an RC network on the gate. The MOSFET begins to conduct when the gate-source voltage ($V_{gs}$) reaches its threshold voltage ($V_{gs(th)}$). According to the Vishay IRFZ44N datasheet, $V_{gs(th)}$ can range from 2.0V to 4.0V. To ensure a reliable turn-on delay, we design for the worst-case upper limit (4.0V) to guarantee the FET stays off until the desired time, but we'll target a nominal 3.0V for a standard logic-level trigger in this example.
| Design Condition | If True / Action | Resulting Value |
|---|---|---|
| Target Supply ($V_s$) | System runs on standard 12V DC. | $V_s = 12V$ |
| Target Delay ($t$) | Need a 2.0-second soft-start delay. | $t = 2.0s$ |
| Target Threshold ($V(t)$) | Logic gate/MOSFET triggers reliably at 3.0V. | $V(t) = 3.0V$ |
| Select Capacitor ($C$) | Pick a standard, physically small electrolytic. 100µF is common and cheap. | $C = 0.0001F$ |
| Calculate Resistor ($R$) | $R = -2.0 / [0.0001 \times \ln(1 - 3/12)]$ $R = -2.0 / [0.0001 \times \ln(0.75)]$ $R = -2.0 / [0.0001 \times -0.28768]$ | $R = 69,521\Omega$ |
| Select Standard E24 Resistor | Nearest E24 values are 68kΩ and 75kΩ. Pick 68kΩ for slightly faster turn-on (1.95s) to ensure it crosses threshold before the 2s mark. | $R = 68,000\Omega$ |
The Final Concrete BOM Pick
Do not leave this to generic bin parts if timing matters. To build this exact 2-second delay circuit with minimal variance, order the following specific components:
- Capacitor: Panasonic EEU-FR1E101 (100µF, 25V, FR Series Low-ESR Aluminum Electrolytic). The low ESR ensures the initial charge curve matches the ideal math closer than a standard cheap capacitor.
- Resistor: Vishay CFR-25JR-52-68K (68kΩ, 1/4W, 5% Carbon Film). If you require <2% timing accuracy, swap this for a Yageo MFR-25FBF52-68K (1% Metal Film).
By mastering the $e^x$ and $\ln$ functions on your calculator and rigorously tracking your SI units, you can translate theoretical time constants into physical, reliable bench prototypes on the first attempt.
