7.51
Dane: Szukane: Wzory:
L = 0,5 H 1 1
Z =
xC = =
ÉC 2Ä„ fC
R = 120 &!
xL = ÉL = 2Ä„ fL
f = 50 Hz
Z = R2 + xL2
xL = ÉL = 2Ä„ fL
2 2
Z = R2 + xL2 = R2 + 2Ä„ fL = 1202 + 2Å"3,14Å"50Å"0,5 =
( ) ( )
= 14400 + 24649 H" 197,6 &!
Z xL
R
______________________________________________________________________
7.52
Dane: Szukane: Wzory:
Z =
1 1
C = 10 nF =10-8 F
xC = =
Õ = ÉC 2Ä„ fC
R = 400 k&! = 4Å"105 &!
xL = ÉL = 2Ä„ fL
f = 50 Hz
Z = R2 + x2
1 1 1 108
xC = = = = = 318471 &!
ÉC 2Ä„ fC 2Å"3,14Å"50Å"10-8 314
2
2
ëÅ‚ öÅ‚
1
Z = R2 + xC 2 = R2 + = 4Å"105 + 3184712 =
( )
ìÅ‚ ÷Å‚
2Ä„ fC
íÅ‚ Å‚Å‚
= 16Å"1010 +101423777841 H" 511Å"103 &! = 0,511 M &!
R
Ć
XC
Z
xC 3,1847 Å"105
tgÕ = = = 0,7962
R 4Å"105
Õ = -38032'
______________________________________________________________________
7.53
Dane: Szukane: Wzory:
L = 0,5 H Z = F( f ) 1 1
xC = =
ÉC 2Ä„ fC
R = 120 &!
xL = ÉL = 2Ä„ fL
f = 0 :1000 Hz
Z = R2 + xL2
xL = ÉL = 2Ä„ fL
2
Z = R2 + xL2 = R2 + 2Ä„ fL
( )
Z=F(f)
©
3500
3000
2500
2000
1500
1000
500
0
0 200 400 600 800 1000 f[Hz]
______________________________________________________________________
7.54
Dane: Szukane: Wzory:
L = 0,51 H Z =
1 1
xC = =
R = 120 &! I = ÉC 2Ä„ fC
xL = ÉL = 2Ä„ fL
U = 230 V UR =
f = 50 Hz UL =
Z = R2 + xL2
U
I =
Z
xL = ÉL = 2Ä„ fL = 2Å"3,14Å"50Å"0,51 H" 160 &!
Z = R2 + xL2 = 1202 +1602 = 40000 = 200 &!
U 230
I = = = 1,15 A
Z 200
UR = IR = 1,15Å"120 = 138V
UL = IxL = 1,15Å"160 = 184 V
U
UL
Z
xL
Ć Ć
UR
I R
______________________________________________________________________
7.55
Dane: Szukane: Wzory:
L = 0,255 H I =
1 1
xC = =
R = 80 &! UR = ÉC 2Ä„ fC
xL = ÉL = 2Ä„ fL
U = 24 V UL =
f = 50 Hz Õ =
Z = R2 + xL2
U
I =
Z
xL = ÉL = 2Ä„ fL = 2Å"3,14Å"50Å"0, 255 H" 80 &!
Z = R2 + xL2 = 802 + 802 = 12800 =113 &!
U 24
I = = = 0, 212 A
Z 113
UR = IR = 0, 212Å"80 = 17 V
UL = IxL = 0, 212Å"80 = 17 V
U
UL
Ć
UR
I
UL 17
tgÕ = = = 1
UR 17
Õ = 450
______________________________________________________________________
7.56
Dane: Szukane: Wzory:
L = 0,255 H I =
1 1
xC = =
R = 80 &! UR = ÉC 2Ä„ fC
xL = ÉL = 2Ä„ fL
U = 24 V UL =
f = 50 Hz Õ =
Z = R2 + xL2
u = Um sinÉt
i =
U
I =
uR =
Z
uL =
xL = ÉL = 2Ä„ fL = 2Å"3,14Å"50Å"0, 255 H" 80 &!
Z = R2 + xL2 = 802 + 802 = 12800 =113 &!
U 24
I = = = 0, 212 A
Z 113
UR = IR = 0, 212Å"80 = 17 V
UL = IxL = 0, 212Å"80 = 17 V
U
UL
Ć
UR
I
UL 17
tgÕ = = = 1
UR 17
Õ = 450
i = Im sin(Ét - 450)
uR = URm sin(Ét - 450)
uL = ULm sin(Ét + 450)
______________________________________________________________________
7.57
Dane: Szukane: Wzory:
xL = ÉL = 2Ä„ fL
U
UL
Ć
UR
I
UL i Å" xL 2Ä„ fL
tgÕ = = =
UR i Å" R R
______________________________________________________________________
7.58
Dane: Szukane: Wzory:
1 1
xC = =
ÉC 2Ä„ fC
UR
I
Ć
UC U
UC i Å" xC 1
tgÕ = = =
UR i Å" R 2Ä„ fCR
______________________________________________________________________
7.59
Dane: Szukane: Wzory:
U = 30 V
L = 1 1
xC = =
I = 0,6 A ÉC 2Ä„ fC
f = 50 Hz xL = ÉL = 2Ä„ fL
R = 40 &!
UR = IR = 0,6Å" 40 = 24V
UR 24
cosÕ = = = 0,8
U 30
Õ = 36008'
UL
sinÕ =
U
UL = U sinÕ = 30Å"0,5897 =17,69 V
UL 17,69
xL = = = 29,485 &!
I 0,6
xL = 2Ä„ fL
xL 29, 485
L = = H" 0,0939 H = 93,9 mH
2Ä„ f 2Å"3,14Å"50
______________________________________________________________________
7.60
Dane: Szukane: Wzory:
U= = 4, 2V xL = 1 1
xC = =
ÉC 2Ä„ fC
I= = 0,14 A L =
xL = ÉL = 2Ä„ fL
U = 75V
f = 50 Hz
I = 1,5 A
U= 4, 2
R = = = 30 &!
I= 0,14
UR = IR = 1,5Å"30 = 45V
UR 45
cosÕ = = = 0,6
U 75
Õ = 53052'
UL
sinÕ =
U
UL = U sinÕ = 75Å"0,8077 = 60,58V
UL 60,58
xL = = = 40,38 &!
I 1,5
xL = 2Ä„ fL
xL 40,38
L = = H" 0,1286 H = 128,6 mH
2Ä„ f 2Å"3,14Å"50
______________________________________________________________________
7.61
Dane: Szukane: Wzory:
R = 10 &!
xL = 1 1
xC = =
L = 200 µH ÉC 2Ä„ fC
Q =
U = 75V xL = ÉL = 2Ä„ fL
xL
f = 0,8 MHz
Q =
R
xL = 2Ä„ fL = 2Å"3,14Å"0,8Å"106 Å"0, 2Å"10-3 H" 1005 &!
xL 1005
Q = = = 100,5
R 10
______________________________________________________________________
7.62
Dane: Szukane: Wzory:
R = 27 &! f ' =
1 1
xC = =
L = 0,1146 H ÉC 2Ä„ fC
I = 2 A xL = ÉL = 2Ä„ fL
f = 50 Hz
I ' = 1,5 A
xL = 2Ä„ fL = 2Å"3,14Å"50Å"0,1146 = 35,9844 H" 36 &!
Z = R2 + xL2 = 272 + 362 = 2023,88 = 45 &!
U = I Å" Z = 2Å" 45 H" 90 V
U 90
Z ' = = = 60 &!
I ' 1,5
2
xL ' = Z - R2 = 602 - 272 = 2871 H" 53,58 &!
xL 53,58
f = = H" 74,5 Hz
2Ä„ L 2Å"3,14Å"0,1146
______________________________________________________________________
7.63
Dane: Szukane: Wzory:
U = 120 V L =
1 1
xC = =
I = 0,5 A L ' = ÉC 2Ä„ fC
f = 100 Hz xL = ÉL = 2Ä„ fL
I ' = 8 A
R = 4 &!
U 120
Z = = = 240 &!
I 0,5
2
xL = Z - R2 = 2402 - 42 H" 240 &!
xL 240
L = = = 0,382 H
2Ä„ f 2Å"3,14Å"100
U 120
Z ' = = = 15 &!
I ' 8
2
xL ' = Z - R2 = 152 - 42 H" 14, 46 &!
xL ' 14, 46
L = = H" 0,023 H
2Ä„ f 2Å"3,14Å"100
______________________________________________________________________
7.64
Dane: Szukane: Wzory:
U = 240 V Z =
1 1
xC = =
I = ÉC 2Ä„ fC
C = 10 µF = 10-5 F
xL = ÉL = 2Ä„ fL
UR =
f = 50 Hz
UC =
R = 500 &!
Õ =
1 1 105
xC = = = = 318 &!
2Ä„ fC 2Å"3,14Å"50Å"10-5 314
Z = R - jxC = (500 - j318) &!
Z = R2 + xC 2 = 5002 + 3182 = 592,6 &!
U 240 240(500 + j318)
I = = = = (0,3418 + j0, 2147) A
Z 500 - j318 5002 + 3182
U 240
I = = = 0, 405 A
Z 592,6
U = RI = 500(0,3418 + j0, 2174) = (170,9 + j108,7) V
R
U = - jxC I = - j318(0,3418 + j0, 2174) = (69,1- j108,7) V
C
0
tgÕU = = 0
240
ÕU = 00
0, 2174
tgÕI = = 0,636
0,3418
ÕI = 32028'
Õ = ÕI -ÕU = 32028'- 00 = 32028'
______________________________________________________________________
7.65
Dane: Szukane: Wzory:
P = 40 W C =
1 1
xC = =
UN = 220 V ÉC 2Ä„ fC
U = 380 V xL = ÉL = 2Ä„ fL
2
f = 50 Hz U
P =
R
UN 2 2202 48400
R = = = = 1210 &!
P 40 40
1
xC =
2Ä„ fC
2
U = UR2 +UC 2
2
UC = U -UR 2 = 3802 - 2202 = 96000 = 309,8V
UC = IxC
UC
I =
xC
U
Z =
I
2
Z = R2 + xC 2
2
U
ëÅ‚ öÅ‚
= R2 + xC 2
ìÅ‚ ÷Å‚
I
íÅ‚ Å‚Å‚
2
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
U
ìÅ‚ ÷Å‚
= R2 + xC 2
UC
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
xC
íÅ‚ Å‚Å‚
2
U xC 2
= R2 + xC 2
UC 2
2
U xC 2 = R2UC 2 + xC 2UC 2
2
U xC 2 - xC 2UC 2 = R2UC 2
R2UC 2
xC 2 =
2
U -UC 2
RUC 1210Å"309,8 374858
xC = = = =1703,9 &!
2
U -UC 2 3802 - 309,82 220
1 1
C = = =1,87 Å"10-6 F =1,87 µF
2Ä„ fxC 2Å"3,14Å"50Å"1703,9
______________________________________________________________________
7.66
Dane: Szukane: Wzory:
1 1
Uwy =
C = 2 µF = 2Å"10-6 F
XC = =
ÉC 2Ä„ fC
R = 5 k&! = 5Å"103 &!
2 2
Z = R2 + X
Uwe = 40V
f1 = 50 Hz
f2 = 10000 Hz
1 1 106
X 'C = = = = 1592 &!
2Ä„ f1C 2Å"3,14Å"50Å" 2Å"10-6 628
Z ' = R2 + X 'C 2 = 50002 +15922 = 5247 &!
Uwe
I ' =
Z '
Uwe 40
U 'wy1 = I ' R = R = 5000 H" 38,1V
Z ' 5247
1 1 106
X ''C = = = = 7,96 &! H" 8 &!
2Ä„ f2C 2Å"3,14Å"10000Å" 2Å"10-6 125600
Z '' = R2 + X ''C 2 = 50002 + 82 H" 5000 &!
Uwe
I '' =
Z ''
Uwe 40
U ''wy1 = I '' R = R = 5000 H" 40 V
Z '' 5000
______________________________________________________________________
7.67
Dane: Szukane: Wzory:
Z =
R1 = 5 &! 1 1
XC = =
UL1 = ÉC 2Ä„ fC
L1 = 1 mH = 10-3 H
UL2 = X = ÉL = 2Ä„ fL
L
R2 = 8 &!
2 2
cosÕ1 = Z = R2 + X
L2 = 3,19 mH = 3,19Å"10-3 H
cosÕ2 =
U = 3,7 V
cosÕ =
f = 500 Hz
2
Z1 = R12 + X = R12 + (2Ä„ fL1)2 = 52 + (2Å"3,14Å"500Å"10-3)2 = 25 + 9,86 = 5,9 &!
L1
R1 5
cosÕ1 = = = 0,8475
Z1 5,9
Z2 = R22 + XL22 = R22 + (2Ä„ fL2)2 = 82 + (2Å"3,14Å"500Å"3,19Å"10-3)2 = 64 +100,33 H"12,82 &!
R2 8
cosÕ2 = = = 0,6240
Z2 12,82
2
2 2 2 2 2
Z = R1 + R2 + X + XL2 = R2 + R2 + 2Ä„ fL1 + 2Ä„ fL2 = R2 + R2 + 2Ä„ f L1 + L2 =
( ) ( ) ( ) ( ) ( ) ( ( )
)
L1
= 132 + (2Å"3,14Å"500Å" 4,19Å"10-3)2 = 169 +173,09 H" 18,5 &!
R1 + R2 5 + 8
cosÕ = = = 0,7027
Z 18,5
U 3,7
I = = = 0, 2 A
Z 18,5
UL1 = I Å" Z1 = 0, 2Å"5,9 = 1,18V
UL2 = I Å" Z2 = 0, 2Å"12,82 = 2,56 V
______________________________________________________________________
7.68
Dane: Szukane: Wzory:
R = 400 &! I ' =
1 1
XC = =
U 'C1 = ÉC 2Ä„ fC
C1 = 16 µF =16Å"10-6 F
U 'R = X = ÉL = 2Ä„ fL
L
C2 = 4 µF = 4Å"10-6 F
2
Õ ' = Z = R2 + XC 2
I = 0, 25 A
f = 50 Hz
2
ëÅ‚ öÅ‚
2 1 1
Z = R2 + XC1 + XC 2 = R2 + + =
( )
ìÅ‚ ÷Å‚
2Ä„ fC1 2Ä„ fC2
íÅ‚ Å‚Å‚
2
ëÅ‚ 1 1 öÅ‚ 2
= 4002 + + = 160000 + 199 + 796 = 115025 H" 1072 &!
( )
ìÅ‚ ÷Å‚
2Å"3,14Å"50Å"16Å"10-6 2Å"3,14Å"50Å" 4Å"10-6
íÅ‚ Å‚Å‚
U = I Å" Z = 0, 25Å"1072 H" 268V
2
2
1 ëÅ‚ 1 öÅ‚
Z ' = R2 + XC12 = R2 + = 4002 + = 4002 +1992 H" 446,8 &!
ìÅ‚ ÷Å‚
2Ä„ fC1 2Å"3,14Å"50Å"16Å"10-6
íÅ‚ Å‚Å‚
U 268
I ' = = H" 0,6 A
Z ' 446,8
U 'R = I 'Å" R = 0,6Å" 400 = 240 V
U 'C1 = I 'Å" XC1 = 0,6Å"199 = 119, 4 V
U 'C1 119, 4
tgÕ = = = 0, 4975
U 'R 240
Õ = 26027 '
______________________________________________________________________
7.69
Dane: Szukane: Wzory:
R = 60 &!
Z = 1 1
XC = =
ÉC 2Ä„ fC
C = 28,6 µF = 28,6Å"10-6 F
X = ÉL = 2Ä„ fL
L = 100 mH = 0,1 H
L
2 2
Z = R2 + X
f = 50 Hz
XL = 2Ä„ fL = 2Å"3,14Å"50Å"0,1 = 31,4 &!
1 1 106
XC = = = H" 111, 4 &!
2Ä„ fC 2Å"3,14Å"50Å"28,6Å"10-6 8980
2
Z = R2 + X - XC 2 = 602 + 31, 4 -111, 4 = 3600 + 6400 = 100 &!
( ) ( )
L
XL
R
Z XC
______________________________________________________________________
7.70
Dane: Szukane: Wzory:
R = 60 &! I =
1 1
XC = =
UR = ÉC 2Ä„ fC
C = 28,6 µF = 28,6Å"10-6 F
UC = X = ÉL = 2Ä„ fL
L =100 mH = 0,1 H
L
2 2
UL = Z = R2 + X
f = 50 Hz
U = 50V
XL = 2Ä„ fL = 2Å"3,14Å"50Å"0,1 = 31,4 &!
1 1 106
XC = = = H" 111, 4 &!
2Ä„ fC 2Å"3,14Å"50Å"28,6Å"10-6 8980
2
Z = R2 + X - XC 2 = 602 + 31, 4 -111, 4 = 3600 + 6400 = 100 &!
( ) ( )
L
U 50
I = = = 0,5 A
Z 100
UR = IR = 0,5Å"60 = 30 V
UL = IX = 0,5Å"31, 4 = 15,7 V
L
UC = IXC = 0,5Å"111, 4 = 55,7 V
UL
UR I
UC
U
______________________________________________________________________
7.71
Dane: Szukane: Wzory:
R =150 &! I =
1 1
XC = =
UC = ÉC 2Ä„ fC
C = 31,9 µF = 31,9Å"10-6 F
URL = X = ÉL = 2Ä„ fL
L = 0,319 H
L
2 2
Õ = Z = R2 + X
f = 50 Hz
U = 240V
X = 2Ä„ fL = 2Å"3,14Å"50Å"0,319 H" 100 &!
L
2
ZL = R2 + X = 1502 +1002 H" 180,3 &!
L
1 1 106
XC = = = H" 99,8 &!
2Ä„ fC 2Å"3,14Å"50Å"31,9Å"10-6 10017
2 2
Z = R2 + X - XC = 1502 + 100 - 99,8 H" 150 &!
( ) ( )
L
U 240
I = = = 1,6 A
Z 150
URL = IZL = 1,6Å"180,3 H" 288,5V
UC = IXC = 1,6Å"99,8 H" 159,7 V
X - XC 100 - 99,8 0, 2
L
sinÕ = = = = 0,0013
Z 150 150
Õ = 0005'
______________________________________________________________________
7.72
Dane: Szukane: Wzory:
R =150 &! I =
1 1
XC = =
Õ = ÉC 2Ä„ fC
C = 31,9 µF = 31,9Å"10-6 F
X = ÉL = 2Ä„ fL
L = 0,319 H
L
2 2
Z = R2 + X
f = 100 Hz
U = 240V
X = 2Ä„ fL = 2Å"3,14Å"100Å"0,319 H" 200 &!
L
2
ZL = R2 + X = 1502 + 2002 = 250 &!
L
1 1 106
XC = = = H" 49,91 &! H" 50 &!
2Ä„ fC 2Å"3,14Å"100Å"31,9Å"10-6 10017
2 2
Z = R2 + X - XC = 1502 + 200 - 50 H" 212,13 &!
( ) ( )
L
U 240
I = = = 1,13 A
Z 212,13
X - XC 200 - 50 150
L
sinÕ = = = = 0,7071
Z 212,13 213,13
Õ = 450
______________________________________________________________________
7.73
Dane: Szukane: Wzory:
R = 50 &! U =
1 1
XC = =
ÉC 2Ä„ fC
C = 6,37 µF = 6,37 Å"10-6 F
X = ÉL = 2Ä„ fL
L = 0,8 H
L
2 2
Z = R2 + X
f = 50 Hz
UC = 200 V
X = 2Ä„ fL = 2Å"3,14Å"50Å"0,8 H" 251 &!
L
1 1
XC = = H" 500 &!
2Ä„ fC 2Å"3,14Å"50Å"6,37 Å"10-6
UC 200
I = = = 0, 4 A
XC 500
UR = IR = 0, 4Å"50 = 20V
UL = IX = 0, 4Å"251 = 100,4 V
L
U = UR2 + (UL -UC )2 = 202 + (100,4 - 200)2 H"102 V
______________________________________________________________________
7.74
Dane: Szukane: Wzory:
R = 160 &! L =
1 1
XC = =
f = 50 Hz C = ÉC 2Ä„ fC
UR = 40 V X = ÉL = 2Ä„ fL
L
2 2
UL = 40V Z = R2 + X
U = 50V
UR 40
I = = = 0, 25 A
R 160
UL 40
X = = = 160 &!
L
I 0, 25
2Ä„ f 160 160
L = = = H" 0,51 H
X 2Å"3,14Å"50 314
L
U 50
Z = = = 200 &!
I 0, 25
2
Z = R2 + (X - XC )2
L
2
XC = X Ä… Z - R2
L
2
XC1 = XL - Z - R2 = 160 - 2002 -1602 = 160 -120 = 40 &!
2
XC1 = XL + Z - R2 = 160 + 2002 -1602 = 160 +120 = 280 &!
1 1
C1 = = = 79,6Å"10-6 F = 79,6 µF
2Ä„ fXC1 314Å" 40
1 1
C2 = = = 11,37 Å"10-6 F = 11,37 µF
2Ä„ fXC 2 314Å" 280
______________________________________________________________________
7.75
Dane: Szukane: Wzory:
I = 0, 2 A XC = 1 1
XC = =
f = 50 Hz ÉC 2Ä„ fC
C =
X = ÉL = 2Ä„ fL
U = 40V
L
2 2
R =120 &! Z = R2 + X
XL = 200 &!
U 40
Z = = = 200 &!
I 0, 2
2
Z = R2 + (X - XC )2
L
2
(X - XC )2 = Z - R2
L
2
XC = X Ä… Z - R2
L
XC1 = 200 + 2002 -1202 = 200 +160 = 360 &!
XC1 = 200 - 2002 -1202 = 200 -160 = 40 &!
1 1 1
C1 = = = H" 8,85Å"10-6 F = 8,85 µF
2Ä„ fXC1 2Å"3,14Å"50Å"360 113040
1 1 1
C2 = = = H" 79,6Å"10-6 F = 79,6 µF
2Ä„ fXC 2 2Å"3,14Å"50Å" 40 12560
______________________________________________________________________
7.76
Dane: Szukane: Wzory:
j00
U = 1 1
I = 0, 2 A C
XC = =
ÉC 2Ä„ fC
U =
f = 50 Hz L
X = ÉL = 2Ä„ fL
U =
L
U = 40 V R
2 2
Z = R2 + X
U =
R = 120 &!
X = 200 &!
L
j00
I = 0, 2e = 0,2(cos00 + j sin 00) = 0,2(1+ j0) = 0, 2 A
j00
e = 1
j 00 j 00
U = RI = 120Å"0, 2e = 24e = 24 V
R
j00
U = jX I = j200Å"0, 2e = j40 V
L L
j00
U = - jXC I = - jXC Å"0, 2e = - jXC Å"0, 2
C
U = U +U +U
R L C
40(cosÕ Ä… j sinÕ) = 24 + j40 - jXC Å"0, 2
40cosÕ Ä… j40sinÕ) = 24 + j(40 - XC Å"0, 2)
40cosÕ = 24
24
cosÕ = = 0,6
40
Õ = 53008'
sin 53008' = 0,8
+ j40sinÕ = j(40 - XC Å"0, 2)
+ j40Å"0,8 = j(40 - XC Å"0, 2)
+ j32 = j(40 - XC Å"0, 2)
32 = 40 - XC Å"0, 2
X 'C = 40
- j40sinÕ = j(-40 + XC Å"0, 2)
- j40Å"0,8 = - j(-40 + XC Å"0, 2)
- j32 = - j(-40 + XC Å"0, 2)
32 = -40 + XC Å"0, 2
X ''C = 360
U 'C = - jXC Å"0, 2 = - j40Å"0, 2 = - j8V
U ''C = - jXC Å"0, 2 = - j360Å"0, 2 = - j72 V
U ' = U +U +U 'C = 24 + j40 - j8 = 24 + j32 V
( )
R L
U '' = U +U +U ''C = 24 + j40 - j72 = 24 - j32 V
( )
R L
y
y
U C
UL
U UR
I
UL
U C
U
UR
I
______________________________________________________________________
7.77
Dane: Szukane: Wzory:
UR = 120 V URC = 1 1
XC = =
ÉC 2Ä„ fC
R : X : XC = 2 : 3: 4 URL =
L
X = ÉL = 2Ä„ fL
L
2 2
Z = R2 + X
UR
R =
I
UR
I =
R
R 2
=
X 3
L
3R
X =
L
2
UR 3R 3UR
UL = IX = Å" =
L
R 2 2
9UR 2 9
URL = UR2 +UL2 = UR2 + = UR 1+ H" 120Å"1,8 = 216 V
4 4
R 2
=
XC 4
4R
XC =
2
UR 4R
UC = IXC = Å" = 2UR
R 2
URL = UR2 +UL2 = UR2 + 4UR2 = UR 1+ 4 H" 120Å"1,8 = 268V
y
UR
UL
UR
I
UC
URC
______________________________________________________________________
7.78
Dane: Szukane: Wzory:
U =
1 1
XC = =
ÉC 2Ä„ fC
X = ÉL = 2Ä„ fL
L
2 2
R = 40 &!
Z = R2 + X
X = 50 &!
L1
X = 80 &!
L2
XC1 = 50 &!
XC 2 = 120 &!
UV = 5V
ZV = R2 + (X - XC1)2 = 402 + 302 = 50 &!
L2
UV 5
I = = = 0,1 A
ZV 50
UL1 = IX = 0,1Å"50 = 5V
L1
UR = IR = 0,1Å" 40 = 4 V
UL2 = IX = 0,1Å"80 = 8V
L2
UC1 = IXC1 = 0,1Å"50 = 5V
UC 2 = IXC 2 = 0,1Å"120 = 12 V
U = IZ = I R2 + (X + X - XC1 - XC 2)2 = 0,1 402 + 402 H" 5,66 V
L1 L2
______________________________________________________________________
7.79
Dane: Szukane: Wzory:
U1 U2 U3 U4 U5 U6
U
______________________________________________________________________
7.80
Dane: Szukane: Wzory:
fr =
R = 12 &!
1 1
XC = =
L = 160 µH ÉC 2Ä„ fC
C = 250 pF X = ÉL = 2Ä„ fL
L
2 2
Z = R2 + X
1
fr =
2Ä„ LC
1 1 1
fr = = = = 796Å"103 = 796 kHz
2Ä„ LC
2Å"3,14 160Å"10-6 Å"250Å"10-12 2Å"3,14Å"10-9 160Å"250
______________________________________________________________________
7.81
Dane: Szukane: Wzory:
fr = 1 1
fr = 465 kHz = 465Å"103 Hz
XC = =
ÉC 2Ä„ fC
C = 1000 pF =10-9 F
X = ÉL = 2Ä„ fL
L
2 2
Z = R2 + X
1
fr =
2Ä„ LC
1 109
L = = H" 117 Å"10-6 H = 117 µH
2
4Å"9,86Å" 216225Å"106
2Ä„ fr C
( )
______________________________________________________________________
7.82
Dane: Szukane: Wzory:
fr = 1 1
fr =1,2 MHz =1,2Å"106 Hz
XC = =
ÉC 2Ä„ fC
L =100 µH =10-4 H
X = ÉL = 2Ä„ fL
L
2 2
Z = R2 + X
1
fr =
2Ä„ LC
1 104 10-9
C = = = H" 0,176Å"10-9 F = 176 pF
2
4Å"9,86Å"1, 44Å"1012 5,679
2Ä„ fr L
( )
______________________________________________________________________
7.83
Dane: Szukane: Wzory:
fr = 1 1
C = 20 nF = 20Å"10-9 F
XC = =
ÉC 2Ä„ fC
I = F( f )
L = 20 mH = 20Å"10-3 H
X = ÉL = 2Ä„ fL
Q1;2;3 =
L
U =10V
2 2
Z = R2 + X
R = 0;20;100 &!
1
fr =
2Ä„ LC
XL XC UL UC
Qr = = = =
R R U U
1 1 106
fr = = = H" 7962 Hz
2Ä„ LC
2Å"3,14 20Å"10-3 Å" 20Å"10-9 6, 28Å" 20
X 2Ä„ fr L 2Å"3,14Å"7962Å"0,02
L
Q1r = = = H" "
R R 0
XL 2Ä„ frL 2Å"3,14Å"7962Å"0,02
Q2r = = = H" 50
R R 20
XL 2Ä„ frL 2Å"3,14Å"7962Å"0,02
Q2r = = = H" 10
R R 100
U
I =
R2 + (X - XC )2
L
I[mA]
600
500
400
300
200
100
0
F[Hz]
______________________________________________________________________
7.84
Dane: Szukane: Wzory:
f = 200 Hz I =
1 1
XC = =
L = 0,318 H UL = ÉC 2Ä„ fC
U = 25V UC = X = ÉL = 2Ä„ fL
L
2 2
R = 50 &! UR = Z = R2 + X
1
C =
fr =
2Ä„ LC
UC;L
=
XL XC UL UC
U Qr = = = =
R R U U
1 1 1
C = = = H" 2Å"10-6 F = 2 µF
2 2
501656
2Ä„ f L 2Å"3,14Å" 200 0,318
( ) ( )
U U U
I = = = =
2 2 2
R2 + XL - XC
( ) ëÅ‚ 1 öÅ‚ ëÅ‚ öÅ‚
R2 + 2Ä„ fL -
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
2Ä„ fC
1
íÅ‚ Å‚Å‚
ìÅ‚ ÷Å‚
R2 + 2Ä„ fL -
1
ìÅ‚ ÷Å‚
2Ä„ f
2
ìÅ‚ ÷Å‚
2Ä„ f L
( )
íÅ‚ Å‚Å‚
U U 25
= = = = 0,5 A
2
2 R 50
ëÅ‚ öÅ‚
2Ä„ f L
( )
R2 + ìÅ‚ 2Ä„ fL - ÷Å‚
ìÅ‚ ÷Å‚
2Ä„ f
íÅ‚ Å‚Å‚
I 0,5 0,5Å"106
UC = I Å" XC = = = H" 200 V
2Ä„ fC 2Å"3,14Å" 200Å" 2Å"10-6 2512
UL = I Å" X = I 2Ä„ fL = 0,5Å" 2Å"3,14Å" 200Å"0,318 H" 200 V
L
UR = I Å" R = 0,5Å"50 = 25V
UL 200
= = 8
U 25
UC 200
= = 8
U 25
______________________________________________________________________
7.85
Dane: Szukane: Wzory:
L = 1 1
CN = 40 nF = 40Å"10-9 F
XC = =
ÉC 2Ä„ fC
f = 3 kHz = 3Å"103 Hz
X = ÉL = 2Ä„ fL
L
2 2
Z = R2 + X
1
fr =
2Ä„ LC
1 1 1
L = = = H" 70, 4Å"10-3 H = 70, 4 mH
2 2
2Ä„ f C
( )
2Å"3,14Å"3Å"103 Å" 40Å"10-9 18,842 Å"106 Å" 40Å"10-9
( )
______________________________________________________________________
7.86
Dane: Szukane: Wzory:
f = 1 1
C1 = 8 nF = 8Å"10-9 F
XC = =
É = ÉC 2Ä„ fC
C2 = 2 nF = 2Å"10-9 F
X = ÉL = 2Ä„ fL
L
L1 = 1 mH = 10-3 H
2 2
Z = R2 + X
L2 = 3 mH = 3Å"10-3 H
1
fr =
2Ä„ LC
XL = XC
XL1 + X = XC1 + XC 2
L1
1 1
ÉL1 +ÉL2 = +
ÉC1 ÉC2
ëÅ‚ öÅ‚
1 1 1
É(L1 + L2) = +
ìÅ‚
É C1 C2 ÷Å‚
íÅ‚ Å‚Å‚
C1 + C2
É2(L1 + L2) =
C1C2
C1 + C2 8Å"10-9 + 2Å"10-9 10Å"10-9
É = = = =
C1C2(L1 + L2) 8Å"10-9 Å" 2Å"10-9(10-3 + 3Å"10-3) 16Å"10-9 Å"10-9 Å"4Å"10-3
10
= H" 0,3952Å"106 rad / s
64Å"10-12
É = 2Ä„ f
É 0,3952Å"106
f = = = 62,9Å"106 Hz H" 63 kHz
2Ä„ 2Å"3,14
______________________________________________________________________
7.87
Dane: Szukane: Wzory:
R =100 &!
fr = 1 1
XC = =
L = 0,1 H ÉC 2Ä„ fC
f1 =
X = ÉL = 2Ä„ fL
Cz =1 F
f2 =
L
2 2
Z = R2 + X
Q =
1
C = 1 µF = 10-6 F
fr =
2Ä„ LC
X =| X - XC |= R
L
Ir
Z d" R 2 I e"
2
W zadaniu jest podana pojemność Cz = 1 F lecz w przykładzie z obliczeniami podstawiana
jest C = 1 µF = 10-6 F
1 1
fr = = = 503 Hz
2Ä„ LC
2Ä„ 0,1Å"10-6
ÉrL 2Ä„ Å"503Å"0,1
Q = = = 3,16
R 100
| XL - XC |= R to X - XC = R lub -( - XC = R
XL
)
L
Rozwiązując jedno równanie:
-( - XC = R
X
)
L
XC - XL = R
1
-Ér L = R
ÉrC
1-Ér LÉrC = RÉrC
-LCÉr 2 - RCÉr +1 = 0
2 2
-(-RC + RC - (-4LC) RC + RC + 4LC
) ( ) ( )
Ér1 = =
-2LC -2LC
Ér1 < 0
2 2
-(-RC - ( ) - (-4LC) RC - ( )
RC RC + 4LC
)
Ér 2 = = H" 2702 rad / s
-2LC -2LC
Ér 2
f1 = = 430 Hz
2Ä„
Rozwiązując drugie równanie:
X - XC = R
L
1
Ér L - = R
ÉrC
Ér LÉrC -1 = RÉrC
LCÉr 2 - RCÉr -1 = 0
2 2
-(-RC + RC - (-4LC) RC + RC + 4LC
) ( ) ( )
É 'r1 = = H" 3702 rad / s
2LC 2LC
2 2
-(-RC RC - (-4LC) RC - ( )
RC + 4LC
)- ( )
É 'r 2 = =
2LC 2LC
É 'r 2 < 0
É 'r1
f2 = H" 589 Hz
2Ä„
B = f2 - f1 = 589 - 430 = 159 Hz
______________________________________________________________________
7.88
Dane: Szukane: Wzory:
R = 50 &!
fr = 1 1
XC = =
L = 5 mH ÉC 2Ä„ fC
f1 =
X = ÉL = 2Ä„ fL
C = 0,5 µF = 0,5Å"10-6 F
f2 =
L
2 2
X =| X - XC |= R
Z = R2 + X
L
1
Ir
fr =
Z d" R 2 I e"
2Ä„ LC
2
1 1 1 105
fr = = = = H" 3184 Hz
2Ä„ LC
2Ä„ 5Å"10-3 Å"0,5Å"10-6 2Ä„ Å" 25 Å"10-5 31, 4
| XL - XC |= R to X - XC = R lub -( - XC = R
XL
)
L
Rozwiązując jedno równanie:
-( - XC = R
X
)
L
XC - XL = R
1
-Ér L = R
ÉrC
1-Ér LÉrC = RÉrC
-LCÉr 2 - RCÉr +1 = 0
2 2
-(-RC + RC - (-4LC) RC + RC + 4LC
) ( ) ( )
Ér1 = =
2Å"(-LC) -2LC
Ér1 < 0
2
2
50Å"0,5Å"10-6 - 50Å"0,5Å"10-6 + 4Å"5Å"10-3 Å"0,5Å"10-6
-(-RC - ( ) - (-4LC) ( )
RC
)
Ér 2 = = =
-2LC -2Å"5Å"10-3 Å"0,5Å"10-6
25Å"10-6 - 625Å"10-12 +10000Å"10-12 -780776Å"103
= = H" 15,616Å"103 = 15616 rad / s
-5Å"10-9 -5
Ér1 58443
f1 = = H" 2487 Hz
2Ä„ 2Å"3,14
Rozwiązując drugie równanie:
X - XC = R
L
1
Ér L - = R
ÉrC
Ér LÉrC -1 = RÉrC
LCÉr 2 - RCÉr -1 = 0
2
2 2
50Å"0,5Å"10-6 + 50Å"0,5Å"10-6 + 4Å"5Å"10-3 Å"0,5
-(-RC + RC - (-4LC) RC + RC + 4LC ( )
) ( ) ( )
É 'r1 = = =
2LC 2LC 2Å"5Å"10-3 Å"0,5Å"10-6
25Å"10-6 + 625Å"10-12 +10Å"10-9 25Å"10-6 + 10625Å"10-12 128,078Å"103
= = = H" 25,616Å"103 = 25616 rad / s
5Å"10-9 5Å"10-9 5
2 2
-(-RC RC - (-4LC) RC - ( )
RC + 4LC
)- ( )
É 'r 2 = =
2LC 2LC
É 'r 2 < 0
Ér1 25616
f2 = = H" 4079 Hz
2Ä„ 2Å"3,14
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