Exam 8: RLC Circuits

Correct Answer:

Verified

$\mathrm { i } ( 0 ) = \mathrm { I } _ { 0 } = \mathrm { I } _ { \mathrm { s } } \times \mathrm { R } _ { 1 } / \left( \mathrm { R } _ { 1 } + \mathrm { R } _ { 2 } \right) = 1 \mathrm {~mA} . \mathrm { v } ( 0 ) = \mathrm { V } _ { 0 } = 0$
$$\mathrm { R } = \mathrm { R } _ { 3 } = 3 \mathrm { k } \Omega$$
$$\frac { d ^ { 2 } v ( t ) } { d t ^ { 2 } } + \frac { R } { L } \frac { d v ( t ) } { d t } + \frac { 1 } { L C } v ( t ) = 0$$ $$a _ { 2 } = 1 , a _ { 1 } = R / L = 15000 , a _ { 0 } = 1 / ( \mathrm { LC } ) = 2.5 \times 10 ^ { 8 }$$
$\alpha = a _ { 1 } / 2 = 7500 , \omega _ { 0 } = \sqrt { a _ { 0 } } = 15811.3883$
Since $\alpha < \omega _ { 0 }$ , this is case 3 (underdamped).
$$D v _ { 0 } = \frac { d v ( 0 ) } { d t } = \frac { i ( 0 ) } { C } = \frac { I _ { 0 } } { C } = 50000$$
$$\beta = \sqrt { \omega _ { o } ^ { 2 } - \alpha ^ { 2 } } = 13919.4109$$
$$s _ { 1 } = - \alpha + j \beta = - 7500 + \mathrm { j } 13919.4109$$
$$s _ { 2 } = - \alpha - j \beta = - 7500 - \mathrm { j } 13919.4109$$
$$B _ { 1 } = V _ { 0 } = 0$$
$$B _ { 2 } = \frac { D v _ { 0 } + \alpha V _ { 0 } } { \beta } = 3.5921$$
$$v ( t ) = 3.5921 \mathrm { e } ^ { - 7500 t } \sin ( 13919.4109 t ) u ( t )$$

Correct Answer:

Verified

$$\begin{array} { l } \mathrm { v } ( 0 ) = \mathrm { V } _ { 0 } = \mathrm { V } _ { \mathrm { s } } \times \mathrm { R } _ { 2 } / \left( \mathrm { R } _ { 1 } + \mathrm { R } _ { 2 } \right) = 3 \mathrm {~V} . \mathrm { i } ( 0 ) = \mathrm { I } _ { 0 } = 0 \\\mathrm { R } = \mathrm { R } _ { 2 } + \mathrm { R } _ { 3 } = 4 \mathrm { k } \Omega \\\frac { d ^ { 2 } v ( t ) } { d t ^ { 2 } } + \frac { R } { L } \frac { d v ( t ) } { d t } + \frac { 1 } { L C } v ( t ) = 0 \\\mathrm { a } _ { 2 } = 1 , \mathrm { a } _ { 1 } = \mathrm { R } / \mathrm { L } = 32000 , \mathrm { a } _ { 0 } = 1 / ( \mathrm { LC } ) = 1.6 \times 10 ^ { 8 } \\\alpha = \mathrm { a } _ { 1 } / 2 = 16000 , \omega _ { 0 } = \sqrt { a _ { 0 } } = 12649.11064\end{array}$$
Since $\alpha > \omega _ { 0 }$ , this is case 1 (overdamped).
$$D v _ { 0 } = \frac { d v ( 0 ) } { d t } = \frac { i ( 0 ) } { C } = \frac { I _ { 0 } } { C } = 0$$
$$s _ { 1 } = - \alpha + \sqrt { \alpha ^ { 2 } - \omega _ { 0 } ^ { 2 } } = - 6202.041$$
$$s _ { 2 } = - \alpha - \sqrt { \alpha ^ { 2 } - \omega _ { 0 } ^ { 2 } } = - 25797.959$$
$$A _ { 1 } = \frac { V _ { 0 } s _ { 2 } - D v _ { 0 } } { s _ { 2 } - s _ { 1 } } = 3.9495$$
$$A _ { 2 } = \frac { D v _ { 0 } - V _ { 0 } s _ { 1 } } { s _ { 2 } - s _ { 1 } } = - 0.9495$$
$$\mathrm { v } ( \mathrm { t } ) = \left[ 3.9495 \mathrm { e } ^ { - 6202.041 \mathrm { t } } - 0.9495 \mathrm { e } ^ { - 25797.96 \mathrm { t } } \right] \mathrm { u } ( \mathrm { t } )$$

Correct Answer:

Verified

Change $I _ { s }$ and $R _ { 1 }$ to a series connection of
$$\mathrm { V } _ { \mathrm { s } } = \mathrm { R } _ { 1 } \mathrm { I } _ { \mathrm { s } } = 0.5 \mathrm {~V}$$
and a resistor $\mathrm { R } _ { 1 }$ .
Let
$$\begin{array} { l } \mathrm { R } = \mathrm { R } _ { 1 } + \mathrm { R } _ { 2 } = 300 \Omega \\- V _ { s } + R i ( t ) + L \frac { d i ( t ) } { d t } + v ( t ) = 0\end{array}$$
$$i ( t ) = C \frac { d v ( t ) } { d t }$$
Substituting Equation (2) into Equation (1), we obtain
$$- V _ { s } + R C \frac { d v ( t ) } { d t } + L C \frac { d ^ { 2 } v ( t ) } { d t ^ { 2 } } + v ( t ) = 0$$
which can be rearranged as $$\frac { d ^ { 2 } v ( t ) } { d t ^ { 2 } } + \frac { R } { L } \frac { d v ( t ) } { d t } + \frac { 1 } { L C } v ( t ) = \frac { 1 } { L C } V _ { s }$$
From Equation (2), we have
$$\begin{array} { l } \frac { d v ( t ) } { d t } = \frac { i ( t ) } { C } \\\frac { d v ( 0 ) } { d t } = \frac { i ( 0 ) } { C } \\\mathrm { a } _ { 2 } = 1 \\a _ { 1 } = \frac { R } { L } = 3000 \\a _ { 0 } = \frac { 1 } { L C } = 5 \times 10 ^ { 8 } \\\alpha = \mathrm { a } _ { 1 } / 2 = 1500 , \omega _ { 0 } = \sqrt { a _ { 0 } } = 22360.6798\end{array}$$
Since $\alpha < \omega _ { 0 }$ , this is case 3 (underdamped).
$$\begin{array} { l } \beta = \sqrt { \omega _ { o } ^ { 2 } - \alpha ^ { 2 } } = 22310.3115 \\s _ { 1 } = - \alpha + j \beta = - 1500 + \mathrm { j } 22310.3115\end{array}$$
$$s _ { 2 } = - \alpha - j \beta = - 1500 - \mathrm { j } 22310.3115$$
$$B _ { 1 } = V _ { 0 } - V _ { \mathrm { s } } = 0 \mathrm {~V}$$
$$B _ { 2 } = \frac { D v _ { 0 } + \alpha \left( V _ { 0 } - V _ { s } \right) } { \beta } = 2.241116 \mathrm {~V}$$
$$\mathrm { v } ( \mathrm { t } ) = \left[ 0.5 + 2.241116 \mathrm { e } ^ { - 1500 \mathrm { t } } \sin ( 22310.3115 \mathrm { t } ) \right] \mathrm { u } ( \mathrm { t } ) \mathrm { V }$$