Purpose

Introducing Time Into Circuits

Real circuits don't change instantly. Capacitors charge, inductors resist change. The time constant tau = RC governs how fast. This is the foundation of ALL dynamic systems -- the bridge from static circuits to control theory.

The Physics

RC Charging

V(t) = V_final * (1 - e^{-t/tau})

tau = R * C (time constant in seconds)

At t = 1*tau: 63.2% charged. At t = 5*tau: 99.3% (settled).

Connection to Control Systems

G(s) = 1/(tau*s + 1) -- this IS a first-order system.

Corbin's fuel metering valve (Chapter 4): tau_FMV = 0.03s -- an RC-like response.

Key Insight: Every first-order control system behaves like an RC circuit. Master the time constant here, and you understand 60% of control engineering.

Historical

Oliver Heaviside (1850-1925)
Self-taught, invented the math for analyzing circuits with time (Laplace transforms, impedance). The first transatlantic telegraph cable (1858) FAILED because of RC time constants -- too long, too much capacitance, signals smeared.

References

Alexander & Sadiku, Ch 7-8 | Feynman Lectures Vol II, Ch 22-23 | AllAboutCircuits.com/textbook/direct-current/chpt-16/