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So I have this RLC circuit: rlc1 And its equations for voltage addition, current conservation, ohm's law and a solution of those equations (Vo).

Then there's this RLC circuit: rlc2

With another set of new equations for current conservation and voltage addition, since a capacitor and inductor have been added in series with the resistor.

I want to obtain new equations for the current conservation, voltage addition and the solution (which is the output voltage Vo) to this other RLC circuit, rlc3

which is just like the 2nd RLC but with another resistor added in parallel with the capacitor and inductor. Thanks in advance!

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i want to obtain new equations for the current conservation, voltage addition and the solution (which is the output voltage Vo) to this other RLC circuit which is just like the 2nd RLC but with another resistor added in parallel with the capacitor and inductor.

Add a term for the conductance of the added resistor into the appropriate equation, and you'll have it.

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You question was geared towards finding the equations, so this will likely not be accepted as an answer. But you can use the Extra-Element Theorem.

This theorem says that the new transfer function with the resistor, is the transfer function without the resistor (\$H_\infty\$) that you already determined, then modified using:

$$ H(s) = H_\infty(s)\frac{1+\frac{Z_n}{R}}{1+\frac{Z_d}{R}} $$

Where \$Z_n\$ and \$Z_d\$ are driving point impedances that are much simpler to find than solving a system of equations.

  • \$Z_n\$ is found by assuming that \$V_i\$ takes on whatever value such that \$V_o=0\$. You usually don't need to know \$V_i\$ to determine \$Z_n\$ though.

schematic

simulate this circuit – Schematic created using CircuitLab

\$Z_n = \frac{V_0}{I} = 0\$

  • \$Z_d\$ is found by applying an input where \$V_i = 0\$. So this is just the impedance you see where you insert the element.

$$ Z_d = \frac{V_o}{I} = (Ls) || \left(\frac{1}{Cs}\right)||\left(R + \frac{1}{Cs} + Ls\right) $$

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