For a RC series circuit (Fig. 19-10), find (a) the time constant of the circuit; (b) vC and vR one time constant after the switch is closed and at 5 s, and (c) vC and vR one time constant after discharge starts, assuming the capacitor is fully charged to 10 V.

Question

For a RC series circuit (Fig. 19-10), find (a) the time constant of the circuit; (b) [latex]v_{C}[/latex] and [latex]v_{R}[/latex] one time constant after the switch is closed and at 5 s, and (c) [latex]v_{C}[/latex] and [latex]v_{R}[/latex] one time constant after discharge starts, assuming the capacitor is fully charged to 10 V.

Accepted Answer

(a) Write Eq. (19-23) for the time constant in a RC series circuit and substitute values for R and C.

T = RC (19-23)

[latex] =\left(100 imes 10^3 ight)\left(20 imes 10^{-6} ight)=2\ s[/latex]

(b) Write the formulas for charging voltage, substitute values, and solve for [latex]v_{C}[/latex] and [latex]v_{R}[/latex].

[latex]v_C=V\left(1-e^{-t / R C} ight)[/latex] (19-17)

When t = T = 2 s:

[latex]v_C=10\left(1-e^{-2 / 2} ight)=10\left(1-e^{-1} ight)=10(1-0.368)=10(0.632)=6.32\ V[/latex]

[latex]V=v_R+v_C[/latex] (19-15)

[latex]v_R=V-v_C=10-6.32=3.68\ V[/latex]

When t = 5 s:

[latex]v_C=10\left(1-e^{-5 / 2} ight)=10\left(1-e^{-2.5} ight)=10(1-0.082)=10(0.918)=9.18\ V[/latex]

[latex]v_R=V-v_C=10-9.18=0.82\ V[/latex]

(c) Write the formula for discharging voltage, substitute values, and solve for [latex]v_{C}[/latex] and [latex]v_{R}[/latex].

[latex]v_C=V e^{-t / R C}[/latex] (19-22)

When t = T = 2 s:

[latex]v_C=V e^{-2 / 2}=V e^{-1}=10(0.368)=3.68\ V[/latex]

[latex]0=v_R+v_C[/latex] (19-20)

[latex]v_R=-v_C=-3.68\ V[/latex]

Question 19.SP.5: For a RC series circuit (Fig. 19-10), find (a) the time cons......

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