Exam 9: Phasors and Impedances

for the circuit shown below, let $\mathrm { Vs } ( \mathrm { t } ) = 396 \cos \left( 2 \pi 60 \mathrm { t } - 30 ^ { \circ } \right) \mathrm { V }$ , $\mathrm { R } _ { 1 } = 15 \Omega , \mathrm { R } _ { 2 } = 35 \Omega , \mathrm { R } _ { 3 } = 46 \Omega , \mathrm { L } _ { 1 } = 33 \mathrm { mH } , \mathrm { L } _ { 2 } = 63 \mathrm { mH } , \mathrm { C } = 126 \mu \mathrm { F }$ (a) Find impedances $Z _ { R 1 } , Z _ { R 2 } , Z _ { R 3 } , Z _ { L 1 } , Z _ { L 2 } , Z _ { C }$ , and phasor $V _ { S }$ for $v ( t )$ . (b) Find the equivalent impedance $Z _ { e q }$ of parallel combination of $Z _ { R 2 } + Z _ { C }$ and $Z _ { R 3 } + Z _ { L }$ . (c) Find the total impedance $Z _ { t }$ seen from the voltage source. (d) Find the phasor for the current $\mathrm { I }$ . (e) Find the phasor for load voltage $V _ { 0 }$ . (f) Find the phasors for $I _ { 1 }$ and $I _ { 2 }$ . $for the circuit shown below, let \mathrm { Vs } ( \mathrm { t } ) = 396 \cos \left( 2 \pi 60 \mathrm { t } - 30 ^ { \circ } \right) \mathrm { V } , \mathrm { R } _ { 1 } = 15 \Omega , \mathrm { R } _ { 2 } = 35 \Omega , \mathrm { R } _ { 3 } = 46 \Omega , \mathrm { L } _ { 1 } = 33 \mathrm { mH } , \mathrm { L } _ { 2 } = 63 \mathrm { mH } , \mathrm { C } = 126 \mu \mathrm { F } (a) Find impedances Z _ { R 1 } , Z _ { R 2 } , Z _ { R 3 } , Z _ { L 1 } , Z _ { L 2 } , Z _ { C } , and phasor V _ { S } for v ( t ) . (b) Find the equivalent impedance Z _ { e q } of parallel combination of Z _ { R 2 } + Z _ { C } and Z _ { R 3 } + Z _ { L } . (c) Find the total impedance Z _ { t } seen from the voltage source. (d) Find the phasor for the current \mathrm { I } . (e) Find the phasor for load voltage V _ { 0 } . (f) Find the phasors for I _ { 1 } and I _ { 2 } .$

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Question 1

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$\begin{array} { l } Z _ { \mathrm { L } 1 } = \mathrm { j } \omega \mathrm { L } _ { 1 } = \mathrm { j } 12.4407 \Omega , \mathrm { Z } _ { \mathrm { L } 2 } = \mathrm { j } \omega \mathrm { L } _ { 2 } = \mathrm { j } 23.7504 \Omega \\\mathrm { Z } _ { \mathrm { C } } = 1 / ( \mathrm { j } \omega \mathrm { C } ) = - \mathrm { j } 21.0522 \Omega \\\mathrm { Z } _ { \mathrm { a } } = \left( \mathrm { R } _ { 2 } + \mathrm { Z } _ { \mathrm { C } } \right) \| \left( \mathrm { R } _ { 3 } + \mathrm { Z } _ { \mathrm { L } 2 } \right) = 25.9642 - \mathrm { j } 2.558 \Omega \\\mathrm { Z } _ { \mathrm { t } } = \mathrm { R } _ { 1 } + \mathrm { Z } _ { \mathrm { L } 1 } + \mathrm { Z } _ { \mathrm { a } } = 40.9642 + \mathrm { j } 9.8828 \Omega \\\mathrm { I } = \mathrm { V } _ { \mathrm { s } } / \mathrm { Z } _ { \mathrm { t } } = 6.8094 - \mathrm { j } 6.4763 \mathrm {~A} \\\mathrm {~V} _ { \mathrm { o } } = \mathrm { Z } _ { \mathrm { a } } \times \mathrm { I } = 160.2351 - \mathrm { j } 185.5697 = 245.1762 \angle - 49.1902 ^ { \circ } \mathrm { V } \\\mathrm { V } _ { \mathrm { o } } ( \mathrm { t } ) = 245.1762 \cos \left( 2 \pi 60 \mathrm { t } - 49.1902 ^ { \circ } \right) \mathrm { V } \\\mathrm { I } _ { 1 } = \mathrm { V } _ { \mathrm { o } } / \left( \mathrm { R } _ { 2 } + \mathrm { Z } _ { \mathrm { C } } \right) = 5.7037 - \mathrm { j } 1.8713 = 6.0028 \angle - 18.1636 ^ { \circ } \mathrm { A } \\\mathrm { I } _ { 2 } = \mathrm { V } _ { \mathrm { o } } / \left( \mathrm { R } _ { 3 } + \mathrm { Z } _ { \mathrm { L } 2 } \right) = 1.1057 - \mathrm { j } 4.605 = 4.7359 \angle - 76.4981 ^ { \circ } \mathrm { A }\end{array}$ clear all;
$\mathrm { R } I = 15 ; \mathrm { R } 2 = 35 ; \mathrm { R } 3 = 46 ; \mathrm { Ll } = 33 \mathrm { e } - 3 ; \mathrm { L } 2 = 63 \mathrm { e } - 3 ; \mathrm { C } = 126 \mathrm { e } - 6 ; \mathrm { f } = 60$ ;
$\mathrm { W } = 2 * \mathrm { p }$ * $\mathrm { f }$
$2 L 1 = j * W ^ { * } L 1$
$\mathrm { ZL } 2 = \mathrm { j } ^ { * } w ^ { \star } \mathrm { L } 2$
$\mathrm { ZC } = 1 / \left( \mathrm { J } ^ { * } \mathrm {~W} ^ { * } \mathrm { C } \right)$
$\mathrm { Vs } = \mathrm { P } 2 \mathrm { Rd } ( 396 , - 30 )$
$\mathrm { Za } = \mathrm { P } ( [ \mathrm { R } 2 + \mathrm { ZC } , \mathrm { R } 3 + \mathrm { ZL } 2 ] )$
$\mathrm { Zt } = \mathrm { R } 1 + \mathrm { ZL } 1 + \mathrm { Za }$
$\mathrm { I } = \mathrm { Vs } / \mathrm { Zt }$
$\mathrm { IP } = \mathrm { R } 2 \mathrm { P } ( \mathrm { I } )$
$V o = Z a * I$
$\mathrm { Vop } = \mathrm { R } 2 \mathrm { P } ( \mathrm { Vo } )$
$I 1 - V \circ / ( R 2 + Z C )$
$I 1 p = R 2 P$ (Il)
$I 2 = \mathrm { VO } / ( \mathrm { R } 3 + \mathrm { ZL } 2 )$
I $2 \mathrm { p } = \mathrm { R } 2 \mathrm { P } ( \mathrm { I } 2 )$

Find V in polar notation. $\mathbf { V } = 11 \angle 110 ^ { 0 } + 17 \angle - 75 ^ { \circ }$

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Question 2

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$\mathbf { V } = 0.6377 - \mathrm { j } 6.0841 = 6.1174 \angle - 84.0164 ^ { \circ } \mathrm { V }$
clear all;
V1-P2Rd (11, 110)
V1p=R2P (V1)
V2=P2Rd (17, -75 )
V2p=R2P (V2)
V=V1+V2
Vp-R2P (V)

for the circuit shown below, let $\mathrm { vs } ( \mathrm { t } ) = 457 \cos \left( 2 \pi 60 \mathrm { t } - 110 ^ { \circ } \right) \mathrm { V } , \mathrm { R } _ { 1 } = 20 \Omega , \mathrm { R } _ { 2 } = 65 \Omega , \mathrm { L } _ { 1 } = 35 \mathrm { mH } , \mathrm { L } _ { 2 } = 85 \mathrm { mH } , \mathrm { C } = 95 \mu \mathrm { F }$ (a) Find impedances $Z _ { \mathrm { R } 1 } , Z _ { L 1 } , Z _ { R 2 }$ , ZL2, Z $Z _ { C }$ and phasor $V _ { S }$ for vs(t). (b) Find the equivalent impedance $Z _ { a }$ of parallel combination of $Z _ { R 2 } + Z _ { L 2 }$ and $Z _ { C }$ . (c) Find the total impedance $Z _ { t }$ seen from the voltage source. (d) Find the phasor for the current I. (e) Find the phasor for $V _ { C }$ across the capacitor. (f) Find $v _ { C } ( t )$ . $for the circuit shown below, let \mathrm { vs } ( \mathrm { t } ) = 457 \cos \left( 2 \pi 60 \mathrm { t } - 110 ^ { \circ } \right) \mathrm { V } , \mathrm { R } _ { 1 } = 20 \Omega , \mathrm { R } _ { 2 } = 65 \Omega , \mathrm { L } _ { 1 } = 35 \mathrm { mH } , \mathrm { L } _ { 2 } = 85 \mathrm { mH } , \mathrm { C } = 95 \mu \mathrm { F } (a) Find impedances Z _ { \mathrm { R } 1 } , Z _ { L 1 } , Z _ { R 2 } , ZL2, Z Z _ { C } and phasor V _ { S } for vs(t). (b) Find the equivalent impedance Z _ { a } of parallel combination of Z _ { R 2 } + Z _ { L 2 } and Z _ { C } . (c) Find the total impedance Z _ { t } seen from the voltage source. (d) Find the phasor for the current I. (e) Find the phasor for V _ { C } across the capacitor. (f) Find v _ { C } ( t ) .$

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Question 3

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$Z _ { R 1 } = R _ { 1 } = 20 \Omega , Z _ { R 2 } = R _ { 2 } = 65 \Omega$ $\begin{array} { l } Z _ { \mathrm { L } 1 } = \mathrm { j } \omega \mathrm { L } _ { 1 } = \mathrm { j } 13.1947 \Omega , \mathrm { Z } _ { \mathrm { L } 2 } = \mathrm { j } \omega \mathrm { L } _ { 2 } = \mathrm { j } 32.0442 \Omega \\\mathrm { Z } _ { \mathrm { C } } = 1 / ( \mathrm { j } \omega \mathrm { C } ) = - \mathrm { j } 27.9219 \Omega \\\mathrm { Z } _ { \mathrm { a } } = \left( \mathrm { R } _ { 2 } + \mathrm { Z } _ { \mathrm { L } 2 } \right) \| \mathrm { Z } _ { \mathrm { C } } = 11.9463 - \mathrm { j } 28.6796 \Omega \\\mathrm { Z } _ { \mathrm { t } } = \mathrm { R } _ { 1 } + \mathrm { Z } _ { \mathrm { L } 1 } + \mathrm { Z } _ { \mathrm { a } } = 31.9463 - \mathrm { j } 15.4849 \Omega \\\mathrm { I } = \mathrm { V } _ { \mathrm { s } } / \mathrm { Z } _ { \mathrm { t } } = 1.3143 - \mathrm { j } 12.8055 \mathrm {~A} \\\mathrm {~V} _ { \mathrm { c } } = \mathrm { Z } _ { \mathrm { a } } \times \mathrm { I } = - 351.5538 - \mathrm { j } 190.6723 = 399.9325 \angle - 151.5259 ^ { \circ } \mathrm { V } \\\mathrm { V } _ { \mathrm { c } } ( \mathrm { t } ) = 399.9325 \cos \left( 2 \pi 60 \mathrm { t } - 151.5259 ^ { \circ } \right) \mathrm { V }\end{array}$ $\begin{array}{l}\text { clear all; }\\\mathrm { R } 1 = 20 ; \mathrm { R } 2 = 65 ; \mathrm { Ll } = 35 \mathrm { e } - 3 ; \mathrm { L } 2 = 85 \mathrm { e } - 3 ; \mathrm { C } = 95 \mathrm { e } - 6 ; \mathrm { f } = 60 \text {; }\\w = 2 ^ { \star } p i * f\\\text { ZL1-j * } W ^ { * } L 1\\\mathrm { ZL } 2 = j ^ { \star } w ^ { \star } \mathrm { L } 2\\\mathrm { ZC } - 1 / \left( j ^ { * } \mathrm { w } ^ { * } \mathrm { C } \right)\\\mathrm { Vs } = \mathrm { P } 2 \operatorname { Rd } ( 457 , - 110 )\\\mathrm { Z } a = \mathrm { P } ( [ \mathrm { R } 2 + \mathrm { ZL } 2 , \mathrm { ZC } ] )\\2 t = R 1 + 2 L 1 + 2 a\\\mathrm { I } = \mathrm { V } s / \mathrm { Zt }\\I p = R 2 P \text { (I) }\\\mathrm { Vc } = 2 \mathrm { a } ^ { \star } \mathrm { I }\\\mathrm { Vcp } = \mathrm { R } 2 \mathrm { P } ( \mathrm { Vc } )\end{array}$

for the circuit shown below, let ()=545 2\pi60-6 ,=20\Omega,=33\Omega,=63,=72 , =128\mu (a) Find impedances $Z _ { L 1 } , Z _ { L 2 }$ , and $Z _ { C }$ . (b) Find the equivalent impedance $\mathrm { Z } _ { \mathrm { a } }$ of parallel combination of $\mathrm { Z } _ { \mathrm { L } 2 }$ and $\mathrm { Z } _ { \mathrm { R } 2 } + \mathrm { Z } _ { \mathrm { C } }$ . (c) Find the total impedance $Z _ { t }$ seen from the voltage source. (d) Find the phasor for the current I. (e) Find the phasor for $V _ { o }$ across the inductor $L _ { 2 }$ . (f) Find the phasors for $I _ { 1 }$ and $I _ { 2 }$ . (g) Find $\mathrm { Vo } _ { \mathrm { o } } ( \mathrm { t } )$ . $for the circuit shown below, let \begin{array} { l } \mathrm { vs } ( \mathrm { t } ) = 545 \cos \left( 2 \pi 60 \mathrm { t } - 60 ^ { \circ } \right) , \mathrm { R } _ { 1 } = 20 \Omega , \mathrm { R } _ { 2 } = 33 \Omega , \mathrm { L } _ { 1 } = 63 \mathrm { mH } , \mathrm { L } _ { 2 } = 72 \mathrm { mH } \text {, } \\ \mathrm { C } = 128 \mu \mathrm { F } \end{array} (a) Find impedances Z _ { L 1 } , Z _ { L 2 } , and Z _ { C } . (b) Find the equivalent impedance \mathrm { Z } _ { \mathrm { a } } of parallel combination of \mathrm { Z } _ { \mathrm { L } 2 } and \mathrm { Z } _ { \mathrm { R } 2 } + \mathrm { Z } _ { \mathrm { C } } . (c) Find the total impedance Z _ { t } seen from the voltage source. (d) Find the phasor for the current I. (e) Find the phasor for V _ { o } across the inductor L _ { 2 } . (f) Find the phasors for I _ { 1 } and I _ { 2 } . (g) Find \mathrm { Vo } _ { \mathrm { o } } ( \mathrm { t } ) .$

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Question 4

for the circuit shown below, let ()=575 2\pi60+3 =35\Omega,=53\Omega,=76,=117\mu,=106\mu (a) Find impedances $\mathrm { Z } _ { \mathrm { L } } , \mathrm { Z } _ { \mathrm { Cl } } , \mathrm { Z } _ { \mathrm { C } 2 }$ , and phasor $\mathrm { V } _ { \mathrm { S } }$ for $\mathrm { vs } ( \mathrm { t } )$ . (b) Find the equivalent impedance $Z _ { a }$ of parallel combination of $Z _ { C 2 }$ and $Z _ { R 2 } + Z _ { L }$ . (c) Find the total impedance $Z _ { t }$ seen from the voltage source. (d) Find the phasor for the current I. (e) Find the phasor for $V _ { o }$ across the capacitor $C _ { 2 }$ . $for the circuit shown below, let \begin{array} { l } \mathrm { vs } ( \mathrm { t } ) = 575 \cos \left( 2 \pi 60 \mathrm { t } + 30 ^ { \circ } \right) \\ \mathrm { R } _ { 1 } = 35 \Omega , \mathrm { R } _ { 2 } = 53 \Omega , \mathrm { L } = 76 \mathrm { mH } , \mathrm { C } _ { 1 } = 117 \mu \mathrm { F } , \mathrm { C } _ { 2 } = 106 \mu \mathrm { F } \end{array} (a) Find impedances \mathrm { Z } _ { \mathrm { L } } , \mathrm { Z } _ { \mathrm { Cl } } , \mathrm { Z } _ { \mathrm { C } 2 } , and phasor \mathrm { V } _ { \mathrm { S } } for \mathrm { vs } ( \mathrm { t } ) . (b) Find the equivalent impedance Z _ { a } of parallel combination of Z _ { C 2 } and Z _ { R 2 } + Z _ { L } . (c) Find the total impedance Z _ { t } seen from the voltage source. (d) Find the phasor for the current I. (e) Find the phasor for V _ { o } across the capacitor C _ { 2 } .$

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Question 5

use phasors to express the following equation as a single cosine. $v ( t ) = - 45 \sin \left( 2 \pi 50 t - 120 ^ { \circ } \right) + 65 \sin \left( 2 \pi 50 t + 150 ^ { \circ } \right)$

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Question 6

Find the phasor I for $\mathrm { i } ( \mathrm { t } ) = - 25 \cos \left( 2 \pi 50 \mathrm { t } - 60 ^ { \circ } \right) \mathrm { A }$

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Question 7

Represent V in polar notation, and find the time domain expression v(t) for the voltage when the frequency is f = 60 Hz. $\mathbf { V } = \frac { 45 \angle 150 ^ { \circ } + 35 \angle 80 ^ { \circ } } { 5 \angle 60 ^ { \circ } + 10 \angle - 15 ^ { \circ } }$

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Question 8

Showing 1 - 8 of 8

Find V in polar notation. $\mathbf { V } = 11 \angle 110 ^ { 0 } + 17 \angle - 75 ^ { \circ }$

use phasors to express the following equation as a single cosine. $v ( t ) = - 45 \sin \left( 2 \pi 50 t - 120 ^ { \circ } \right) + 65 \sin \left( 2 \pi 50 t + 150 ^ { \circ } \right)$

Find the phasor I for $\mathrm { i } ( \mathrm { t } ) = - 25 \cos \left( 2 \pi 50 \mathrm { t } - 60 ^ { \circ } \right) \mathrm { A }$

Represent V in polar notation, and find the time domain expression v(t) for the voltage when the frequency is f = 60 Hz. $\mathbf { V } = \frac { 45 \angle 150 ^ { \circ } + 35 \angle 80 ^ { \circ } } { 5 \angle 60 ^ { \circ } + 10 \angle - 15 ^ { \circ } }$

Voltage, Current, Power, and Sources

Circuit Laws

Circuit Analysis Methods

Circuit Theorems

Operational Amplifier Circuits

Capacitors and Inductors

RL and RC Circuits

RLC Circuits

Analysis of Phasor Transformed Circuits

AC Power

Three-Phase Systems

Magnetically Coupled Circuits

The Laplace Transform

Circuits Analysis in the S-Domain

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Exam 9: Phasors and Impedances

Find V in polar notation. V=11∠1100+17∠−75∘\mathbf { V } = 11 \angle 110 ^ { 0 } + 17 \angle - 75 ^ { \circ }V=11∠1100+17∠−75∘

use phasors to express the following equation as a single cosine. v(t)=−45sin⁡(2π50t−120∘)+65sin⁡(2π50t+150∘)v ( t ) = - 45 \sin \left( 2 \pi 50 t - 120 ^ { \circ } \right) + 65 \sin \left( 2 \pi 50 t + 150 ^ { \circ } \right)v(t)=−45sin(2π50t−120∘)+65sin(2π50t+150∘)

Find the phasor I for i(t)=−25cos⁡(2π50t−60∘)A\mathrm { i } ( \mathrm { t } ) = - 25 \cos \left( 2 \pi 50 \mathrm { t } - 60 ^ { \circ } \right) \mathrm { A }i(t)=−25cos(2π50t−60∘)A

Voltage, Current, Power, and Sources

Circuit Laws

Circuit Analysis Methods

Circuit Theorems

Operational Amplifier Circuits

Capacitors and Inductors

RL and RC Circuits

RLC Circuits

Analysis of Phasor Transformed Circuits

AC Power

Three-Phase Systems

Magnetically Coupled Circuits

The Laplace Transform

Circuits Analysis in the S-Domain

Filters

Find V in polar notation. $\mathbf { V } = 11 \angle 110 ^ { 0 } + 17 \angle - 75 ^ { \circ }$

use phasors to express the following equation as a single cosine. $v ( t ) = - 45 \sin \left( 2 \pi 50 t - 120 ^ { \circ } \right) + 65 \sin \left( 2 \pi 50 t + 150 ^ { \circ } \right)$

Find the phasor I for $\mathrm { i } ( \mathrm { t } ) = - 25 \cos \left( 2 \pi 50 \mathrm { t } - 60 ^ { \circ } \right) \mathrm { A }$