# דף נוסחאות מבוא להנדסת חשמל ואלקטרוניקה אלי 2

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femto (f)=10-15 pico (p) = 10-12 nano (n) = 10-9 micro (m) = 10-6 milli (m) = 10 -3 Kilo (k) = 103 Mega (M) = 1069

qe !

1 ! 1.6 v 10 19 C 6.24 v 1018-

1 amper !

1 coulomb 1 second

:

q0 A.

-

B

f(t)=f(t+nT);n=1,2,3 . Cycle time (or period)- T [s ] Frequency- f = 1/T [ Hz = 1/s] Amplitude-A Angular frequency[rad/s]a(t)T-cycle time (or period)

.(

A )

1 volt !

1 joule 1 coulomb

N B,g=2pf=2p/T

WgB q0

A t

.

:

Pav !

1 p(t )dt T 0[W] p(t)

T

1 watt !

1 joule ! 1 volt v1 a pere 1 second dW dq dW p t ! v (t )i(t ) ! ! dq dt dtt2

t2

[s]

T

t1

. .

" ) (

vs(t)

Vs

[W]

(average)

Pav

p(X )dX v(X )i(X )dXt1

B

A

W-[J] t-[s] P-[W] V-[V] I-[A]

is(t) " )( " )

vs(t)

Is

Vs

,+ i + V VA

:(l -

)i(t)

1 f ! [ Hz ] Im T 2T rad [ ! 2Tf ! [ ] T s

v Vl ! !R i A

V [V] i - [A] R [W]

l [m] A [mm ] [ mm2/m]2

Ji [8

R!

v i

i(t) (amplitude) length of conductorl [s] [Hz] [rad/s]( , , )

Im T f

Resistivity -

r (characteristic of a material) A

Surface Area of conductor-

(phase)

. :( )

.

V(V)

1 [ Hz ] T V v Im 2T rad [ ! 2Tf ! [ ] T s f !

vi(t) V

Ji

v(t ) ! Vm sin( [t Jv )slope=R I(A)

v[8

v(t) (amplitude) [s], , ,

Vm T f

(cycle time) (frequency) (angular frequency)

R(; )

I + V _

(Electric conductance)

(phase)

v

1 G- !? A S ; G! 1 Rv1(t),v2(t),v3(t)

v1 (t ) ! Vm1 sin([t J1 )

v 2 (t ) 2 ! i (t ) R p (t ) ! v (t )i(t ) ! R

Vm1 Vm3 Vm2

v2 (t ) ! Vm 2 sin([t ) v3 (t ) ! Vm 3 sin([t J3 )[t [rad] J J [8

/

m tgF RT

R0 m (T T0 ) R0 m R0 (T T0 ) R00(T-T0)]

i(t ) ! is (t )

RT

RT=R0[1+ [V] [ ] [V]( " " )

es(t)T= 0[1+ 0(T-T0)]

r V

T0

T

T ( C)

short circuit

e(t )

L

V !H A/ s

;

Volt v sec ! Henry Amper

:

L inductivity [H] N number of turns l length of coil [m] A area of coil [m2] m0=4pX10-7 [H/m]

:

v(t ) ! L

di (t ) dtt t

1 1 i (t ) ! v ( x )dx ! i (t 0 ) v ( x )dx L g L t0 di(t ) p (t ) ! (t )i(t ) ! L i(t ) dt t 1 1 di( x) i( x)dx ! L i( x)di( x) ! Li 2 ( x) ! Li 2 (t ) wL (t ) ! L g 2 2 dx g g

t

di (t ) ; v (t ) dt di(t ) v(t ) L dt 2 Q0 N A L l

e(t )

Rab ! 0 Vab ! 0open circuit

-B

Rab p g Gab ! 0, I oc ! 0

(V

)

[Wb/m2] [Wb ] -

:

Rv p gdN e(t ) ! N dt( " ) Electro Motive Force (EMF) [V] - e(t) (A

:

(inductivity)

-L [H]

) : :

y

Q NA d ( 0 i (t )) dN (t ) Q N 2 A di(t ) l e(t ) ! N ! N ! 0 dt l dt dt Q0 N 2 A e(t ) L! ! di(t ) / dt l -N

RA p 0

: ( ) .1 .2

[m2] [m]

-A -l - m0 [Wb/(A(T)m)]

1) T ?_x1 (t ) x2 (t )a ! T ?_x1 (t )a T ?_x2 (t )a A A A 2) T ? Ex (t )a ! ET ? x (t )a ; E - constant _ A _ A

t

1 wL (t ) ! Li 2 (t ) [J] 2

v(t ) r

v(t ) ! es (t ) i (t ) r

RT R0 T T0

E0 !

m R0

Temperature coefficient of resistance

a [1/0C]R(;) RT linear region R0

F

y

i!

dV dq d (VC ) ! !C dt dt dt t 1 v(t ) ! i( x )dx C g v(t ) ! 1 1 1 gi( x)dx C t i( x)dx ! v(t0 ) C t i( x)dx C 0 0:t0 t t

coloumb voltC

C!

dq (q ! dV (V

+C

-

C!

q q p V ! p q ! C V C V dv dq d (C v) i! ! !C dt dt dt(Capacity) [F] (charge) [C] (voltage) [V]

C

dV (t ) p(t ) ! v(t )i(t ) ! v(t )C dtt

qt t

wC (t ) ! v( x)Cg

1 1 dv ( x) dx ! C v( x)dv( x) ! C v 2 ( x) ! Cv 2 (t ) 2 2 dx g g 1 2 1 q 2 (t ) 1 Cv (t ) ! ! q (t ) v(t ) [J] 2 2 C 2

V i!

wC (t ) !

dV dq d (VC ) ! !C dt dt dt t 1 v(t ) ! i ( x)dx C g v(t ) ! 1 Ct0

i ( x)dx

g

1 1 i( x) dx ! v(t0 ) i( x)dx C t C t0 0

t

t

:

Kirchhoff s Current Law (KCL)-

A[m2].

d[m]

k

k 1

Kirchhoff s Voltage Law (KVL). ,V1 V81 V8 V78 V7 V57 V67 V6 V5 V36 V15 V85 V45 V53 V3 V21 V14 V42 V4 V43 V2

Ir !

!Ir

V81 V78 V67 V36 V43 V14 ! 0 V45 V14 V15 ! 0

r

V23

C C0

i

n

0 iin

i

out

C0 !

i1 in

i2

q I0 A ! V d

i3 i4

C0q0 0=

i7 i6

i5

(

)

8.85X10-12 F/m

:

0

0

! I rI 0

A d

];[

R

v(t ) ! Ri(t )

.

i (t ) !

v(t ) R

w ! pdtt1

t2

i(t)+

v1(t)+ _ +

v2(t) R2 vn- (t)

_ +

v3(t) R3

_ +

]H[ ]F[

L C

v (t ) ! L v (t ) ! v(0)

di dtt

i(t ) ! i(0)

_

_

+

_

i(t)

i(t ) ! i1 (t ) ! i2 (t ) ! ...... ! in (t )2

2

n

(t )

n

i(t )v (t ) ! p (t )n

Veq ! Vkk !1

pT ! pkk !1

v (t ) ! i(t ) R1 i(t ) R 2 i (t ) R3 ...... i (t ) Rn v (t ) ! i(t )R1 R2 R3 ........ Rn v (t ) ! RT ! Req ! R1 R2 R3 ....... Rn i (t ):

.

vk (t ) ! i (t ) Rk !

v(t ) Rk Req v(t ) Rk R1 R2 R3 ..... Rn,:i(t)+

, . , , . , . .R1 v (t)_ _

v k (t )

v 1 (t)+

v 2 (t)+

i(t)

R2

:i(t)

, .+

Req ! RKk !1

n

+

A_

v(t) i(t)

Req=R 1+R2 +R3+.......+R n_

, . , , . . , , . , ,

v1 (t ) R1 ! v2 (t ) R2

(t ) ! 1 (t )

(t ) 3 (t ) ......

KVL : (t ) 1 (t )

(t ) 3 (t ) ......

n

(t ) ! 0

v1 (t )i(t ) v2 (t )i (t ) v3 (t )i (t ) .... v (t )i(t ) ! i(t )?v1 (t ) v2 (t ) v3 (t ) ..... v (t ) A!

Rn

Rn-

A v(t) vn(t)

R

1 v(t ' ) dt ' L 0 dv dt

t

p(t ) ! v(t )i(t ) 1 w(t ) ! Li 2 (t ) 2 1 w(t ) ! Cv 2 (t ) 2

R4_ +

v4(t)

1 i(t ' ) dt ' C 0

i (t ) ! C

:

p1 (t ) p2 (t ) p3 (t ) ..... p (t ) !

-PT

k

- Pk

. , ,

.

:

a i(t)

i1(t) vab(t) b i(t) R1k n

a R1 a a a

a R2 b b

a R3 b b R9 a a

a R4 R5 b b R8 a R6 R7

a

eq

k 1

:

R R2 Req ! R R2

a

i(t) i2(t) R2

:

n-n -R :

R Req ! nn

.....in (t ) !i (t ) !

vab (t )n

vab (t ) vab (t ) vab (t ) v (t ) ......... ab R1 R2 R3 Rn

pT ! pkk !1v (t) i(t) v2(t)

-PT - Pk

v3(t)

1 1 1 1 ......... i (t ) ! vab (t ) Rn R1 R2 R3 i(t ) 1 1 1 1 1 ! ! ......... vab (t ) Req R1 R2 R3 Rn :

A

v(t)

C vn(t)

C2 vn- (t)

C3

i(t)

Req !Cn

Cn-

R1 R2 R1 R2 R1 R2 R1 R2

abi(t)

(t ) ! i (t ) Req ! i (t )ab

A

v(t)

Ceq

i(t)

i1 (t ) !

(t ) i (t ) R2 ! R1 R1 R2a i(t) i 1 (t) v ab (t) b i(t) R1 i 2 (t) R2

i (t ) R2 i1 (t ) ! R1 R2 i2 (t ) ! i (t ) R1 R1 R2

1 1 1 1 1 ! ........ Ceq C1 C 2 C3 Cn

1

2

i1 (t ) R2 ! i2 (t ) R1

Ohm s Law:

i1 (t ) !

i (t )

i1 (t ) i2 (t ) i3 (t ) ....... in (t )

vab (t )

1 n 1 k 1 k

Geq

G

n

k

vab(t) i(t) b R10 b b b R11 Rn

a a

a

vab (t ) ! v (t ) ! v2 (t ) ! v3 (t ) ! v4 (t ) ! .... ! vn (t )

; i2 (t ) !

vab (t )

; i (t ) !

vab (t )

1 i1

a i(t)

v1 R

2

+

i2 v2

i

R1 !

R12 R13 R12 R13 R23

2

3

n

i(t)

A n eq

v(t)

Ceq

R12 R23 R2 ! R12 R13 R23 R13 R23 R3 ! R12 R13 R23:

n

i(t)

k

1

2

3

k !1

i(t)

1 i1

i(t)

-

-

-

v1 R

2

+

i2 v2

i

R R R2 R3 R3 R1 R12 ! 1 2 R3 R13 ! R23 ! R1 R2 R2 R3 R3 R1 R2 R1 R2 R2 R3 R3 R1 R1:(

!3

Y

).

"

(

50

Leq ! L L2 L3 ........ Ln

!

'

(

v2

& '

)

-

'

(

i2

(

i2

(

-

R2

-

i

R2

'

(

R12

(

i12 i1

R1

(

v12

+

+

v1

1 i1 v12 R1

i(t ) ! i1 (t ) i2 (t ) i3 (t ) ...... in (t )n

1 1 1 1 1 ! ....... Leq L1 L2 L3 Ln

+ v(t) -

i(t) +

L1 v1(t)

- +

L2 v2(t)

i(t) -

Ln vn(t)

+

v(t ) ! v1 (t ) v2 (t ) v3 (t ) ..... vn (t )

di n di v(t ) ! Lk ! Le dt k !1 dt

4

RY !

R( 3

v(t)

L1

L2

L

5

+

+

i1(t)

+

i2(t)

+

i (t)

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