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EENG223 Circuit Theory I / INFE221 Electric Circuits – Final Exam

 Faculty of Engineering

ELECTIC!L !N" ELECT#NIC E NGINEEING "E$!T%ENT

EENG223 Circuit Theory I

INFE221 – Electrical Circuits

FINAL EXAM 

Sprin 2!!"#!$

23 &une 2''(

"uration) 12' minutes

 Instructor%  #* +u,rer 

-ol.e all $ro0lems

-T"ENT-

 N%E  N!%E

-N!%E

G#$ N#*

 &ro'le( &oints

1 2'

2 2'

3 2'

4 2'

2

T#T!L 1'

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i o

5 !

12 6

/ 6

+

v  x

12 78 6

su9erno:e

v 1   v 2   v 3

/   v  x

;

i o

+

v  x

127

12 6

5!

/ 6

8 6

i 1

i 2 i 3

/  v  x

;

EENG223 Circuit Theory I / INFE221 Electric Circuits – Final Exam

1) In the circuit of Figure 1<

=a> Fin: the current io using no:al analysis* *1! pts+ =0> Fin: the current io using mesh analysis* *1! pts+

*a+

  +CL for the su9erno:e )

  1 1 15 ' 1*2 7

  x x

vv v v v− − = = ⇒ = −  

2 1 12 13*2 7v v= − = −  +CL for no:e 3 )

3 2 3 1 1 23

122 3 ' 38*4 7

8 12  x

v v v v v vv v

− − +− + + = ⇒ = = −   1 3' 3*1 !

12

v vi

  −= =

*'+

+7L e?n* can 0e @ritten only for mesh 1)

 

1 1 3

3 2 2

3 2 2 3

1 '

12 12 8= > ' =1>

= > 2

5 ! '*2 ! 5*2 !

-u0stitute in =1> 3*1 !

 x

i i i

i v i i

i i i i

i i

− + + − =

= − = − − =− = ⇒ = ⇒ =

⇒ = =

 

Fiure 1

i o

5 !

12 6

/ 6+

v  x

12 78 6

/   v  x

;

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EENG223 Circuit Theory I / INFE221 Electric Circuits – Final Exam

3) In the circuit in Figure 3< the o9Bam9s are i:eal

=a> Fin: the .oltage v' @hen =o9en circuit> f   R   = ∞ * *1! pts+

=0> Fin: the .oltage v' @hen 12 ,  f   R   = Ω * *1! pts+

=a>Ahen  f   R   = ∞ 

'1

2 '1

' 2

2' '*4 8 7

1'  4 7

2'

4 7

v

v v

v v

+

+

= − × = −

⇒ = = −

∴ = = −

=0>Ahen 12 ,  f   R   = Ω  < the first am9lifier may 0e consi:ere: as a summing am9lifier< @ith in9uts @hichare the '*4 7 an: v'*

 

'1 ' '

2 '1 '

' 2 ' '

4' 4' 1''*4 =8 > 7

2 12 3

1'

  =4 > 72' 3

84 7 1* 7

3

v v v

v v v

v v v v

+

+

 = − × + = − + ÷  

⇒ = = − +

∴ = ⇒ = − ⇒ = −

 

Fiure 3

;

2 , 6

4' , 6

1' , 6

1' , 6

 R  f  

v '

'*4 7

;

2 , 6

4' , 6

1' , 6

1' , 6

v '

v '1

v ;2

'*4 7

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EENG223 Circuit Theory I / INFE221 Electric Circuits – Final Exam

-* The s@itch in the circuit in Figure 4 has 0een close: for a long time* It is o9ene: at t   '* Fin:the ca9acitor .oltage v=t > for t D '*

!tt

'

B

 the circuit is un:er :c con:itions)

+7L for the loo9)

' '

' '

' 3''=3 > 1'' '

'*' ! =' > 1'' 7

i i

i v i−

− + + =

⇒ = ⇒ = =

Ahen the s@itch is o9ene: at t ')

  =' > =' > / 7v v+ −

= =  

+CL at no:e !) '3C i i= −   !lso< C dv

i C dt 

=  < '1''

vi   =

 '*3 '*3

3'*1 '*3

1''

= > ='>* 7t t 

dv dvv v

dt dt  

v t v e e− −

⇒ = − ⇒ = −

∴ = =

 

# the solution coul: 0e foun: 0y :etermining the time constant as   eq R C τ   = < @here eq R is foun: fromthe circuit

  +CL) ' '1''

3 1''3

t t t eq

t 

vi i v i R

i= = ⇒ = = Ω

 / 3 /1'1'' 1''*1 s = > ='>* 7

3 3

t t v t v e e

τ  

τ    − −∴ = × = ⇒ = =

 

;

 E

/' 7

Fiure -

t  C '

;

v = t  >

1'' 6

'*1 F

i '2   i '

3'' 6

;

 E

/' 7 i '2   i '

;

v ='  B

>

3   i ' 1'' 6

3'' 6

;

v = t  >

1'' 6

'*1 F

i '2   i '

i C

!

1'' 6

i '2   i '

!

i t 

;

v t 

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EENG223 Circuit Theory I / INFE221 Electric Circuits – Final Exam

*The s@itch in the circuit in Figure has 0een close: for a long time* It is o9ene: at t '*

*a+ Fin: the current i=t > for t D '* *2! pts+*'+ Fin: the total energy :issi9ate: in the 12' resistor after the s@itch is o9ene:* *, pts+

=a> !t t 'B the circuit is un:er :c con:itions)

=' > 1' ! =' > ' 7i v− −= − =  

Ahen the s@itch is o9ene: at t '< the resulting circuit is a sourceBfree LC circuit)

 

!t t '; =' > =' > 1' !i i+ −= = − *

3

' 5

'

1 11' ra:*/s

'*1 1' 1'

12'5'' / s<

2 2 '*1 H

 LC 

 R

 L

ω 

α α ω 

−

= = =× ×

Ω= = = ⇒

× 

Therefore< the res9onse is un:erB:am9e:* The solution for the current has the form

( )  2 2

1 2 '= > cos= > sin= > 8'' ra:*/st 

d d d i t e B t B t  α 

ω ω ω ω α  

−

= + = − =

To fin: the un,no@n coefficients< the initial con:itions are a99lie:* First the initial con:ition for the time

:eri.ati.e of current must 0e :etermine:*

( )1

+7L) = > = > ' =' > =' > =' > 1'*=' 12''> 12''' !/sdi di

 L v t Ri t v Ridt dt L

+ + ++ + = ⇒ = − + = − − =

1='> 1' !i B= = −  

t  C '1' J F

i = t  >1'' mH

2'' 6

12' 61' !

Fiure ,

12' 6

; v =' >

i =' >

1' !

2'' 6

1' J F

1'' mH12' 6

; v = t  >

i = t  >

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EENG223 Circuit Theory I / INFE221 Electric Circuits – Final Exam

( ) ( ) ( )1 2 1 2

1 2 2

cos= > sin= > sin= > cos= >

12''' 5''= 1'>' ='> K* !

8''

t t 

d d d d d d  

d 

die B t B t e B t B t  

dt 

dit B B B

dt 

α α 

α ω ω ω ω ω ω  

α ω 

− −= − + + − +

+ −= ⇒ = − + ⇒ = =

( )5''= > 1' cos=8'' > K*/sin=8'' > !t i t e t t  −= − +

=0> The energy :issi9ate: in the resistor is the total energy store: in the in:uctor an: the ca9acitor att   '; <

 2 2 21 1 1=' > =' > '*1 1' &

2 2 2total W Li Cv

+ += + = × × =