How to define initial conditions for complex RC circuit?

Question

I need to calculate \$U_2 (t)\$. See the circuit below:

Given \$E\$, \$R_1\$, \$R_2\$, \$C_1\$, \$C_2\$, \$C_3\$.

The first Kirchhoff's rule:

$$ i_1 = i_2 + i_3 = i_4 $$

Now find the all currents:

$$ i_1 = \frac{E-U_1}{R_1} \\ i_2 = (C_2 + C_3)\frac{d(U_1-U_2)}{dt} \\ i_3 = \frac{U_1 - U_2}{R_2} \\ i_4 = C_1 \frac{dU_2}{dt} $$

Now I made the substitution. The currents in the Kirchhoff's rule are substituted with their actual expressions, and \$C=C_2+C_3\$. I've got the differential equation:

$$ \frac{d^2U_2}{dt^2} + \frac{(C_1+C+\frac{R_1 C_1}{R_2})}{R_1C_1C} \frac{dU_2}{dt} + \frac{U_2}{R_1R_2C_1C}=\frac{E}{R_1R_2C_1C} $$

The solution is:

$$ U_2(t)=E+K_1e^{t(\varphi+\lambda)} + K_2e^{t(\varphi-\lambda)} $$

where

$$ \varphi=-\frac{1}{2}(\frac{1}{C_1R_1}+\frac{1}{CR_1}+\frac{1}{CR_2}) \\ \lambda=\frac{\sqrt{C^2R_2^2-2CC_1R_1R_2+2CC_1R_2^2+C_1^2R_1^2+2C_1^2R_1R_2+C_1^2R_2^2}}{2CC_1R_1R_2} $$

The solution has two integration constants, so I need two initial conditions. The first is trivial - \$U_2(0)=0\$. And what about the second condition? How can I define it? Measuring real value in simulation or real circuit? I wouldn't do this, I prefer to calculate real equation.

Answer 1

When circuit is switched on, \$U_1(0^+)=0\$. Let's find current i1=i4:

$$ i_4(0^+)=\frac{E}{R_1} $$

On the other hand,

$$ i_4=C_1\frac{dU_2}{dt} $$

We can substitute \$dU_2/dt\$ with found expression and make the equality:

$$ C_1\frac{d(E+K_1e^{t(\varphi-\lambda)}+K_2e^{t(\varphi+\lambda)})}{dt}=\frac{E}{R_1} $$

Now find the derivative:

$$ C_1((\varphi-\lambda)K_1e^{t(\varphi-\lambda)}+(\varphi+\lambda)K_2e^{t(\varphi+\lambda)})=\frac{E}{R_1} $$

Now substitute \$t=0\$:

$$ C_1((\varphi-\lambda)K_1+(\varphi+\lambda)K_2)=\frac{E}{R_1} $$

Finally, there is the system of equations with unknown integration constants K1, K2:

$$ \begin{cases} (\varphi-\lambda)K_1+(\varphi+\lambda)K_2=\frac{E}{R_1C_1}\\ K_1+K_2=-E \end{cases} $$