In the circuit shown below, the switch is closed at {t}=0. The value of \theta in degrees which will give the maximum value of DC offset of the current at the time of switching is

GATE Electrical Engineering (EE) 2019 Solved Paper

Question : 23 of 65

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In the circuit shown below, the switch is closed at

{t}=0

. The value of

θ

in degrees which will give the maximum value of DC offset of the current at the time of switching is

Solution:

{i}( {t})=\;{V_{ {w}}}/{|Z|} \sin (ω t+θ-φ)+A e^{\;{-t}/{τ}}

|Z|=√{R^{2}+(ω L)^{2}}

φ=\tan ^{-1}(\;{ω L}/{R})

Now

{i}(0^{-})= {i}(0)= {i}(0^{+})=0 {A}

So, at

{t}=0 ⇒ {i}=0

So,

{A}=\;{-V_{ {w}}}/{|Z|} \sin (θ-φ)

So, Total solution

{i}( {t})=\;{V_{ {s}}}/{|Z|} \sin (ω t+θ-φ)-\;{V_{ {s}}}/{|Z|} \sin (θ-φ) e^{\;{-t}/{ {τ}}}

DC offset value is

\;{-V_{\;\text "m "\;}}/{|Z|} \sin (θ-φ) e^{\;{-t}/{τ}}

Maximum when,

\sin (θ-φ)=-1

So,

θ-φ=-90^{∘}

θ=-90^{∘}+45

θ=-45^{∘}

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