Solving this electronic problem with impedance ?

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Solving this electronic problem with impedance ?

There is two of this image in series, two capacitors $C_{1}$ $C_1$ $C_{2}$ $C_2$ and two resistors $R_{1}$ $R_1$ $R_{2}$ $R_2$

To be more precise this is two real capacitors (capacitor and resistor in parallel) in series.

So i have the differential equation etablished by Kirchhoff's laws :

$i (\frac{1}{R_{1}} + \frac{1}{R_{2}}) + (C_{1} + C_{2}) \frac{d i}{d t} = C_{1} C_{2} \frac{d^{2} u}{{d t}^{2}} + (\frac{C_{2}}{R_{1}} + \frac{C_{1}}{R_{2}}) \frac{d u}{d t} + \frac{u}{R_{1} R_{2}}$ $i(1/R_1 + 1/R_2) + (C_1 + C_2)(di)/(dt) = C_1C_2(d^2u)/(dt^2) + (C_2/R_1 + C_1/R_2)(du)/(dt) + u/(R_1R_2)$

i tried to do it by impedance because i thought it will take less time.

Impedance of the first real capacitor :

$Z_{1} = {(\frac{1}{Z_{C_{1}}} + \frac{1}{Z_{R_{1}}})}^{- 1} = \frac{Z_{R_{1}}}{1 + \frac{Z_{R_{1}}}{Z_{C_{1}}}}$ $Z_1 = (1/Z_(C_1) + 1/Z_(R_1))^(-1) = Z_(R_1)/(1+Z_(R_1)/(Z_(C_1))$

Impedance of the second :

$Z_{2} = {(\frac{1}{Z_{C_{2}}} + \frac{1}{Z_{R_{2}}})}^{- 1} = \frac{Z_{R_{2}}}{1 + \frac{Z_{R_{2}}}{Z_{C_{2}}}}$ $Z_2 = (1/Z_(C_2) + 1/Z_(R_2))^(-1) = Z_(R_2)/(1+Z_(R_2)/(Z_(C_2))$

the equivalent impedance of the entire circuit is :

$Z_{e q} = Z_{1} + Z_{2}$ $Z_(eq) = Z_1 + Z_2$

and then $Z_{e q} = \frac{u}{i}$ $Z_(eq) = u/i$

but it lead to

$i = \frac{1}{R_{1}} (u + R_{1} C_{1} j w u) + \frac{1}{R_{2}} (u + R_{2} C_{2} j w u)$ $i = 1/R_1(u + R_1C_1jwu) + 1/R_2(u + R_2C_2jwu)$

and :

$i = u (\frac{1}{R_{1}} + \frac{1}{R_{2}}) + \frac{d u}{d t} (C_{1} + C_{2})$ $i = u(1/R_1 + 1/R_2) + (du)/dt(C_1+C_2)$

$j$ $j$ is the imaginary unit

Obviously it's not the same, i know i'm good with Kirchkoff's law because i checked the answer, but i'm not good with impedance, why ?

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Answer 1 · 2018-03-29T08:57:46

See below.

Explanation:

When you solve using impedances you are assuming that the circuit is submitted to a sinusoid periodic input. You are solving without considering the transient modes. So be careful with this approach.

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Solving this electronic problem with impedance ?

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