To calculate the forced response of an RL (Resistor-Inductor) circuit, you'll need to apply an external sinusoidal voltage or current source to the circuit. The forced response is the steady-state response of the circuit after transient effects have died out. In an RL circuit, the transient response occurs when the current through the inductor is changing, and it eventually settles to a steady-state value due to the presence of the resistor.
The differential equation governing the behavior of an RL circuit is:
L di/dt + R i = V(t)
where:
L is the inductance (measured in henries, H),
R is the resistance (measured in ohms, Ω),
i is the current through the circuit (measured in amperes, A),
t is time (measured in seconds, s), and
V(t) is the time-varying external voltage source (measured in volts, V).
To calculate the forced response, follow these steps:
Determine the sinusoidal voltage or current source: Express the external voltage or current source as a function of time in sinusoidal form. For example, a sinusoidal voltage source can be represented as V(t) = V_m sin(ω t), where V_m is the peak voltage and ω is the angular frequency (2π times the frequency in hertz).
Assume a sinusoidal current response: Assume that the current through the inductor also follows a sinusoidal form with the same frequency as the external source. This is the steady-state response we are interested in, and we'll represent it as I(t) = I_m sin(ω t + φ), where I_m is the peak current amplitude, and φ is the phase angle.
Substitute the assumed current response into the differential equation: Replace i(t) in the differential equation with I(t) and solve for I_m and φ.
Solve for I_m and φ: By solving the differential equation with the assumed sinusoidal current response, you will get expressions for I_m and φ in terms of circuit parameters (L, R) and the characteristics of the external voltage or current source (V_m, ω).
Calculate the forced response: Once you have the values of I_m and φ, you can use them to write the expression for the forced response current through the inductor, I(t) = I_m sin(ω t + φ).
Remember that the forced response only represents the steady-state behavior of the circuit. To obtain the complete response, you need to consider the transient response, which occurs during the initial period when the current is changing and hasn't reached the steady state yet.