Chemistry - RL circuit

An RL circuit consists of a resistor R and an inductor L. An RL circuit has the time constant τ (tau), the time it takes the current in the circuit to reach $1 - e - 1$ of its equilibrium value, calculated with

τ = L / R

When a voltage is applied to the circuit, the current increases from 0 to an equilibrium value of $V / R$ . The current will have reached roughly 63% of its equilibrium value after τ, and will be at essentially full strength (99.3%) after about 5τ.

Derivation

For a series RL circuit attached to a voltage source solved with nondimensionalization,

$L \frac{dI}{dt} + IR = V(t) \Rightarrow \frac{d \chi}{d \tau} + \chi = F(\tau)$

with substitutions

$I = \chi x_c, t = \tau t_c, x_c = \frac{R}{L}, \ t_c = \frac{L}{R}, \ F = V.$

The first characteristic unit corresponds to the total current in the circuit. The second characterstic unit corresponds to the time constant for the system.

RL circuit

Derivation

See also