User:Egm4313.s12.team20.ibrahim/Report 1

Given
Circuit Equation:
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$$ V = LC\frac{d^2v_{c}}{dt^2} + RC\frac{dv_{c}}{dt} + v_{c}$$
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 * (4.0)
 * }

Capacitance:
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$$ Q=Cv_{c} \rightarrow \int idt=Cv_{c} \rightarrow i = C\frac{dv_{c}}{dt}$$
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 * (4.1)
 * }

Solution
Alternate Circuit Equation 1

Current Equation:
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$$ I = C\frac{dv_{c}}{dt}$$
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 * (4.2)
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Derive Equation 4.2:
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$$ I' = C\frac{d^2v_{c}}{dt^2}$$
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 * (4.3)
 * }

Rearrange the Current Intergral from Equation 4.1:
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$$ \int idt=Cv_{c} \rightarrow v_{c}=\frac{1}{C}\int idt $$
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 * (4.4)
 * }

Substitute Equations 4.2, 4.3 and 4.4 into Equation 4.0:
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$$ V = LC\frac{d^2v_{c}}{dt^2} + RC\frac{dv_{c}}{dt} + v_{c}$$ $$ \Downarrow $$
 * style="width:95%" |
 * }
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$$ V = LI'+ RI + \frac{1}{C}\int idt $$
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 * (4.5)
 * }

Derive Equation 4.5 to find Alternate Circuit Equation 1:
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$$ V = LI'+ RI + \frac{1}{C}\int idt $$ $$ \Downarrow $$
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 * }
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Alternate Circuit Equation 2

Charge Equation:
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$$ Q = Cv_{c} \ $$
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 * (4.7)
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Derive Equation 4.7:
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$$ Q'=C\frac{dv_{c}}{dt}$$
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 * (4.8)
 * }

Derive Equation 4.8:
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$$ Q''=C\frac{d^2v_{c}}{dt^2}$$
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 * (4.9)
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Rearrange the Charge equation from Equation 4.1:
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$$ Q = Cv_{c} \rightarrow v_{c}=\frac{Q}{C}$$
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 * (4.10)
 * }

Substitute Equations 4.8, 4.9 and 4.10 into Equation 4.0:
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$$ V = LC\frac{d^2v_{c}}{dt^2} + RC\frac{dv_{c}}{dt} + v_{c}$$ $$ \Downarrow $$
 * style="width:95%" |
 * }
 * }