Übungsaufgaben zu Wechselstromelemente

Lösung zur
Aufgabe 6.7.1
(Spule)

Induktiver Widerstand der Spule
$Xl = ωL = 2πfL = 2π ⋅ 50Hz ⋅ 0,1H = 31,4Ω-$ (B.4.1)
Strom
$U 230V I = ---= ------ = 7,3A- XL 31,4Ω$ (B.4.2)
Phasenverschiebungswinkel
$U- U 230V I- = Z-- = jX---= j31,4Ω- -L L ∘ = −-j7,3A--= (7,3A-⁄ −-90-) (B.4.3)$

Lösung zur
Aufgabe 6.7.2
(Kondensator)

Kapazitiver Widerstand des Kondensators
$1 1 XC = − ----= − ------ ωC 2πf C = − ----------1------------= − 19,9kΩ (B.4.4) 2π ⋅ 800Hz ⋅ 10 ⋅ 10−9F ---------$
Strom
$U 50V I = -----= --------3--= 2,51mA--- |XC | 19,9 ⋅ 10 Ω$ (B.4.5)
Phasenverschiebungswinkel
$U- U 50V I- = ----= ------= -----------3-- ZC − jXC − j19,9 ⋅∘ 10 Ω = j2,51mA--= (2,51mA--⁄ 90-) (B.4.6)$

Lösung zur
Aufgabe 6.7.3
(RL-Reihe)

Die Reihenschaltung eines ohmschen Widerstandes R = 50Ω und einer Spule mit der Induktivität L = 0,2H liegt an einer Wechselspannung von U = 230V , f = 50Hz.

Scheinwiderstand der Reihenschaltung
$Z- = Z ⁄ φZ = R + jωL = 50Ω + j2 π ⋅ 50Hz ⋅ 0,2H = 50 Ω + j62,8Ω = (80,27-Ω⁄ 51,5∘) (B.4.7) ---------------$
Strom
$U 230V ∘ I-= --= --------⁄----∘- = (2,87A-⁄-−-51,5-)- Z- (80,27Ω - 51,5 )$ (B.4.8)
Teilspannung am Widerstand
$U-R = I ⋅ ZR = I ⋅ R = (2,87A ⁄ − 51,5∘) ⋅ 50Ω = (143,26V ⁄ − 51,5∘) (B.4.9) -------------------$
Teilspannung an der Spule
$U = I⋅ Z = I ⋅ jX --L ---L -- L∘ ∘ = (2,87A ⁄ − 51,5 )(62,8Ω⁄-90 ) = (179,9V ⁄-38,5 ∘) (B.4.10) ---------------$
Vorwiderstand
$∘ --------- Rv = Z′2 − X2L − R ∘ ------2----------2- = (115Ω ) − (62,8Ω ) − 50Ω = 46,33Ω- (B.4.11) -------$

Lösung zur
Aufgabe 6.7.4
(RL-Reihe)

Ohmsche Anteil

U0- 100V-- R = I = 1A = 100Ω-

(B.4.12)

Induktivität

∘ ------------------- √ --2----2- (230Ω )2 − (100Ω)2 L = --Z--−-R-- = -------------------- = 0,659H ω 2π ⋅ 50Hz --------

(B.4.13)

Lösung zur
Aufgabe 6.7.5
(RC-Reihe)

Scheinwiderstand
$⁄ Z- = Z- φZ = R − j-1--= 20Ω − j ----------1------------ ωC 2π ⋅ 1000Hz ⋅ 8 ⋅ 10−6F = 20Ω − j19,9Ω = (28,2Ω⁄-− 44,9∘) (B.4.14) -----------------$
Strom
$U 10V ∘ I-= --= -------⁄-------∘-= (0,355A-⁄ 44,9-) Z- (28,2Ω - − 44,9 )$ (B.4.15)
Teilspannung am Widerstand
$U-R = I ⋅ ZR = I-⋅ R = (0,355A ⁄ 44,9∘) ⋅ 20Ω = (7,10⁄-44,9∘) (B.4.16) ------------$
Teilspannung am Kondensator
$UC = I-⋅ ZC = I-⋅ (− jXC ) = (0,355A ⁄ 44,9∘) ⋅ (19,9Ω ⁄ − 90∘) ⁄ ∘ = (7,06V---−-45,1-) (B.4.17)$

Lösung zur
Aufgabe 6.7.6
(RC-Reihe)

Kapazität
$1 1 C = − -----= ------------------= 2,54μF-- ωXC 2π ⋅ 50Hz ⋅ 1254 Ω$ (B.4.18)
Induktivität
$−-XC- -1254-Ω--- L = ω = 2π ⋅ 50Hz = 3,99H--$ (B.4.19)

Lösung zur
Aufgabe 6.7.7
(RL-Parallel)

Die Parallelschaltung eines ohmschen Widerstandes R = 40Ω und einer Spule mit der Induktivität L = 0,1H wird an eine Wechselspannungsquelle mit der Spannung U = 100V , f = 50Hz, angeschlossen.

Scheinwleitwert
$Y- = Y ⁄ φY 1- -1- -1-- --------1-------- = R − j ωL = 40Ω − j2π ⋅ 50Hz ⋅ 0,1H = 0,025S − j0,0318S = (0,0405S-⁄-−-51,8∘) (B.4.20) -------------------$
Scheinwiderstand
$1- --------1---------- ⁄ ∘ Z-= Y-= (0,0405S ⁄-− 51,8∘) = (24,8Ω-- 51,8-)$ (B.4.21)
Strom
$⁄ ∘ I- = U ⋅ Y = 100V ⋅ (0,0405S- − 51,8 ) = (4,05A⁄-−--51,8-∘)- (B.4.22) -----------------$
Widerstand
$R = 15,3Ω r ------$ (B.4.23)

Induktivität

$X 19,5Ω Lr = --L- = ---------- = 62mH--- 2πf 2π ⋅ 50Hz$ (B.4.24)

Lösung zur
Aufgabe 6.7.8
(Spule)

Widerstand
$R = 38,4Ω- ------$ (B.4.25)

Induktivität

$XL 66,4Ω L = ---- = ---------- = 211mH--- 2πf 2π ⋅ 50Hz$ (B.4.26)

Widerstand

1 Rp = ---------= 153Ω- 0,00652S

(B.4.27)

Induktivität

1 1 Lp = − ---------= ---------------------= 282mH--- BL ⋅ 2πf 0,01129S ⋅ 2π ⋅ 50Hz

(B.4.28)

Lösung zur
Aufgabe 6.7.9
(Kondensator)

Verlustfaktor

1- -1-- tan δ = G--= -R--= -------106Ω--------= 3,18 ⋅ 10−3 BC ωC 2π ⋅ 50Hz ⋅ 10−6F -----------

(B.4.29)

Verlustwinkel

− 3 360 ∘ ∘ δ ≈ tanδ = 3,18 ⋅ 10 ⋅----- = 0,182-- 2π

(B.4.30)

Widerstand
$Rr = 10,1Ω-$ (B.4.31)

Kondensator

$--1-- --------1--------- C = − ωX = 2π ⋅ 50Hz ⋅ 3180Ω = 1,0μF- C$ (B.4.32)

Lösung zur
Aufgabe 6.7.10
(Kondensator)

Verlustwiderstand
$1 1 R = ---------= ------------------------------= 15,9kΩ- BC tan δ 2π ⋅ 50Hz ⋅ 10 ⋅ 10−6F ⋅ 2 ⋅ 10− 2------$ (B.4.33)
Widerstand
$Rr = 6,37Ω- ------$ (B.4.34)

Kondensator

$1 1 C = − ωX--- = 2π-⋅ 50Hz-⋅ 318Ω-= 10,0μF-- C$ (B.4.35)

Phasenverschiebungswinkel

φ = 90∘ − δ = 90∘ − 1,15 ∘ = 88,85-∘

(B.4.36)

B.4 Übungsaufgaben zu Wechselstromelemente