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I would like to do a transient analysis on this circuit but I don't know how to use those formulas $$V(t)=V_0*(1-e^{\frac{-t}{R*C}})$$ $$I(t)=\frac{V_0}{R}*e^{\frac{-t}{R*C}}$$ in this case.
There are no values given for the components as I want to understand the process and not just know the result.
Thank you.
You have a major misunderstanding. You wanted to apply formulas which are valid for a circuit which has one resistor and one capacitor. To solve the behaviour of more complex circuits you must learn to derive equations for it starting from the circuit and general circuit laws. Systematic methods for it - general ones and methods for limited cases such as single frequency sinusoidal voltages and currents - have been well known well over 100 years. Then you must be able to solve the equations to get the formulas which are valid for your circuit (=a common feedback circuit in low frequency oscillators, the Wien Bridge). Circuit analysis books present those known methods and have plenty of practicing material.
If it happens that you are trying to analyze this to get the frequency of a Wien Bridge oscillator you should learn how to apply phasor calculus with complex numbers or Laplace domain transfer functions. Or search the result from handbooks or web. Sorry!
Setup two equations using KVL and try solving:
\$C_1\dfrac{dV_{C_1}}{dt}R_1+V_{C_1}+V_{C_2}=V_0\$
\$V_{C_2}+C_2\dfrac{dV_{C_2}}{dt}R_2=0 \$
\$Z = R_1+Z_{C_1} + R_2//Z_{C_2} =R_1+Z_{C_1} + \dfrac{R_2*Z_{C_2}}{R_2+Z_{C_{2}}} \$
\$I_o=V_o/Z = \dfrac{R_2+Z_{C_2}}{R_1+Z_{C_1}+R_2*Z_{C_2}}\$
You can also simulate it and interactively change any component value to correlate your theoretical understanding of Bode Plots and transient Time response.
It's just an equivalent impedance ratio using capacitor impedances and initial conditions using KVL KCL. Normally one might use a current source to measure voltage as impedance. I started with 10k and a voltage source. As all the Caps short out at HF & what's left is the single series 10R as my flat curve.
