Circuito nel dominio del tempo
Inviato: 2 ott 2012, 23:07Salve ragazzi, secondo voi questo esercizio è giusto?
Calcolare la![\[{v_C}\left( t \right)\]](/forum/latexrender/pictures/27cf8da7e59829f97a3d55ee1aed1bda.png "\[{v_C}\left( t \right)\]")
per ![\[t > 0\]](/forum/latexrender/pictures/6bae9030b5a5154b4390d0129dd45e04.png "\[t > 0\]")
![\[\begin{array}{l}
{R_1} = {R_2} = 3\Omega ;\hspace{1cm}
C = \frac{1}{3}F ;\hspace{1cm}
L = \frac{3}{2}H ;\hspace{1cm}
{g_m} = \frac{1}{3}S \\\\
{v_{g1}}\left( t \right) = u\left( { - t} \right)V ;\hspace{1cm}
{v_{g2}}\left( t \right) = u\left( t \right)V
\end{array}\]](/forum/latexrender/pictures/ca36a03f32ae3b5f22f8ec9e38eb467b.png "\[\begin{array}{l}
{R_1} = {R_2} = 3\Omega ;\hspace{1cm}
C = \frac{1}{3}F ;\hspace{1cm}
L = \frac{3}{2}H ;\hspace{1cm}
{g_m} = \frac{1}{3}S \\\\
{v_{g1}}\left( t \right) = u\left( { - t} \right)V ;\hspace{1cm}
{v_{g2}}\left( t \right) = u\left( t \right)V
\end{array}\]")
![\[t < 0\]](/forum/latexrender/pictures/e86b85a558af892cb96e22ce68865ce1.png "\[t < 0\]")
![\[\begin{array}{l}
{v_C}\left( {{0^ - }} \right) = {v_{g1}}\left( t \right)\\\\
{i_L}\left( {{0^ - }} \right) = \frac{{{v_{g1}}\left( t \right)}}{{{R_2}}}
\end{array}\]](/forum/latexrender/pictures/44329c54cc949188ff8deeb414dacaa7.png "\[\begin{array}{l}
{v_C}\left( {{0^ - }} \right) = {v_{g1}}\left( t \right)\\\\
{i_L}\left( {{0^ - }} \right) = \frac{{{v_{g1}}\left( t \right)}}{{{R_2}}}
\end{array}\]")
![\[\frac{{{v_{g2}}\left( t \right)}}{{{R_2}}} = {i_C}\left( t \right) + {i_L}\left( t \right) + \frac{{{v_{g1}}\left( t \right)}}{{{R_1}}} + {g_m}{v_C}\left( t \right)\]](/forum/latexrender/pictures/1df0e9b08cfdfe6f6c0ca329420dd4c8.png "\[\frac{{{v_{g2}}\left( t \right)}}{{{R_2}}} = {i_C}\left( t \right) + {i_L}\left( t \right) + \frac{{{v_{g1}}\left( t \right)}}{{{R_1}}} + {g_m}{v_C}\left( t \right)\]")
![\[\frac{{{v_{g2}}\left( t \right)}}{{{R_2}}} = \frac{1}{L}\int\limits_0^t {{v_C}\left( t \right)} dt + C\frac{{d{v_C}\left( t \right)}}{{dt}} + \frac{{{v_C}\left( t \right)}}{{{R_1}}} + {g_m}{v_C}\left( t \right)\]](/forum/latexrender/pictures/d8976a704f0c6a5a31c7f2e631acb0c5.png "\[\frac{{{v_{g2}}\left( t \right)}}{{{R_2}}} = \frac{1}{L}\int\limits_0^t {{v_C}\left( t \right)} dt + C\frac{{d{v_C}\left( t \right)}}{{dt}} + \frac{{{v_C}\left( t \right)}}{{{R_1}}} + {g_m}{v_C}\left( t \right)\]")
Derivo in modo da calcolare la soluzione generale:
![\[C\frac{{{d^2}{v_C}\left( t \right)}}{{d{t^2}}} + \left( {\frac{1}{{{R_1}}} + {g_m}} \right)\frac{{d{v_C}\left( t \right)}}{{dt}} + \frac{1}{L}{v_C}\left( t \right) = 0\]](/forum/latexrender/pictures/391cc4c9a3eb4feb5cb687a52c4af6d8.png "\[C\frac{{{d^2}{v_C}\left( t \right)}}{{d{t^2}}} + \left( {\frac{1}{{{R_1}}} + {g_m}} \right)\frac{{d{v_C}\left( t \right)}}{{dt}} + \frac{1}{L}{v_C}\left( t \right) = 0\]")
![\[\frac{{{d^2}{v_C}\left( t \right)}}{{d{t^2}}} + \frac{1}{C}\left( {\frac{1}{{{R_1}}} + {g_m}} \right)\frac{{d{v_C}\left( t \right)}}{{dt}} + \frac{1}{{CL}}{v_C}\left( t \right) = 0\]](/forum/latexrender/pictures/8c611decf2fd58de5bcd8fc02a3c0518.png "\[\frac{{{d^2}{v_C}\left( t \right)}}{{d{t^2}}} + \frac{1}{C}\left( {\frac{1}{{{R_1}}} + {g_m}} \right)\frac{{d{v_C}\left( t \right)}}{{dt}} + \frac{1}{{CL}}{v_C}\left( t \right) = 0\]")
Cerco le radici basandomi sullo studio dell'equazione caratteristica:
![\[\begin{array}{l}
{s^2} + 2s + 2 = 0\\
{s_1} = - 1 - j\\
{s_2} = - 1 + j
\end{array}\]](/forum/latexrender/pictures/282b57c84e34ac37718e59602feafc1d.png "\[\begin{array}{l}
{s^2} + 2s + 2 = 0\\
{s_1} = - 1 - j\\
{s_2} = - 1 + j
\end{array}\]")
a questo punto scrivo la:
![\[{v_C}\left( t \right) = {e^{ - t}}\left[ {{c_1}\sin \left( t \right) + {c_2}\cos \left( t \right)} \right] + {v_{{C_p}}}\left( t \right)\]](/forum/latexrender/pictures/ae319cd5d39a56a6ade480f813e1a812.png "\[{v_C}\left( t \right) = {e^{ - t}}\left[ {{c_1}\sin \left( t \right) + {c_2}\cos \left( t \right)} \right] + {v_{{C_p}}}\left( t \right)\]")
la soluzione particolare dovrebbe essere la seguente (spero di non sbagliarmi)
![\[{v_{{C_p}}}\left( t \right) = \frac{{{v_{g2}}\left( t \right)}}{{{R_2}}} = \frac{1}{3} = 0.33V\]](/forum/latexrender/pictures/a0b4aefe9ad8a7d686f503ecc54f1670.png "\[{v_{{C_p}}}\left( t \right) = \frac{{{v_{g2}}\left( t \right)}}{{{R_2}}} = \frac{1}{3} = 0.33V\]")
derivo la
![\[{{\dot v}_C}\left( t \right) = {e^{ - t}}\left[ {\sin \left( t \right)\left( { - {c_1} - {c_2}} \right) + \cos \left( t \right)\left( { - {c_2} - {c_1}} \right)} \right]\]](/forum/latexrender/pictures/3f88ad419804a922fb4c8b624ab3f41c.png "\[{{\dot v}_C}\left( t \right) = {e^{ - t}}\left[ {\sin \left( t \right)\left( { - {c_1} - {c_2}} \right) + \cos \left( t \right)\left( { - {c_2} - {c_1}} \right)} \right]\]")
Adesso calcolo le condizioni iniziali:
![\[\left\{ \begin{array}{l}
{v_C}\left( 0 \right) = {c_2}\\
{{\dot v}_C}\left( 0 \right) = - {c_2} - {c_1}
\end{array} \right.\]](/forum/latexrender/pictures/e4bc3592737ecaf036bab2cd2a428523.png "\[\left\{ \begin{array}{l}
{v_C}\left( 0 \right) = {c_2}\\
{{\dot v}_C}\left( 0 \right) = - {c_2} - {c_1}
\end{array} \right.\]")
![\[\left\{ \begin{array}{l}
{c_2} = 2\\
{c_2} - {c_1} = 2
\end{array} \right.\]](/forum/latexrender/pictures/692deb8f5d0bae569d55b39445cd273f.png "\[\left\{ \begin{array}{l}
{c_2} = 2\\
{c_2} - {c_1} = 2
\end{array} \right.\]")
![\[\left\{ \begin{array}{l}
{c_2} = 2\\
{c_1} = 0
\end{array} \right.\]](/forum/latexrender/pictures/6e5d448e950ca8bb013bd6f2aa7f9953.png "\[\left\{ \begin{array}{l}
{c_2} = 2\\
{c_1} = 0
\end{array} \right.\]")
dulcis in fundo
![% MathType!Translator!2!1!LaTeX.tdl!LaTeX 2.09 and later!
\[{v_C}\left( t \right) = {e^{ - t}}\left[ {2\cos \left( t \right)} \right] + 0.33V\]% MathType!End!2!1!](/forum/latexrender/pictures/c647cdd62a419069926e993743a9cfdf.png "% MathType!Translator!2!1!LaTeX.tdl!LaTeX 2.09 and later!
\[{v_C}\left( t \right) = {e^{ - t}}\left[ {2\cos \left( t \right)} \right] + 0.33V\]% MathType!End!2!1!")
Spero di non aver fatto errori di calcolo, l'ho rifatto 3 volte, secondo voi quanti errori ho fatto?
Calcolare la
per 
Derivo in modo da calcolare la soluzione generale:
Cerco le radici basandomi sullo studio dell'equazione caratteristica:
a questo punto scrivo la
:la soluzione particolare dovrebbe essere la seguente (spero di non sbagliarmi)
derivo la
Adesso calcolo le condizioni iniziali:
dulcis in fundo
Spero di non aver fatto errori di calcolo, l'ho rifatto 3 volte, secondo voi quanti errori ho fatto?
![\[\begin{array}{l}
{v_C}\left( {{0^ - }} \right) = {V_{g1}} \\
{i_L}\left( {{0^ - }} \right) = \frac{{{V_{g1}}}}{{{R_2}}} \\
\end{array}\]](/forum/latexrender/pictures/798293c6024edbfb04a07adf94519f7c.png "\[\begin{array}{l}
{v_C}\left( {{0^ - }} \right) = {V_{g1}} \\
{i_L}\left( {{0^ - }} \right) = \frac{{{V_{g1}}}}{{{R_2}}} \\
\end{array}\]")
![\[\frac{{{v_{C}}\left( t \right)}}{{{R_1}}}\]](/forum/latexrender/pictures/7bfe826da88a7b71b1deb68e5812771b.png "\[\frac{{{v_{C}}\left( t \right)}}{{{R_1}}}\]")
o sbaglio?![\[\frac{{{v_{g2}}\left( t \right)}}{{{R_2}}} = {i_C}\left( t \right) + {i_L}\left( t \right) + \frac{{{v_C}\left( t \right)}}{{{R_1}}} + {g_m}{v_C}\left( t \right)\]](/forum/latexrender/pictures/05e140ed07a291a5c35304c8768d34df.png "\[\frac{{{v_{g2}}\left( t \right)}}{{{R_2}}} = {i_C}\left( t \right) + {i_L}\left( t \right) + \frac{{{v_C}\left( t \right)}}{{{R_1}}} + {g_m}{v_C}\left( t \right)\]")
