7.1
Dane: Szukane: Wzory:
l = 20 cm = 0, 2 m T =
É = 2Ä„ f
D = 10 cm = 0,1 m
f =
1
T =
É ' = 314 rad / s(n = 3000 obr / min)
É = f
B = 0, 4 T
Em = Blv
e1...7 =
Ä„ Ä„ Ä„ Ä„ 3 7
e = Em sinÄ…
Ä… = 0; ; ; ; ; Ä„ ; Ä„
6 4 3 2 4 4
(liczba par biegunów) p =1
pÉ ' 1Å"314
f = = = 50 Hz
2Ä„ 2Ä„
1 1
T = = = 0,02 s
f 50
É = pÉ ' = 3Å"314 = 314 rad / s
2Ä„ r D D
Em = 2Å" Blv = 2Å" B Å"l Å" = 2Å" B Å"l Å" 2Ä„ fr = 2Å" B Å"l Å" 2Ä„ f Å" = 2Å" B Å"l Å"É Å"
T 2 2
0,1
Em = 2Å"0, 4Å"0,2Å"314Å" H" 2,51V
2
e1 = Em sinÄ… = 2,51Å"sin 0o = 0V
Ä„
e2 = Em sin = 2,51Å"sin 30o = 2,51Å"0,5 = 1, 26 V
6
Ä„
e3 = Em sin = 2,51Å"sin 45o = 2,51Å"0,707 = 1,77 V
4
Ä„
e4 = Em sin = 2,51Å"sin 60o = 2,51Å"0,866 = 2,17 V
3
Ä„
e5 = Em sin = 2,51Å"sin 90o = 2,51Å"1 = 2,51V
2
3
e6 = Em sin Ä„ = 2,51Å"sin135o = 2,51Å"sin(180o - 45o ) = 2,51Å"sin 45o = 2,51Å"0,707 = 1,77 V
4
7
e7 = Em sin Ä„ = 2,51Å"sin 315o = 2,51Å"sin(360o - 45o ) = 2,51Å"(-sin 45o ) = -2,51Å"0,707 = -1,77 V
4
______________________________________________________________________
7.2
Dane: Szukane: Wzory:
f = 500 Hz 1
T =
T =
f
1 1
T = = = 0,002 s
f 500
______________________________________________________________________
7.3
Dane: Szukane: Wzory:
T = 0,004 s f = 1
T =
f
É = 2Ä„ f
1 1
f = = = 250 Hz
T 0,004
É = 2Ä„ f = 6, 28Å" 250 = 1570 rad / s
______________________________________________________________________
7.4
Dane: Szukane: Wzory:
f = 50 Hz T = 1
T =
f
É =
É = 2Ä„ f
 =
c
 =
f
1 1
f = = = 250 Hz
T 0,004
É = 2Ä„ f = 6, 28Å" 250 = 1570 rad / s
c 3Å"108
 = = = 6Å"106 m = 6000 km
f 50
______________________________________________________________________
7.5
Dane: Szukane: Wzory:
T = 1
f1 = 227 kHz = 227 Å"103 Hz
T =
f
 =
f2 = 818 kHz = 227 Å"103 Hz
c
 =
f3 = 67,94 MHz = 67,94Å"106 Hz
f
1 1
T1 = = = 4, 4Å"10-6 s = 4, 4 µs
f1 227 Å"103
c 3Å"108
1 = = = 1322 m
f1 227 Å"103
1 1
T2 = = = 1, 222Å"10-6 s = 1, 222 µs
f2 818Å"103
c 3Å"108
2 = = = 367 m
f2 818Å"103
1 1
T3 = = = 14,72Å"10-9 s = 14,72 ns
f3 67,94Å"106
c 3Å"108
3 = = = 4,42 m
f3 67,94Å"106
______________________________________________________________________
7.6
Dane: Szukane: Wzory:
 = 1 m f = c
 =
f
c 3Å"108
f = = = 3Å"108 Hz = 300 MHz
 1
______________________________________________________________________
7.7
Dane: Szukane: Wzory:
f = 50 Hz É = É = 2Ä„ f
É = 2Ä„ f = 6, 28Å"50 = 314 rad / s
______________________________________________________________________
7.8
Dane: Szukane: Wzory:
p =
f = 50 Hz
pÉ '
f =
É ' = 20,9 rad / s (n = 200 obr / min) 2Ä„
2Ä„ f 6, 28Å"50
p = = = 15
É ' 20,9
______________________________________________________________________
7.9
Dane: Szukane: Wzory:
p = 2 f = pÉ '
f =
É ' = 52,3 rad / s (n = 500 obr / min) 2Ä„
pÉ ' 2Å"52,3
f = = = 16,66 Hz
2Ä„ 6, 28
______________________________________________________________________
7.10
Dane: Szukane: Wzory:
p1 = 2 pÉ '
É1...4 ' =
f =
2Ä„
p2 = 3
p3 = 4
p4 = 5
f = 50 Hz
2Ä„ f 6, 28Å"50
É1 ' = = = 157 rad / s
p1 2
2Ä„ f 6, 28Å"50
É2 ' = = = 104,7 rad / s
p2 3
2Ä„ f 6, 28Å"50
É3 ' = = = 78,5 rad / s
p3 4
2Ä„ f 6, 28Å"50
É4 ' = = = 62,8 rad / s
p3 5
______________________________________________________________________
7.11
Dane: Szukane: Wzory:
et = É
f =
2Ä„
it =
u = 310sinÉt
Ä„
i = 2sin(Ét - )
4
t = 0,005 s
f = 50 Hz
Ét 2Ä„ ft
Ä… = Å"360o = Å"360o = 50Å"0,005Å"360o = 90o
2Ä„ 2Ä„
ut = 310sinÉt = 310sinÄ… = 310sin 90o = 310 V
Ä„
Ét -
Ä„ 8Å"50Å"0,005 -1
4Ét -Ä„ 4Å" 2Ä„ ft -Ä„ ( )
4
Ä… = Å"360o = Å"360o = Å"360o = Å"360o = 0,125Å"360o = 45o
2Ä„ 4Å" 2Ä„ 4Å" 2Ä„ 8Ä„
Ä„
ëÅ‚Ét öÅ‚
it = 2sin - = 2sinÄ… = 2sin 45o = 1, 41 A
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
lub
Ä„ Ä„
ëÅ‚Ét öÅ‚
it = 2sin - = 2cos = 2cos 45o = 1, 41 A
ìÅ‚ ÷Å‚
4 4
íÅ‚ Å‚Å‚
______________________________________________________________________
7.12
Dane: Szukane: Wzory:
t =
É
Ä„
f =
Õ =
2Ä„
6
f = 500 Hz
Õ t
=
2Ä„ T
Õ
= tf
2Ä„
Ä„
Õ 1
6
t = = = = 1,667 Å"10-4 s = 166,7 µs
2Ä„ f 2Ä„ f 12Å"500
______________________________________________________________________
7.13
Dane: Szukane: Wzory:
Õ =
É
l = 300 km = 3Å"105 m
f =
t
2Ä„
f = 50 Hz
=
c
T
 =
f
c 3Å"108
 = = = 6Å"106 m
f 50
W ten sposób lub troszkę inaczej
Õ l Õ l
= =
360o  2Ä„ 
l 3Å"105 l 3Å"105 Ä„
Õ = 360o = 360o = 18o Õ = 2Ä„ = 2Ä„ =
 6Å"106  6Å"106 10
Õ Ä…
=
2Ä„ 360o
t Õ 18o 1
= = =
Ä„
T 360o 360o 20
Õ Å"360o 10 Å"360o
Ä… = = = 18o
2Ä„ 2Ä„
Õ
Ä„
t 2Ä„ f 1 1
10
= = = =
1
T 2Ä„ 10Å" 2 20
f
______________________________________________________________________
7.14
Dane: Szukane: Wzory:
i =
Ä„ i = I sin(Ét +Õ)
i1 = 10sin(Ét + ) [A]
c
4
 =
Ä„
f
i2 = 10sin(Ét + ) [A]
2
i3 = 5sinÉt [A]
Ä„
4
Õ1 = Å"360o = 45o
2Ä„
Ä„
2
Õ2 = Å"360o = 90o
2Ä„
0
Õ3 = Å"360o = 0o
2Ä„
i = i1 + i2 + i3
y
i2 i
i1
x
i3
2
i1x = I1 cosÕ1 = 10cos 45o = 10 = 10Å"0,707 = 7,07 A
2
2
i1y = I1 sinÕ1 = 10cos 45o = 10 = 10Å"0,707 = 7,07 A
2
ix = i1x + i2x + ix3 = 7,07 + 0 + 5 = 12,07 A
iy = i1y + i2 y + ixy = 7,07 +10 + 0 = 17,07 A
i = ix2 + iy2 = 12,072 +17,072 = 20,91 A
iy 17,07
tgÄ… = = = 1, 4142
ix 12,07
Ä… = 54o44'
i = 20,91sin(Ét + 54o44')
______________________________________________________________________
7.15
Dane: Szukane: Wzory:
Um1 = Um2 = 200 V
Um = u = Um sin(Ét +Õ)
Odświeżyć wzory
f = 50 Hz
trygonometryczne i zwiÄ…zki
od 0 do 2Ä„
pomiędzy kątami
Ä… = 00; 600; 900;1200;1800
1sposób
y
u
u2
Ä… Ä…
x
u1
Teraz dodawanie wektorów jak na geometrii.
u2 = (u1 + u2 cosÄ…)2 + (u2 sinÄ…)2 = u12 + 2u1u2 cosÄ… + u22 cos2 Ä… + u22 sin2 Ä… =
= u12 + 2u1u2 cosÄ… + u22(cos2 Ä… + sin2 Ä…) = u12 + 2u1u2 cosÄ… + u22 Å"1
u = u12 + 2u1u2 cosÄ… + u22
um1 = Um1 2 + 2Um1Um2 cosÄ…1 + Um2 2 = 40000 + 80000Å"1+ 40000 = 400V
( ) ( )
1
um2 = Um1 2 + 2Um1Um2 cosÄ…2 + Um2 2 = 40000 + 80000Å" + 40000 = 346,4 V
( ) ( )
2
um3 = Um1 2 + 2Um1Um2 cosÄ…3 + Um2 2 = 40000 + 80000Å"0 + 40000 = 282,8V
( ) ( )
um4 = Um1 2 + 2Um1Um2 cosÄ…4 + Um2 2 = Um1 2 + 2Um1Um2 cos(900 + 300) + Um2 2 =
( ) ( ) ( ) ( )
1
= Um1 2 + 2Um1Um2(-sin 300) + Um2 2 = 40000 -80000Å" + 40000 = 200 V
( ) ( )
2
um5 = Um1 2 + 2Um1Um2 cosÄ…5 + Um2 2 = Um1 2 + 2Um1Um2 cos(1800) + Um2 2 =
( ) ( ) ( ) ( )
= 40000 + 80000Å"(-1) + 40000 = 0V
2sposób
Dodawanie wektorów z wykorzystaniem Twierdzenia Cosinusów a2 = b2 + c2 - 2bc cosą
y
u
u2
²
Ä… Ä…
x
u1
² = 1800 -Ä…
u2 = u12 + u2 - 2u1u2 cos ²
um1 = Um12 +Um22 - 2Um1Um2 cos ² = Um12 +Um22 - 2Um1Um2 cos(1800 -Ä… ) =
= Um12 +Um22 - 2Um1Um2(-cosÄ…) = 40000 + 40000 + 80000 Å"1 = 400 V
um2 = Um12 +Um22 - 2Um1Um2 cos ²2 = Um12 +Um22 - 2Um1Um2 cos(1800 -Ä…2) =
1
= Um12 +Um22 - 2Um1Um2(- cosÄ…2) = 40000 + 40000 + 80000Å" = 346, 4 V
2
um3 = Um12 +Um22 - 2Um1Um2 cos ²3 = Um12 +Um22 - 2Um1Um2 cos(1800 -Ä…3) =
= Um12 +Um22 - 2Um1Um2(- cosÄ…3) = 40000 + 40000 + 80000Å"0 = 282,8V
um4 = Um12 +Um22 - 2Um1Um2 cos ²4 = Um12 +Um22 - 2Um1Um2 cos(1800 -Ä…4) =
1
= Um12 +Um22 - 2Um1Um2 cos(1800 -1200 ) = 40000 + 40000 - 80000Å" = 200 V
2
um5 = Um12 +Um22 - 2Um1Um2 cos ²5 = Um12 +Um22 - 2Um1Um2 cos(1800 -Ä…5) =
= Um12 +Um22 - 2Um1Um2 cos(1800 -1800 ) = 40000 + 40000 - 80000Å"0 = 0 V
______________________________________________________________________
7.16
Dane: Szukane: Wzory:
Ä… =
Um1 = Um2 = 200 V u = Um sin(Ét +Õ)
Odświeżyć wzory
f = 50 Hz
trygonometryczne i zwiÄ…zki
um = 220 V
pomiędzy kątami
1sposób
y
u
u2
Ä… Ä…
x
u1
Teraz dodawanie wektorów jak na geometrii.
u2 = (u1 + u2 cosÄ… )2 + (u2 sinÄ… )2 = u12 + 2u1u2 cosÄ… + u22 cos2 Ä… + u22 sin2 Ä… =
= u12 + 2u1u2 cosÄ… + u22(cos2 Ä… + sin2 Ä… ) = u12 + 2u1u2 cosÄ… + u22 Å"1
um2 = u12 + 2u1u2 cosÄ… + u22
um2 - u12 - u22 2202 - 2002 - 2002 48400 - 40000 - 4000
cosÄ… = = = = -0,395
2u1u2 2Å" 200Å" 200 80000
Cosinus przyjmuje wartość ujemną w II i III ćwiartce
-cosÄ… = cos(1800 -Õ)
Ä… = 66014' Õ =1800 -Ä… =113046'
lub
-cosÄ… = cos(1800 +Õ)
Ä… = 66014' Õ =1800 +Ä… = 226014'
2sposób
Dodawanie wektorów z wykorzystaniem Twierdzenia Cosinusów a2 = b2 + c2 - 2bc cosą
y
u
u2
²
Ä… Ä…
x
u1
² = 1800 -Ä…
u2 = u12 + u2 - 2u1u2 cos ²
um2 = Um12 +Um22 - 2Um1Um2 cos ²
um2 -Um12 -Um22 2202 - 2002 - 2002 48400 - 40000 - 40000
cos ² = = = = 0,395
-2Um1Um2 -2Å" 200Å" 200 -80000
Cosinus przyjmuje wartość dodatnia w I i IV ćwiartce
Ä… = 66014' ² = 1800 -Ä… = 113046'
lub
cosÄ… = cos(-Õ)
Ä… = -66014' Õ = 1800 -Ä… = 1800 + 66014' = 246014'
______________________________________________________________________
7.17
Dane: Szukane: Wzory:
Ä… =
Um1 = Um2 = 200 V u = Um sin(Ét +Õ)
Um
f = 50 Hz
Usk =
2
Usk = 220 V
Odświeżyć wzory
trygonometryczne i zwiÄ…zki
pomiędzy kątami
1sposób
y
u
u2
Ä… Ä…
x
u1
Teraz dodawanie wektorów jak na geometrii.
u2 = (u1 + u2 cosÄ…)2 + (u2 sinÄ…)2 = u12 + 2u1u2 cosÄ… + u22 cos2 Ä… + u22 sin2 Ä… =
= u12 + 2u1u2 cosÄ… + u22(cos2 Ä… + sin2 Ä…) = u12 + 2u1u2 cosÄ… + u22 Å"1
um = Usk 2
um2 = u12 + 2u1u2 cosÄ… + u22
2
( )
um2 - u12 - u22 220Å" 2 - 2002 - 2002 96800 - 40000 - 4000
cosÄ… = = = = 0, 21
2u1u2 2Å" 200Å"200 80000
Cosinus przyjmuje wartość dodatnią w I i IV ćwiartce
Ä… = 78054'
lub
cosÄ… = cos(-Õ)
Ä… = 78058' Õ = -78058'
2sposób
Dodawanie wektorów z wykorzystaniem Twierdzenia Cosinusów a2 = b2 + c2 - 2bc cosą
y
u
u2
²
Ä… Ä…
x
u1
² = 1800 -Ä…
u2 = u12 + u2 - 2u1u2 cos ²
um = Usk 2
um2 = Um12 +Um22 - 2Um1Um2 cos ²
2
( )
um2 -Um12 -Um22 220 2 - 2002 - 2002 96800 - 40000 - 40000
cos ² = = = = -0, 21
-2Um1Um2 -2Å" 200Å" 200 -80000
Cosinus przyjmuje wartość ujemną w II i III ćwiartce
-cosÕ = cos(1800 - ² )
² =1800 - 78054' Ä… =1800 - ² = 78054'
lub
- cosÕ = cos(-² )
² = 78054' Ä… = -² = -78054'
______________________________________________________________________
7.18
Dane: Szukane: Wzory:
2
Isk = 5,55 A Im =
Isr (Ä„ ) = Im
Ä„
Isr (Ä„ ) =
Im
Isk =
2
Im = Isk 2 = 5,55 2 = 7,82 A
2 2 2 2
Isr (Ä„ ) = Im = Isk 2 = 5,55 = 4,98 A
Ä„ Ä„ Ä„
______________________________________________________________________
7.19
Dane: Szukane: Wzory:
Um
Usk = 230 V Um =
Usk =
2
Um = Usk 2 = 230 2 H" 324,3V
______________________________________________________________________
7.20
Dane: Szukane: Wzory:
2Ä„ Im
Isk =
i = Im sin(Ét + ) Isk =
6
2
i = 1,3 A
t = 0
2Ä„ 2Ä„
i = Im sin(Ét + ) = Im sin( ) = Im sin(600)
6 6
i 1,3
Im = = H" 1,5 A
sin(600)
3
2
Im
Isk = = 1,06 A
2
______________________________________________________________________
7.21
Dane: Szukane: Wzory:
Um
Um1 = Um2 = 1,8V Usk =
Usk =
Ä„ 2
Ä… = 900 =
2
Ä„
u = u1 + u2 = Um1 sin(Ét) +Um2 sin(Ét + )
2
1sposób
y
u
u2
Ä… Ä…
x
u1
Teraz dodawanie wektorów jak na geometrii.
Przy tym kÄ…cie najlepiej od razu z Pitagorasa
Um2 = Um12 +Um22
Um = Um12 +Um22 = 1,82 +1,82 =1,8Å" 2 = 2,55V
Um
U = =1,8V
2
______________________________________________________________________
7.22
Dane: Szukane: Wzory:
2
Isr = Im
i = 5sinÉt
Isr (Ä„ ) =
Ä„
i = Im sinÉt
Im = 5 A
2 2
Isr = Im = 5 H" 3,18 A
Ä„ Ä„
______________________________________________________________________
7.23
Dane: Szukane: Wzory:
Q
I =
Q1 =
Im = 5 2 A t
2
Q2 =
t = 8 h
Isr = Im
Ä„
Do obliczeń bierzemy prąd średni. (patrz definicja wartości średniej)
Przy prostowanie dwupołówkowym, wartość średnia prądu dla każdego półokresu jest taka
sama.
2 2
Q2 = Isrt = Imt = 5 2 Å"8Å"3600 = 129711C
Ä„ Ä„
lub
2 2
Q2 = Isrt = Imt = 5 2 Å"8 = 36 Ah
Ä„ Ä„
Przy prostowanie jednopołówkowym, w każdym okresie jeden półokres prądu jest blokowany
czyli wartość średnia w okresie jest o połowę mniejsza.
Isr 2 1
Q1 = t = Imt = 5 2 Å"8Å"3600 = 64855 C
2 2Ä„ Ä„
lub
Isr 2 1
Q1 = t = Imt = 5 2 Å"8 = 18 Ah
2 2Ä„ Ä„
______________________________________________________________________
7.24
Dane: Szukane: Wzory:
I2 = 0, 222 A I1a = 2
Isr = Im
Ä„
I1b =
Im
Isk =
2
Miernik elektromagnetyczne wyskalowane są w wartości skutecznej.
Miernik magnetoelektryczne reagują na wartość średnią.
I2a = I2b = Im = Isk 2 = 0,222 2 = 0,314 A
Prostowanie dwópołówkowe
2Im 2Isk 2 2Å"0, 222 2
I1a = Isr = = = = 0, 2 A
Ä„ Ä„ 3,14
Prostowanie jednopołówkowe
Isr 2Im 2Isk 2 0, 222 2
I1b = = = = = 0,1 A
2 2Ä„ 2Ä„ 3,14
______________________________________________________________________
7.25
Dane: Szukane: Wzory:
Um = 180 V Usr =
Um
0 + Im Im R Um
U1sr = I1sr R = R = R = R =
2 2 2 2
Im + Im 2Im Um
U2sr = I2srR = R = R = ImR = R = Um
2 2 R
Um
Im + 0 Im R Um
U3sr = I3sr R = R = R = R =
2 2 2 2
Um Um
U1sr +U2sr +U3sr 2 +Um + 2 4 4
Usr = = = Um = 180 = 120 V
3 3 6 6
______________________________________________________________________
7.26
Dane: Szukane: Wzory:
m = kIt
mCu = 10 g WR =
t = 3 h = 10800 s
R = 12 &!
k = 0,3294 mg / C = 0,3294Å"10-3 g / C
m 10 10
Isr = = = = 2,8109 A
kt 0,3294Å"10-3 Å"10800 3,55752
2
Isr = Im
Ä„
IsrĄ
Im 2 IsrÄ„ 2,8109Å"3,14
Isk = = = = = 3,1298 A
2 2 2 2 2 2
W = Pt = Isk 2Rt = 1269570 J H" 1, 26Å"106 J
______________________________________________________________________
7.27
Dane: Szukane: Wzory:
2 + j2 ; I ćwiartka
3 - j ; IV
-1+ j 3 ; II
-1- j ; III
______________________________________________________________________
7.28
Dane: Szukane: Wzory:
r = a2 + b2
z = a + jb
b
Õ = arctg dla a > 0 i b e" 0
jÕ
z = re
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
2
jarctg
j450
2
2 + j2 ;I ćwiartka; 22 + 22 Å"e = 2 2 Å"e
1
- jarctg
j (-450 +3600 ) j3150
1
1- j ;IV ćwiartka; 12 +12 Å"e = 2 Å"e- j 450 = 2 Å"e = 2 Å"e
-1- j 3 ;III ćwiartka;
- 3
jarctg
j 3-Ä„
(arctg )
j (600 -1800 ) j(-1200 ) j(-1200 +3600 ) j 2400
-1
(-1)2 + (- 3)2e = 2e = 2e = 2e = 2e = 2e
______________________________________________________________________
7.29
Dane: Szukane: Wzory:
z = a + jb
r = a2 + b2
jÕ
z = re b
Õ = arctg
a
a = r cosÕ
b = r sinÕ
jÕ
re = r cosÕ + j sinÕ
3 1
j300
10e = 10cos300 + j10sin 300 = 10 + j10 = 8,65 + j5
2 2
2 2
2e- j 450 = 2cos(-450) + j2sin(-450) = 2cos 450 + j2(-sin 450 ) = 2Å" - j2Å" = 2 - j 2
2 2
j900
3e = 3cos900 + j3sin 900 = 3Å"0 + j3Å"1 = j3
Ä„
j
1 3
3
18e = 18cos 600 + j18sin 600 = 18Å" + j18Å" = 9 + j9 3
2 2
2
j Ä„
3
4e = 4cos1200 + j4sin1200 = 4cos(1800 - 600) + j4sin(1800 - 600) =
1 3
= 4(-cos 600) + j4sin 600 = -4Å" + j4Å" = -2 + j2 3
2 2
j1350
16e = 16cos1350 + j16sin1350 = 16Å"cos(1800 - 450) + j16sin(1800 - 450) =
2 2
= 16Å"(- cos 450) + j16sin 450 = -16Å" + j16Å" = -8 2 + j8 2
2 2
______________________________________________________________________
7.30
Dane: Szukane: Wzory:
r = a2 + b2
z = a + jb
b
Õ = arctg dla a > 0 i b e" 0
jÕ
z = re
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
Ä„ Ä„
öÅ‚
I = 10ëÅ‚ cos + j sin [A]
ìÅ‚ ÷Å‚
6 6
íÅ‚ Å‚Å‚
Ä„ Ä„ 3 1
öÅ‚
I = 10ëÅ‚ cos + j sin =10 cos300 + j sin 300 = 10 + j10 = 5 3 + j5
( )
ìÅ‚ ÷Å‚
6 6 2 2
íÅ‚ Å‚Å‚
r = 10
Ä„
j
6
I = 10e
______________________________________________________________________
7.31
Dane: Szukane: Wzory:
z = a + jb
I = jÕ
I1 = 2 + j3 [A]
( )
z = re
I2 = 2,5 - j2,5 [A]
( )
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
I1 = 2 + j3 [A]
( )
I2 = 2,5 - j2,5 [A]
( )
I = I1 + I2 = 2 + j3 + 2,5 - j2,5 = 4,5 + j0,5 [A]
r1 = 22 + 32 = 13
r2 = 2,52 + (-2,5)2 = 6, 25 + 6, 25 = 12,5 [A]
r1 > r2
______________________________________________________________________
7.32
Dane: Szukane: Wzory:
z = a + jb
VA = 230 V
UAB = jÕ
z = re
VB = 230e- j1200 V
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
VA = 230 V
VB = 230e- j1200 = 230(cos(-1200) + j sin(-1200)) =
= 230(cos1200 + j(-sin1200)) = 230(cos(1800 - 600) + j(-sin(180 - 600))) =
1 3
= 230(-cos 600) + j(-sin 600 )) = 230(- ) + j230(- )) = -115 - j115 3 [V ]
2 2
U = VA -VB = 230 - (-115 - j115 3) = 345 + j115 3 [V ]
AB
2
rAB = 3452 + 115 3 = 119025 + 39675 H" 398, 4
( )
______________________________________________________________________
7.33
Dane: Szukane: Wzory:
z = a + jb
jÕ
z = re
j2 = -1
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
1+ j 1- j = 1- j + j - j2 = 1-(-1 = 2
( )( ) )
2 2
1+ j 1- j = îÅ‚ 1+ j 1+ j Å‚Å‚ îÅ‚ 1- j 1- j Å‚Å‚ = 1+ j + j + j2 1- j - j + j2 =
( ) ( ) ( )( )ûÅ‚ ðÅ‚( )( )ûÅ‚
( )( )
ðÅ‚
= 1+ j2 + 1- j2 + = j2 j2 = - j2 4 = -(-1 4 = 4
) (-1
( (-1
)( ) (- ) )
)
j(1- j) = j - j2 = j -(-1 = 1+ j
)
______________________________________________________________________
7.34
Dane: Szukane: Wzory:
z = a + jb
jÕ
z = re
j2 = -1
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
1+ j 1+ j 1+ j 1+ j + j + j2 1+ j2 -1
= = = = j
1- j 1- j 1+ j 1- j2 2
1- j 1- j 1- j 1- j - j + j2 1- j2 -1
= = = = - j
1+ j 1+ j 1- j 1- j2 2
1 1 1- j 1- j 1- j 1
= = = = 1- j
( )
1+ j 1+ j 1- j 1- j2 2 2
1 1 1+ j 1+ j 1+ j 1
= = = = 1+ j
( )
1- j 1- j 1+ j 1- j2 2 2
1 1 - j - j
= Å" = = - j
j j - j - j2
2 - j 2 - j -2 - j6 -4 - j12 + j2 + j26 -10 - j10 1
= Å" = = = - ( )
1+ j
-2 + j6 -2 + j6 -2 - j6 4 - j236 40 4
______________________________________________________________________
7.35
Dane: Szukane: Wzory:
z = a + jb
I = 3 + j4
jÕ
z = re
Z = 2 - j4
j2 = -1
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
I Å" Z = 3+ j4 2 - j4 = 6 - j12 + j8 - j216 = 6 - j4 -(-1 16 = 22 - j4
( )( ) )
IV ćwiartka
2
r = 222 + = 500 = 10 5 H" 22,36
(-4
)
b 4
sinÕ = = H" 0,1789
r 22,36
Ä… = -Õ = -10018'
a 22
cosÕ = = H" 0,9839
r 22,36
Ä… = -Õ = -10018'
lub
b
Ä… = -arctg dla a > 0 i b < 0
a
4
Ä… = -arctg = -10018'
22
I Å" Z = 22,36e- j10018'
______________________________________________________________________
7.36
Dane: Szukane: Wzory:
z = a + jb
j150
jÕ
I = 5e
z = re
Z = 30e- j75
j2 = -1
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
j150 j(150 +(-750 ))
I Å" Z = 5e 30e- j75 = 5Å"30e =150e- j 600
I Å" Z =150(cos(-600) + j sin(-600)) =150(cos 600 + j(-sin 600)) =
ëÅ‚ öÅ‚
1 3
=150ìÅ‚ - j = 75 - j75 3
÷Å‚
ìÅ‚ ÷Å‚
2 2
íÅ‚ Å‚Å‚
______________________________________________________________________
7.37
Dane: Szukane: Wzory:
z = a + jb
jÕ
z = re
j2 = -1
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
5 - j5 ; sprzężona to 5 + j5
- j ; sprzężona to j
2 ; sprzężona to 2
10 + j ; sprzężona to 10 - j
j 200
3e ; sprzężona to 3e- j 200
j300
1,5e- j300 ; sprzężona to 1,5e
______________________________________________________________________
7.38
Dane: Szukane: Wzory:
z = a + jb
j900
jÕ
U = 380e
z = re
j300
Z = 76e j2 = -1
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
j900
U 380e 380
j(900 -300 ) j600
= = e = 5e
j300
Z 76
76e
______________________________________________________________________
7.39
Dane: Szukane: Wzory:
z = a + jb
Z1 = 2 + j4 jÕ
z = re
Z2 = 2 - j6
j2 = -1
r = a2 + b2
b
Õ = arctg dla a > 0 i b e" 0
a
b
Õ = -arctg dla a > 0 i b < 0
a
b
Õ = arctg + Ä„ dla a < 0 i b e" 0
a
b
Õ = arctg -Ä„ dla a < 0 i b < 0
a
Ä„
Õ = + dla a = 0 i b > 0
2
Ä„
Õ = - dla a = 0 i b < 0
2
a = r cosÕ
b = r sinÕ
Z1 Å" Z2
2 + j4 Å" 2 4
( ) ( - j6 - j12 + j8 - j224 - j4 -(-1 24
) 4 )
= = = =
Z1 + Z2 2 + j4 + 2 - j6 4 - j2 4 - j2
28 - j4 4 + j2 112 - j16 + j56 - j28 120 + j40
= = = = 6 + j2
4 - j2 4 + j2 42 + 22 20
______________________________________________________________________
7.40
Dane: Szukane: Wzory:
R = 200 &! isk = u
i =
R
u = 311sin 314t
i =
u 311sin 314t 311
i = = = sin 314t = 1,555sin 314t [A]
R 200 200
Im = 1,555 [A]
Im 1,555
isk = = H" 1,1[A]
1, 41
2
______________________________________________________________________
7.41
Dane: Szukane: Wzory:
R = 40 &!
u
Pm =
i =
I = 2,5 A R
P = ui
2
Pm = Im2R = I 2 R = 6, 25Å" 2Å" 40 = 500 W
( )
______________________________________________________________________
7.42
Dane: Szukane: Wzory:
L = 0,6 H xL =
xL = ÉL = 2Ä„ fL
U = 220 V
I =
f = 50 Hz
xL = ÉL = 2Ä„ fL = 2Å"3,14Å"50Å"0,6 = 188, 4 &!
U 220
I = = H" 1,168 A
xL 188, 4
U
Ć=Ą/2
I
______________________________________________________________________
7.43
Dane: Szukane: Wzory:
R H" 0 &!
L = xL = ÉL = 2Ä„ fL
I = 0, 23 A
U = I Å" xL
U = 150 V
f = 50 Hz
U 150
xL = = = 652 &!
I 0, 23
xL xL 652 652
L = = = = &!s H" 2,08 H
É 2Ä„ f 2Å"3,14Å"50 314
______________________________________________________________________
7.44
Dane: Szukane: Wzory:
x =
L =12 mH
xL = ÉL = 2Ä„ fL
B =
f1 =1000 Hz
U = I Å" xL
f2 = 20 kHz 1
B =
xL
xL1 = 2Ä„ f1L = 75,36 &!
1 1
B1 = = H" 0,01226 S = 12, 26 mS
xL1 75,36
xL2 = 2Ä„ f2L = 1507, 2 &!
1 1
B2 = = = 0,00066 S = 0,66 mS
2Ä„ f2L 1507, 2
______________________________________________________________________
7.45
Dane: Szukane: Wzory:
i = 22sin 6280t [A]
xL = ÉL = 2Ä„ fL
L = 0,25 H
U = I Å" xL
i = 0, 22sin 6280t
i = Im sinÉt
xL = ÉL = 6280Å"0, 25 = 1570 &!
u = ixL = 0, 22sin 6280t Å"1570 = 345, 4sin 6280t + 900 V
( )
Ponieważ napięcie na elemencie indukcyjnym jest przyspieszone o 900 w stosunku do prądu.
Ä„
ëÅ‚6280t + öÅ‚
czyli u = 345, 4sin V
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
U
Ć=Ą/2
I
______________________________________________________________________
7.46
Dane: Szukane: Wzory:
C = 10 µF
xC = 1 1
xC = =
f = 50 Hz ÉC 2Ä„ fC
BC =
1
BC =
xC
1 1 1 106
xC = = = = H" 318, 47 &!
ÉC 2Ä„ fC 2Å"3,14Å"50Å"10Å"10-6 3140
1 1
BC = = = 3140Å"10-6 S = 3,14Å"10-3 S
xC 318, 47
______________________________________________________________________
7.47
Dane: Szukane: Wzory:
C = 5 µF i =
1 1
xC = =
Wm =
u = 400sin 314t [V ] ÉC 2Ä„ fC
1
BC =
xC
2
CU
W =
2
u = 400sin 314t [V ]
U = Um sinÉt
1 1 106
xC = = = H" 636,94 &!
ÉC 314Å"5Å"10-6 1570
1 1
BC = = = 1570Å"10-6 S = 1,57Å"10-3 S
xC 636,94
CUm2 5Å"10-6 Å" 4002 5Å"16Å"10-2
Wm = = = = 0, 4 J
2 2 2
Ponieważ w idealnym kondensatorze prąd jest przyspieszony w stosunku do napięcia o 900
u 400sin 314t
i = = = 0,628sin 314t + 900 [A]
( )
xC 636,94
I
Ć=Ą/2
U
______________________________________________________________________
7.48
Dane: Szukane: Wzory:
I = 0,35 A C = 1 1
xC = =
ÉC 2Ä„ fC
U = 135V
f = 50 Hz
U
I =
xC
U
xC =
I
1 U
=
ÉC I
I I 0,3 0,3
C = = = = H" 7,08Å"10-6 F = 7,08 µF
ÉU 2Ä„ fU 2Å"3,14Å"50Å"135 42390
______________________________________________________________________
7.49
Dane: Szukane: Wzory:
L = 0,1 H C = 1 1
xC = =
ÉC 2Ä„ fC
C = 0,1 µF
xL = ÉL = 2Ä„ fL
f = 1000 : 5000 Hz
1 1 1 103
xC = = = = = 796, 2 &!
ÉC 2Ä„ fC 2Å"3,14Å" 2000Å"0,1Å"10-6 1, 256
xL = ÉL = 2Ä„ fL = 2Å"3,14Å" 2000Å"0,1 = 1256 &!
3500
3000
2500
2000
1500
1000
500
0
f [Hz]
1000 2000 3000 4000 5000 6000
______________________________________________________________________
7.50
Dane: Szukane: Wzory:
I = 1 1
C = 0,75 µF = 0,75Å"10-6 F
xC = =
ÉC 2Ä„ fC
U =15 kV = 15Å"103 V
U
f = 50 Hz
I =
xL
1 1 1 106
xC = = = = = 4246 &!
ÉC 2Ä„ fC 2Å"3,14Å"50Å"0,75Å"10-6 235,5
U U
I = = = U 2Ä„ fC = 3,53 A
xL 1
2Ä„ fC
______________________________________________________________________