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

See also: RL circuit, Current, Inductor, Nondimensionalization, Power supply, RC circuit, RLC circuit, Resistor, Tau