User:Egm4313.s12.team17.deaver.md/Report 1

= Problem 4 - Modeling: Electric Circuits =

Given
Charge Equation:
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$$  \displaystyle Q = \int I dt = C\nu_C $$     (Eq.1)
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Circuit Equations:
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$$  \displaystyle V = LC\frac{d^2\nu_C}{dt^2} + RC\frac{d\nu_C}{dt} + \nu_C $$     (Eq.2)
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$$  \displaystyle V^\prime = LI^{\prime\prime} + RI^\prime + \frac{1}{C}I $$     (Eq.3)
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$$  \displaystyle V = LQ^{\prime\prime} + RQ^\prime + \frac{1}{C}Q $$     (Eq.4)
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Problem Statement
Use Eq. 2 to derive the alternative forms of the circuit equation 3 and 4.[2]

Solution
Solve for \nu_C \, in terms of I for Eq. 1.
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$$  \displaystyle \nu_C = \frac{1}{C}\int I dt = \frac{1}{C}Q $$     (Eq.5)
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Substitute Eq. 5 into Eq. 2.
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$$  \displaystyle V = LC\frac{d^2\nu_C}{dt^2} + RC\frac{d\nu_C}{dt} + \frac{1}{C}\int I dt $$ (Eq.6)
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Take the differential of Eq. 1 twice with respect to t \, in terms of Q and \nu_C \,.
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$$  \displaystyle Q^\prime = C\frac{d\nu_C}{dt} $$     (Eq.7)
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$$  \displaystyle Q^{\prime\prime} = C\frac{d^2\nu_C}{dt^2} $$     (Eq.8)
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Substitute Eq. 5, 7, and 8 into Eq. 6.
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$$  \displaystyle V = LQ^{\prime\prime} + RQ^\prime + \frac{1}{C}Q $$     (Eq.9)
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Take the differential of Eq. 9 with respect to t \,.
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$$  \displaystyle V^\prime = LQ^{\prime\prime\prime} + RQ^{\prime\prime} + \frac{1}{C}Q^\prime $$     (Eq.10)
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Take the differential of Eq. 1 three time with respect to t \, in terms of Q and I.
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$$  \displaystyle Q^\prime = I $$ (Eq.11)
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$$  \displaystyle Q^{\prime\prime} = I^\prime $$     (Eq.12)
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$$  \displaystyle Q^{\prime\prime\prime} = I^{\prime\prime} $$     (Eq.13)
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Substitute Eq. 11, 12, and 13 into Eq. 10.
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$$  \displaystyle V^\prime = LI^{\prime\prime} + RI^\prime + \frac{1}{C}I $$     (Eq.14)
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