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Capacitor Discharge with Exponential Functions

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Overview

This example demonstrates using mathematical functions with dimensional analysis by modeling an RC circuit's capacitor discharge. It showcases:

  • How exponential functions work with dimensionless quantities
  • Automatic SI prefix selection for readable output across wide value ranges
  • Dimensional analysis ensuring correct physics

Circuit Model

The example simulates an RC discharge circuit where a charged capacitor discharges through a resistor, following the exponential decay equation:

\[V(t) = V_0 \cdot e^{-\frac{t}{RC}}\]
V₀ R C
  • Parameters:
    • Capacitance: 0.47 μF
    • Initial voltage: 5.0 V
    • Resistance: 4.7 kΩ
    • Time constant: τ = RC = 2.209 ms
5 V 4 V 3 V 2 V 1 V 0 V 0 τ 2τ 3τ 4τ 5τ Time Voltage ≈1.84 V @ τ
  • Expected behavior:
    • At t = 0: V = 5.0 V (initial)
    • At t = τ: V ≈ 1.84 V (37% of initial)
    • At t = 5τ: V ≈ 0.034 V (< 1%)

Key Concepts

Dimensional Analysis in Exponential Functions

The library provides exp() that works with dimensionless quantities:

capacitor_time_curve.cpp
{
  using namespace mp_units;
  using namespace mp_units::si::unit_symbols;

  constexpr auto CC = isq::capacitance(0.47 * uF);
  constexpr auto V0 = isq::voltage(5.0 * V);
  constexpr auto RR = isq::resistance(4.7 * si::kilo<si::ohm>);

  std::cout.setf(std::ios_base::fixed, std::ios_base::floatfield);
  std::cout.precision(3);
  std::cout << "Capacitor time curve example for the following parameters:\n";
  std::cout << "Capacitance: " << CC << "\n";
  std::cout << "Initial Voltage: " << V0 << "\n";
  std::cout << "Resistance: " << RR << "\n";

  std::cout << "Time curve:\n";
  for (auto tt = 0 * ms; tt <= 50 * ms; ++tt) {
    const QuantityOf<isq::voltage> auto Vt = V0 * exp(-tt / (RR * CC));
    std::cout << "- at " << tt << " voltage is ";
    si::invoke_with_prefixed([](auto q) { std::cout << q; }, Vt, V);
    std::cout << "\n";
  }
}

How it works:

  1. RC time constant: RR * CC results in a time quantity (Ω·F = s)
  2. Dimensionless ratio: tt / (RR * CC) divides time by time → dimensionless
  3. Exponential function: exp() accepts the dimensionless quantity directly
  4. Dimension restoration: Multiplying by V0 gives the result being a voltage quantity

The library automatically verifies dimensional correctness at compile time, preventing common physics errors.

Automatic SI Prefix Selection

Rather than manually selecting output units, the example uses invoke_with_prefixed():

capacitor_time_curve.cpp
    const QuantityOf<isq::voltage> auto Vt = V0 * exp(-tt / (RR * CC));
    std::cout << "- at " << tt << " voltage is ";
    si::invoke_with_prefixed([](auto q) { std::cout << q; }, Vt, V);

This function automatically:

  • Calculates the appropriate SI prefix (V, mV, μV, nV, pV, etc.)
  • Ensures values stay in readable range (typically [1.0, 1000) for engineering mode)
  • Handles the output formatting in a single call

The lambda receives the quantity scaled to the optimal prefix, ready for output.

Sample Output

Capacitor time curve example for the following parameters:
Capacitance: 0.470 µF
Initial Voltage: 5.000 V
Resistance: 4.700 kΩ
Time curve:
- at 0 ms voltage is 5.000 V
- at 1 ms voltage is 3.180 V
- at 2 ms voltage is 2.022 V
- at 3 ms voltage is 1.286 V
- at 4 ms voltage is 817.638 mV    ← Transition to millivolts
- at 5 ms voltage is 519.946 mV
...
- at 15 ms voltage is 5.623 mV
- at 18 ms voltage is 1.446 mV
- at 19 ms voltage is 919.446 µV   ← Transition to microvolts
- at 20 ms voltage is 584.688 µV
...
- at 30 ms voltage is 6.323 µV
- at 34 ms voltage is 1.034 µV
- at 35 ms voltage is 657.491 nV   ← Transition to nanovolts
...
- at 45 ms voltage is 7.110 nV
- at 49 ms voltage is 1.163 nV
- at 50 ms voltage is 739.358 pV   ← Transition to picovolts

The output demonstrates seamless transitions between SI prefixes (V → mV → μV → nV → pV) as the voltage decays over 9 orders of magnitude, keeping values in a readable range throughout.

Why This Matters

Compile-Time Dimensional Safety

The library ensures dimensional correctness at compile time:

// ✓ Correct: time/time is dimensionless
QuantityOf<dimensionless> auto dimensionless_arg = -tt / (RR * CC);

// ✗ Won't compile: can't take exp of a dimensional quantity
// auto wrong = exp(-tt);  // Error: exp requires dimensionless!

This prevents common physics/engineering errors where equations are dimensionally invalid.

Practical Applications

This exponential decay pattern applies across many domains:

  • Electrical: RC/RL circuits, signal decay, filter response
  • Thermal: Newton's law of cooling, heat dissipation
  • Mechanical: Damped oscillations, friction
  • Chemistry: First-order reactions, radioactive decay
  • Biology: Drug metabolism, population decay

Output Readability

Instead of choosing fixed units (forcing users to read 0.000005 V or 5000000 nV), automatic prefix selection maintains readability across 9+ orders of magnitude.