Grundlagen Elektrotechnik 2 (GET2)18 Bode DiagramProf. Dr. Jörg Vollrath17 Übertragungsfunktion |
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Länge: 1:22:04 |
0:0:0 Evaluierung 0:0:0 Differenzverstärker 0:2:0 Eingangs und Ausgangswiderstand |
Version 4 SHEET 1 880 680 WIRE 336 48 304 48 WIRE 368 48 336 48 WIRE 496 48 448 48 WIRE 528 48 496 48 WIRE 496 112 496 48 WIRE 496 208 496 192 WIRE 496 208 304 208 WIRE 528 208 496 208 WIRE 496 224 496 208 FLAG 496 224 0 FLAG 336 48 Ue FLAG 496 48 Ua SYMBOL ind 464 32 R90 WINDOW 0 5 56 VBottom 2 WINDOW 3 32 56 VTop 2 SYMATTR InstName L1 SYMATTR Value 35m SYMBOL res 512 208 R180 WINDOW 0 36 76 Left 2 WINDOW 3 36 40 Left 2 SYMATTR InstName R1 SYMATTR Value 147 TEXT 304 232 Left 2 !;tran 1m TEXT 304 264 Left 2 !.ac dec 50 10 100k TEXT 296 -8 Left 2 !V1 UE 0 SINE(0 1 16k) AC 1 TEXT 264 112 Left 2 ;UE TEXT 544 104 Left 2 ;UA LINE Normal 304 192 288 176 LINE Normal 320 176 304 192 LINE Normal 304 192 304 64 LINE Normal 528 184 512 168 LINE Normal 544 168 528 184 LINE Normal 528 184 528 56 |
\frac{\underline{U}_{A}}{\underline{U}_E} = \frac{R_1}{L_1} \frac{1}{j \omega + \frac{R_1}{L_1} } |
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\underline{T} (j\omega) = K \frac{(j \omega - s_{N1})(j \omega - s_{N2})...(j \omega - s_{Nm})}
{(j \omega - s_{P1})(j \omega - s_{P2})...(j \omega - s_{Pn}) } |
\frac{U_{out}}{U_E} = \frac{j \omega + \frac{R2}{L1} }{j \omega + \frac{R1 + R2}{L1} } 20 log_{10} | j \omega + \frac{R2}{L1} | - 20 log_{10} | {j \omega + \frac{R1 + R2}{L1} } | \frac{R2}{2 \pi L1 } = 86 kHz \frac{R1 + R2}{2 \pi L1} = 934 kHz 20 log | \frac{R2}{L1} | = 114 dB und - 20 log | {\frac{R1 + R2}{L1} } | = - 135 dB |
Gesucht ist die Übertragungsfunktion folgenden Netzes:Version 4 SHEET 1 880 680 WIRE 336 48 304 48 WIRE 368 48 336 48 WIRE 496 48 448 48 WIRE 592 48 496 48 WIRE 656 48 592 48 WIRE 496 112 496 48 WIRE 592 112 592 48 WIRE 496 208 496 192 WIRE 496 208 304 208 WIRE 592 208 592 176 WIRE 592 208 496 208 WIRE 656 208 592 208 WIRE 496 224 496 208 FLAG 496 224 0 FLAG 336 48 Ue FLAG 496 48 Ua SYMBOL res 352 64 R270 WINDOW 0 32 56 VTop 2 WINDOW 3 0 56 VBottom 2 SYMATTR InstName R1 SYMATTR Value 100 SYMBOL res 480 96 R0 SYMATTR InstName R2 SYMATTR Value 300 SYMBOL cap 576 112 R0 SYMATTR InstName C1 SYMATTR Value 1� TEXT 304 232 Left 2 !;tran 1m TEXT 304 264 Left 2 !.ac dec 50 10 100k TEXT 296 -8 Left 2 !V1 UE 0 SINE(0 1 16k) AC 1 TEXT 264 112 Left 2 ;UE TEXT 672 112 Left 2 ;UA LINE Normal 304 192 288 176 LINE Normal 320 176 304 192 LINE Normal 304 192 304 64 LINE Normal 656 192 640 176 LINE Normal 672 176 656 192 LINE Normal 656 192 656 64
Wir wollen den Amplituden und Phasengang der Spannung U(j ω),
den Frequenzgang F(jω) der auf die Quellenspannung bezogenen Spannung
bestimmen.
\underline{F} \left( j \omega \right) = \frac{\underline{U}_A \left( j \omega \right)}{\underline{U}_E \left( j \omega \right)} = Spannungsteilerregel, da der Strom der Gleiche ist: \underline{I} = \frac{ \underline{U}_E}{R_1 + \frac{1}{\frac{1}{R_2} + j \omega C_1}} = \underline{U}_A \left( \frac{1}{R_2} + j \omega C_1 \right) \underline{F} \left( j \omega \right) = \frac{\underline{U}_A}{\underline{U}_E} = \frac{ 1}{ \left( R_1 + \frac{1}{\frac{1}{R_2} + j \omega C_1} \right) \left( \frac{1}{R_2} + j \omega C_1 \right)} Die Brüche unter dem Bruchstrich sollen eliminiert werden: Ausmultiplizieren \underline{F} \left( j \omega \right) = \frac{\underline{U}_A}{\underline{U}_E} = \frac{ 1}{ \frac{R_1}{R_2} + j \omega C_1 R_1 + 1} = \frac{ 1}{ j \omega C_1 R_1 + \frac{R_1}{R_2} + 1} = \frac{ 1}{C_1 R_1} \frac{ 1}{ j \omega + \frac{R_1 + R_2 }{C_1 R_1 R_2}} K = \frac{ 1}{C_1 R_1} s_{p1} = - \frac{R_1 + R_2 }{C_1 R_1 R_2} Amplitudengang: U_A ( \omega ) = U_E K \frac{1}{\sqrt{ \left( \frac{R_1 + R_2 }{C_1 R_1 R_2} \right)^2 + \omega^2 } } Phasengang: \phi_A ( \omega ) = \phi_E - arctan \left( \frac{\omega}{\frac{R_1 + R_2 }{C_1 R_1 R_2}} \right) = \phi_E - arctan \left( \frac{\omega C_1 R_1 R_2 }{R_1 + R_2 } \right) |
Faktor | log10 | dB | ln |
1 | 0 | 0 | 0 |
2 | 0.3 | 6 | 0.69 |
2.72 | 0.43 | 8.6 | 1 |
7.39 | 0.87 | 17.4 | 2 |
10 | 1 | 20 | 2.3 |
100 | 2 | 40 | 4.6 |
Version 4 SHEET 1 880 680 WIRE 144 48 80 48 WIRE 256 48 224 48 WIRE 288 48 256 48 WIRE 384 48 352 48 WIRE 80 80 80 48 WIRE 256 80 256 48 WIRE 384 80 384 48 WIRE 80 176 80 160 WIRE 256 176 256 144 WIRE 256 176 80 176 WIRE 384 176 384 160 WIRE 384 176 256 176 WIRE 256 192 256 176 FLAG 256 192 0 SYMBOL voltage 80 64 R0 WINDOW 0 24 16 Invisible 2 WINDOW 3 24 104 Invisible 2 WINDOW 123 0 0 Left 2 WINDOW 39 0 0 Left 2 SYMATTR InstName Vq SYMATTR Value SINE(0 1 1k) SYMATTR Value2 AC 1 SYMBOL res 240 32 R90 WINDOW 0 0 56 VBottom 2 WINDOW 3 32 56 VTop 2 SYMATTR InstName R1 SYMATTR Value 50 SYMBOL cap 240 80 R0 SYMATTR InstName C1 SYMATTR Value 2n SYMBOL cap 352 32 R90 WINDOW 0 0 32 VBottom 2 WINDOW 3 32 32 VTop 2 SYMATTR InstName C2 SYMATTR Value 2n SYMBOL res 368 64 R0 SYMATTR InstName R2 SYMATTR Value 50 TEXT 304 120 Left 2 ;U2 TEXT 192 120 Left 2 ;U1 TEXT 8 120 Left 2 ;Uq TEXT 40 192 Left 2 !;ac dec 10 1 1G TEXT 288 200 Left 2 !.tran 4ms LINE Normal 336 160 336 64 LINE Normal 336 160 336 160 LINE Normal 320 144 336 160 LINE Normal 352 144 336 160 LINE Normal 224 160 224 64 LINE Normal 224 160 224 160 LINE Normal 208 144 224 160 LINE Normal 240 144 224 160 LINE Normal 40 160 40 64 LINE Normal 40 160 40 160 LINE Normal 24 144 40 160 LINE Normal 56 144 40 160
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Version 4 SHEET 1 880 680 WIRE 144 0 80 0 WIRE 256 0 224 0 WIRE 288 0 256 0 WIRE 384 0 352 0 WIRE 80 80 80 0 WIRE 256 80 256 0 WIRE 384 80 384 0 WIRE 80 176 80 160 WIRE 256 176 256 144 WIRE 256 176 80 176 WIRE 384 176 384 160 WIRE 384 176 256 176 WIRE 256 192 256 176 FLAG 256 192 0 FLAG 80 0 Uq FLAG 256 0 U1 FLAG 384 0 U2 SYMBOL voltage 80 64 R0 WINDOW 0 24 16 Invisible 2 WINDOW 3 24 104 Invisible 2 WINDOW 123 8 8 Left 2 WINDOW 39 0 0 Left 2 SYMATTR InstName Vq SYMATTR Value SINE(0 1 1E3) SYMATTR Value2 AC 1 SYMBOL res 240 -16 R90 WINDOW 0 0 56 VBottom 2 WINDOW 3 32 56 VTop 2 SYMATTR InstName R1 SYMATTR Value 4.7k SYMBOL cap 240 80 R0 SYMATTR InstName C1 SYMATTR Value 2.2n SYMBOL cap 352 -16 R90 WINDOW 0 0 32 VBottom 2 WINDOW 3 32 32 VTop 2 SYMATTR InstName C2 SYMATTR Value 22n SYMBOL res 368 64 R0 SYMATTR InstName R2 SYMATTR Value 47k TEXT 304 120 Left 2 ;U2 TEXT 192 120 Left 2 ;U1 TEXT 8 120 Left 2 ;Uq TEXT 32 200 Left 2 !.ac dec 10 1 1E9 TEXT 104 24 Left 2 ;Iq TEXT 264 56 Left 2 ;IC1 TEXT 400 56 Left 2 ;IR2 LINE Normal 336 160 336 16 LINE Normal 336 160 336 160 LINE Normal 320 144 336 160 LINE Normal 352 144 336 160 LINE Normal 224 160 224 16 LINE Normal 224 160 224 160 LINE Normal 208 144 224 160 LINE Normal 240 144 224 160 LINE Normal 40 160 40 16 LINE Normal 40 160 40 160 LINE Normal 24 144 40 160 LINE Normal 56 144 40 160 LINE Normal 80 16 80 16 LINE Normal 96 32 80 16 LINE Normal 64 32 80 16 LINE Normal 256 48 256 48 LINE Normal 240 32 256 48 LINE Normal 272 32 256 48 LINE Normal 384 48 384 48 LINE Normal 368 32 384 48 LINE Normal 400 32 384 48 |
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3.1. Geben Sie die Pole und Nullstellen der Übertragungsfunktion \frac{\underline{U}_1}{\underline{U}_q} von folgendem Netz an. R_1 = 10 \Omega, R_2 = 2490 \Omega, L = 40 mH, C = 40nF. 3.2. Geben Sie die Spannung \underline{U}_1 für \omega = 0 und \omega \rightarrow \infty an. |
Version 4 SHEET 1 880 680 WIRE 128 -96 112 -96 WIRE 144 -96 128 -96 WIRE 240 -96 208 -96 WIRE 368 -96 320 -96 WIRE 432 -96 368 -96 WIRE 368 -80 368 -96 WIRE 368 16 368 0 WIRE 208 112 128 112 WIRE 368 112 368 96 WIRE 368 112 208 112 WIRE 432 112 368 112 WIRE 208 128 208 112 FLAG 208 128 0 FLAG 128 -96 Uq SYMBOL res 384 16 R180 WINDOW 0 36 76 Left 2 WINDOW 3 36 40 Left 2 SYMATTR InstName R1 SYMATTR Value 10 SYMBOL cap 208 -112 R90 WINDOW 0 0 32 VBottom 2 WINDOW 3 32 32 VTop 2 SYMATTR InstName C SYMATTR Value 40n SYMBOL ind 384 0 M0 SYMATTR InstName L1 SYMATTR Value 40m SYMBOL res 336 -112 R90 WINDOW 0 0 56 VBottom 2 WINDOW 3 32 56 VTop 2 SYMATTR InstName R2 SYMATTR Value 2490 TEXT 88 -8 Left 2 ;Uq TEXT 248 144 Left 2 !U1 Uq 0 AC 1 TEXT 448 8 Left 2 ;U1 LINE Normal 128 96 128 -80 LINE Normal 144 80 128 96 LINE Normal 128 96 112 80 LINE Normal 432 96 432 -80 LINE Normal 416 80 432 96 LINE Normal 432 96 448 80 |
Version 4 SHEET 1 932 680 WIRE 128 -96 112 -96 WIRE 144 -96 128 -96 WIRE 240 -96 208 -96 WIRE 368 -96 320 -96 WIRE 432 -96 368 -96 WIRE 368 -80 368 -96 WIRE 368 16 368 0 WIRE 208 112 128 112 WIRE 368 112 368 96 WIRE 368 112 208 112 WIRE 432 112 368 112 WIRE 208 128 208 112 WIRE 304 176 192 176 WIRE 192 208 192 176 WIRE 144 224 112 224 WIRE 144 272 128 272 WIRE 128 304 128 272 WIRE 192 304 192 288 WIRE 304 352 192 352 WIRE 192 384 192 352 WIRE 144 400 112 400 WIRE 144 448 128 448 WIRE 128 480 128 448 WIRE 192 480 192 464 FLAG 208 128 0 FLAG 128 -96 Uq FLAG 192 304 0 FLAG 128 304 0 FLAG 112 224 Uq FLAG 432 -96 U1 FLAG 304 176 U11 FLAG 192 480 0 FLAG 128 480 0 FLAG 112 400 Uq FLAG 304 352 U12 SYMBOL res 384 16 R180 WINDOW 0 36 76 Left 2 WINDOW 3 36 40 Left 2 SYMATTR InstName R1 SYMATTR Value 10 SYMBOL cap 208 -112 R90 WINDOW 0 0 32 VBottom 2 WINDOW 3 32 32 VTop 2 SYMATTR InstName C SYMATTR Value 40n SYMBOL ind 384 0 M0 SYMATTR InstName L1 SYMATTR Value 40m SYMBOL res 336 -112 R90 WINDOW 0 0 56 VBottom 2 WINDOW 3 32 56 VTop 2 SYMATTR InstName R2 SYMATTR Value 2490 SYMBOL e 192 192 R0 SYMATTR InstName E1 SYMATTR Value Laplace = (10 + s * 40E-3)/(1/(s*40E-9)+ 2500 + s * 40E-3) SYMBOL e 192 368 R0 SYMATTR InstName E2 SYMATTR Value Laplace = s * (s+250)/(s+50E3)/(s+12.5E3) TEXT 88 -8 Left 2 ;Uq TEXT 248 144 Left 2 !Vq Uq 0 DC 1 AC 1 TEXT 448 8 Left 2 ;U1 TEXT 60 328 Left 2 !.ac dec 10 1 1G LINE Normal 128 96 128 -80 LINE Normal 144 80 128 96 LINE Normal 128 96 112 80 LINE Normal 432 96 432 -80 LINE Normal 416 80 432 96 LINE Normal 432 96 448 80
Version 4 SymbolType CELL LINE Normal -48 32 -32 32 LINE Normal -32 32 -24 36 LINE Normal -48 80 -32 80 LINE Normal -32 80 -24 76 LINE Normal 0 16 0 24 LINE Normal 0 96 0 88 LINE Normal -48 72 -40 72 LINE Normal -48 40 -40 40 LINE Normal -44 36 -44 44 LINE Normal -4 72 4 72 LINE Normal -4 40 4 40 LINE Normal 0 36 0 44 CIRCLE Normal -32 24 32 88 WINDOW 0 24 16 Left 2 WINDOW 3 24 96 Left 2 SYMATTR Value E SYMATTR Prefix E SYMATTR Description Voltage dependent voltage source PIN 0 16 NONE 0 PINATTR PinName + PINATTR SpiceOrder 1 PIN 0 96 NONE 0 PINATTR PinName - PINATTR SpiceOrder 2 PIN -48 32 NONE 0 PINATTR PinName P PINATTR SpiceOrder 3 PIN -48 80 NONE 0 PINATTR PinName N PINATTR SpiceOrder 4