fundamentals of electrical engineering i 9.71

8/10/2019 Fundamentals of Electrical Engineering i 9.71

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Fundamentals of Electrical Engineering I

By:

Don Johnson


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Fundamentals of Electrical Engineering I

By:

Don Johnson

Online:

OpenStax-CNX


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s (x, t) x x0 s (x0, t)


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-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Am

plitude


12/333

s (x) = (r (x) , g (x) , b (x))T

97 65


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<

>

\

Source Transmitter Channel Receiver Sink

message modulated

message

corrupted

modulatedmessage

demodulated

message

s(t) x(t) r(t) s(t)


14/333

System

x(t) y(t)

x (t) y (t)

s (t)

s (t) x (t)

s (t) x (t)

r (t)

s (t) s (t)


15/333

s (t) = Acos(2f t + ) Acos(t + )

A

f s1 t

106 = 2f t= 0

= 2

Asin(2f t + ) = Acos

2f t + 2

Acos(2f n + ) n

cos (2f n) = cos (2 (f+ 1) n)


16/333

A = A0(1 + kT)

A0 k

T T

s=

1

T

T0

s2 (t) dt

T s (t) = s (t + T)

s (t) = Asin (2f0t + )

sq (t)


17/333

A

A

t

sq(t)

22

x (t) = Asin(2f0t) 0 t T

T A

f0

N

T

x (t) = A1sin(2f1t) + A2sin(2f2t) 0 t T

T


18/333

A1

A2

N1

A1

N2 A2 N1N2 T

log2(N1N2)


19/333

cos( + ) = cos ()cos()

sin()sin() cos(2 (f+ 1) n) = cos (2f n)cos(2n)

sin(2f n)sin(2n) = cos(2f n)

1 1


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1

1 i

i

i

j jb =

b2 z ab a b j jb 0 b a b

a x b y


22/333

a

b

r

x y z z = a+ jb j y z

z = a+jb Re (z) a

j

z Im (z) b

j

z

z z

z = Re (z) +jIm (z)

z = Re (z) jIm (z)

a1+jb1+ a2+jb2 = a1+ a2+j (b1+ b2)

j ja j j (jb) = b j2 = 1 j 90

Re (z) = z+z

2 Im (z) = zz

2j


23/333

a +jb =

a2 + b2

aa2 + b2

+j ba2 + b2

a b

z = a +jb = r

r= |z| = a2 + b2a= rcos()

b= rsin()

= arctanba

r z |z|

3 2j

z

z = rej

ej = cos() +jsin()

cos() =ej + e(j)

2

sin() =ej e(j)

2j

ex = 1 + x

1!+

x2

2! +

x3

3! + . . .

j x

e

j

= 1 + j

1!2

2!j3

3! + . . .

j2 = 1 j3 = j j4 = 1

ej = 1 2

2! + +j

1!

3

3! + . . .

cos() sin()


24/333

z1 z2 = (a1 a2) +j (b1 b2)

z1z2 = (a1+jb1) (a2+jb2)

= a1a2 b1b2+j (a1b2+ a2b1)

z1z2

= a1+jb1a2+jb2= a1+jb1a2+jb2

a2jb2a2jb2

= (a1+jb1)(a2jb2)a22+b22

= a1a2+b1b2+j(a2b1a1b2)a22+b22

z1z2 = r1ej1r2ej2

= r1r2ej(1+2)

z1z2

= r1ej1

r2ej2=

r1r2

ej(12)

s= j2f

s + 2

s2 + s + 1

s= j2f


25/333

s + 2

s2 + s + 1=

j2f+ 2

42f2 +j2f+ 1

=

4 + 42f2ejarctan(f)

(1 42f2)2 + 42f2ejarctan

2f

142f2

=

4 + 42f2

1 42f2 + 164f4 ejarctan(f)arctan

2f

142f2

x y

s (t) = Acos(2f0t + )

A f0

s (t) = Aej(2f0t+)

= Aejej2f0t

j 1 Aej

A


26/333

f0 f0

cos(2f t) =ej2ft + e(j2ft)

2

sin(2f t) =ej2ft e(j2ft)

2j

ej2ft = cos (2f t) +jsin(2f t)

Acos(2f t + ) = Re

Aejej2ft

Asin(2f t + ) = Im

Aejej2ft


27/333

f T T = 1f

A t= 0 A f T 1f


28/333

s (t) = et

Exponential

t

e1

1

1e

0.368

s (t) = Aejet ej2ft

= Aeje(1+j2f)t

t

u (t)

u (t) =

0 t < 01 t > 0

t

u(t)

1


29/333

s (t) = Asin(2f t) u (t)

p(t) =

0 t < 0

1 0< t <

0 t >

1

t

p(t)

sq (t)

t

Square Wave

A

T


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p(t)

p(t) = u (t) u (t )

T

A


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s (n) n ={. . . , 1, 0, 1, . . . }

(n m) n= m

n

sn

1

s (n) = ej2fn

s (n) = Acos(2f n + )

f 12 , 12

ej2(f+m)n = ej2fnej2mn

= ej2fn

2

(n) =

1 n= 00


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1

n

n

m s (m) m (n m)

s (n) =

m=

s (m) (n m)

s (n) {a1, . . . , aK} A


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y (t) = S(x (t)) x y

Systemx(t) y(t)

x (t) y (t)

y (t) =S(x (t)) S()

S1[] S2[]x(t) y(t)w(t)

w (t) =S1(x (t)) y (t) =S2(w (t)) x (t)


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x(t)

x(t)

x(t)

+y(t)

S1[]

S2[]

x (t)

x (t)

y (t) y (t) = S1(x (t))+ S2(x (t))

x (t)

S1[]x(t) e(t) y(t)

S2[]

+

y (t) = S1(e (t)) e (t)

y (t) e (t) = x (t) S2(y (t))


35/333

x (t) y (t)

y (t) = Asin(2f0t) u (t) t = 0 A f0

y (t) = Gx (t)

AmplifierG

1 G

y (t) = x (t )


36/333

Delay

t= t= 0

y (t) = x (t)

Time

Reverse


37/333

y (t) = ddtx (t)

t =

y (t) =

t

x () d

S() S(G1x1(t) + G2x2(t)) = G1S(x1(t)) + G2S(x2(t))

S(Gx (t)) = GS(x (t))

2x (t) x (t) +x (t)

S(0) = 0 S(0) =S(x (t)

x (t)) = S(x (t))

S(x (t)) = 0

x (t) = et + sin (2f0t)

S(x (t))

y (t) = S

et

+ S(sin (2f0t))

(y (t) = S(x (t))) (y (t ) = S(x (t )))


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y (t) = ddt(x)

y (t) = d2

dt2 (x)

y (t) =ddt(x)

2

y (t) = dxdt + x

y (t) = x1+ x2

y (t) = x (t

)

y (t) = cos(2f t) x (t)

y (t) = x (t) y (t) = x2 (t)

y (t) = |x (t) | y (t) = mx (t) + b

1

1+3j

2 1 +j+ ej

2

ej3 + ej + e(j

3 )

2713

315


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jj

j2j

jjj

3sin (24t)

2cos

260t + 4

2cos

t + 6

+ 4sin

t 3

S

Re

Aej2ft

=Re

S

Aej2ft

Re[] S[]

Re[]S[]

Aej2ft

Aej2ft

S[Re[Aej2ft]]

Re[S[Aej2ft]]

v (t) = Re (V est) V s V s

v (t) = cos(5t) v (t) = sin

8t + 4

v (t) = et

v (t) = e(3t)sin

4t + 34

v (t) = 5e(2t)sin(8t + 2) v (t) = 2

v (t) = 4sin (2t) + 3cos (2t) v (t) = 2cos

100t + 6

3sin 100t + 2


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10

t1

s(t)

10

t1

s(t)

2

10

t1

s(t)

2

t1

s(t)

11

2

t1

s(t)

-1

1

2 3 4


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x (t) y (t)

t1

1

x(t)

2 3

1

1

1 2 3 t

y(t)

t1

1

x(t)

2 3

0.5

x (t) y (t)


43/333

x(t)

t1

y(t)

t1

1/2

1/2

1

12

u (t)

x(t)

t1

2

2 3 4

2

x (t) r (t)

x(t)

t1

1r(t)

t1

1

2

x1(t)


44/333

x2(t)

x1(t)

t1

1

2 3

t1

1

1/2

x2(t)

dy (t)

dt + ay (t) = x (t)

a

x (t) = u (t) y (t) =

1 e(at)u (t)

t= 10


45/333

z+ z = a +jb + a

jb = 2a= 2Re (z) z

z = a +jb

(a

jb) = 2jb = 2 (j, Im (z))

3 2j r

13 =

32 + (2)2

arctan 23 0.588 33.7 13 (33.7)

zz = (a +jb) (a jb) = a2 + b2 zz = r2 = (|z|)2

sq (t) =

n=(1)nApT/2

t nT2

t t

t t

y (t) = Gx (t

)

y (t) = x ( (t )) = x (t + ) y (t) =x ((t) )


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+

v

i

p (t) = v (t) i (t)

t

E(t) = t

p () d


49/333

R

+

v

i

v = Ri

R

v (t) = Ri (t)

i = Gv G 1R

p (t) = Ri2 (t) = 1

Rv2 (t)

C

+

v

i

i= Cdv(t)dt


50/333

q = Cv

i (t) = Cdv (t)

dt v (t) =

1

C

t

i () d

p (t) = C v (t)dv (t)

dt

t

E(t) =1

2Cv2 (t)

t=

L+

v

i

v = L di(t)dt

v (t) = Ldi (t)

dt i (t) =

1

L

t

v () d

p (t) = Li (t)di (t)

dt t

E(t) =1

2Li2 (t)


51/333

+

v

i

vs+

+

v

i

is

v = vs i i= is v

v = vs i =is


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v

(t)

vin+

R1

R2

+

vout

vin +

R1

R2

+

vout

iout

+ v1i1

+

v

i

Systemvin(t) vout(t)

Source

v

v

R2

R2 v


53/333

R2

(i) i1 = 0i1 i2 = 0

i + i2 = 0

vin+

R1

R2

+

vout

vin+

R1

R2

+

vout

iout

+ v1i1

+

v

i

v

v

R2

n


54/333

v1+ v2 v= 0

v= v

v1 = R1i1

v

=R2i

(i) i1 = 0i1 i

= 0

v+ v1+ v

= 0

v

v

v

=v1+ v

v

= R1i1 + R2i

i1 = i

v = R1i +R2i = (R1+ R2) i i

= v R1+R2

v

= R2

R1+ R2v


55/333

P = k

vkik

vkik P = 0

i

i

=v

R1+R2 R2 v

= R2R1+R2 v

P2 = R2

(R1+ R2)2 v

2

P2

v2

R

i2R

L

A

R=

L

A

R1

i

(v

i

) =

1

R1+ R2v

2


56/333

vin+

R1

R2

+

vout

vin+

R1

R2

+

vout

iout

+ v1i1

+

v

i

vin vout

R2

voutvin

= R2

R1+ R2

i1 i2 vini1

=R1+ R2


57/333

vin+

R1

R2 vin+

R1+R2

R1 + R2 R2

R1 R2

iout

iin R1 R2

iouti1

+

v1

+

v2iin

+

v

v1 = v v2 = v

iin i1 i2 = 0

iout= R1

R1+ R2iin

iout

R1 R2 1R1

+ 1R2

1= R1R2R1+R2

R1 R2


58/333

R1 R2 R1R2

R1+R2

i2 = G2G1+G2

i

i2 = R1R1+R2

i

R1 R2

i

i2


59/333

vin+

R1

R2

+

vout RL

source system sink

vout = R2R1+R2

vin

RL

R2 RL

i2 = voutR2

iL= voutRL

R1

i1 = i2+ iL i1 = vout 1R2 + 1RL Req R2 RL R1

v1 = R1vout

Req

vin= v1+ vout v1

vin= vout

R1Req

+ 1

voutvin

= Req

R1+ Req

Req R2

Req = 1R2

+ 1RL

1

1R2

1RL R2 RL


60/333

R1

R2 R3

R4

R2 R3 R4 R1

RT =R1 (R2 R3+ R4)

RT = R1R2R3+ R1R2R4+ R1R3R4

R1R2+ R1R3+ R2R3+ R2R4+ R3R4

3 2

RL R1 R2

R2R1+R2

=

1 + R1R2

1


61/333

R1

R2

RN

+

v2

+

v

R

T

G1 G2 GNGT

i2

i

RT =PN

n=1Rn vn = RnRT

v GT =PN

n=1Gn

in= GnGT

i


62/333

vin+

R1

R2

+

v

i

v i

v= (R1 R2) i + R2R1+ R2

vin

veq+

Req

+

v

i

v= Reqi + veq

Req = R1 R2 veq =R2

R1+R2vin

v= Reqi + veq


63/333

i v = Ri voc

isc = veqReq

veq = voc

Req=

voc

isc

vin+

R1

R2

+

v

i

R1

R2

R3iin

voc


64/333

R1 R2 R3 voc =

iinR3R1R1+R2+R3

R3

R3

isc =

iinR1

R1+R2

R3(R1+R2)R1+R2+R3

R3 R1 R2 Req= R3 R1+ R2

veq+

Req

+

v

i

Sourcesand

Resistors

+

v

i

+

v

i

Reqieq

Thvenin Equivalent Mayer-Norton Equivalent

v= Reqi + veq

i= v

Req ieq

ieq = veqReq


65/333

R1

R2

R3iin

RL v =

veqRLRL+Req

i = ieqReqRL+Req


66/333

vin+

R

C vout

+

RC

vin = vR+ vout i = Cdvoutdt

vR= Ri

RCdvout

dt + vout= vin


67/333

vin+

R

C vout

+

RC

RC vin = Vinej2ft

Vin

v = Ri v = V ej2ft i = VRej2ft

I=

V

R

i = Cdvdt i= C V j2f ej2ft I=CV j2f v = Ldidt v= LI j2f ej2ft V =LI j2f

Z= VI


68/333

R

+

v

i

C+

v

i

L +

v

i

ZR= R ZC= 1j2fC

ZL = j2fL

2

nn vn =

nn Vne

j2ft

= 0

nn

Vn= 0

v (t) = Ki2 (t) i (t) = I ej2ft v (t) = KI2ej22ft

V = ZI


69/333

time-domainroom

f

Onlysignals

differential equationsKVL, KCLsuperposition

frequency-domainroom

Onlycomplex amplitudes

impedancestransfer functionsvoltage & current dividerKVL, KCL

superposition

t

vin+

R

C v

out

+

vout(t) = Vout= VinH(f)

v(t) = Vej2ft

i(t) = Iej2ft

vin= p (t) vin= Vinej2ft


70/333

Vin

RC vin= Vine

j2ft

vin+

R

C vout

+

Vin+

Vout

+

ZC

ZR

RC

RC

Vout

Vout = ZC

ZC+ ZRVin

Vout=1j2fC

1j2fC+ R

Vin

Vout = 1

j2fRC+ 1Vin

RCdvoutdt +vout = v in

Vout(j2fRC+ 1) = Vin

RCj 2f Voutej2ft + Voute

j2ft =Vinej2ft

j2f

RCdvout

dt + vout= vin


71/333

j2f

p (t) =v (t) i (t)

f

v (t) = |V|cos (2f t + )i (t) = |I|cos (2f t + )

V |V|ej |I|ej

v (t) = 12

V ej2ft + Ve(j2ft)

i (t) = 12

Iej2ft + Ie(j2ft)

p (t) 14V I+ VI+ V Iej4ft + VIe(j4ft)

12Re (V I

) + 12Re

V Iej4ft

12

Re (V I) + 12 |V||I|cos(4f t + + )

12V I

P

=1

2Re (V I)


72/333

P

=

1

2 Re (Z) (|I|)2

=

1

2 Re 1Z (|V|)2

2

V

I

V =ZeqI+ Veq

I= V

Zeq Ieq

Veq =ZeqIeq


73/333

veq+

Req

+

v

i

Sourcesand

Resistors

+

v

i

+

v

i

Reqieq

Thvenin Equivalent Mayer-Norton Equivalent

+

V

I

ZeqIeq

Mayer-Norton Equivalent

Veq+

Zeq

+

V

I

Thvenin Equivalent

Sources,Resistors,

Capacitors,Inductors

+

V

I


74/333

Vin+

R

C

+

V

I

Isc VoutR

Zeq = R 1j2fC = R1+j2fRC

Veq = 1

1 +j2fRCVin

Ieq= 1

RVin

Zeq= R1 +j2fRC

j2fRC

Vout

Vin = H(f)= 1j2fRC+1


75/333

vin+

R

C vout

+

RC

-1 0 1

1

1 / 2

|H(f)|

1

2RC

1

2RC

f

-1 1

/ 4

/ 4

/ 2

/ 2

H(f)

f0 1

2RC

1

2RC

RC= 1 |H(f) | = 1(2fRC)2+1

(H(f)) = arctan (2fRC)


76/333

|H(f) | = |H(f) | (H(f)) = (H(f))

1

2

f = 0 2fRC = 1

fc = 12RC

fc

12RC = 10

3

RC= 1032 = 1.59 104

4

2

vin(t) = Asin(2f t)

= A2j

ej2ft e(j2ft)

vout(t) = A2j

H(f) ej2ft A2j

H(f) e(j2ft)

H(f) =|H(f) |ej(H(f)) |H(f) | =|H(f) | (H(f)) = (H(f))

vout(t) = A2j |H(f) |ej2ft+(H(f)) A2j |H(f) |e((j2ft))(H(f))

= A|H(f) |sin(2f t + (H(f)))

vin(t) = Im

V ej2ft

vout(t) = Im

V H(f) ej2ft


77/333

vin

+

R

+

v

i

L

iout

Vin = 2cos(260t) + 3


78/333

Vin V

Iout

Vin= j2fL

R+j2fL

1

j2fL= 1j2fL+R

= H(f)

voltage divider = j2f L

R+j2f L

inductor admittance = 1

j2f L

3|H(0) | = 3

R

2|H(60) |cos(260t + (H(60)))

iout = 2|H(60) |cos (260t + (H(60))) + 3 H(0)

2fcL = R fc = R2L

RL = 20= 62.8

| 1j260L + R

| = 1R

| 16j+ 1

| = 1R

137

0.16 1R

1/6

0.3R

3R

2 0.3Rcos

260t 2

=0.3R sin(260t)


79/333

0 0.10

1

2

3

4

5

Time (s)

Voltag

e(v)orCurrent(A)

input

voltage

outputcurrent

RL

R= 6.28 L= 100mH


80/333

vin+

R1

R2 R3

e1 e2

en

e1 = v in

e2 vinR1

+ e2R2

+ e2R3

= 0

e2 = R2R3

R1R2+ R1R3+ R2R3vin

R3 e2R3


81/333

R1

R2

R3

e1 e2

iin

i

e1R1

+e1 e2

R2 iin= 0

e2 e1R2

+ e2R3

= 0

e1 1R1 + 1R2 e2 1R2 =iine1

1

R2+ e2

1

R2+

1

R3

= 0

e1 =R2+ R3

R3e2

e2 = R1R3

R1+ R2+ R3iin

i= e2R3


82/333

e1 e2

i

vin

+

1 1

1 1

2

e1 vin1

+e1

1 +

e1 e21

= 0

e2 vin2

+e2

1 +

e2 e11

= 0

e1 = 613vin e2 =

513vin

e21 =

513vin

+

C

R1

Vin R2

+

Vout

E

R2


83/333

E

Vin

R1 + Ej 2f C+

E

R2 = 0

E= R2

R1+ R2+j2f R1R2CVin

EVin

|H(f) | = R2(R1+ R2)

2+ (2f R1R2C)

2

(H(f)) = arctan2f R1R2C

R1+ R2 R2

fc = 12R1C

0 10

1

f

|H(f)|

No R2

R1=1, R2=1

12RC

1

2

R1C

R1+R2

R2

R1 = 1 R2 = 1 C= 1


84/333

R2 = R1

R2

R2

a

bc

1 2

3i3

i2

i1

v

=e

e

v

i

= (e

e

) i

=e

i

e

i

v

i

e

i

e

i

v

i

e

i

e

i

e

(i

i

)

k

v

i

= 0


85/333

kV I = 0

kV I = 0

k

1

2Re (V

I

) = 0

kv

(t1) i

(t2) = 0


86/333

+

+

v kv

+a

b

c+Rin G(eaeb)

a

b

cRout


87/333

+vin

+

RL

RFR

vout

+

+

RL

RF

R

+Rin Gv

Rout

vout

+

v

+

v

vout

v vinR

+ v

Rin+

v voutRF

= 0

vout (G) vRout

+vout v

RF

+vout

RL

= 0

vout vin RFRout

Rout GRF

1

Rout+

1

Rin+

1

RL

1

R+

1

Rin+

1

RF

1

RF

vout =

1

Rvin

G


88/333

+a

b

c

+Rin G(eaeb)

a

b

cRout

Rin Rout G 105

(15V)


89/333

+vin

+

RL

RFR

vout

+

+

RL

RFR

+Rin Gv

Rout

vout

+

v

+

RFRout

Rout GRF

1

Rout+

1

Rin+

1

RL

1

R+

1

Rin+

1

RF

1

RF

vout =

1

Rvin

RL Rout 1RL

R Rin

1

Rin

RF

Rout GRF

1

R+

1

RF

1

RF

vout =

1

Rvout

Rout 1G 1

G

1

R+

1

RF

1

RF

vout=

1

Rvin


90/333

RF R GR RF 1RF

vout =

RFR

vin

RF R

ZFZ

Vin

Vout

+

+

+

VoutVin

= ZFZ

H(f) = K

1 + jffc

K fc

ZF =K Z= 1 + jffc


91/333

ZF = 11+ jffc

Z = 1K

ZF1 = 1 + jffc

1fc

RC

1RF

+j2f C R

H(f) = RF

R

1+j2fRFC

RFR

1RFC

RFC= 5.3105 RFR = 10 R= RF10 R < Rin R

|ZF|R

< 105

RF|1+j2fR FC|

R < 105

RFR = 10

r10d r

r


92/333

vin+

RL

RFR

+Rin Gv

Routiin

i

iF

vout

+

+

v

++

RL

RFR

e

iin=0

+

vout

+ v

iin 105

v v iin= vRin

v= 105 iin = 1011 iin

Rin v

e


93/333

R vinR

R

iF = i v vinRF

R

vout = v = vinRFR

++

RL

RF

R1

R2+

vin

(2)

vout

+

e

i

+ v

vin(1)

vout =

RFR1

v(1)in

RF

R2v(2)in

e 0 i RF

v(1)inR1

+v (2)inR2

v= vout


94/333

0.5 v (V)

i ( A)

10

20

30

40

50

+

v

i

I0 = 1 A

i (t) = I0

e

qkTv(t) 1

q k

T

K

kT

q = 25

I0

I0


95/333

vin

+

R

+

vout

voutR

=I0

e

qkT(vinvout) 1

vin vout vin vout vin

vout= (RI0)

vout

R

vin' vin

"vout

idiode

vout'

vout"

vout

vin

t


96/333

vout vin

vin

vout vin vin vin vout vin

vin (RI0)

+vin

+

R

+

vout

vout =

kT

q ln

v inRI0

+ 1


97/333

+

v

1

2

+

v

1

1

+

v

L

C

i cos (2t)

v v

i1

R

R

vin

R2

R1

i1+

iin

RL2015 A


98/333

R1

R2

R3 R4

R5

R1

R2 R3

R4

R1R2

R3

R4

1

1

1

1

1 1 1

vin+

iin

1/2

1

1/2

1/2

i


99/333

i

i =C1vin+C2iin

i

7sin 5t+

3 6

1

4

+

vout

6 6

2

1 i

120 +

6

180

12

5

i

48

20


100/333


101/333

+

R1

iin

vout

R2

R3

R4

i

vout

R1 = 1

R2 = 2

R3 = 2

R4 = 4

i

iin

Im

(4 + 2j) ej220t

1 +

32 3 +j4

2j 63

2+j 63

4 j3 1 +j 12

3ej + 4ej2

3 +j 2 2e(j 4 )

31+j3

z u

z= 1 + eju

z |z| z1z+1

x

ejx

1 + ejx

exejx


102/333

ejx + ej(x+4 )

sin(2u) = 2sin (u)cos(u)

cos2 (u) = 1+cos(2u)2 cos2 (u) + sin2 (u) = 1

ddu(sin (u)) = cos (u)

vin+

1

1 2

+

v

vin+

1

2

+

v

ioutvin

+

1

1

4

ioutvin

+

1

1

1


103/333

iout

vin+

R1

L

C

R2

vin

+

R1

L

iout

iinC

R2

+

v

L1

iin CR L2

iin 21 1

1

1

i

sin(2f0t) i (t) 23 sin 2f0t + 4 v2(t) = 13sin(2f0t)

vin(t)

+ +

v2(t)

Z1i(t)

Z2

+ v1(t)

v1(t) Z1 Z2


104/333

vin(t) = 10sint2

+

+

vout

1

2 4vin

1

iout

iin

1

2

1

21

2

cos(2t)


105/333

L

R

Cvin

+

+

vout

R1 R2 R1 R1

R1 R2

R


106/333

IL

Vg+

Rs RT100

power generator lossy powertransmission line

load

100 1

Vg+

Zg

ZL


107/333

8

xi(t) = Asin(2fit) , 0 t T A fi T 1

T

sin(t) 1

2sin

t + 4

vin = 2


108/333

+ Resistors

+

v

i

vin

v = 10

v= 1

i = 2 v

vin(t) sin(t)

Circuit

+

v

i

+

1

vin

v (t) 1

2cos t + 4 i (t) sin(t)

v (t) i (t) = sin(t)


109/333

S(x (t)) = y (t) y (t) S() x (t)

S() S(x (t )) =y (t ) x (t)

y (t) = x (t)sin(2f0t) y (t) = x (t 0)

y (t) = x (t) + N(t)

Z

H(f) = 1

j10f+ 2

Z

+

C

R

vin

iout

Z


110/333

1vin+

+

vout

Zi(t)

5sin (t) i (t) =

2cos

t arctan (2) 4

vout(t)

Z

Circuit

+

v(t)

i(t)

sin(t) 4sin

t + 4

2sin (4t) Asin(4t + )

A

3sin (t) 3 2sin t 4


111/333

H(f) = 82f2

82f2 + 4 +j6f

1/2

4vin

+

+

voutZL

2

Vin

+

Z2

Z1

Vout

+


112/333

Z1

Z2

C

C

L

L

6

iin

+ vout

2

3

1

iin= 5sin (2000t)


113/333

1

iin

+

vout

1

1

1

1

sin(t)

1 + sin (t)

R1

C1

probe

C2R2

+

vin

oscilloscope

+

vout

R2 = 1M C2 = 30pF


114/333


115/333

+

+

vcar

1

vroad 1 1

+

+

vcar

1

vroad 1 1

vroad(t) vcar(t)

vroad(t) = 1 + sin (2t)

+

+

vout

1/4

2vin 3/4

Vin Vout

vout(t) vin(t) = sin t2 + 4

sin

t + 4

2cos(t)


116/333

vout

iin R1 R2 RL

+

vout

ib

ib

++

61

1/3 3+

vout 3i

i


117/333

+

R1

R2

+

vin

+

vout

+

V(1)in

+

R1R2

R3

R4

Vout

+

+

V(2)in

+

+

10

5

5Vin Vout

+

+

vin

+

+

vout4

4

1

2

2

1

1

2


118/333

+

Vin

C

C

R

R

RL

Iout

Iout

Vin

RL

vin= V0e t

Z2

Z1

Z3

Z4

VinVout

+

+

+

+


119/333

vout(t)

+

vin

+

vout4.7 k

1 k 10 nF

1 F

+

++

+

R1

R2

Vout

Vin

C1

R3

R4 C2

+

vout(t) R1 = 530 C1 = 1F R2 = 5.3k C2 = 0.01F R3 = R4 = 5.3k


120/333


121/333

+

R2 C2Vin

+

R1

C1

R

R

+

R1

R2

+

Rf

Vout

+

R1

R2

+

Vin

+

Vout

C

Rin

sin(2f0t) f0


122/333

+

Zf

Vout

vout

vout Zf

Zin

+

1Zin

Vout


123/333

+

+

R2

Vout

R1 C

+

Vin

R1= 1 k

R2= 1 k

C = 31.8 nF

sin(2f0t)


124/333

n 1

R2R1+R2

R1

(v v ) i =

R1

(R1+ R2)2 v

2

1

R1+ R2v

2 = R1

(R1+ R2)2

v

2 + R2

(R1+ R2)2

v

2

vin= R2iout iout = vinR2

R1 R2

Req= R21+

R2RL

R2RL

R2RL


125/333

P

=V

I

cos( )

Im

V ej2ft

=V ej2ft Ve(j2ft)

2j

V ej2ft

V H(f) ej2ft Ve(j2ft) VH(f) e(j2ft)

H(f) = H(f) Im

V H(f) ej2ft

Re

V ej2ft Re V H(f) ej2ft

e11 =

613vin

1113vin

1311

2


126/333


127/333


128/333

s (t) T

T

1

T

kT

k= {. . . , 1, 0, 1, . . .} s (t) =

k=

ckej 2kt

T

ck = 12(ak jbk) ck

c0 = a0

ej2kt

T

T k

T0

ej2kt

T e(j)2lt

T dt=

T k= l0 k =l

e(j2lt) [0, T]

ck = 1T

T0

s (t) e(j2kt

T )dt

c0 = 1T

T0

s (t) dt

sqT(t)

sqT(t) =

1 0< t < T2

1 T2 < t < T

ck = 1

T

T2

0

e(j2kt

T )dt 1T

TT2

e(j2kt

T )dt

j


129/333

ck = 2j2k

(1)k 1

= 2jk k 0 k

sq (t) =

k{...,3,1,1,3,... }

2

jke(j)

2ktT

1T

k

k

ck

s (t)

ck = ck

ck Re (ck) = Re (ck) Im (ck) =Im (ck)

ck =ck

s (t) = s (t) ck = ck

s (t) = s (t) ck = ck

s (t ) cke j2kT ck s (t) 2kT

1T

T

0 s (t

) e(j)

2ktT dt = 1T

T

s (t) e(j)2k(t+)

T dt

= 1Te(j) 2kT T s (t) e(j) 2ktT dt

T () dt=

T0

() dt


130/333

1

T T

0 s2

(t) dt=

k= (|ck|)

2

T

t

p(t)A

T A

ck = 1

T

0

Aej2kt

T dt=

A

j2k

e

j2kT 1

1 e(j) =e j2

ej2 e j2

=e

j2 2jsin

2

ck = Ae jk

Tsin

kT

k

ck =ck

|ck| =A|sin kT k

|

(ck) = kT

+ neg

sin

kT

k

sign(k)

neg()


131/333

T

= 0.2

A= 1

(ck) k

T

2

c0

2

(2) 2

[, ) 2


132/333

T

s (t) = a0+

k=1

akcos

2kt

T

+

k=1

bksin

2kt

T

ak bk ck

ck =1

2(ak jbk)

T0

sin

2kt

T

cos

2lt

T

dt= 0 , k Z l Z

T0

sin

2kt

T

sin

2lt

T

dt=

T2 (k= l) and (k = 0) and (l = 0)

0 (k =l) or (k= 0 =l)

T0

cos2ktT cos2ltT dt= T2

(k= l) and (k = 0) and (l = 0)T k= 0 = l

0 k =l

sin()sin() = 12(cos ( ) cos( + ))cos()cos() = 12(cos ( + ) + cos ( ))sin()cos() = 12(sin ( + ) + sin ( ))

T T2 l

cos2ltT

al T0 s (t)cos

2ltT

dt =

T0 a0cos

2ltT

dt +

k=1ak

T0 cos

2ktT

cos

2ltT

dt +

k=1bk

T0 sin

2ktT

cos

2ltT

dt


133/333

k

l

alT2

k = 0 = l a0T

al = 2

T T0 s (t)cos

2ltT

dt , l = 0

a0 = 1T

T0

s (t) dt

ak = 2T

T0

s (t)cos2ktT

dt , k = 0

bk = 2T

T0

s (t)sin2ktT

dt

a0 s (t)

s (t) =

sin

2tT

0 t < T2

0 T2 t < T

bk

bk = 2

T

T2

0

sin

2t

T

sin

2kt

T

dt

T2

0 sin

2tT

sin

2ktT

dt = 12

T2

0 cos

2(k1)t

T

cos

2(k+1)t

T

dt

=

12

k = 1

0

b1 =1

2

b2= b3 = = 0

a0

1

ak =

2

1k21

k {2, 4, . . . }

0


134/333

k

kT k = 1

k= 2

k

ak

-0.5

0

0.5

bk

0 2 4 6 8 10

0

0.5

k


135/333

rms(s) = 1

T T

0

s2 (t) dt

power (s) = rms2 (s)

= 1TT0

s2 (t) dt

power (s) = 1

T T

0 a0+

k=1 akcos2kt

T +

k=1 bksin2kt

T 2

dt

power (s) = a02 +

1

2

k=1

ak2 + bk

2

0 2 4 6 8 10

Ps(k)

0

0.1

0.2

k

k

ak2+bk

2

2 Ps(k)


136/333

sK(t)

K+ 1

sK(t) = a0+

Kk=1

akcos

2kt

T

+

Kk=1

bksin

2kt

T


137/333

k

ak

-0.5

0

0.5

bk

0 2 4 6 8 10

0

0.5

k

0

0.5

1

K=0

t

0

1

K=1

t

0.5

0

0.5

1

K=2

t

0 0.5 1 1.5 2

0

0.5

1

K=4

t

K+ 1

K+ 1


138/333

K+ 1

K(t) =

k=K+1akcos

2kt

T +

k=K+1bksin

2kt

T

rms(K) =

12

k=K+1

ak2 + bk2

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Relativermserror

K


139/333

0 2 4 6 8 100

0.5

1Ps(k)

k

Relativerm

serror

0 2 4 6 8 100

0.5

1

K

K

1k2

1k


140/333

0

1K=49

-1

t

-1

0

1K=1

t

-1

0

1K=5

t

-1

0

1 K=11

t

sq (t)

t


141/333

t

limitK

rms(K) = 0

s1(t) s2(t) s1(t) =s2(t) t rms(s1 s2) = 0

x (t) T T

x (t) =K

k=Kcke

j 2ktT

ck

ck x (t)

N {a1, . . . , aN} an cn= 1 ck = 0 k= n nth 1T

NT

a13 13 13 = 11012


142/333

N

T

T

0 T 2T 3T

-2

-1

0

1

2

t

x(t)

T


143/333

T

L (a1s1(t) + a2s2(t)) = a1L (s1(t)) +a2L (s2(t))

x (t) = ej2kt

T y (t) = H

kT

ej

2ktT

f = kT x (t)

y (t) =

k=ckH

kT

ej

2ktT

ckHkT


144/333

T

t

p(t)

A

0

0

0.2

0 1 20

1

0

0

0.2

0 1 20

1

0 1 20

1

0 10 20

0

0.2

pectra

agntu

efc: 100 Hz fc: 1 kHz fc: 10 kHz

Frequency (kHz) Frequency (kHz) Frequency (kHz)

Time (ms) Time (ms) Time (ms)

Amplitude

10 2010 20

T

= 0.2

RC

RC

H(f) = 1

1 +j2fRC


145/333

RC

sT(t) T

ck(T)

ck(T) = 1

T

T2

T2sT(t) e

j2ktT dt

fundamentals of electrical engineering i 9.71

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