Exam 15: Circuits Analysis in the S-Domain

Assume $\mathrm { L } _ { 1 } = 1 \mathrm { H } , \mathrm { L } _ { 2 } = 2 \mathrm { H } , \mathrm { C } = 0.5 \mathrm {~F} , \mathrm { R } = 1 \Omega$ in the circuit shown below. Find the transfer function $\mathrm { H } ( \mathrm { s } ) = \mathrm { V } _ { \mathrm { o } } ( \mathrm { s } ) / \mathrm { V } _ { \mathrm { in } } ( \mathrm { s } )$ $Assume \mathrm { L } _ { 1 } = 1 \mathrm { H } , \mathrm { L } _ { 2 } = 2 \mathrm { H } , \mathrm { C } = 0.5 \mathrm {~F} , \mathrm { R } = 1 \Omega in the circuit shown below. Find the transfer function \mathrm { H } ( \mathrm { s } ) = \mathrm { V } _ { \mathrm { o } } ( \mathrm { s } ) / \mathrm { V } _ { \mathrm { in } } ( \mathrm { s } )$

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Question 1

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$\begin{array} { l } \frac { V _ { a } - V _ { i n } } { s L _ { 1 } } + \frac { V _ { a } } { \frac { 1 } { s C } } + \frac { V _ { a } - V _ { o } } { R } = 0 \\\\\\frac { V _ { o } - V _ { a } } { R } + \frac { V _ { o } } { s L _ { 2 } } = 0 \\\\H ( s ) = \frac { 2 s } { s ^ { 3 } + 0.5 s ^ { 2 } + 3 s + 1 }\end{array}$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { \left( s + 10 ^ { 3 } \right) ^ { 2 } \left( s + 10 ^ { 6 } \right) ^ { 2 } } { 100 \left( s + 10 ^ { 4 } \right) ^ { 2 } \left( s + 10 ^ { 5 } \right) ^ { 2 } }$

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Question 2

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Find the convolution y(t) of x(t) and h(t); that is, find $y ( t ) = h ( t ) * x ( t )$ $x ( t ) = u ( t ) - u ( t - 5 ) ,$ $h ( t ) = \delta ( t - 10 ) + \delta ( t - 12 )$

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Question 3

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$y ( t ) = \left\{ \begin{array} { l l } 1 , & 10 \leq t < 12 \\2 , & 12 \leq t < 15 \\1 , & 15 \leq t < 17 \\0 , & \text { elsewhere }\end{array} \right.$

In the circuit shown below, let $\mathrm { R } _ { 1 } = 2 \Omega , \mathrm { R } _ { 2 } = 3 \Omega , \mathrm { C } = 0.1 \mathrm {~F} , \mathrm { v } \left( 0 ^ { - } \right) = 2 \mathrm {~V} , \mathrm { v } _ { \mathrm { s } } ( \mathrm { t } ) = 2 \mathrm { u } ( \mathrm { t } )$ Draw the circuit in the s-domain and find $V ( s )$ and $v ( t )$ for $t \geq 0$ . $In the circuit shown below, let \mathrm { R } _ { 1 } = 2 \Omega , \mathrm { R } _ { 2 } = 3 \Omega , \mathrm { C } = 0.1 \mathrm {~F} , \mathrm { v } \left( 0 ^ { - } \right) = 2 \mathrm {~V} , \mathrm { v } _ { \mathrm { s } } ( \mathrm { t } ) = 2 \mathrm { u } ( \mathrm { t } ) Draw the circuit in the s-domain and find V ( s ) and v ( t ) for t \geq 0 .$

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Question 4

Find the convolution y(t) of x(t) and h(t); that is, $\text { find } y ( t ) = h ( t ) ^ { * } x ( t )$ $x ( t ) = \left\{ \begin{array} { c c } 3 , & 0 \leq t < 7 \\0 , & \text { otherwise }\end{array} , \quad h ( t ) = \left\{ \begin{array} { c c } 2 , & 0 \leq t < 3 \\0 , & \text { otherwise }\end{array} \right. \right.$

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Question 5

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = 4 e ^ { - 3 t } u ( t ) , \quad h ( t ) = 2 e ^ { - 5 t } u ( t )$

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Question 6

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { \left( s + 10 ^ { 3 } \right) ^ { 2 } } { \left( s + 10 ^ { 4 } \right) ^ { 2 } }$

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Question 7

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { s + 1000 } { s + 100 }$

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Question 8

Find the convolution y(t) of x(t) and h(t); that is, find $y ( t ) = h ( t ) ^ { * } x ( t )$ $x ( t ) = 2 \cos ( 3 t ) u ( t ) , \quad h ( t ) = 4 \cos ( 4 t ) u ( t )$

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Question 9

In the circuit shown below, let $\mathrm { R } _ { 1 } = 2 \Omega , \mathrm { R } _ { 2 } = 3 \Omega , \mathrm { L } = 2 \mathrm { H } , \mathrm { i } \left( 0 ^ { - } \right) = 2 \mathrm {~A} , \mathrm { v } _ { \mathrm { s } } ( \mathrm { t } ) = 3 \mathrm { u } ( \mathrm { t } )$ Draw the circuit in the s-domain and find $\mathrm { V } ( \mathrm { s } )$ and $\mathrm { v } ( \mathrm { t } )$ for $\mathrm { t } \geq 0$ . $In the circuit shown below, let \mathrm { R } _ { 1 } = 2 \Omega , \mathrm { R } _ { 2 } = 3 \Omega , \mathrm { L } = 2 \mathrm { H } , \mathrm { i } \left( 0 ^ { - } \right) = 2 \mathrm {~A} , \mathrm { v } _ { \mathrm { s } } ( \mathrm { t } ) = 3 \mathrm { u } ( \mathrm { t } ) Draw the circuit in the s-domain and find \mathrm { V } ( \mathrm { s } ) and \mathrm { v } ( \mathrm { t } ) for \mathrm { t } \geq 0 .$

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Question 10

Find the convolution y(t) of x(t) and h(t); that is, $\text { find } y ( t ) = h ( t ) ^ { * } x ( t )$ $x ( t ) = t ^ { 2 } u ( t ) , \quad h ( t ) = u ( t )$

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Question 11

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = t u ( t ) , \quad h ( t ) = t u ( t )$

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Question 12

In the circuit shown below, let $\mathrm { R } _ { 1 } = 1 \Omega , \mathrm { R } _ { 2 } = 4 \Omega , \mathrm { C } = 0.5 \mathrm {~F} , \mathrm {~L} = 1 \mathrm { H } , \mathrm { v } \left( 0 ^ { - } \right) = 3 \mathrm {~V} , \mathrm { i } \left( 0 ^ { - } \right) = 1 \mathrm {~A} , \mathrm { v } _ { \mathrm { s } } ( \mathrm { t } ) = 4 \mathrm { u } ( \mathrm { t } )$ Draw the circuit in the s-domain and find $V ( s )$ and $v ( t )$ for $t \geq 0$ . $In the circuit shown below, let \mathrm { R } _ { 1 } = 1 \Omega , \mathrm { R } _ { 2 } = 4 \Omega , \mathrm { C } = 0.5 \mathrm {~F} , \mathrm {~L} = 1 \mathrm { H } , \mathrm { v } \left( 0 ^ { - } \right) = 3 \mathrm {~V} , \mathrm { i } \left( 0 ^ { - } \right) = 1 \mathrm {~A} , \mathrm { v } _ { \mathrm { s } } ( \mathrm { t } ) = 4 \mathrm { u } ( \mathrm { t } ) Draw the circuit in the s-domain and find V ( s ) and v ( t ) for t \geq 0 .$

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Question 13

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = 3 \sin ( 6 t ) u ( t ) , \quad h ( t ) = 2 \cos ( 6 t ) u ( t )$

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Question 14

Assume $\mathrm { L } = 1 \mathrm { H } , \mathrm { C } _ { 1 } = 0.2 \mathrm {~F} , \mathrm { C } _ { 2 } = 0.5 \mathrm {~F} , \mathrm { R } = 1 \Omega$ in the circuit shown below. Find the transfer function H(s) = Vo(s)/Vin(s). $Assume \mathrm { L } = 1 \mathrm { H } , \mathrm { C } _ { 1 } = 0.2 \mathrm {~F} , \mathrm { C } _ { 2 } = 0.5 \mathrm {~F} , \mathrm { R } = 1 \Omega in the circuit shown below. Find the transfer function H(s) = Vo(s)/Vin(s).$

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Question 15

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { 10 ^ { 5 } ( s + 100 ) } { ( s + 1000 ) ( s + 10000 ) }$

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Question 16

In the circuit shown below, let $\mathrm { R } _ { 1 } = 3 \Omega , \mathrm { R } _ { 2 } = 1 \Omega , \mathrm { C } = 0.2 \mathrm {~F} , \mathrm {~L} = 2 \mathrm { H } , \mathrm { v } \left( 0 ^ { - } \right) = 1 \mathrm {~V} , \mathrm { i } \left( 0 ^ { - } \right) = 3 \mathrm {~A} , \mathrm { i } _ { \mathrm { s } } ( \mathrm { t } ) = 3 \mathrm { u } ( \mathrm { t } )$ Draw the circuit in the s-domain and find $V _ { 0 } ( s )$ and $v _ { 0 } ( t )$ for $t \geq 0$ . $In the circuit shown below, let \mathrm { R } _ { 1 } = 3 \Omega , \mathrm { R } _ { 2 } = 1 \Omega , \mathrm { C } = 0.2 \mathrm {~F} , \mathrm {~L} = 2 \mathrm { H } , \mathrm { v } \left( 0 ^ { - } \right) = 1 \mathrm {~V} , \mathrm { i } \left( 0 ^ { - } \right) = 3 \mathrm {~A} , \mathrm { i } _ { \mathrm { s } } ( \mathrm { t } ) = 3 \mathrm { u } ( \mathrm { t } ) Draw the circuit in the s-domain and find V _ { 0 } ( s ) and v _ { 0 } ( t ) for t \geq 0 .$

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Question 17

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { s + 100 } { s + 1000 }$

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Question 18

In the circuit shown below, let $\mathrm { R } _ { 1 } = 3 \Omega , \mathrm { R } _ { 2 } = 1 \Omega , \mathrm { R } _ { 3 } = 2 \Omega , \mathrm { C } = 0.1 \mathrm {~F} , \mathrm {~L} = 1 \mathrm { H } , \mathrm { v } \left( 0 ^ { - } \right) = 2 \mathrm {~V} , \mathrm { i } \left( 0 ^ { - } \right) = 2 \mathrm {~A} , \mathrm { v } _ { \mathrm { s } } ( \mathrm { t } ) = 2 \mathrm { u } ( \mathrm { t } )$ Draw the circuit in the s-domain and find $V ( s )$ and $v ( t )$ for $t \geq 0$ . $In the circuit shown below, let \mathrm { R } _ { 1 } = 3 \Omega , \mathrm { R } _ { 2 } = 1 \Omega , \mathrm { R } _ { 3 } = 2 \Omega , \mathrm { C } = 0.1 \mathrm {~F} , \mathrm {~L} = 1 \mathrm { H } , \mathrm { v } \left( 0 ^ { - } \right) = 2 \mathrm {~V} , \mathrm { i } \left( 0 ^ { - } \right) = 2 \mathrm {~A} , \mathrm { v } _ { \mathrm { s } } ( \mathrm { t } ) = 2 \mathrm { u } ( \mathrm { t } ) Draw the circuit in the s-domain and find V ( s ) and v ( t ) for t \geq 0 .$

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Question 19

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { \left( s + 10 ^ { 4 } \right) ^ { 2 } \left( s + 10 ^ { 5 } \right) ^ { 2 } } { \left( s + 10 ^ { 3 } \right) ^ { 2 } \left( s + 10 ^ { 6 } \right) ^ { 2 } }$

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Question 20

Showing 1 - 20 of 26

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { \left( s + 10 ^ { 3 } \right) ^ { 2 } \left( s + 10 ^ { 6 } \right) ^ { 2 } } { 100 \left( s + 10 ^ { 4 } \right) ^ { 2 } \left( s + 10 ^ { 5 } \right) ^ { 2 } }$

Find the convolution y(t) of x(t) and h(t); that is, find $y ( t ) = h ( t ) * x ( t )$ $x ( t ) = u ( t ) - u ( t - 5 ) ,$ $h ( t ) = \delta ( t - 10 ) + \delta ( t - 12 )$

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = 4 e ^ { - 3 t } u ( t ) , \quad h ( t ) = 2 e ^ { - 5 t } u ( t )$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { \left( s + 10 ^ { 3 } \right) ^ { 2 } } { \left( s + 10 ^ { 4 } \right) ^ { 2 } }$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { s + 1000 } { s + 100 }$

Find the convolution y(t) of x(t) and h(t); that is, find $y ( t ) = h ( t ) ^ { * } x ( t )$ $x ( t ) = 2 \cos ( 3 t ) u ( t ) , \quad h ( t ) = 4 \cos ( 4 t ) u ( t )$

Find the convolution y(t) of x(t) and h(t); that is, $\text { find } y ( t ) = h ( t ) ^ { * } x ( t )$ $x ( t ) = t ^ { 2 } u ( t ) , \quad h ( t ) = u ( t )$

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = t u ( t ) , \quad h ( t ) = t u ( t )$

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = 3 \sin ( 6 t ) u ( t ) , \quad h ( t ) = 2 \cos ( 6 t ) u ( t )$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { 10 ^ { 5 } ( s + 100 ) } { ( s + 1000 ) ( s + 10000 ) }$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { s + 100 } { s + 1000 }$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { \left( s + 10 ^ { 4 } \right) ^ { 2 } \left( s + 10 ^ { 5 } \right) ^ { 2 } } { \left( s + 10 ^ { 3 } \right) ^ { 2 } \left( s + 10 ^ { 6 } \right) ^ { 2 } }$

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Exam 15: Circuits Analysis in the S-Domain

Find the convolution y(t) of x(t) and h(t); that is, find y(t)=h(t)∗x(t)y ( t ) = h ( t ) * x ( t )y(t)=h(t)∗x(t) x(t)=u(t)−u(t−5),x ( t ) = u ( t ) - u ( t - 5 ) ,x(t)=u(t)−u(t−5), h(t)=δ(t−10)+δ(t−12)h ( t ) = \delta ( t - 10 ) + \delta ( t - 12 )h(t)=δ(t−10)+δ(t−12)

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). x(t)=4e−3tu(t),h(t)=2e−5tu(t)x ( t ) = 4 e ^ { - 3 t } u ( t ) , \quad h ( t ) = 2 e ^ { - 5 t } u ( t )x(t)=4e−3tu(t),h(t)=2e−5tu(t)

Plot the magnitude and phase Bode diagrams of the transfer function given by H(s)=(s+103)2(s+104)2H ( s ) = \frac { \left( s + 10 ^ { 3 } \right) ^ { 2 } } { \left( s + 10 ^ { 4 } \right) ^ { 2 } }H(s)=(s+104)2(s+103)2​

Plot the magnitude and phase Bode diagrams of the transfer function given by H(s)=s+1000s+100H ( s ) = \frac { s + 1000 } { s + 100 }H(s)=s+100s+1000​

Find the convolution y(t) of x(t) and h(t); that is, find y(t)=h(t)∗x(t)y ( t ) = h ( t ) ^ { * } x ( t )y(t)=h(t)∗x(t) x(t)=2cos⁡(3t)u(t),h(t)=4cos⁡(4t)u(t)x ( t ) = 2 \cos ( 3 t ) u ( t ) , \quad h ( t ) = 4 \cos ( 4 t ) u ( t )x(t)=2cos(3t)u(t),h(t)=4cos(4t)u(t)

Find the convolution y(t) of x(t) and h(t); that is, find y(t)=h(t)∗x(t)\text { find } y ( t ) = h ( t ) ^ { * } x ( t ) find y(t)=h(t)∗x(t) x(t)=t2u(t),h(t)=u(t)x ( t ) = t ^ { 2 } u ( t ) , \quad h ( t ) = u ( t )x(t)=t2u(t),h(t)=u(t)

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). x(t)=tu(t),h(t)=tu(t)x ( t ) = t u ( t ) , \quad h ( t ) = t u ( t )x(t)=tu(t),h(t)=tu(t)

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). x(t)=3sin⁡(6t)u(t),h(t)=2cos⁡(6t)u(t)x ( t ) = 3 \sin ( 6 t ) u ( t ) , \quad h ( t ) = 2 \cos ( 6 t ) u ( t )x(t)=3sin(6t)u(t),h(t)=2cos(6t)u(t)

Assume L=1H,C1=0.2 F,C2=0.5 F,R=1Ω\mathrm { L } = 1 \mathrm { H } , \mathrm { C } _ { 1 } = 0.2 \mathrm {~F} , \mathrm { C } _ { 2 } = 0.5 \mathrm {~F} , \mathrm { R } = 1 \OmegaL=1H,C1​=0.2 F,C2​=0.5 F,R=1Ω in the circuit shown below. Find the transfer function H(s) = Vo(s)/Vin(s).

Plot the magnitude and phase Bode diagrams of the transfer function given by H(s)=105(s+100)(s+1000)(s+10000)H ( s ) = \frac { 10 ^ { 5 } ( s + 100 ) } { ( s + 1000 ) ( s + 10000 ) }H(s)=(s+1000)(s+10000)105(s+100)​

Plot the magnitude and phase Bode diagrams of the transfer function given by H(s)=s+100s+1000H ( s ) = \frac { s + 100 } { s + 1000 }H(s)=s+1000s+100​

Voltage, Current, Power, and Sources

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Circuit Analysis Methods

Circuit Theorems

Operational Amplifier Circuits

Capacitors and Inductors

RL and RC Circuits

RLC Circuits

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Analysis of Phasor Transformed Circuits

AC Power

Three-Phase Systems

Magnetically Coupled Circuits

The Laplace Transform

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Find the convolution y(t) of x(t) and h(t); that is, find $y ( t ) = h ( t ) * x ( t )$ $x ( t ) = u ( t ) - u ( t - 5 ) ,$ $h ( t ) = \delta ( t - 10 ) + \delta ( t - 12 )$

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = 4 e ^ { - 3 t } u ( t ) , \quad h ( t ) = 2 e ^ { - 5 t } u ( t )$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { \left( s + 10 ^ { 3 } \right) ^ { 2 } } { \left( s + 10 ^ { 4 } \right) ^ { 2 } }$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { s + 1000 } { s + 100 }$

Find the convolution y(t) of x(t) and h(t); that is, find $y ( t ) = h ( t ) ^ { * } x ( t )$ $x ( t ) = 2 \cos ( 3 t ) u ( t ) , \quad h ( t ) = 4 \cos ( 4 t ) u ( t )$

Find the convolution y(t) of x(t) and h(t); that is, $\text { find } y ( t ) = h ( t ) ^ { * } x ( t )$ $x ( t ) = t ^ { 2 } u ( t ) , \quad h ( t ) = u ( t )$

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = t u ( t ) , \quad h ( t ) = t u ( t )$

Find the convolution y(t) of x(t) and h(t); that is, find y(t) = h(t)*x(t). $x ( t ) = 3 \sin ( 6 t ) u ( t ) , \quad h ( t ) = 2 \cos ( 6 t ) u ( t )$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { 10 ^ { 5 } ( s + 100 ) } { ( s + 1000 ) ( s + 10000 ) }$

Plot the magnitude and phase Bode diagrams of the transfer function given by $H ( s ) = \frac { s + 100 } { s + 1000 }$