Question

The particular current obtained from the solution of i in the sinusoidal response of R-L-C circuit is?

(a) ip = V/√(R^2+(1/ωC+ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC+ωL)/R))

(b) ip = V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R))

(c) ip = V/√(R^2+(1/ωC+ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R))

(d) ip = V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC+ωL)/R))

The question was posed to me in unit test.

My doubt stems from Sinusoidal Response of an R-L-C Circuit in portion Transients of Network Theory

Answer 1

Correct option is (b) ip = V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R))

To explain I would say: The characteristic equation consists of two parts, viz. complementary function and particular integral. The particular integral is ip =  V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R)).