HAL: I'm sorry, Dave. I'm afraid I can't do that.Dave Bowman: lVhat's the problem?

HAL: I think you know what the problem is just as well as I do.- 2007: A Space Odyssey, Arthur C. Clarke

Prohlem L (15 points)

For all of the problems below,

V _0V for f<01 -LV f or r >0

a) Consider the circuit below. (5 points)

Rl

u

Provide a symbolic expression for the following at t = co:

{t t;

Solution:fvl

' : .., : ::,1, :

L.-

i

Krt IC r

vct =

Vtt =

Ycz =

Vt"z =

vc3 =

Rn*Ft

(i

Rz

lct =

I-IL1 -

icz =

irz =

icE =

(4 _)

\"t Rz

Cr

Rz

f. l C

L'

&ta LV -I t.,r

C,'f t=,}.,-, C, $p<..* ,

*, *\

t"- st"**-l

(rLL

\V =V,

f)f\l

r.:)L

b) Consider the circuit below. {5 points)

S*

q

Provide a symbolic expression for the following at t = oo:

t=-*

frtlIT I

-L

rc1 -

:-tL1 -

icz =

irz =

ics =

-_l-_Ri+F.

e

C

Ll, ]

(,+CatCa fvl

ul'tvc+

c) Since real capacitors have a leakage resistance, let's add that in. Consider the circuit below. [5 points)

*c

Provide a symbolic expression for the following at t = oo:

xtra Space

t*L,e*';*!.iJtf-t'h r -Ht z_:

f-:::* :

P.* /l tr ff{ig ir ( fi4+po)

t?; l/ R1;1/(fli + b )t-*'"tt-*.,')'#t--.r'

p- L3 4 Rp

U

,t)[ry{€ r+.

f1+ R+{^ f 3+ -.flu,e

LI

&tr - *.-tF*lnr-t Kq ) Kg

U

tc1 -

Itt =

rc? -

rL? -

IC3 -

f,+L-.,vcr = ffi-o*; i'v I

vlt= il

F*L,iYcz= tr-il- Lv-J

Y;;z = ti

fu[^ ' ScKthr_ ,S., ,. frlVcg = [d., + t,, - Rc+ ki',

i1$'

"Minor magicians take pains to fit this traditional wizardly bill. By contrast,the really powerful magicians take pleasure in looking like accountants."

- f onathan Strou{ The Amulet of Samarkand

Problem 3 (25 points)Consider the circuit below.

{}fqt

R3*( v'.'V*-Vo UC

tsti

=i Vi vt 1eI +€ ) h*t.*E

l--Vr12" *6

f - iv{.E;+

The switch is closed until t = 20 s, then opened. \t." V*, $"u, ; &.+_u *:r, t_. s,rs- ? !'1-r€ 11

% €*

vs

v ar-

lri+L raci.Shcv-t

rl Vr

al Provide a bolic expression for ir at t < 20 s?

't<:.c

t* -n t* jr- ---) i =Rz F3

V" -vr-Rr-

-tc€"€)Tft;E

ion for iz at t < Z0 s? {2.5

i(ro) =:\

| - t-tc-/*r 1*\tv; _\Ll i -\ Rz*Gtr( B, -+ q?3rfr"t*h

cl Provide a

tZ zo:

dl Provide a svmbolic

ion for ir at t > 20 s? [10

for iz at t > 20 s? [L0 points

fl). t<-)

V:t 4LSi_____l

[(o^) = -- ha,LL' fr.bp]

i(*) * i(*t r fi(zo) * ic*)],- (t *zo) 1z

L=-flr)

'And yet, will we ever come to an end of discussion and talk if we think we must always reply to replies? Forreplies come from those who either cannot understand what is said to them, or are so stubborn and contentious

that they refuse to give in even if they do understand."

- St. Augustine of Hippo , City of God

Problem 4 (30 points) i

For all the circuits below,

V _0Vforf<0h-LV for t>0

Provide an expression forvo"t[t) for t> 0.' '

a) (5 points)

fi,

v;n -f ? u*rt

vout(t)= I - e-*/Gc) frrll

b) (5 points) V"(o)-e>-_t

J-=

, i--\v

Vo.-t ( o)=OVV*.^,t-(*) ^lV

Ve.r*: V"u": V'a

= l: (+d({c

f",tr-&{

{in

veql(t)= ;tltT

^*t-le c)-L/ t-l

t.o

o',;'5-rt- Vc(ao)= lv{{: -+-.

Yct: = 1- E:V=

c) (5 pinx)

t-o-*{1.---*it

d) (5 poinx).L+Hr

Vi* T- L UoutnH-

t<a- * l*- V6to)=c;r\-vt-zo

.--\\-A-rr! I r ,G/ "(.)ti

Vot^t -

V.tra)= 0

-'-vL$-):cVi*,- vc-

vo"t[t]= tV

voulft] = L V

NLtz , ' 5 w>" C {- Dtror-.'{ti+".

i,.ftt.^tt< ( .r1 .fLa' aiJt

6 Lt\. t "tt *!

[^i<, i: C:it\'1 tru{t t,,h!t'. '1-,,...J{r,r,u.ch\1ur {r-u"r,,+1 C- I6tt) ,'s riu ,tt"$:Ly t^o?l

,

,'T t=c ; '1-at)atsgir / t r-t. =

-* ?v=t[iAu=ti].*x3

Fj.- .r.,uCL. A (-l;) r r,"t lr:3c,.\ tt*

t< o {co)*or---'-'-1 ,!'i + 'V;+6J-\'-\./ v

t'z o -.-*-,-L'

.K-a *Y f--vt

P:o'!.<p-c= o

V,"<:v.. *r/o= L+ rlo-il e L -

=lV

.61'O= U

1

,r,'k*

l

,1 "l'V'

lt\o):.,

R

+ t>o \.---{ t (r--'(b)t-r,"Q *

"->t

vsut(t)= I * ^:tRftq Cv3

vo,t

*

itr). t- t u-ur,T= LR./uor+(+) = i(+).R

= [- <- rl'e

Il"dt(v

vo"t(t)=.-tRlr-

I

x1 f rz]

t..,5*3 l1o)-'5

trot'-Vr-t**-

P rrlrr"4 =f u*1:=

ift) =+k".*'t , *rot-

LK

{-"t-(f) -- Ld_lffi

= r (tt. t1=)

+

= €'-rl-r-