In the context of electrical circuits, the forced response of an RL circuit refers to the behavior of the circuit in the presence of a time-varying external voltage or current source. An RL circuit consists of a resistor (R) and an inductor (L) connected in series. When an external voltage or current source is applied to the circuit, it causes a current to flow through the inductor and the resistor.
The forced response of the RL circuit is the steady-state behavior of the current through the inductor and the voltage across the components after the transient effects have died out. Transients are the temporary behaviors that occur when the circuit is first energized, and they eventually decay over time.
In the steady-state, the current flowing through the inductor and the voltage across the inductor and resistor will be sinusoidal if the external source is sinusoidal. The forced response is determined by the frequency and amplitude of the external source and the circuit's characteristics (resistance and inductance).
Mathematically, the forced response of an RL circuit to an external sinusoidal voltage source can be expressed using phasor notation, where the current phasor (I) and the voltage phasor (V) are related by the impedance of the inductor and resistor in series. For a sinusoidal voltage source V(t) = Vm * cos(ωt), the forced response of the current is given by:
I(t) = Im * cos(ωt + φ)
Where:
Im is the amplitude of the current,
ω is the angular frequency of the external source (2π times the frequency),
φ is the phase angle difference between the voltage and current waveforms.
The phase angle φ is determined by the relative values of the inductive reactance (XL = ωL) and the resistance (R) in the circuit and can be calculated using trigonometric functions.
Understanding the forced response is crucial in the analysis and design of RL circuits, especially when they are used in filters, power supplies, or other applications where the steady-state behavior is essential.