Podstawowe równania opisujące obwód:

\begin{aligned} &u_{L 1}=L_{1} \cdot \frac{d i_{L 1}}{d t} \\ &u_{L 5}=L_{5} \cdot \frac{d i_{L 5}}{d t} \\ &i_{C 2}=C_{2} \cdot \frac{d u_{C 2}}{d t} \\ &i-i_{L 1}+i_{C 2}=0 \\ &-i_{R 3}+i_{L 1}-i_{R 4}=0 \\ &i_{R 4}-i_{L 5}+i_{C 2}=0 \\ &u_{L 1}+u_{C 2}+i_{C 2} \cdot R_{2}-E+i_{R 4} \cdot R_{4}=0 \\ &i_{R 3} \cdot R_{3}-i_{R 4} \cdot R_{4}-u_{L 5}=0 \end{aligned}

Przekształcamy

\begin{aligned} \begin{array}{lll} i-i_{L 1}+C_{2} \cdot \frac{d u_{C 2}}{d t}=0 & i-i_{L 1}+C_{2} \cdot \frac{d u_{C 2}}{d t}=0 & \frac{d u_{C 2}}{d t}=\frac{-i+i_{L 1}}{C_{2}} \\ -i_{R 3}+i_{L 1}-i_{R 4}=0 & i_{R 3}=i_{L 1}-i_{R 4} & i_{R 3}=i_{L 1}-i_{L 5}+i_{C 2} \\ i_{R 4}-i_{L 5}+i_{C 2}=0 & i_{R 4}=i_{L 5}-i_{C 2} & \end{array} \end{aligned} \begin{aligned} &L_{1} \cdot \frac{d i_{L 1}}{d t}+u_{C 2}+C_{2} \cdot \frac{d u_{C 2}}{d t} \cdot R_{2}-E+\left(i_{L 5}-C_{2} \cdot \frac{d u_{C 2}}{d t}\right) \cdot R_{4}=0 \\ &\left(i_{L 1}-i_{L 5}+C_{2} \cdot \frac{d u_{C 2}}{d t}\right) \cdot R_{3}-\left(i_{L 5}-C_{2} \cdot \frac{d u_{C 2}}{d t}\right) \cdot R_{4}-L_{5} \cdot \frac{d i_{L 5}}{d t}=0 \end{aligned} \begin{aligned} \frac{d i_{L 1}}{d t} &=\frac{\left(C_{2} \cdot R_{4}-C_{2} \cdot R_{2}\right) \cdot \frac{d u_{C 2}}{d t}+\left(E-\left(u_{C 2}+R_{4} \cdot i_{L 5}\right)\right)}{L_{1}} \\ \frac{d i_{L 1}}{\mathrm{dt}} &=\frac{\left(C_{2} \cdot R_{4}-C_{2} \cdot R_{2}\right) \cdot \frac{-i+i_{L 1}}{C_{2}}+\left(E-\left(u_{C 2}+R_{4} \cdot i_{L 5}\right)\right)}{L_{1}} \\ \frac{d i_{L 1}}{\mathrm{dt}} &=\left(-\frac{1}{L_{1}}\right) \cdot u_{C 2}-\frac{R_{4}}{L_{1}} \cdot i_{L 5}+\frac{R_{4}-R_{2}}{L_{1}} \cdot i_{L 1}+\frac{1}{L_{1}} \cdot E+\frac{R_{2}-R_{4}}{L_{1}} \cdot i \end{aligned} \begin{aligned} &\frac{d i_{L 5}}{d t}=\frac{\left(C_{2} \cdot R_{4}+C_{2} \cdot R_{3}\right) \cdot \frac{d u_{C 2}}{d t}+\left(R_{3} \cdot i_{L 1}-\left(R_{4}+R_{3}\right) \cdot i_{L 5}\right)}{L_{5}} \\ &\frac{d i_{L 5}}{\mathrm{dt}}=\frac{-R_{4}-R_{3}}{L_{5}} \cdot i_{L 5}+\frac{R_{4}+2 \cdot R_{3}}{L_{5}} \cdot i_{L 1}-\frac{R_{4}+R_{3}}{L_{5}} \cdot i \end{aligned} \begin{aligned} \frac{d}{d t}\left[\begin{array}{c} i_{L 1} \\ i_{L 5} \\ u_{C 2} \end{array}\right]=\left[\begin{array}{ccc} \frac{R_{4}-R_{2}}{L_{1}} & -\frac{R_{4}}{L_{1}} & -\frac{1}{L_{1}} \\ \frac{R_{4}+2 \cdot R_{3}}{L_{5}} & \frac{-R_{4}-R_{3}}{L_{5}} & 0 \\ \frac{1}{C_{2}} & 0 & 0 \end{array}\right] \cdot\left[\begin{array}{c} i_{L 1} \\ i_{L 5} \\ u_{C 2} \end{array}\right]+\left[\begin{array}{cc} \frac{1}{L_{1}} & \frac{R_{2}-R_{4}}{L_{1}} \\ 0 & -\frac{R_{4}+R_{3}}{L_{5}} \\ 0 & \frac{-1}{C_{2}} \end{array}\right] \cdot\left[\begin{array}{l} e \\ i \end{array}\right] \end{aligned} \begin{aligned} &i_{R 4}=i_{L 5}-C_{2} \cdot \frac{-i+i_{L 1}}{C_{2}}=i_{L 5}+i-i_{L 1} \\ &u_{L 1}+u_{C 2}+C_{2} \cdot \frac{-i+i_{L 1}}{C_{2}} \cdot R_{2}-E+\left(i_{L 5}+i-i_{L 1}\right) \cdot R_{4}=0 \\ &u_{L 1}=-u_{C 2}-R_{4} \cdot i_{L 5}+\left(R_{4}-R_{2}\right) \cdot i_{L 1}+E+\left(R_{2}-R_{4}\right) \cdot i \end{aligned} \begin{aligned} \left[\begin{array}{c} u_{L 1} \\ i_{R 4} \end{array}\right]=\left[\begin{array}{ccc} R_{4}-R_{2} & -R_{4} & -1 \\ -1 & 0 & 1 \end{array}\right] \cdot\left[\begin{array}{c} i_{L 1} \\ i_{L 5} \\ u_{C 2} \end{array}\right]+\left[\begin{array}{cc} 1 & R_{2}-R_{4} \\ 0 & 1 \end{array}\right] \cdot\left[\begin{array}{l} e \\ i \end{array}\right] \end{aligned}

Stany nieustalone

Teoria obwodów - Stany nieustalone - Przykład 1

Przykład 1