Elektrotechnik 1 PDF Übungsaufgaben

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! I I = 36,45 A. I 10 3 6= 1 m2 kg W s3 A A As m2 kg Nm Ws s2 m2 kg J VA s3 s 1 s kg...

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! I I = 36,45 A. I 10 3 6= 1 m2 kg W s3 A A As m2 kg Nm Ws s2 m2 kg J VA s3 s 1 s kg Vs s2 A m2 s 4 A2 C kg m2 V m kg J s2 m m2 kg Vs s 2 A2 A  1012 109 106 103 102 101 10 1 10 2 10 3 µ 10 6 10 9 10 12 10 15 10 18 Q 19 QElektron = 1,62 · 10 C. [Q] = C = As. 19 QProton = QElektron = 1,62 · 10 C. e e = QProton = QElektron. QD d QD dt QD (t)  QD I A I QD d QD dt QD,0 t t I QD d QD (t) I= dt C [I] = = A. s Zt QD (t) = I(⌧ ) d ⌧ + QD,0 ⌧ =0 t ⌧ QD,0 = QD (t = 0) t=0 I A I J= A A [J] = 2 m A A mm2 (mm)2 10 3  + U + = ˆ Wne I, QD Wel , P U U I, QD Wne Wne Wel dt dW  W U QD dt dW d QD dt dt dW U= d QD t t W QD P dW P = dt J [P ] = = W = Watt s W P QD I dW U= d QD J [U ] = = V = Volt. C W QD U dW = P · dt d QD = I · d t P · dt P U= = I · dt I Q1  ~eQ1,Q2 a F~Q2 Q2 ~eQ2,Q1 F~Q1 Q1 Q2 1 Q1 · Q2 F~Q1 = · · ~eQ2,Q1. 4⇡✏0 a2 a ✏0 ~eQ2,Q1 Q2 Q1 Q2 Q1 1 Q2 · Q1 F~Q2 = · · ~eQ1,Q2. 4⇡✏0 a2 = F~Q1 ~eQ1,Q2 Q1 Q2 ~eQ2,Q1 ~eQ2,Q1 = ~eQ1,Q2 Q1 Q2 Q1 Q2 q n Q1 ,... , Qn Xn 1 q · Qi F~q = · 2 · ~eQi,q,. 4⇡✏0 aQi,q, i=1 E q F~ q  F~ E=. q l + Z2 U12 = E(~s⇤ ) d ~s⇤. 1 ~s⇤ W12 Z2 W12 = dW 1 Z2 = q · E(~s⇤ ) d ~s⇤ 1 Z2 W12 = q · E(~s⇤ ) d ~s⇤ 1 = q · U12. ' P E(P ) 'B B 'B = 0 P ZB 'P = E(~s⇤ ) d ~s⇤ + 'B. P  ZB '1 = E(~s⇤ ) d ~s⇤ + 'B. 1 ZB '2 = E(~s⇤ ) d ~s⇤ + 'B. 2 0 B 1 ZB Z '1 '2 = E(~s⇤ ) d ~s⇤ + 'B @ E(~s⇤ ) d ~s⇤ + 'B A 1 2 ZB Z2 ⇤ ⇤ = E(~s ) d ~s + E(~s⇤ ) d ~s⇤ 1 B B B Z2 '1 '2 = E(~s⇤ ) d ~s⇤ = U12 1  l U ~ E + l I U + '(n 1) U = 'n '0 'n '(n '2 ' 1 '0 2)... + ~ E l I U +  4 I 3 2 1 t 50 100 150 200 250 I(t) QD (t) QD,0 = QD (t = 0) = 0 C 1A 0 < t  100 s 100 s t > 100 s I(t) = 2 A ( 1A 1A + 100 s ·t 0 < t  100 s I(t) = 2A t > 100 s  0 < t  100 s Zt QD (t) = I(⌧ ) d ⌧ + QD,0 0 Zt 1A = 1A + · ⌧ d⌧ + 0 100 s 0  t 1A 1 2 = 1A · ⌧ + · ⌧ 100 s 2 0 ✓ ◆ 1A 2 = 1A · t + ·t (0 + 0) 200 s 1A = 1A · t + · t2 200 s t = 100 s 1A QD (t = 100 s) = 1 A · 100 s + · (100 s)2 200 s 1A = 100 A · s + · 10000 s2 200 s A · s2 = 100 C + 50 s = 150 C. QD,0 t > 100 s Zt QD (t) = I(⌧ ) d ⌧ + QD (t = 100 s) 0 Zt = 2 A d ⌧ + 150 C ⌧ =100 s = [2 A · ⌧ ]t100 s + 150 C = (2 A · t) (2 A · 100 s) + 150 C = 2A · t 200 C + 150 C = 2A · t 50 C. t Q1 = Q2 = Q3 = 10 10 C z z=0 ✏0 = 0,885 · 10 11 C Vm Q3  y Q1 Q3 x Q2 z Q3 Q1 Q2 aQ1,Q3 = aQ2,Q3 p x 2 Xn 1 q · Qi F~q = · 2 · ~eQi,q, 4⇡✏0 aQi,q, i=1 n=2 q = Q3 Q3 X2 1 Q3 · Qi F~Q3 = · 2 · ~eQi,Q3 4⇡✏0 aQi,Q3 i=1  1 Q1 · Q3 1 Q3 · Q2 F~Q3 = · · ~eQ1,Q3 + · · ~eQ2,Q3. 4⇡✏0 a2Q1,Q3 4⇡✏0 a2Q2,Q3 Q1 Q3 p p aQ1 ,Q3 = (1 m)2 + (1 m)2 = 2 m = 1,41 m, Q2 Q3 p p aQ2 ,Q3 = (1 m)2 + (1 m)2 = 2 m = 1,41 m. a⇤ p a⇤ = aQ1 ,Q3 = aQ2 ,Q3 = 2 m = 1,41 m. Q1 = Q2 = Q3 = Q⇤ = 10 · 10 10 C 1 Q3 · Q1 1 Q3 · Q2 F~Q3 = · 2 · ~eQ1,Q3 + · 2 · ~eQ2,Q3 4⇡✏0 aQ1,Q3 4⇡✏0 aQ2,Q3 1 Q⇤ · Q⇤ 1 Q⇤ · Q⇤ = · · ~ e Q1,Q3 + · · ~eQ2,Q3 4⇡✏0 (a⇤ )2 4⇡✏0 (a⇤ )2 1 (Q⇤ )2 = · ⇤ 2 · (~eQ1,Q3 + ~eQ2,Q3 ) 4⇡✏0 (a ) x, y, z z Q3 Q1 0 1 0 1 a⇤ 1 ~rQ1,Q3 = @ a⇤ A = a⇤ · @ 1 A. 0 0 | {z } Richtungsvektor 0 1 1 p p @ 1 A = (1)2 + ( 1)2 + 0 = 2. 0  p 2 2 0 13 p1 6p B 2 C7 ~rQ1,Q3 = a⇤ · 4 2 · @ p1 2 A5 0 p ~rQ1,Q3 = a⇤ · 2 · ~eQ1,Q3 , 0 1 p1 2 B p1 C ~eQ1,Q3 = @ 2 A. 0 0 1 p1 2 B p1 C ~eQ2,Q3 = @ 2 A. 0 1 (Q⇤ )2 F~Q3 = · ⇤ 2 · (~eQ1,Q3 + ~eQ2,Q3 ) 4⇡✏0 (a ) 20 1 1 0 13 p p1 1 (Q⇤ )2 6B 2 C B 2 C7 = · · 4@ p1 A+@ p1 A5 4⇡✏0 (a⇤ )2 2 2 0 0 0 1 p2 1 (Q⇤ )2 2 = · ⇤ 2 ·@ 0 A 4⇡✏0 (a ) 0 0 1 ⇤ 2 1 1 (Q ) 2 @ A = · ·p · 0 4⇡✏0 (a⇤ )2 2 0 1 (Q⇤ )2 2 = · ⇤ 2 · p · ~ex 4⇡✏0 (a ) 2 1 (10 10 As)2 2 = As · p · p · ~ex 4 · ⇡ · 0,885 · 10 11 Vm ( 2 m)2 2 11 = 6,358 · 10 N · ~ex.  Vm C2 V·A·s J N= · 2 = = = N C m m m a= ~ = E · ~ex E E = 250 V/m q = 10 · 10 11 C x=0 x = 4 cm ✏0 = 0,885 · 10 11 As Vm Z2 W12 = q · ~ s⇤ ) d ~s⇤ E(~ 1 = q · U12. Z~rb Uab = ~ s⇤ ) d ~s⇤ E(~ ~s⇤ =~ ra y a '1 '2 ~ E x  ~ra ~rb 0 1 0 1 xa xb ~ra = @ ya A bzw. ~rb = @ yb A za zb U ~r = (x, y, z) Z~rb Uab = ~ s⇤ ) d ~s⇤ E(~ ~s⇤ =~ ra 0 1 0 1 Z~rb Ex dx = @ 0 A · @ dy A ~s⇤ =~ ra 0 dz Z~rb = Ex · d x. ~s⇤ =~ ra y z x Zxb = Ex · d x x=xa = [x · Ex ]xxba = Ex · (xb xa ) m 2 3 ~ rZn+1 m X1 6 ~ s⇤ ) d ~s⇤ 7 U12 = 4 E(~ 5. n=0 ~s⇤ =~ rn m X1 U12 = [Ex · (xn+1 xn )] n=0 0 m+1  xa xb Uab xa xb Uab xa xb Uab Uab = Ex · (xb xa ) m = 8 x0 = 0 cm y0 = 1,9 cm x1 = 2 cm y1 = 1,9 cm x2 = 2 cm y2 = 0,9 cm x3 = 1,5 cm y3 = 0,9 cm x4 = 1,5 cm y4 = 1,4 cm x5 = 2,5 cm y5 = 1,4 cm x6 = 2,5 cm y6 = 1,9 cm x7 = 3,5 cm y7 = 1,4 cm x8 = 4 cm y8 = 1,4 cm U12 = UWeg1 + UWeg2 + UWeg3 + UWeg4 + UWeg5 + UWeg6 + UWeg7 + UWeg8 = 5V + 0V 1,25 V + 0 V + 2, 5 V + 0 V + 2,5 V + 1,25 V = 10 V. 11 9 W12 = q · U12 = 10 · 10 As · 10 V = 10 VAs (x0 = 0 cm, y0 = 1 cm) (x1 = 1 cm, y1 = 1 cm) (x2 = 1 cm, y2 = 1,5 cm) (x3 = 3 cm, y3 = 1,5 cm) (x4 = 3 cm, y4 = 0,3 cm)  (x5 = 4 cm, y5 = 0,3 cm) U12 = UWeg1 + UWeg2 + UWeg3 + UWeg4 + UWeg5 = 2,5 V + 0 V + 5 V + 0 V + 2, 5 V = 10 V. 11 9 W12 = q · U12 = 10 · 10 As · 10 V = 10 VAs 2 U12 = E · (4 cm 0 cm) = 250 V/m · 4 · 10 m = 10 V. 11 9 W12 = q · U12 = 10 · 10 As · 10 V = 10 VAs. PZP = UZP · IZP. IZP = 0 UZP = 0  I I U U P Uq R Iq P >0 P 0 P 0 P 0 U R=. I V [R] = =⌦ A  UP UP P I U P R ( dU P U dI U I I ! U R=. I 1 I G= = , R U A [G] = = S V R U (I) I(U ) U2 P =U ·I = = I2 · R R dU rD =. dI rD = R  I A l ⇢ d⇢ d# ⇢0 = ⇢(#0 ) ⌧ #0 # l A ⇢·l R=. A [⇢] = ⌦ m ⇢ ⌦ mm 2  1 = ⇢ 1 S [] = = ⌦m m  ⇢(#) = ⇢0 · [1 + ↵0 · (# #0 )]. # ↵0 #0 ⇢0 = ⇢(#0 ) ⌧ 1 ↵0 =. #0 ⌧ #0 = 20 C = 293 K ↵20 = ↵0 (#0 = 20 C) ⇢20 = ⇢(#0 = 20 C) ⇢(#) · l R(#) = A = R0 · [1 + ↵0 · (# #0 )] R0 ⇢(#0 ) · l R0 =. A 3 1 1 8 #0 = 20 C ↵20,Cu = 3,9 · 10 1/K ⇢0,Cu = 56·106 ⌦m = 1,78 · 10 ⌦m  UP Pq = P I Uq = U P ( Pq ( Uq U I I=0 Uq Uq U Ri U (I) = Uq P = U · I = Uq · I. Uq P Pq,Spannungsquelle = Uq · I. Ri U (I) = Uq Ri · I,  P Uq2 Ri U I Uq Pq = P + PV Ui PV PV Uq2 P ) Ri 2 Ri ( Uq2 Pq Ui U 4 Ri ( P Uq U I IKS IKS I IKS 2 U (I = 0) ULL ULL = U (I = 0) = Uq. IKS U =0 Uq IKS = I(U = 0) =. Ri P =U ·I = (Uq Ui ) · I = Uq · I Ui · I = Uq · I Ri · I 2 = Pq PV.  I(U ) = Iq. P = U · Iq = U · I. Pq,Stromquelle = U · Iq Ri Gi I(U ) = Iq Gi · U. IKS = Iq , Iq ULL = = I q · Ri. Gi P =U ·I = U · (Iq Ii ) = U · Iq U · Ii U2 = U · Iq U 2 · Gi = U · I q Ri = Pq PV.  IP Pq I = Iq Iq = I P Pq ( ( U U U =0 P Iq2 Gi I Iq Pq = P + PV PV I Ii Iq2 Iq 2 Gi Pq PV I P Iq2 ( ) ) Ri 4 Gi P U U ULL ULL I ULL 2  ULL UZP (IZP ) = ULL · IZP. IKS IZP (UZP ) IKS IZP (UZP ) = · UZP + IKS , ULL Uq Iq ULL Uq = Uq IKS Uq = Ri,Uq ULL Iq = Gi,Iq IKS Uq = Iq ULL Uq = ULL Iq Iq , Uq = = Iq · Ri,Iq Gi,Iq IKS Uq = IKS Iq Uq , = Iq. Ri,Uq Ri,Uq = Ri,Iq = Ri Uq Iq = U q = I q · Ri Ri  I Pab Pzu Pab ⌘= Pzu Pab = Pab + PV Pzu PV = Pzu PV I = 2A U U = 20 V P = U · I. P = 20 V · 2 A = 40 W. PVZS < 0  I U I = 38 mA U = 67 V R U R=. I U 67 V R= = = 1763,2 ⌦ I 38 · 10 3 A U = 0,5 V R1 R2 R3 A1 A2 A3 R1 R2 R3 I(U ) U (I) U = 0,5 V = 500 mV I = 10 mA U R= , I  800,00 700,00 A3 600,00 U 500,00 400,00 A2 300,00 200,00 100,00 A1 0,00 0 10 20 30 40 50 60 I 100 mV A1 : R1 = = 10 ⌦, 10 mA 400 mV A2 : R2 = = 25 ⌦, 16 mA 650 mV A3 : R3 = = 21,67 ⌦. 30 mA Ua Ub Ia Ib (Ia , Ua ) (Ib , Ub ) dU Ua Ub 0 600 mV A1 : R1 = = = = 10 ⌦ dI Ia Ib 0 60 mA Ua Ub 610 0 mV A2 : R3 = = = 13,26 ⌦ Ia Ib 0 46 mA Ua Ub 560 740 mV A3 : R3 = = = 3⌦ Ia Ib 0 60 mA A1 A2 R2 < 0  800,00 700,00 A3 600,00 U 500,00 400,00 A2 300,00 200,00 100,00 A1 0,00 0 10 20 30 40 50 60 I U I R ⌦ I = f (U ) R = f (U ) R = f (U )  70,00 60,00 50,00 40,00 I 30,00 20,00 10,00 0,00 0 100 200 300 400 500 600 700 800 U 120,00 100,00 80,00 ⌦ R 60,00 40,00 20,00 0,00 0 100 200 300 400 500 600 700 800 U  28 m⌦ ISpule ↵20,Cu = ↵0 (20 C) = 3,9 · 10 3 1/K R(#) = R0 · [1 + ↵0 · (# #0 )]. R(382 K) = R20 · [1 + ↵20 · (382 K 293 K)] ⇥ ⇤ = 28 · 10 3 ⌦ · 1 + 3,9 · 10 3 K · 69 K 3 = 28 · 10 ⌦ · 1,269 = 35,535 m⌦. Rwarm Rkalt =c c 2 Pwarm = ISpule · Rwarm ISpule 2 Pkalt = ISpule · Rkalt. 2 Pwarm Pkalt ISpule = = Rwarm Rkalt Rwarm Pwarm = Pkalt ·. Rkalt Rwarm = c · Rkalt ,  c · Rkalt Pwarm = Pkalt · = Pkalt · c. Rkalt c = 1,269  X X i= i. zu ab X u=0 Masche X Uq,ges = Uq,⌫ ⌫  Uq,1 Uq,2 Uq,⌫ Uq,ges I I... = ˆ Iq,1 Iq,2 Iq,ges Iq,µ U X Iq,ges = Iq,µ µ n I n X Rgesamt = Rµ µ=1 U µ = I · Rµ n X n X U = I · Rgesamt = I · Rµ = Uµ. µ=1 µ=1  U U I R1 R2 Rµ I Rges... = ˆ U1 U2 Uµ U 1 I 1 R1 = G1 1 I 2 R2 = G2 U 1 1 Rges = I I 3 R3 = G3 I Gges = ˆ 1 I ⌫ R⌫ = G⌫ Uµ Rµ = U Rgesamt m U m X 1 Ggesamt = = G⌫. Rgesamt ⌫ I n X n X I= I⌫ = U · G⌫ ⌫ ⌫ I⌫ G⌫ = I Ggesamt  1 1 1 R20 = R12 = R23 = G20 G12 G23 1 1 R10 = R30 = G10 G30 1 R31 = G31 G10 · G20 G12 = , G10 + G20 + G30 G20 · G30 G23 = , G10 + G20 + G30 G10 · G30 G31 = , G10 + G20 + G30 R12 · R31 R10 = , R12 + R23 + R31 R12 · R23 R20 = , R12 + R23 + R31 R23 · R31 R30 =. R12 + R23 + R31 R = 3RY , R = R12 = R23 = R31 GY = G10 = G20 = G30  Uq1 Uq2 Uq1 Uq2 R3 Iq R1 R4 R1 ⌦ Iq R2 ⌦ R2 R3 ⌦ R4 ⌦ k k 1 z k z m=z (k 1) k=3 z=5 m=3  Uq1 Uq2 1 I3 1 2 3 R3 R1 UR3 I2 4 R4 UR1 Iq UR4 5 2 R2 I1 UIq UR2 3 I4 ! Ma 1 2 5 ! Mb 1 2 4 ! Mc 1 3 ! Md 2 3 4 ! Me 2 3 5 ! Mf 4 5 UR1 = R1 · I1 , UR2 = R2 · I2 , UR3 = R3 · I3 , UR4 = R4 · I4 , I1 I2 I3 I4 UIq  R1 R2 I1 R3 I2 R4 I3 Iq I4 Uq1 UIq Uq2 m=z (k 1), k 1 m k 1 x=m+k 1 UIq = UR2 K1 K2 Ma Md Mf  K1 I1 + I3 + I4 = 0, K2 I3 + Iq I2 = 0. I3 Iq I2 I2 + I3 = Iq. Ma Z1 Z2 Z5 Uq1 + UR3 + UIq UR1 = 0. Uq1 + I3 · R3 + UIq I1 · R1 = 0. I1 · R1 + I3 · R3 + UIq = Uq1. Mf UR2 = UIq , I 2 · R2 UIq = 0 Md Uq2 + UR4 UR2 UR3 = 0 , Uq2 + I4 · R4 I 2 · R2 I 3 · R3 = 0 , I 2 · R2 I3 · R3 + I4 · R4 = Uq2 0 1 0 1 0 1 1 0 1 1 0 I1 0 B 0 1 1 0 0 C B I2 C B Iq C B C B C B C B R1 0 R3 0 1 C·B I3 C=B Uq1 C. B C B C B C @ 0 R2 0 0 UIq A @ I4 A @ 0 A 0 R2 R3 R4 0 UIq Uq2  x=m nStromquellen , x U R = IR · R UR = IR · R x x m=z (k 1) = 5 (3 1) = 3, x=m nStromquellen = 3 1 = 2.  Iq Mf Z5 Z5 Mf IM1 Mb Ma Z5 I1 Z1 Z1 Z5 Md Z2 Z3 Z4 IM2 ! I1 = IM1 ! I2 = I M 1 + Iq IM2 ! I3 = I M 1 IM2 ! I4 = I M 2 Mb 0 = Uq1 + UR3 + UR2 UR1 0 = Uq1 + I M 1 I M 2 · R3 + I M 1 + I q I M 2 · R2 I M 1 · R1 0= Uq2 + UR4 UR2 UR3 = Uq2 + I M 2 · R4 I M 1 + Iq I M 2 · R2 IM1 I M 2 · R3. ✓ ◆ ✓ M1 ◆ ✓ ◆ R1 + R2 + R3 R2 R3 I Uq1 Iq · R2 · =. R2 R3 R2 + R3 + R4 IM2 Uq2 + Iq · R2 IM1 = 1 A IM2 = 8 A  IM1 IM2 Iq I1 = IM1 = 1A ) UR1 = I1 · R1 = 1A · 1⌦ = 1 V, M1 M2 I3 = I I = 1A 8A = 7A ) UR3 = I3 · R3 = 7A · 2⌦ = 14 V I4 = I M 2 = 8 A, ) UR4 = I4 · R4 = 8 A · 2 ⌦ = 16 V, I2 = I M 1 + Iq I M 2 = 1 A + 10 A 8 A = 3 A, ) UR2 = I2 · R2 = 3 A · 3 ⌦ = 9 V Mf UIq = UR2 = 9 V x=k 1 nUq,ideal , nUq,ideal 'B = 0 '1 '2 '1 '2 IR =. R  x=2 K3 '=0 K1 '1 K2 '2 Ka Kb Uab = 'a 'b Kb Ka Uba = 'b 'a Uq1 R1 'hilf,1 'hilf,1 = '1 + Uq1 UR1 = 'hilf,1 'B = '1 + Uq1. I1 UR1 '1 + Uq1 I1 = =. R1 R1 UR3 = '1 '2 , I3 UR3 '1 '2 I3 = = R3 R3 'hilf,2 = '1 + Uq2 ,  UR4 = 'hilf,2 'B = '1 + Uq2. UR4 '1 + Uq2 I4 = = R4 R4 UR2 '2 'B '2 I2 = = =. R2 R2 R2 K1 I1 I4 I3 = 0 , I1 + I4 + I3 = 0 '1 + Uq1 '1 '2 '1 + Uq2 , + + =0 R1 R3 R4 ✓ ◆ ✓ ◆ 1 1 1 1 Uq1 Uq2 $ '1 · + + + '2 · = , R1 R3 R4 R3 R1 R4 K2 I3 + Iq I2 = 0 '1 '2 '2 , + Iq =0 R3 R2 ✓ ◆ ✓ ◆ 1 1 1 , '1 · + '2 · = Iq R3 R3 R2 ✓ ◆ ✓ ◆ ! 1 1 1 1 Uq1 Uq2 R1 + R3 + R4 R3 '1 R1 R4 1 1 1 · = R3 R3 R2 '2 Iq '1 = 5V '2 = 9 V  UR1 = '1 + Uq1 = 4 V + ( 5 V = 1 V, UR1 1V ) I1 = = 1A R1 1⌦ UR3 = '1 '2 = 5 V 9 V = 14 V, UR3 14 V ) I3 = = = 7A R3 2⌦ UR4 = '1 + Uq2 = 5 V + 21 V = 16 V, UR4 16 V ) I4 = = = 8 A, R4 2⌦ UR2 = '1 = 9 V, UR2 9V ) I2 = = = 3 A. R2 3⌦ Mf UIq = UR2 = 9 V R L C a, b,... q= a Uq = 0 Iq = 0 I Ia , Ib ,... q X I= Im. m  Uq = 0 = ˆ Iq = 0 = ˆ I1 I2 I3 I4 Mf a : Uq1 b : Uq2 c : Iq a Uq1 ✓ ◆ 1 1 1 17 Rges,a = R1 + + = ⌦ R4 R2 + R3 7 I1,a Uq1 4V 28 I1,a = = 17 = A. Rges,a 7 ⌦ 17 Z2 Z4 I3,a = I2,a I1,a K1 1 I3,a R2 +R3 R4 2 = 1 1 = =. I1,a R4 + R2 +R3 R2 + R3 + R4 7 I1 I3 ✓ ◆ 2 28 8 I3,a = · A = A = I2,a. 7 17 17 K1 28 8 20 I4,a = I1,a I3,a = A+ A= A. 17 17 17 b I4,b ✓ ◆ 1 1 1 17 Rges,b = R4 + + = ⌦. R 1 R2 + R3 6 Uq2 21 V 126 I4,b = = 17 = A. Rges,b 6 ⌦ 17 I1,b 1 R1 R 2 + R3 126 5 105 I1,b = I4,b · 1 1 = I4,b · = A· = A. R1 + R2 +R3 R2 + R3 + R1 17 6 17 105 126 21 I2,b = I3,b = I1,b I4,b = A A= A. 17 17 17 c I2 = I3 0 ! 1 1 ✓ ◆ 1 1 1 1 1 Rges,c =@ + R3 + + A R2 R1 R4 R2 R3 (R1 + R4 ) + R1 R2 R4 24 = = ⌦ R3 (R1 + R4 ) + R1 R4 + R2 (R1 + R4 ) 17 1 1 R2 3⌦ 1 24 ⌦ 8 80 I2,c = Iq · 1 = Iq · 17 = Iq · · = · 10 A = A. Rges,c 24 ⌦ 3 ⌦ 17 17 17 K2 Iq + I3,c = I2,c 80 90 , I3,c = I2,c Iq = A 10 A = A. 17 17 1 R4 R1 90 1 30 I4,c = I3,c · 1 1 = I3,c · = A· = A. R4 + R1 R 1 + R4 17 3 17  Uq1 1 I3,a R3 R1 I2,a R4 2 R2 I1,a 3 I4,a Uq1 Uq2 1 I3,b R3 R1 I2,b R4 2 R2 I1,b 3 I4,b Uq2 K1 I1,c I1,c + I3,c + I4,c = 0 30 90 60 , I1,c = I4,c I3,c = A+ A= A. 17 17 17 28 105 60 I1 = I1,a + I1,b + I1,c = A A+ A = 1A 17 17 17 8 21 80 I2 = I2,a + I2,b + I2,c = A A+ A = 3A 17 17 17 8 21 90 I3 = I3,a + I3,b + I3,c = A A A = 7A 17 17 17 20 126 30 I4 = I4,a + I4,b + I4,c = A+ A+ = 8A 17 17 17 Mf  1 I3,c 2 R3 1 R3 I3,c Iq I2,c I1,c I4,c R1 I2,c R4 , Iq 2 R2 R2 R1 R4 I1,c 3 I4,c 3 Iq R3 R1 R2 R5 R4 R6 R7 R1 R R2 2R R8 R3 R R4 R R5 2R R6 R R7 2R R8 R ✓ ◆ ! ⇣ ⌘ Rges = R1 R2 R3 || R5 || (R6 R7 ) R4 || R8 || R3 R5 R6 + R7  Ra ✓ ◆ 1 ✓ ◆ 1 1 1 1 1 1 1 6 Ra = + + = + + = R R 3 R5 R6 + R7 R 2R 3R 11 R2 R a R4 R8 6 39 Rb = R2 + Ra + R4 = 2R + R + R = R. 11 11 ✓ ◆ 1 ✓ ◆ 1 1 1 1 11 39 Rc = + = + = R. Rc R8 R 39R 50 R1 Rc 39 89 Rges = R1 + Rc = R + R = R. 50 50 00 1 1 1 1 BB C C BB C C BB ✓ ◆ 1 C C BB 1 1 1 C 1 C Rges B B = R1 + BBR2 + + + +R4 C + C R3 R5 R6 + R7 C R8 C BB | {z } C C BB C C @@ Ra A A | {z } Rb | {z } Rc  ~ B(r) = B' · ~e' r ~ B(r) ' x I z ~v q P B~ = µ0 · q · ~v ⇥ ~e 4⇡ a2 Vs [B] = 2 = T = Tesla m µ0 7 Vs µ0 = 4 · ⇡ · 10 , Am ~e P a P Z ~ = µ0 · J~ d V ⇥ ~e B 4⇡ a2 Strömungsgebiet Z µ0 · I d ~s ⇥ ~e = ·. 4⇡ a2 Leiterschleife d ~s I z ~ = µ0 · I ~e'. B 2⇡r A ~ dA Z = ~ d A. B ~ A [ ] = Vs = Tm2 = Wb = Weber = B · A. I ~ dA B ~ = 0. Hülle H ~ B ~ = H µ0 · µr A [H] = m µr,Vakuum = 1 s Z2 V = ~ d ~s. H 1  I V̊ = ~ d ~s. H Umlaufweg A Z ⇥= J~ d A. ~ Fläche [⇥] = A X ⇥= Iµ. µ B H B H µr µr < 1 4 9 1 10 < µr,diamagnetisch < 1 10 ! µr ⇡ 1.  6 4 1 + 10 < µr,paramagnetisch < 1 + 10 ! µr ⇡ 1. 104 B(H) H B B Br H Hc Hc > 1000 A/m H B  B H B µ= H B(H)  nA s q ~v ~ B ⇣ ⌘ F~ = q · ~v ⇥ B ~. ~l l I B ⇣ ⌘ F~ = I · ~l ⇥ B ~. F~ ~ B I I Z ~ d ~s = H ~ = V̊ = ⇥. J dA Rand der Fläche A Fläche A ~nA A ~ = d (A·~nA ) dA d ~s  I Z ~ ~ d ~s = @B ~ E d A. @t Rand der Fläche A Fläche A H ~ d ~s sE I X ~ d ~s = E U s Ui N d u(t) = N · dt =N·. [ ] = Vs = Wb = Weber L d i(t) u(t) = L ·. dt L=. I Vs [L] = = H = Henry. A  L 1 (I1 ) N1 · 1 (I1 ) L1 = = I1 I1 2 = N2 · (I1 ) 2 (I1 ) M21 = I1 [M ] = H. 1 (I2 ) M12 = I2 M12 = M21 = M. 1 = 1 (I1 ) + 1 (I2 ) = L1 · I1 + M · I2 falls µ = const. 2 = 2 (I1 ) + 2 (I2 ) = M · I1 + L2 · I2 falls µ = const. d 1 d I1 d I2 u1 = = L1 · +M · dt dt dt d 2 d I2 d I1 u2 = = L2 · +M ·. dt dt dt  R u(t) i(t) = R u(t) = R · i(t) d u(t) = ( (i(t))). dt L d u(t) = (L · i(t)). dt Z 1 i(t) = · u(t) d t + K. L K d i(t) = Q. dt  C d i(t) = (C · u(t)). dt Z 1 u(t) = · i(t) d t + K. C K d u(t)|Gleichstrom = (L · i(t)) = 0 dt d i(t)|Gleichspannung = (C · u(t)) = 0 dt  x x(t) = x(t + n · T ) T n = 0, ±1, ±2,... x(t) x xmax xmin x̄ T 1 f= , T ! = 2⇡f. xmin xmax x = xmax xmin tZ A +T 1 x̄ = x d t. T tA |x̄| tZ A +T 1 |x̄| = |x| d t. T tA  x(t) xmax x xmin t T x x X v u tZA +T u u1 X=t x2 d t. T tA tA xmax Scheitelfaktor = X X Formfaktor =. |x̄| x(t) = x̂ cos(!t + '0 ). ! '0 !t + '0 '0 = !t0.  x(t) x̂ t t0 T Schwingungsbreite x = 2x̂, Gleichanteil x̄ = 0, 2 Gleichrichtwert |x̄| = x̂, ⇡ 1 E↵ektivwert X = p x̂, 2 p Scheitelfaktor = 2, ⇡ Formfaktor = p = 1,11. 2 2 f (t) T 1 X f (t) = A0 + Cn · sin (n!1 t + 'n ) , n=1 ! f 1 A0 'n n Cn n = 30 n sin (n!t) f1 f1 !1 = 2⇡f1  1 X f (t) = A0 + (An cos n!1 t + Bn sin n!1 t) , n=1 p Cn = A2n + Bn2 An 'n = arctan. Bn ZT 1 A0 = f (t) d t T 0 ZT 2 An = f (t) cos n!1 t d t T 0 ZT 2 Bn = f (t) sin n!1 t d t T 0 A0 = 0 f (t) f (t) = f ( t) f (t) Bn = 0 f (t) = f ( t) An = 0 T f (t) = f (t + 2) n B n = An = 0 n = gerade i(t) ī |i| I  i(t) Iˆ T 2T 3T 4T t T 00 I1 I1 Im P