Z NPK dla pierwszego oczka

−U_C1 + U_R3 + U_R2 = 0

Pomijamy napięcie na rezystorze R₃

−U_C1 + U_R2 = 0

U_C1 = U_R2

U_C1 = i_R2 • R₂

Z PPK

$$\left\{ \begin{matrix} i_{1} = i_{C1} + i_{2} \\ i_{2} = i_{C2} + i_{3} \\ \end{matrix} \right.\ $$

Jako że pomineliśmy napięcie na R₃ pomijamy natężenie prądu i₃.

$$\left\{ \begin{matrix} i_{1} = i_{C1} + i_{2} \\ i_{2} = i_{C2} \\ \end{matrix} \right.\ $$

Wiadomo, że prąd elektryczny to zmiana ładunku w czasie

$$i = \frac{\text{dQ}}{\text{dt}} = C \bullet \frac{dU_{c}}{\text{dt}}$$

Czyli

$$i_{C1} = C_{1} \bullet \frac{dU_{C1}}{\text{dt}}$$

$$i_{C2} = C_{2} \bullet \frac{dU_{C2}}{\text{dt}}$$

Zatem

$$\left\{ \begin{matrix} i_{1} = C_{1} \bullet \frac{dU_{C1}}{\text{dt}} + i_{2} \\ i_{2} = C_{2} \bullet \frac{dU_{C2}}{\text{dt}} \\ \end{matrix} \right.\ $$

$$\left\{ \begin{matrix} i_{1} = C_{1} \bullet \frac{dU_{C1}}{\text{dt}} + C_{2} \bullet \frac{dU_{C2}}{\text{dt}} \\ i_{2} = C_{2} \bullet \frac{dU_{C2}}{\text{dt}} \\ \end{matrix} \right.\ $$

Wracając

U_C1 = i₂ • R₂

$$U_{C1} = C_{2} \bullet \frac{dU_{C2}}{\text{dt}} \bullet R_{2}$$

$$\frac{dU_{C2}}{\text{dt}} = \frac{U_{C1}}{C_{2} \bullet R_{2}}$$

Otrzymaliśmy pierwsze równanie stanu

Z NPK dla drugiego oczka.

U_we − U_R1 − U_R2 − U_C2 = 0

U_we − i₁ • R₁ − U_R2 − U_C2 = 0

U_R2 = U_C1

$$U_{\text{we}} - \left( C_{1} \bullet \frac{dU_{C1}}{\text{dt}} + C_{2} \bullet \frac{dU_{C2}}{\text{dt}} \right) \bullet R_{1} - U_{C1} - U_{C2} = 0$$

$$U_{\text{we}} - \left( C_{1} \bullet \frac{dU_{C1}}{\text{dt}} + C_{2} \bullet \frac{U_{C1}}{C_{2} \bullet R_{2}} \right) \bullet R_{1} - U_{C1} - U_{C2} = 0$$

$$U_{\text{we}} - \left( C_{1} \bullet \frac{dU_{C1}}{\text{dt}} + \frac{U_{C1}}{R_{2}} \right) \bullet R_{1} - U_{C1} - U_{C2} = 0$$

$$U_{\text{we}} - C_{1}R_{1} \bullet \frac{dU_{C1}}{\text{dt}} - \frac{R_{1}}{R_{2}}{\bullet U}_{C1} - U_{C1} - U_{C2} = 0$$

$$U_{\text{we}} - \frac{R_{1}}{R_{2}}{\bullet U}_{C1} - U_{C1} - U_{C2} = C_{1}R_{1} \bullet \frac{dU_{C1}}{\text{dt}}$$

$$\frac{U_{\text{we}}}{C_{1}R_{1}} - \frac{\frac{R_{1}}{R_{2}}{\bullet U}_{C1}}{C_{1}R_{1}} - \frac{U_{C1}}{C_{1}R_{1}} - \frac{U_{C2}}{C_{1}R_{1}} = \frac{dU_{C1}}{\text{dt}}$$

$$\frac{U_{\text{we}}}{C_{1}R_{1}} - \frac{U_{C1}}{C_{1}R_{2}} - \frac{U_{C1}}{C_{1}R_{1}} - \frac{U_{C2}}{C_{1}R_{1}} = \frac{dU_{C1}}{\text{dt}}$$

Zatem otrzymaliśmy drugie równanie stanu

$$\frac{dU_{C1}}{\text{dt}} = \frac{1}{C_{1}R_{1}} \bullet U_{\text{we}} - \frac{R_{1} + R_{2}}{C_{1}R_{1}R_{2}} \bullet U_{C1} - \frac{1}{C_{1}R_{1}} \bullet U_{C2}$$