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Following the answers received to my previous question, I obtained many ideas and I realized that what I was really looking for was something else and needed a new post: I would like to analyze the following circuit. In particular I would like to derive an expression that describes \$i_3(t)\$ or \$V_{C1}(t)\$ or \$V_{R1}(t)\$.

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I started by solving simpler systems like \$R+C\$ and \$R+R \parallel C\$ without problems. The typical $$i(t)=\frac{V}{R}\exp\left(\frac{-t}{RC}\right)$$ So I wrote the following equations:

$$V=i_0R_0+\frac{Q_3}{C_1}+i_2R_2$$ $$V=i_0R_0+i_1R_1+i_2R_2$$ $$V=i_0R_0+i_1R_1+\frac{Q_4}{C_2}$$ $$V=i_0R_0+\frac{Q_3}{C_1}+\frac{Q_4}{C_2}$$ $$i_0=i_1+i_3$$

I differentiated them with respect to time considering that $$\frac{dQ}{dt}=i$$

I tried to solve the system of 5 equations in 5 variables (\$i_0\$, \$i_1\$, \$i_2\$, \$i_3\$, \$i_4\$) to find \$i_3(t)\$ but I can get only identities.

I suppose one or more equations are redundant and some are missing. Maybe a second order differential?

Null
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Theiden
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  • Would it help you to consider KCL instead?$$\begin{align*}\frac{V_a}{R_0}+\frac{V_a}{R_1}+C_1\frac{\text{d}}{\text{d}t}V_a&=\frac{V}{R_0}+\frac{V_b}{R_1}+C_1\frac{\text{d}}{\text{d}t}V_b\\\\\frac{V_b}{R_1}+\frac{V_b}{R_2}+C_1\frac{\text{d}}{\text{d}t}V_b+C_2\frac{\text{d}}{\text{d}t}V_b&= \frac{V_a}{R_1}+C_1\frac{\text{d}}{\text{d}t}V_a\\\\V_{a\left(t=0\right)}&=0\:\text{V}\\\\V_{b\left(t=0\right)}&=0\:\text{V}\end{align*}$$ – jonk Sep 28 '22 at 18:42
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    Are you familiar with impedances? The parallel combinations of R and C can be simplified into a single impedance each, which would make it easier to find \$V_{C1}(t) = V_{R1}(t)\$. – Null Sep 28 '22 at 19:03

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