1 OHMSCHES GESETZ.........................................................

2
1
OHMSCHES GESETZ..................................................................................................... 7
1.1
1.2
1.3
1.4
1.5
6.2
6.3
Abgeglichen .....................................................................................................................................................................................................................7
Spannungsstabilisierung Allgemein.................................................................................................................................................................................8
Spannungsstabilisierung mit Z-Diode..............................................................................................................................................................................8
6.4
GLEICHSTROMMASCHINE ..........................................................................................................................................10
3.1.1
3.1.1.1
3.1.1.2
3.1.1.3
3.1.2
3.1.2.1
3.1.2.2
3.1.2.3
3.1.2.4
3.1.3
3.1.4
3.1.5
3.1.6
3.1.6.1
3.1.6.2
3.2
4
6.6
6.7
6.8
6.9
ELEKTROWÄRME ......................................................................................................................................................13
WÄRMEWIDERSTAND................................................................................................................................................13
WÄRMEBEDINGTE WIDERSTANDSÄNDERUNG............................................................................................................13
4.3.1.1
4.3.1.2
4.4
5
PTC ........................................................................................................................................................................................................................13
NTC .......................................................................................................................................................................................................................13
LINEARISIERUNG.......................................................................................................................................................13
6.10
FREQUENZ ................................................................................................................................................................14
SINUSFÖRMIG............................................................................................................................................................14
MITTELWERTE ..........................................................................................................................................................14
5.3.1
5.3.2
5.3.2.1
5.3.3
5.3.4
5.4
5.5
7
Sternschaltung................................................................................................................................................................................................................16
Zeigerdiagramm.....................................................................................................................................................................................................16
symmetrische Belastung ........................................................................................................................................................................................16
unsymmetrische Belastung ....................................................................................................................................................................................16
Dreieckschaltung ...........................................................................................................................................................................................................16
symmetrische Belastung ........................................................................................................................................................................................16
unsymmetrische Belastung ....................................................................................................................................................................................16
Stern zu Dreieck (T zu Π)..............................................................................................................................................................................................17
Dreieck zu Stern (Π zu T)..............................................................................................................................................................................................17
Wechselstromgeneratoren..............................................................................................................................................................................................17
Einphasenwechselstrom.........................................................................................................................................................................................17
KOMPLEXE LEISTUNG ...............................................................................................................................................18
5.6.1
5.6.2
5.6.3
5.6.3.1
5.6.3.2
5.6.4
5.6.4.1
5.6.4.2
Leistung einer Impedanz................................................................................................................................................................................................18
Leistung einer Admittanz...............................................................................................................................................................................................18
Leistungsanpassung .......................................................................................................................................................................................................18
Leistungsanpassung mit reeller Last ......................................................................................................................................................................18
Leistungsanpassung mit komplexer Last ...............................................................................................................................................................18
Lastfaktor .......................................................................................................................................................................................................................18
Einphasig ...............................................................................................................................................................................................................18
Dreiphasig..............................................................................................................................................................................................................18
Daniel Arnold
KOMPLEXE OPERATOREN ......................................................................................................................................... 29
DIODEN........................................................................................................................... 30
7.1
HALBLEITERKONSTANTEN ........................................................................................................................................ 30
7.1.1
7.1.2
7.1.3
GLEICHRICHTER ........................................................................................................................................................15
DREHSTROM (3 PHASENSTROM) ...............................................................................................................................16
5.5.1
5.5.1.1
5.5.1.2
5.5.1.3
5.5.2
5.5.2.1
5.5.2.2
5.5.3
5.5.4
5.5.5
5.5.5.1
5.6
Gleichwert (arithmetischer Mittelwert)..........................................................................................................................................................................14
Effektivwert (RMS) .......................................................................................................................................................................................................14
Effektivwert einer Mischgrösse .............................................................................................................................................................................14
Gleichrichtwert ..............................................................................................................................................................................................................14
Verhältniszahlen ............................................................................................................................................................................................................14
Transformator.................................................................................................................................................................................................................27
Selbstinduktivität ...........................................................................................................................................................................................................27
Gegeninduktivität...........................................................................................................................................................................................................27
Kopplungsfaktor.............................................................................................................................................................................................................27
Ideal........................................................................................................................................................................................................................27
Real ........................................................................................................................................................................................................................27
Serieschaltung gekoppelter Spulen ................................................................................................................................................................................27
gleichsinnig ............................................................................................................................................................................................................27
gegensinnig ............................................................................................................................................................................................................27
Induzierte Spannung.......................................................................................................................................................................................................27
Verlustloser Übertrager ..................................................................................................................................................................................................28
Übersetzungsverhältnis im Leerlauf.......................................................................................................................................................................28
Übersetzungsverhältnis bei Belastung....................................................................................................................................................................28
Übersetzungsverhältnis bei ideal fester Kopplung .................................................................................................................................................28
Stromübersetzung bei ideal fester Kopplung .........................................................................................................................................................28
Netzwerktransformation.........................................................................................................................................................................................28
6.10.1
Impedanz........................................................................................................................................................................................................................29
6.10.2
Admittanz.......................................................................................................................................................................................................................29
6.10.3
Operatoren der Netzwerkelemente.................................................................................................................................................................................29
6.10.3.1
Widerstand .............................................................................................................................................................................................................29
6.10.3.2
Kondensator / Kapazität .........................................................................................................................................................................................29
6.10.3.3
Spule / Induktivität .................................................................................................................................................................................................29
WECHSELSPANNUNG .................................................................................................14
5.1
5.2
5.3
Reihenschwingkreis .......................................................................................................................................................................................................26
Parallelschwingkreis ......................................................................................................................................................................................................26
Bandbreite und Güte ......................................................................................................................................................................................................27
GEKOPPELTE SPULEN ............................................................................................................................................... 27
6.9.1
6.9.2
6.9.3
6.9.4
6.9.4.1
6.9.4.2
6.9.5
6.9.5.1
6.9.5.2
6.9.6
6.9.7
6.9.7.1
6.9.7.2
6.9.7.3
6.9.7.4
6.9.7.5
WÄRME ...........................................................................................................................13
4.1
4.2
4.3
Kondensator ...................................................................................................................................................................................................................25
Spule ..............................................................................................................................................................................................................................25
UMWANDLUNGEN..................................................................................................................................................... 25
LC-SCHWINGKREIS .................................................................................................................................................. 26
6.8.1
6.8.2
6.8.3
ERSATZSCHALTBILD DER VOLLPOLMASCHINE ..........................................................................................................11
Serieschaltung ................................................................................................................................................................................................................23
RL Schaltung..........................................................................................................................................................................................................23
LR Schaltung..........................................................................................................................................................................................................24
Parallelschaltung ............................................................................................................................................................................................................24
VERLUSTE ................................................................................................................................................................ 25
6.6.1
6.6.2
Induzierte Spannung ......................................................................................................................................................................................................11
Frequenz f ..............................................................................................................................................................................................................11
Längsreaktanz ................................................................................................................................................................................................................11
Serieschaltung ................................................................................................................................................................................................................21
RC Schaltung .........................................................................................................................................................................................................21
CR Schaltung .........................................................................................................................................................................................................22
Parallelschaltung ............................................................................................................................................................................................................22
RL SCHALTUNGEN ................................................................................................................................................... 23
6.5.1
6.5.1.1
6.5.1.2
6.5.2
SYNCHRONMASCHINE ...............................................................................................................................................11
3.2.1
3.2.1.1
3.2.2
3.3
6.5
die Rotorwicklung..........................................................................................................................................................................................................10
Polteilung ...............................................................................................................................................................................................................10
Aktive Leiterzahl ...................................................................................................................................................................................................10
Unfangsgeschwindigkeit........................................................................................................................................................................................10
Induzierte Spannung ......................................................................................................................................................................................................10
induzierte Spannung...............................................................................................................................................................................................10
Mittelwert der induzierten Spannung.....................................................................................................................................................................10
total induzierte Spannung ......................................................................................................................................................................................10
total induzierter Spannungsmomentanwert............................................................................................................................................................10
Drehmoment ..................................................................................................................................................................................................................10
Drehzahl.........................................................................................................................................................................................................................11
Ersatzschaltbild..............................................................................................................................................................................................................11
Stationärer Betrieb .........................................................................................................................................................................................................11
Leerlaufdrehzahl ....................................................................................................................................................................................................11
Lastkennlinie..........................................................................................................................................................................................................11
Widerstand .....................................................................................................................................................................................................................20
Kondensatoren ...............................................................................................................................................................................................................20
Impulsverformung..................................................................................................................................................................................................20
Wechselstromverhalten ..........................................................................................................................................................................................20
Spule ..............................................................................................................................................................................................................................20
Impulsverformung..................................................................................................................................................................................................20
Wechselstromverhalten ..........................................................................................................................................................................................20
RC SCHALTUNGEN ................................................................................................................................................... 21
6.4.1
6.4.1.1
6.4.1.2
6.4.2
MASCHINEN...................................................................................................................10
3.1
Plattenkondensator .........................................................................................................................................................................................................19
Plattenkondensator an konstanter Spannung ..........................................................................................................................................................19
Plattenkondensator mit konstantem Q....................................................................................................................................................................19
INDUKTIVITÄTEN ...................................................................................................................................................... 19
NETZWERKELEMENTE .............................................................................................................................................. 20
6.3.1
6.3.2
6.3.2.1
6.3.2.2
6.3.3
6.3.3.1
6.3.3.2
LEISTUNG, ARBEIT....................................................................................................... 9
3
ELEKTRISCHE FELDER ............................................................................................................................................... 19
6.1.1
6.1.1.1
6.1.1.2
SPANNUNGSERZEUGER ............................................................................................................................................... 8
STABILISIERUNG VON U UND I.................................................................................................................................... 8
1.7.1
1.7.2
2
C UND L .......................................................................................................................... 19
6.1
KIRCHHOFF ................................................................................................................................................................. 7
SERIESCHALTUNG ....................................................................................................................................................... 7
PARALLELSCHALTUNG ................................................................................................................................................ 7
SPANNUNGSTEILER ..................................................................................................................................................... 7
BRÜCKENSCHALTUNG................................................................................................................................................. 7
1.5.1
1.6
1.7
6
7.2
DIODENKENNLINIEN ................................................................................................................................................. 30
7.2.1
7.2.2
7.2.3
8
Boltzmann-Konstante.....................................................................................................................................................................................................30
Elektrische Elementarladung..........................................................................................................................................................................................30
Absoluter Nullpunkt.......................................................................................................................................................................................................30
Ideale Diodenkennlinie ..................................................................................................................................................................................................30
Reale Diodenkennlinie ...................................................................................................................................................................................................30
Diodensperrstrom...........................................................................................................................................................................................................30
TRANSISTOR................................................................................................................. 31
8.1
BIPOLARE TRANSISTOREN ........................................................................................................................................ 31
8.1.1
8.1.2
8.1.3
8.1.3.1
8.1.3.2
8.1.3.3
8.1.3.4
8.1.4
8.1.4.1
8.1.4.2
8.1.4.3
8.1.5
8.1.5.1
8.1.5.2
© Daniel Arnold
www.lanny.ch
Kennlinie........................................................................................................................................................................................................................31
Millerkapazität ...............................................................................................................................................................................................................31
Effekte............................................................................................................................................................................................................................31
Steilheit der Diodenkennlinie.................................................................................................................................................................................31
Eigenwiderstand des Emitters rE ............................................................................................................................................................................31
Temperaturabhängigkeit von uBE............................................................................................................................................................................31
Early-Effekt............................................................................................................................................................................................................31
Verhalten eines Bipolartransistors .................................................................................................................................................................................31
Grossignalstromverstärkung...................................................................................................................................................................................31
Grossignalgleichung...............................................................................................................................................................................................31
Leistung bei Grossignalverstärkung.......................................................................................................................................................................32
Arbeitspunkt und Kleinsignalverhalten..........................................................................................................................................................................32
Kleinsignalersatzschaltbild.....................................................................................................................................................................................32
Kleinsignalgleichungen..........................................................................................................................................................................................32
09.09.2004
3
8.1.5.3
8.1.6
8.1.6.1
8.1.6.2
8.1.6.3
8.1.6.4
8.1.6.5
8.1.6.6
8.1.6.7
8.1.7
8.1.8
8.1.9
8.2
8.3
Darlington-Schaltung.....................................................................................................................................................................................................35
Transistor als Temperaturmesselement..........................................................................................................................................................................35
Bootstrapping.................................................................................................................................................................................................................35
Emitterschaltung ............................................................................................................................................................................................................35
Reine Emitterschaltung ohne Lastwiderstand ........................................................................................................................................................35
Emitterschaltung mit Stromgegenkopplung...........................................................................................................................................................35
Emitterschaltung mit Spannungsgegenkopplung ...................................................................................................................................................36
FET ..........................................................................................................................................................................37
8.3.1
8.3.2
8.3.3
8.3.3.1
8.3.3.2
8.3.3.3
8.3.4
8.4
8.5
12.2
13
9.3
9.3.1
9.3.2
9.3.3
9.3.3.1
9.3.3.2
9.3.3.3
9.3.3.4
9.3.3.5
9.3.3.6
9.3.3.7
13.2
9.4
REALER OP...............................................................................................................................................................45
9.4.1
9.4.2
9.4.3
9.4.3.1
9.4.4
9.4.5
GBW (Gain-Bandwith-Product) ....................................................................................................................................................................................45
Slew Rate .......................................................................................................................................................................................................................45
Biasstrom .......................................................................................................................................................................................................................45
Biasstromkompensation .........................................................................................................................................................................................45
Offsetstrom ....................................................................................................................................................................................................................45
Reale Verstärkung bei Rückkopplung ...........................................................................................................................................................................45
HF-TECHNIK..................................................................................................................46
10.1
KO............................................................................................................................................................................46
10.1.1
Rise Time, Bandbreite ...................................................................................................................................................................................................46
10.1.1.1
<= 1GHz ................................................................................................................................................................................................................46
10.1.1.2
6GHz......................................................................................................................................................................................................................46
10.2
HF-LEITUNGEN ........................................................................................................................................................46
10.2.1
10.2.2
Wellenwiderstand ≥10kHz.............................................................................................................................................................................................46
Ausbreitungsgeschwindigkeit ........................................................................................................................................................................................46
OSZILLATOREN............................................................................................................47
11.1
11.2
SCHWINGBEDINGUNG IM FREQUENZBEREICH ...........................................................................................................47
QUARZE ....................................................................................................................................................................47
11.2.1
© Daniel Arnold
www.lanny.ch
Serieresonanz .................................................................................................................................................................................................................47
09.09.2004
Klirrfaktor ......................................................................................................................................................................................................................54
ZEITMITTELWERTE ................................................................................................................................................... 54
13.2.1
13.2.2
13.3
Gleichwert (arithmetischer Mittelwert)..........................................................................................................................................................................54
Effektivwert (RMS) .......................................................................................................................................................................................................54
KOMPLEXE ZAHLEN ................................................................................................................................................. 54
13.3.1
Darstellungen .................................................................................................................................................................................................................54
13.3.1.1
Normalform............................................................................................................................................................................................................54
13.3.1.2
Polarform ...............................................................................................................................................................................................................54
13.3.1.3
Exponentialform.....................................................................................................................................................................................................54
13.3.2
Umrechnungen ...............................................................................................................................................................................................................54
13.4
Verstärkung....................................................................................................................................................................................................................39
ohne Last................................................................................................................................................................................................................39
mit Last ..................................................................................................................................................................................................................39
Differenzierer.................................................................................................................................................................................................................42
Integrierer ......................................................................................................................................................................................................................42
Schaltungen und ihre Übertragungsfunktionen..............................................................................................................................................................43
P .............................................................................................................................................................................................................................43
I ..............................................................................................................................................................................................................................43
D ............................................................................................................................................................................................................................43
P und I....................................................................................................................................................................................................................43
Hochpass................................................................................................................................................................................................................43
Tiefpass..................................................................................................................................................................................................................44
D und P ..................................................................................................................................................................................................................44
QUALITÄT ................................................................................................................................................................ 54
13.1.1
Differenzverstärkung .....................................................................................................................................................................................................39
Gleichtaktverstärkung ....................................................................................................................................................................................................39
Gleichtaktunterdrückung (Common Mode Rejection Ratio) .........................................................................................................................................39
GRUNDSCHALTUNGEN MIT SPEICHER MIT IDEALEM OP............................................................................................42
ZUSTANDSRAUMDARSTELLUNG................................................................................................................................ 52
SIGNALE......................................................................................................................... 54
13.1
FET-Typen.....................................................................................................................................................................................................................37
Arbeitspunkt ..................................................................................................................................................................................................................38
Kleinsignalverhalten ......................................................................................................................................................................................................38
Kleinsignalersatzschaltbild ....................................................................................................................................................................................38
Kleinsignalgleichungen..........................................................................................................................................................................................39
Kleinsignalparameter .............................................................................................................................................................................................39
Kleinsignalverstärkung in Sourceschaltung...................................................................................................................................................................39
Invertierende Verstärkervarianten..................................................................................................................................................................................40
Invertierender Verstärker .......................................................................................................................................................................................40
Limes Verstärker....................................................................................................................................................................................................40
Summierverstärker.................................................................................................................................................................................................40
Nichtinvertierender Verstärkervarianten........................................................................................................................................................................40
Nichtinvertierender Verstärker ..............................................................................................................................................................................40
Impedanzwandler...................................................................................................................................................................................................40
Differenzverstärker ........................................................................................................................................................................................................41
Schmitttrigger ................................................................................................................................................................................................................41
Invertierender Schmitttrigger .................................................................................................................................................................................41
nicht-invertierender Schmitttrigger........................................................................................................................................................................41
P .....................................................................................................................................................................................................................................48
P-Regler .........................................................................................................................................................................................................................48
I-Regler ..........................................................................................................................................................................................................................49
D-Regler.........................................................................................................................................................................................................................49
PI-Regler ........................................................................................................................................................................................................................50
PD-Regler.......................................................................................................................................................................................................................50
PID-Regler .....................................................................................................................................................................................................................51
Phasenreserven...............................................................................................................................................................................................................51
12.2.1
Regelungsnormalform....................................................................................................................................................................................................52
12.2.1.1
Zusandsgleichung...................................................................................................................................................................................................52
12.2.1.2
Spezialfall: Eingang wirkt nicht durch seine Ableitungen auf den Ausgang .........................................................................................................52
KORRELATION .......................................................................................................................................................... 54
13.4.1
Autokorrelationsfunktion ...............................................................................................................................................................................................54
13.4.1.1
Periodische Signale ................................................................................................................................................................................................54
13.4.1.2
Stochastische Signale .............................................................................................................................................................................................55
13.4.1.3
Signale mit begrenzter Signalenergie (zeitbegrenzt)..............................................................................................................................................55
13.4.2
Kreuzkorrelationsfunktion .............................................................................................................................................................................................55
13.5
13.6
13.7
FREQUENZKOMPENSATION .......................................................................................................................................40
GRUNDSCHALTUNGEN OHNE SPEICHER MIT IDEALEM OP.........................................................................................40
9.2.1
9.2.1.1
9.2.1.2
9.2.1.3
9.2.2
9.2.2.1
9.2.2.2
9.2.3
9.2.4
9.2.4.1
9.2.4.2
REGELTECHNISCHE GLIEDER .................................................................................................................................... 48
12.1.1
12.1.2
12.1.3
12.1.4
12.1.5
12.1.6
12.1.7
12.1.8
OPERATIONSVERSTÄRKER......................................................................................40
9.1
9.2
Parallelresonanz .............................................................................................................................................................................................................47
Güte................................................................................................................................................................................................................................47
STEUER- UND REGELUNGSTECHNIK ................................................................... 48
12.1
GEGENTAKTSTUFE ....................................................................................................................................................39
8.5.1
8.5.1.1
8.5.1.2
11
12
DIFFERENZVERSTÄRKER ...........................................................................................................................................39
8.4.1
8.4.2
8.4.3
10
11.2.2
11.2.3
GRUNDSCHALTUNGEN MIT BIPOLAREM TRANSISTOR ................................................................................................35
8.2.1
8.2.2
8.2.3
8.2.4
8.2.4.1
8.2.4.2
8.2.4.3
9
4
Graphische Arbeitspunktbestimmung....................................................................................................................................................................32
Kleinsignalparameter .....................................................................................................................................................................................................33
Graphische Ermittlung ...........................................................................................................................................................................................33
Kleinsignalstromverstärkung .................................................................................................................................................................................33
Steilheit ..................................................................................................................................................................................................................33
Kleinsignaleingangswiderstand .............................................................................................................................................................................33
Kleinsignalausgangswiderstand .............................................................................................................................................................................33
Rückwärtssteilheit..................................................................................................................................................................................................33
Vierpol-Matrizen ...................................................................................................................................................................................................33
Verlustleistung ...............................................................................................................................................................................................................33
Gegenkopplung..............................................................................................................................................................................................................34
Parallelgegenkopplung nur AC......................................................................................................................................................................................34
FALTUNG .................................................................................................................................................................. 55
FOURIERTRANSFORMATION ...................................................................................................................................... 56
HARMONISCHE SIGNALE IN LTI-SYSTEMEN ............................................................................................................. 56
13.7.1
13.7.2
13.7.3
13.7.4
14
Widerstand .....................................................................................................................................................................................................................56
Spule ..............................................................................................................................................................................................................................56
Kondensator ...................................................................................................................................................................................................................56
Spannungsquelle ............................................................................................................................................................................................................56
NACHRICHTENTECHNIK.......................................................................................... 57
14.1
LEITUNGSGEBUNDENE ÜBERTRAGUNG .................................................................................................................... 57
14.1.1
14.1.1.1
14.1.1.2
14.1.1.3
14.1.1.4
14.1.2
14.1.2.1
14.1.2.2
14.1.3
14.1.3.1
14.1.3.2
14.1.3.3
14.1.3.4
14.1.4
14.1.4.1
14.1.4.2
14.1.4.3
14.1.4.4
14.2
Leitungsmodell...............................................................................................................................................................................................................57
Beläge.....................................................................................................................................................................................................................57
Ausbreitungskoeffizient .........................................................................................................................................................................................57
Wellenwiderstand...................................................................................................................................................................................................57
Wellendämpfungsmass ..........................................................................................................................................................................................57
Anpassung......................................................................................................................................................................................................................57
Maximale Wirkleistung..........................................................................................................................................................................................57
Wellenanpassung....................................................................................................................................................................................................57
Reflexionen und stehende Wellen..................................................................................................................................................................................58
Überlagerung der vor- und rücklaufenden Signale.................................................................................................................................................58
Reflexionsfaktor.....................................................................................................................................................................................................58
Rückflussdämpfung................................................................................................................................................................................................58
Anpassungsfaktor / Welligkeitsfaktor ....................................................................................................................................................................58
Glasfasern.......................................................................................................................................................................................................................58
Lichtausbreitung in Glasfasern...............................................................................................................................................................................58
Numerische Apertur (Lichteinsammlungszahl) .....................................................................................................................................................58
Eingekoppelte Leistung..........................................................................................................................................................................................58
Übertragungsdämpfung..........................................................................................................................................................................................58
DRAHTLOSE ÜBERTRAGUNG .................................................................................................................................... 59
14.2.1
14.2.2
14.2.2.1
14.2.2.2
14.2.2.3
14.2.2.4
14.2.2.5
14.2.2.6
14.2.3
14.2.3.1
14.2.3.2
14.2.3.3
14.2.3.4
14.2.3.5
14.2.3.6
14.2.3.7
14.2.4
14.2.4.1
14.2.4.2
© Daniel Arnold
www.lanny.ch
Komponenten einer drahtlosen Übertragung .................................................................................................................................................................59
Übertragungsverluste im freien Raum ...........................................................................................................................................................................59
isotroper Strahler (Referenz für Gewinne) .............................................................................................................................................................59
Empfangene Leistung bei einem isotropen Strahler...............................................................................................................................................59
Gewinn ...................................................................................................................................................................................................................59
Empfangene Leistung allgemein ............................................................................................................................................................................59
free space path loss.................................................................................................................................................................................................59
Fernfeldbedingung .................................................................................................................................................................................................59
Effektiv abgestrahlte Leistung .......................................................................................................................................................................................60
Kabelverluste .........................................................................................................................................................................................................60
Reflexionsfaktor.....................................................................................................................................................................................................60
Verfügbare Leistung an der Antenne .....................................................................................................................................................................60
Strahlungswirkungsgrad.........................................................................................................................................................................................60
Richtfaktor .............................................................................................................................................................................................................60
Gewinn ...................................................................................................................................................................................................................60
ERP Effective Radiated Power ..............................................................................................................................................................................60
Empfangene Leistung.....................................................................................................................................................................................................60
Reflexionsfaktor.....................................................................................................................................................................................................61
Leistung am Eingang des Antennenzubringers ......................................................................................................................................................61
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6
14.2.4.3
Kabelverluste .........................................................................................................................................................................................................61
14.2.4.4
Leistung am Empfängereingang ............................................................................................................................................................................61
14.2.5
Antennengrundlagen ......................................................................................................................................................................................................61
14.2.6
Antennenimpedanz ........................................................................................................................................................................................................61
14.2.6.1
Strahlungseffizienz ................................................................................................................................................................................................61
14.2.6.2
Reflexionsgrad .......................................................................................................................................................................................................61
14.2.6.3
Welligkeitsfaktor ...................................................................................................................................................................................................61
14.3
ANALOGE FILTER .......................................................................................................64
15.1
DIE ÜBERTRAGUNGSFUNKTION ................................................................................................................................64
15.1.1
Darstellungsarten ...........................................................................................................................................................................................................64
15.1.2
Berechnungen mit TI-89 ................................................................................................................................................................................................64
15.1.2.1
Nullstellen..............................................................................................................................................................................................................64
15.1.2.2
Pole ........................................................................................................................................................................................................................64
15.1.2.3
Residuen ................................................................................................................................................................................................................64
15.2
STANDARD ÜBERTRAGUNGSFUNKTIONEN 2. ORDNUNG ...........................................................................................64
15.2.1.1
15.2.1.2
15.2.2
15.2.2.1
15.2.3
15.2.4
15.2.5
15.2.6
15.3
Allgemeine Übertragungsfunktion eines Filters 2. Ordnung .................................................................................................................................64
Übertragungsfunktion eines Filters 2. Ordnung mit konjugiert komplexem Polpaar ............................................................................................64
Tiefpassübertragungsfunktion........................................................................................................................................................................................65
Amplitudenmaximum ............................................................................................................................................................................................65
Hochpassübertragungsfunktion......................................................................................................................................................................................65
Bandpassübertragungsfunktion......................................................................................................................................................................................65
Bandstoppübertragungsfunktion ....................................................................................................................................................................................66
Allpassübertragungsfunktion .........................................................................................................................................................................................66
FILTERBERECHNUNGEN HÖHERER ORDNUNG ...........................................................................................................67
15.3.1
Vorgehen beim Filterentwurf höherer Ordnung ............................................................................................................................................................67
15.3.2
Approximationen ...........................................................................................................................................................................................................67
15.3.2.1
Butterworth ............................................................................................................................................................................................................67
15.3.2.2
Tschebyscheff ........................................................................................................................................................................................................67
15.3.2.3
Inverse Tschebyscheff ...........................................................................................................................................................................................67
15.3.2.4
Cauer (elliptische Approximation).........................................................................................................................................................................67
15.3.2.5
Bessel.....................................................................................................................................................................................................................68
15.3.3
Matlab Funktonen ..........................................................................................................................................................................................................68
15.3.3.1
Ordnung .................................................................................................................................................................................................................68
15.3.3.2
Nullstellen, Pole, Verstärkungsfaktor ....................................................................................................................................................................68
15.3.3.3
Bestimmung der Zähler- und Nennerpolynome normiert auf 1 rad/s ....................................................................................................................68
15.3.3.4
Vektor mit logarithmische verteilten Werten.........................................................................................................................................................68
15.3.3.5
Frequenzgang berechnen .......................................................................................................................................................................................68
15.3.3.6
Amplitudengang darstellen in [dB]........................................................................................................................................................................68
15.3.3.7
Phasengang darstellen............................................................................................................................................................................................68
15.4
RAUSCHEN ............................................................................................................................................................... 80
16.3.1
Rauschquellen ................................................................................................................................................................................................................80
16.3.1.1
Widerstandsrauschen..............................................................................................................................................................................................80
16.3.1.2
Schrotrauschen (gilt nicht in Metallen) ..................................................................................................................................................................80
16.3.1.3
1/f Rauschen (Flickerrauschen)..............................................................................................................................................................................80
16.3.2
SNR................................................................................................................................................................................................................................80
16.3.2.1
Addition von Rauschen ..........................................................................................................................................................................................80
CODIERUNG ..............................................................................................................................................................62
14.3.1
Informationsgehalt .........................................................................................................................................................................................................62
14.3.1.1
Information ............................................................................................................................................................................................................62
14.3.1.2
Informationsgehalt .................................................................................................................................................................................................62
14.3.1.3
Information des gesamten Ereignisses ...................................................................................................................................................................62
14.3.1.4
Mittlere Informationsgehalt pro Element (Entropie H)..........................................................................................................................................62
14.3.2
Kanalkapazität ...............................................................................................................................................................................................................62
14.3.2.1
Informationsfluss ...................................................................................................................................................................................................62
14.3.2.2
Symbolrate .............................................................................................................................................................................................................62
14.3.2.3
Maximale Symboldauer .........................................................................................................................................................................................62
14.3.2.4
Maximale Auflösung .............................................................................................................................................................................................62
14.3.2.5
Kanalkapazität .......................................................................................................................................................................................................63
14.3.3
Codewortlänge ...............................................................................................................................................................................................................63
14.3.3.1
Mittlere Codewortlänge .........................................................................................................................................................................................63
14.3.3.2
Redundanz .............................................................................................................................................................................................................63
15
16.3
17
MODULATION UND DEMODULATION .................................................................. 81
17.1
ANALOGE VERFAHREN ............................................................................................................................................. 81
17.1.1
17.1.1.1
17.1.2
17.1.2.1
17.1.2.2
17.1.2.3
17.1.2.4
17.1.3
17.1.4
17.1.5
17.1.5.1
17.1.5.2
17.1.5.3
17.2
Trägerschwingung..........................................................................................................................................................................................................81
Funktionsparameter................................................................................................................................................................................................81
Amplitudenmodulation AM ...........................................................................................................................................................................................81
Modulationsgrad ....................................................................................................................................................................................................81
DSB........................................................................................................................................................................................................................81
DSBSC ...................................................................................................................................................................................................................81
SSB ........................................................................................................................................................................................................................81
Phasenmodulation PM ...................................................................................................................................................................................................81
Frequenzmodulation FM ................................................................................................................................................................................................81
Pulsmodulation...............................................................................................................................................................................................................82
Pulsamlitudenmodulation PAM .............................................................................................................................................................................82
Pulsdauermodulation PDM ....................................................................................................................................................................................82
Pulspositionsmodulation (Pulsphasenmodulation) PPM........................................................................................................................................82
DIGITALE VERFAHREN ............................................................................................................................................. 83
17.2.1
17.2.2
17.2.2.1
17.2.2.2
17.2.2.3
17.2.2.4
17.2.2.5
17.2.2.6
17.2.2.7
17.2.2.8
17.2.3
17.2.3.1
17.2.3.2
17.2.4
17.2.4.1
17.2.4.2
17.2.4.3
17.2.5
17.2.5.1
17.2.5.2
Q-Funktion .....................................................................................................................................................................................................................83
Digitale Übertragung analoger Signale ..........................................................................................................................................................................84
Benötigte Bandbreite für einen Puls.......................................................................................................................................................................84
Begriffe ..................................................................................................................................................................................................................84
Impulsnebensprechen (Inter-Symbol Interference ISI) ..........................................................................................................................................84
Nyquist Impulsformer (ISI -> 0) ............................................................................................................................................................................85
Mittlere Rauschleistung..........................................................................................................................................................................................86
Mittlere Symbolfehler-WSK ..................................................................................................................................................................................86
PAM Signale ..........................................................................................................................................................................................................86
PCM (Pulse Code Modulation) ..............................................................................................................................................................................86
Digitale Modulation von sinusförmigen Trägern ...........................................................................................................................................................87
Amplitude Shift Keying ASK ................................................................................................................................................................................87
Phase Shift Keying PSK.........................................................................................................................................................................................87
Binäre Empfänger ..........................................................................................................................................................................................................88
Allgemeines ...........................................................................................................................................................................................................88
Korrelationsempfänger...........................................................................................................................................................................................88
Nichtkohärente binäre Empfänger..........................................................................................................................................................................89
Vergleich von digitalen Modulationsverfahren..............................................................................................................................................................90
Binär.......................................................................................................................................................................................................................90
M-ary......................................................................................................................................................................................................................90
AKTIVE FILTER .........................................................................................................................................................69
15.4.1
Aktive RC-Filter 1. Ordnung .........................................................................................................................................................................................69
15.4.1.1
Differentiator .........................................................................................................................................................................................................69
15.4.1.2
Integrator................................................................................................................................................................................................................69
15.4.1.3
Tiefpass 1. Ordnung mit invertierendem Verstärker..............................................................................................................................................69
15.4.1.4
Tiefpass 1. Ordnung mit nichtinvertierendem Verstärker......................................................................................................................................69
15.4.1.5
Hochpass 1. Ordnung mit invertierendem Verstärker............................................................................................................................................70
15.4.1.6
Hochpass 1. Ordnung mit nichtinvertierendem Verstärker....................................................................................................................................70
15.4.1.7
Allpass 1. Ordnung 1. Variante..............................................................................................................................................................................70
15.4.1.8
Allpass 1. Ordnung 2. Variante..............................................................................................................................................................................70
15.4.2
Aktive Filter 2. Ordnung................................................................................................................................................................................................71
15.4.2.1
KRC Filter (Sallen-Key Filter) ..............................................................................................................................................................................71
15.4.2.2
Multiple Feedback Filter........................................................................................................................................................................................73
15.4.3
Universalfilter Filter ......................................................................................................................................................................................................74
15.4.3.1
Zustandvariablen Filter ..........................................................................................................................................................................................74
15.4.3.2
Biquad Filter ..........................................................................................................................................................................................................75
15.5
KASKADENENTWURF ................................................................................................................................................77
15.5.1
15.5.2
15.5.3
15.5.4
16
Ent-Normierung .............................................................................................................................................................................................................77
Koeffiziententabelle 1....................................................................................................................................................................................................77
Koeffiziententabelle 2....................................................................................................................................................................................................78
Ent-Normierung auf Hochpass ......................................................................................................................................................................................78
SIGNALÜBERTRAGUNG.............................................................................................79
16.1
SIGNALVERZERRUGEN ..............................................................................................................................................79
16.1.1
16.1.2
16.1.3
16.1.4
16.1.4.1
16.2
Klirrfaktor ......................................................................................................................................................................................................................79
Verzerrungsfreie Übertragung .......................................................................................................................................................................................79
Amplitudenverzerrungen ...............................................................................................................................................................................................79
Phasen- oder Laufzeitverzerrungen ...............................................................................................................................................................................79
Bandpassähnlicher Kanal mit Laufzeitverzerrungen .............................................................................................................................................79
EQUALIZATION .........................................................................................................................................................79
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09.09.2004
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www.lanny.ch
09.09.2004
7
1
8
Ohmsches Gesetz
U
R
1
R =
G
I =
R =
ρ ∗l
A
l
R =
γ ∗ A
I
J =
A
1.1
l
γ ∗ A
=
[I ] = A
1.6
[U ] = V
U = U 0 − I ∗ Ri
[ R] = Ω
IK =
[G ] = S
[ρ] =
[γ ] =
Ω ∗ mm 2
m
m
∆U 1
∆U 1 ∗ U 2
U
S= 1 =
∆U 2 ∆U 2 ∗ U1
U2
I2
I3
Serieschaltung
U = U1 + U 2 + ...
U1 R1
=
U 2 R2
Parallelschaltung
I = I1 + I 2 + ...
G = G1 + G2 + ...
R=
1
1
1
+
+ ...
R1 R2
∆U Zu ≈
U2 =
1.5
∆U Zθ ≈ U Z ∗ α Z ∗ ∆θ
Spannungsteiler
U R2 =
∆U1 ∗ rZ
RV
∆U Zi ≈ − ∆I L ∗ rZ
I1 R2
=
I 2 R1
1.4
∆U 2
∆I 2
1.7.2
Spannungsstabilisierung mit Z-Diode
∆U Z
rZ =
∆I Z
U min − U Z
RV max =
I Z min + I L max
U −UZ
RV min = max
I Z max + I L min
R = R1 + R2 + ...
1.3
Spannungsstabilisierung Allgemein
∆U1
G=
∆U 2
ra =
1.2
Stabilisierung von U und I
1.7.1
Kirchhoff
I1
U0
Ri
1.7
Ω ∗ mm 2
[ J ] = A m2
∑I =0
∑U = 0
Spannungserzeuger
R2
∗U
R1 + R2
Rp
R1 + R p
∗U
S≈
RV U 2
U
∗
=G∗ 2
rZ U1
U1
G≈
RV
rZ
Brückenschaltung
1.5.1
Abgeglichen
R1 R3
=
R2 R4
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09.09.2004
9
2
Leistung, Arbeit
P =U ∗I
P = I R2 ∗ R
P=
10
2
R
U
R
[ S ] = VA
[ P] = W
[Q] = var
W = P ∗t
W = F ∗s
3
Maschinen
3.1
Gleichstrommaschine
3.1.1
die Rotorwicklung
p:
a:
D:
τ:
Z:
Z’:
l:
Ω:
v:
uiL:
uim:
Ui:
ui(t):
Polpaarzahl
Anzahl parallele Rotorwicklungszweigpaare
mittlerer Spulendurchmesser
Polteilung
totale Leiterzahl auf dem Rotor
In Serie geschaltete Leiter innerhalb eines Rotorwicklungszweigpaares
aktive Leiterlänge (Leiterlänge, welche dem magnetischen Feld ausgesetzt ist)
Winkelgeschwindigkeit des Rotors
Umfangsgeschwindigkeit der Leiter
Induzierter Spannungsmomentanwert in einem Leiter
Mittlerer induzierter Spannungswert über eine Polteilung
Total induzierte Spannung bei konstanter Winkelgeschwindigkeit Ω
Total induzierter Spannungsmomentanwert bei zeitlich veränderlicher Winkelgeschwindigkeit Ω
3.1.1.1
Polteilung
D ⋅π
τ=
2⋅ p
3.1.1.2
Aktive Leiterzahl
Z
Z′=
2⋅a
3.1.1.3
v=
Unfangsgeschwindigkeit
D
⋅Ω
2
3.1.2
Induzierte Spannung
3.1.2.1
U
induzierte Spannung
u iL = v ⋅ B ( x ) ⋅ l
3.1.2.2
u im =
p
π
Mittelwert der induzierten Spannung
⋅ I ⋅Ω
3.1.2.3
total induzierte Spannung
Ui = K ⋅ Φ ⋅ Ω
Z′⋅ p
K=
2 ⋅ a ⋅π
3.1.2.4
total induzierter Spannungsmomentanwert
u i (t ) = K ⋅ Φ ⋅ Ω (t )
3.1.3
Drehmoment
m = K ⋅Φ ⋅i
M = K ⋅Φ ⋅I
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09.09.2004
11
3.1.4
Ω~
=
12
Drehzahl
U Rotor
K ⋅Φ
3.1.5
Ersatzschaltbild
∑R
RR =
i
Rotorkreis
LRotor =
LR =
N Rotor ⋅ Φ quer
∑L
I
=
i
Rotor
3.1.6
Stationärer Betrieb
3.1.6.1
Leerlaufdrehzahl
Ud
Ω0 ~
=
K ⋅Φ
U −U R
Ω0 = d
K ⋅Φ
3.1.6.2
Ω = Ω0 −
3.2
Lastkennlinie
RR
(K ⋅ Φ )2
⋅ M Last
Synchronmaschine
3.2.1 Induzierte Spannung
ˆ ⋅ cos (Ω ⋅ t )
Φ Spule (t ) = Φ
ˆ ⋅ Ω ⋅ k w1
Uˆ = N ⋅ Φ
ˆ ⋅ k w1 ⋅ N 2 ⋅ π
2 ⋅π ⋅ f ⋅ Φ
ˆ ⋅ k w1 ⋅ N = 4.44 ⋅ f ⋅ N ⋅ Φ
ˆ ⋅ k w1 =
U =
=
f ⋅Φ
2
2
3.2.1.1
Xd =
IF
Frequenz f
Ω⋅ p
ω
f =
=
2 ⋅π
2 ⋅π
3.2.2
U0
IK
IK
Längsreaktanz
Un
IK0
Ω
3.3
Ersatzschaltbild der Vollpolmaschine
RS
Xd=Xq
~
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www.lanny.ch
09.09.2004
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www.lanny.ch
09.09.2004
13
4
Wärme
4.1
5
Elektrowärme
Wechselspannung
5.1
[Q ] = J = Ws
Q ab = η ∗ W Eauf
η=
14
Pab
Pauf
Frequenz
1
f =
T
[T ] = s
[ϕ ] ≅ rad
5.2
4.2
Wärmewiderstand
∆θ
Rth =
PV
4.3
5.3
[α ] = 1 K
5.3.1
wärmebedingte Widerstandsänderung
∆R = α 20 ∗ R 20 ∗ ∆ϑ
Rϑ = R 20 + ∆R = R 20 + α 20 ∗ R 20 ∗ ∆ϑ
4.3.1.1
[ ∆ϑ ] = K
i=
4.3.1.2
[ω ] = 1 s
Mittelwerte
Gleichwert (arithmetischer Mittelwert)
t1 +T
1
T
∫ i ⋅ dt
t1
PTC
1
u=
T
RPTC = RN ⋅ eα (ϑ −ϑN )
t1 +T
∫ u ⋅ dt
t1
NTC
RNTC = RN ⋅ e
4.4
Sinusförmig
u = uˆ sin (ω ⋅ t + ϕ u )
K
W
[ ∆θ ] = K
[P] = W
[ R th ] =
[ f ] = Hz
5.3.2
1 1 

b  −
 ϑ ϑN 
Effektivwert (RMS)
1
T
I=
Linearisierung
∫i
2
⋅ dt
t1
1
T
U=
Parallel zu Rm
R ⋅ (Ru + Ro ) − 2 Ru Ro
Rlin = m
Ru + Ro − 2 Rm
t1 +T
t1 +T
∫u
5.3.2.1
U= u
2
⋅ dt
t1
Effektivwert einer Mischgrösse
2
eff
+ u~eff2
2
~
I = i eff + ieff2
5.3.3
i=
1
T
1
u=
T
5.3.4
Gleichrichtwert
t1 +T
∫ i ⋅ dt
t1
t1 +T
∫ u ⋅ dt
t1
Verhältniszahlen
iˆ
uˆ
k s = oder
I
U
I
U
F = oder
i
u
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09.09.2004
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www.lanny.ch
09.09.2004
15
16
5.4
Gleichrichter
Q = CG ∗ u pss ≈ t E ∗ I d
Up ≈
u pss ≈
5.5
Drehstrom (3 Phasenstrom)
5.5.1
u pss
2∗ 3
0.75 ∗ Tp ∗ I d
Sternschaltung
5.5.1.1
Zeigerdiagramm
L2
U S = Stern - bzw. Strangspannung
U 1N = U S ∠0°
CG
U 2 N = U S ∠ − 120°
N
U 3 N = U S ∠120°
L1
U 13 = 3 ⋅ U S ∠30°
P P
I
3.1
1.5
1.5
1.23
1.1
1
1/2
1/3
1/2
1/3
T
d
Z
I
d
U
di
U
1
2^0.5
2^0.5/2
2^0.5/3^0.5
2^0.5
2^0.5
uˆ
R
U
U23
U31
U 23 = 3 ⋅ U S ∠ − 90°
Schaltung:
M1
M2
M3
B2
B6
U2N
U12
U1N
U 31 = 3 ⋅ U S ∠150°
L3
U3N
1
2*2^0.5
2^0.5
2*2^0.5/3^0.5
2^0.5
2^0.5
5.5.1.2
symmetrische Belastung
I N = I1 + I 2 + I 3 = 0
P = 3 ⋅ U S ⋅ I S ⋅ cos ϕ
Q = 3 ⋅ U S ⋅ I S ⋅ sin ϕ
S = P + jQ = 3 ⋅ U S ⋅ I S ∠ϕ
S = 3 ⋅ U ⋅ I∠ϕ
5.5.1.3
unsymmetrische Belastung
I N = I1 + I 2 + I 3 ≠ 0
S = U 1N I 1 + U 2 N I 2 + U 3 N I 3
*
U KN =
*
*
Y 1U 1 N + Y 2 U 2 N + Y 3 U 3 N
Y1 + Y 2 + Y 3
S = U 13 I 1 + U 23 I 2
*
*
*
1
*
S = U 12 I + U 32 I 3
S = U 21 I + U 31 I 3
*
2
5.5.2
*
Dreieckschaltung
5.5.2.1
symmetrische Belastung
I = 3 ⋅ I∆
S = 3 ⋅U ∆ ⋅ I ∆ ∠ϕ = 3 ⋅U ⋅ I∠ϕ
5.5.2.2
unsymmetrische Belastung
S = U 13 I 1 + U 23 I 2
*
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www.lanny.ch
*
09.09.2004
17
18
Stern zu Dreieck (T zu Π)
5.5.3
RS = R1 ⋅ R2 + R2 ⋅ R3 + R3 ⋅ R1
5.6
R
R12 = S
R3
S = P2 +Q2
R23 =
RS
R1
R31 =
RS
R2
5.5.4
ϕ = ϕ u − ϕ i = arctan
Q
P
S = P + jQ = S∠ϕ
2
2
S =U ⋅I
5.6.1
Dreieck zu Stern (Π
Π zu T)
RD = R12 + R23 + R31
1
3
1
3
R ⋅R
R1 = 12 31
RD
R2 =
R23 ⋅ R12
RD
R3 =
R31 ⋅ R23
RD
5.5.5
Komplexe Leistung
Leistung einer Impedanz
U2
S = * = I2 ⋅Z
Z
5.6.2
Leistung einer Admittanz
S = U ⋅Y * =
2
5.6.3
Wechselstromgeneratoren
5.5.5.1
*
I2
Y
Leistungsanpassung
5.6.3.1
Leistungsanpassung mit reeller Last
RL = Z i
Einphasenwechselstrom
f = p∗n
5.6.3.2
Leistungsanpassung mit komplexer Last
Z L = Z i*
YL = Yi *
5.6.4
Lastfaktor
5.6.4.1
Einphasig
Re(S ) P
λ = cos(ϕ ) =
=
S
S
5.6.4.2
Dreiphasig
Re(S1 + S 2 + S3 ) P1 + P2 + P3
λ = cos(ϕ ) =
=
S1 + S 2 + S3
S1 + S 2 + S3
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09.09.2004
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www.lanny.ch
09.09.2004
19
6
20
C und L
6.1
elektrische Felder
U
E=
l
D(Flussdichte) = ε 0 ∗ ε r ∗ E
EKreisfläche um Punktladung
Q
=
4π ∗ ε 0 ∗ ε r ∗ r 2
Q ∗Q
1
∗ 1 2 2
l
4π ∗ ε
Q1 ∗ Q2
1
∗
FLinienladungen =
l∗s
2π ∗ ε
FFreie Ladung im Feld = E ∗ Q
Fzwischen Q1und Q 2 =
Q = I ∗ t = C ∗U
[F ] = N
V
m
As
[ε ] =
Vm
[Q ] = As = C
As
[C ] =
=F
V
[W ] = Ws
[E] =
6.3.1
Widerstand
u R = R ⋅ iR
6.3.2
Kondensatoren
6.3.2.1
Impulsverformung
τ = R ⋅C
−
W=
t
U 0 −τ
⋅e
R
iC ( laden ) =
u C ( entladen ) = U 0 ⋅ e
As
Vm
Wechselstromverhalten
Spule
Impulsverformung
L
τ=
R
Plattenkondensator an konstanter Spannung
t
−
U
⋅ (1 − e τ )
R
t
U −
= ⋅e τ
R
i Ein =
i Aus
Plattenkondensator mit konstantem Q
1 Q2
⋅
2 ε⋅A
6.3.3.2
Wechselstromverhalten
X L = ω ⋅ L = 2π ⋅ f ⋅ L
uL = L ⋅
Induktivitäten
L = N 2 ∗ AL
iL =
L∗I2
W=
2
© Daniel Arnold
www.lanny.ch
t
τ
U 0 −τ
⋅e
R
6.3.3.1
1 ε⋅A
F = ⋅ 2 ⋅U 2
2 s
6.1.1.2
As
=F
V
[L ] = Vs = H
A
[C ] =
t
iC ( entladen ) = −
6.3.3
Plattenkondensator
6.1.1.1
−
[τ ] = s
[X ] = Ω
1
1
Xc =
=
ω ⋅ C 2π ⋅ f ⋅ C
du
iC = C ⋅ C
dt
1
u C = ∫ iC ⋅ dt + U C 0
C
Q ∗U C ∗U 2
=
2
2
6.1.1
t
u C ( laden ) = U 0 ⋅ (1 − e τ )
6.3.2.2
l
ε 0 = 8.854 E - 12
6.2
Netzwerkelemente
ε0 ∗εr ∗ A
C=
F=
6.3
09.09.2004
1
L
di L
dt
∫ u L ⋅ dt + I L 0
© Daniel Arnold
www.lanny.ch
09.09.2004
21
6.4
U
CR Schaltung
I
Serieschaltung
UR
=
Z
U = U
Z =
6.4.1.2
RC Schaltungen
6.4.1
I =
22
=
R
2
R
+U
UC
C
UE
XC
UR,A
I
UA
R
UE
R
UC
XC
Z
2
C
R 2 + X C2
tan ϕ =
UC
UR
=
fg
XC
R
R = Z ∗ cos ϕ
X C = Z ∗ sin ϕ
6.4.1.1
Im (Z)
V
[dB]
RC Schaltung
log f
Re (Z)
I
I
R
UE
Ua/Ue
UR
UA
UE
C
R
UC,A
Z
45°
XC
log f
6.4.2
I =
fg
V
[dB]
90°
Parallelschaltung
I R2 + I C2
Y = G 2 + B C2
Im (Z)
1
Z =
1
R2
log f
Re (Z)
tan ϕ =
Ua/Ue
IC
IR
+
=
1
X C2
BC
G
=
R
XC
-45°
I
-90°
UE
IR
R
IC
UE
© Daniel Arnold
www.lanny.ch
09.09.2004
© Daniel Arnold
www.lanny.ch
IC
I
Y
BC
C
IR
G
09.09.2004
23
6.5
U
Z
LR Schaltung
I
Serieschaltung
UR
=
U = U
Z =
6.5.1.2
RL Schaltungen
6.5.1
I =
24
=
R
2
R
+U
L
UL
UE
XL
UE
UA
R
UR,A
R 2 + X L2
tan ϕ =
UL
UR
=
UL
XL
Z
I
2
L
R
XL
R
6.5.1.1
fg
V
[dB]
R = Z ∗ cos ϕ
X L = Z ∗ sin ϕ
Im (Z)
RL Schaltung
Re (Z)
log f
Ua/Ue
I
R
UE
UE
UA
L
UL,A
-45°
XL
Z
-90°
I
UR
R
6.5.2
fg
I =
Im (Z)
V
[dB]
Parallelschaltung
I R2 + I L2
Y = G 2 + B L2
1
Z =
1
R2
log f
Re (Z)
Ua/Ue
tan ϕ =
90°
+
IL
IR
=
1
X L2
BL
G
=
R
XL
45°
I
log f
IR
IR
IL
G
UE
UE
© Daniel Arnold
www.lanny.ch
09.09.2004
© Daniel Arnold
www.lanny.ch
R
L
I
IL
Y
BL
09.09.2004
25
6.6
Verluste
6.6.1
1
=
d
6.8
Kondensator
IR
d = tan δ =
Q=
26
=
IC
IC
BC
=
XC
Rp
6.8.1
Reihenschwingkreis
U = U R2 + (U L − U C ) 2
R
C
Z =
R 2 + (X L − X C )2
Rp
=
IR
Gp
LC-Schwingkreis
f0 =
XC
1
2π ∗
L∗C
I
6.6.2
d = tan δ =
Q=
1
d
ω →∞
Spule
=
UR
=
UL
UL
induktiv
UL
L
Umwandlungen
C
ZS =
6.8.2
R 2S + X S2
1
R 2P
+
I =
1
ϕ P = arctan
Z
XS
=
Rp ≈
RS
RP
f0 ≈
XP
f0 =
R S = Z S ∗ cos ϕ S
RP =
UL
I
ZP
45°
− 45°
R
kapazitiv
Parallelschwingkreis
I R2 + ( I L − I C ) 2
1
R2
+(
1
XL
−
1
ω →∞
)2
XC
Y
2π
kapazitiv
L∗C
1
∗
L∗C
cos ϕ P
IC
I
RV ∗ C
1
1
ω Cu
IL
L
2π ∗
ω Lo
XL
Z
Z = G 2 + ( B L − BC ) 2
X P2
1
ϕ S = arctan
U
U
UC
ZS = ZP
1
XC
RV
ϕS = ϕP
ZP =
UC
Z
XL
L
6.7
UR
R
XL
=
UR
R
RV
−
UR
R V2
ω Co
IR
L2
45°
− 45°
BL
IR
IC
IL
X S = Z S ∗ sin ϕ S
XP =
ZP
U
sin ϕ P
R
C
L
BC
Y
induktiv
ω Lu
G
© Daniel Arnold
www.lanny.ch
09.09.2004
© Daniel Arnold
www.lanny.ch
09.09.2004
27
6.8.3
28
Bandbreite und Güte
6.9.7
QParallel =
Verlustloser Übertrager
U 1 = j ⋅ ω ⋅ L1 ⋅ I 1 + j ⋅ ω ⋅ L12 ⋅ I 2
f
B = ∆f = 0 = f og − f ug
Q
X
QSerie = 0
RV
U 2 = j ⋅ ω ⋅ L21 ⋅ I 1 + j ⋅ ω ⋅ L2 ⋅ I 2
L12 = L21 = k ⋅ L1 ⋅ L2
I1 =
Rp
X0
U1
6.9.7.1
6.9
Gekoppelte Spulen
6.9.1
Transformator
X L1  N 1 

=
X L1  N 2 
L1  N 2
=
L2  N 1
6.9.2
L1



U1 1
=
U2 k
U1 1
=
U2 k
2
L1 ⋅ L2
1
Z2
Übersetzungsverhältnis bei ideal fester Kopplung
k =1
Selbstinduktivität
U1
L1
N12 ⋅ Gm
N
= sgn k ⋅
= sgn k ⋅
= sgn k ⋅ 1 = sgn k ⋅ ü12
U2
L2
N 22 ⋅ Gm
N2
6.9.3
6.9.7.4
Gegeninduktivität
N ⋅ Φ 21
L12 = L21 = 2
I1
I1
1
=
− I 2 ü12
6.9.4
6.9.7.5
Kopplungsfaktor
Stromübersetzung bei ideal fester Kopplung
Netzwerktransformation
U
ü 2 ⋅U 2
Z 1 = 1 = 12
= ü122 ⋅ Z 2
I1
− I2
1
Y1 = 2 Y 2
ü12
Ideal
k =1
L1 ⋅ L 2
6.9.4.2
U2
Übersetzungsverhältnis bei Belastung
L1
1− k 2
+ j ⋅ω ⋅
L2
k
L1 = N 12 ⋅ G m1
L12 = L 21 =
jωL21I1
L1
L2
6.9.7.3
6.9.4.1
jωL12I2
Übersetzungsverhältnis im Leerlauf
6.9.7.2
2
L2
1
1 L2
U2 − ⋅
I2
k
L1
j ⋅ ω ⋅ k ⋅ L1 ⋅ L2
Real
k <1
L12 = L 21 = k ⋅ L1 ⋅ L 2
6.9.5
Serieschaltung gekoppelter Spulen
6.9.5.1
gleichsinnig
L E = L1 + L2 + 2 ⋅ L12
6.9.5.2
gegensinnig
L E = L1 + L2 − 2 ⋅ L12
6.9.6
Induzierte Spannung
u1 = i1 ⋅ R1 + L1 ⋅
© Daniel Arnold
www.lanny.ch
di1
di
+ L12 ⋅ 2
dt
dt
09.09.2004
© Daniel Arnold
www.lanny.ch
09.09.2004
29
30
7
6.10
6.10.1 Impedanz
7.1
Halbleiterkonstanten
7.1.1
Boltzmann-Konstante
jϕ u
U U ⋅e
=
= Z ⋅ e jϕ Z = Z (cos ϕ Z ⋅ j sin ϕ Z )
I
I ⋅ e jϕ i
Z = Re (Z ) + j Im (Z ) = R + jX
Z=
TI89: _k
Z=
R2 + X 2
Im (Z ) X
tan ϕ Z =
=
Re (Z ) R
ϕ Z = ϕ u − ϕ i = arctan
k = 1.380658 ⋅10 −23
X
R
7.1.2
I
I ⋅ e jϕ i
=
= Y ⋅ e jϕ Y = Y (cos ϕ Y ⋅ j sin ϕ Y )
U U ⋅ e jϕ u
1
Y = = Re (Y ) + j Im (Y ) = G + jB
Z
Y=
7.1.3
Im (Y ) B
=
Re (Y ) G
B
= −ϕ Z
G
Widerstand
YR = G
ZR = R
ZR = R
YR = G
ϕZ = 0
ϕY = 0
6.10.3.2
6.10.3.3
ZL = j ⋅ω ⋅ L
XL = ω ⋅ L
ϕZ = π 2
© Daniel Arnold
www.lanny.ch
7.2
Diodenkennlinien
7.2.1
Ideale Diodenkennlinie
7.2.2
Reale Diodenkennlinie
 UD

I = I 0 ⋅  e n ⋅U T − 1




n ≈ 1..2
Kondensator / Kapazität
1
− j
=
ZC =
ω ⋅ C j ⋅ω ⋅ C
−1
XC =
ω ⋅C
ϕZ = −π 2
Absoluter Nullpunkt
 UD

I = I 0 ⋅  e U T − 1




k ⋅T
UT =
≈ 26mV
q
6.10.3 Operatoren der Netzwerkelemente
6.10.3.1
Elektrische Elementarladung
0 K = −273.15°C
Y = G2 + B2
ϕ Y = ϕ i − ϕ u = arctan
J
K
TI89: _q
q = e = 1.60217733 ⋅ 10 −19 C
6.10.2 Admittanz
tan ϕ Y =
Dioden
Komplexe Operatoren
YC = j ⋅ ω ⋅ C
BC = ω ⋅ C
ϕY = π 2
7.2.3
Diodensperrstrom
∆ϑ
I ϑ = I 0 ⋅ 2 8..10 K
Spule / Induktivität
YL =
BL =
−j
ω ⋅L
=
1
j ⋅ω ⋅ L
−1
ω ⋅L
ϕY = − π 2
09.09.2004
© Daniel Arnold
www.lanny.ch
09.09.2004
31
8
32
Transistor
Halbleiterkonstanten unter Dioden
8.1
Bipolare Transistoren
8.1.1
Kennlinie
8.1.4.3
Leistung bei Grossignalverstärkung
2
U eff = U DC
+
uˆ 2
2
2
= U DC
+ u AC
2
2
I eff = I DC
+
PRE = I C2 _ DC ∗ R E
(
2
2
PRL = I RL
_ eff ∗ R L = i RL _ AC ∗ R L
t
PDC ≈ I C ( DC ) ∗ U CE ( DC )
PAC ≈ I Ceff ∗ U CEeff =
T1
Cin
=>
@ ≈ 20°C
RC
Kleinsignalersatzschaltbild
RE
T1
re
Eigenwiderstand des Emitters rE
8.1.5.2
Temperaturabhängigkeit von uBE
Kleinsignalgleichungen
1
iB =
⋅ u BE + S r ⋅ uCE
rBE
iC = S ⋅ u BE +
Early-Effekt
1
⋅ uCE
rBE
8.1.5.3
du BE
≈ 0.0001
duCE
Graphische Arbeitspunktbestimmung
Verhalten eines Bipolartransistors
8.1.4.1
B=
UCC
R1
T1
du BE
≈ −2 mV K
dT
8.1.4
− (R1 ⋅ U BE + R2 ⋅ (U BE − U CC ))
⋅ (R1 + R 2 ) ⋅ R E + R1 ⋅ (R 2 + R E ) + R 2 ⋅ R E
R2
du
25mV
rE = BE ≈
diC
IC
8.1.3.4
hFE
Arbeitspunkt und Kleinsignalverhalten
8.1.5.1
60mV
lg(∆I C )
8.1.3.3
8.1.5
IB =
Steilheit der Diodenkennlinie
8.1.3.2
î C ∗ û CE
2
PV = PDC − PAC
T1
Effekte
8.1.3.1
∆u BE ≈
CMiller
CMiller
Millerkapazität
Cin = C Miller ⋅ (1 + β )
8.1.3
)
PRC = I C2 _ eff ∗ RC = I C2 _ DC + iC2 _ AC ∗ R C
 U BE

I C = I S ⋅  e UT − 1




U T ≈ 26mV
8.1.2
U
oder
I
iˆ 2
2
2
= I DC
+ i AC
2
Grossignalstromverstärkung
IC
IB
8.1.4.2
IC = IS ∗ e
Grossignalgleichung
U BE
UT
 U CE
1 +
UA




U BE
IC =
IS
∗ e UT
B0
© Daniel Arnold
www.lanny.ch
09.09.2004
© Daniel Arnold
www.lanny.ch
09.09.2004
33
8.1.6
34
Kleinsignalparameter
8.1.6.1
8.1.8
Vu ' =
Graphische Ermittlung
Vu
1 + K ∗ Vu
8.1.9
vu =
Rp ' =
8.1.6.2
β = h21,e
Parallelgegenkopplung nur AC
− Phasenvers chiebung β ∗ (RC || R L )
(1 + β ) ∗ RE
+ rBE
Rp
1 − Phasenvers chiebungnu etralisati on vu
Kleinsignalstromverstärkung
y
dI
= 21,e = C
y11,e dI B
8.1.6.3
U CE = konst
Steilheit
h
∂I C
S = 21,e = y 21,e =
h11,e
∂U BE
8.1.6.4
=
A
IC,A
UT
Kleinsignaleingangswiderstand
rBE = h11,e =
1
=
y11,e
8.1.6.5
rCE =
Gegenkopplung
∂U BE
∂I B
=
A
∂U BE
∂I C
⋅
A
∂I C
∂I B
=
A
β
S
Kleinsignalausgangswiderstand
h11, e
h11, e ⋅ h 22 , e − h12 , e ⋅ h21, e
=
∂U CE
1
=
y 22 , e
∂I C
=
A
U A + U CE , A
U CE , A <<U A
I C,A
≈
UA
I C,A
rCE ≈ h 22 , e
8.1.6.6
Rückwärtssteilheit
∂I B
Sr =
∂U CE
≈0
A
8.1.6.7
Vierpol-Matrizen
 1
i B   rBE
i  = 
C  S


Sr 
u 
u   y11,e
 ⋅ BE = Ye ⋅  BE  = 
1  u CE 
u CE   y 21,e
rCE 
u BE 
 i B   h11, e
 i  = H e ⋅ u  =  h
 C 
 CE   21, e
8.1.7
y12 ,e  u BE 
⋅
y 22 , e  u CE 
h12 , e   i B 
⋅
h22 , e  u CE 
Verlustleistung
Ptot = I B ∗ U BE + I C ∗ U CE
© Daniel Arnold
www.lanny.ch
09.09.2004
© Daniel Arnold
www.lanny.ch
09.09.2004
35
36
8.2.4.3
8.2
Grundschaltungen mit bipolarem Transistor
Emitterschaltung mit Spannungsgegenkopplung
Vu =
8.2.1 Darlington-Schaltung
β = β1 ⋅ β 2
T1
Transistor als Temperaturmesselement
 e⋅U BE 
I = I S ⋅  e k⋅T − 1


k ⋅T  I
U BE (T ) ≈
⋅ ln
e
 IS
ue
≈ R1
ie
ra =
ua
≈ RC
ia
R1 +
R2
R1 + R2
S ⋅ RC
S ⋅ RC >>1+ R 2 R1
≈
−
R2
R1
1 R  R 
||  1 + 2  + 2 
R1  β 
S
I
T1

 ≈ TKSi ⋅ T

TKSi = −2.25mV / K
Bootstrapping
RIN − > ∞
10.0k
8.2.3
≈−
ia = 0
re =
T2
8.2.2
ua
ue
10.0n
10.0k
10.0u
8.2.4
1.0k
4.7k
!NPN
Emitterschaltung
8.2.4.1
Reine Emitterschaltung ohne Lastwiderstand
Vu =
8.2.4.2
© Daniel Arnold
www.lanny.ch
ua
ue
= − S ⋅ (R C || rCE
)
rCE >> R C
≈
− S ⋅ RC
ia = 0
re =
ue
= rBE
ie
ra =
ua
= R C || rCE
ia
Vu =
ua
ue
rCE >> R C
≈
RC
Emitterschaltung mit Stromgegenkopplung
≈
ia = 0
− S ⋅ RC
1 + S ⋅ RE
S ⋅ RE >> RC
≈
− RC
RE
re =
ue
≈ rBE + β ⋅ RE = rBE (1 + S ⋅ RE )
ie
ra =
ua
≈ RC
ia
09.09.2004
© Daniel Arnold
www.lanny.ch
09.09.2004
37
8.3
FET
8.3.1
FET-Typen
38
8.3.2
Arbeitspunkt
8.3.3
Kleinsignalverhalten
8.3.3.1
© Daniel Arnold
www.lanny.ch
09.09.2004
© Daniel Arnold
www.lanny.ch
Kleinsignalersatzschaltbild
09.09.2004
39
8.3.3.2
40
Kleinsignalgleichungen
9
iG = 0
i D = S ⋅ u GS +
1
⋅ u DS
rDS
8.3.3.3
∂I D
S=
∂U GS
rDS =
8.3.4
VU = S
9.1
A
A
Kleinsignalverstärkung in Sourceschaltung
9.2
Grundschaltungen ohne Speicher mit idealem OP
9.2.1
Invertierende Verstärkervarianten
Rk
9.2.1.1
rDS ∗ R D
rDS + R D
Ua =
VU ≈ S ∗ R D
V =
8.4
Differenzverstärker
vUdiff =
Invertierender Verstärker
Re
− RK
⋅U e
Re
Ue
Ua
+
U a − RK
=
Ue
Re
UCC
RC
8.4.1
Frequenzkompensation
f
fg = T
V
Kleinsignalparameter
∂U DS
∂I D
Operationsverstärker
RC
Uaus
Differenzverstärkung
T1
h fe ⋅ RC
U aus
1
= ⋅
∆U ein 2 hie + (1 + h fe )⋅ R E
T2
V =
∆U
R2
9.2.1.2
Ua
Ue
Limes Verstärker
− [R 2 (R3 + R 4 ) + R3 ⋅ R4 ]
=
R4
R3
R1 ⋅ R3
R1
Ue
Ua
+
8.4.2
Gleichtaktverstärkung
vUcom = − h fe ⋅
8.4.3
RE = 0 ; h fe >>1;hie = 0
RC
− RC
≈
hie + (1 + h fe )⋅ (R E + 2 ⋅ R EE )
2 ⋅ R EE
UEE
Summierverstärker
R
R
R
− U a = K ⋅ U e1 + K ⋅ U e 2 + K ⋅ U e 3 + ...
Re1
Re 2
Re 3
8.5.1
Verstärkung
9.2.2
Re3
Ue3
Ua
(h fe + 1)⋅ R4
u30
=
u1
hie + (1 + h fe )⋅ R4
Nichtinvertierender Verstärkervarianten
9.2.2.1
UCC
R1
Nichtinvertierender Verstärker
+
Ue
R
V = 1+ 2
R1
R3
u2
-
V0
∗ Z ie
V
V
Za ≈
∗ Z ai
V0
Ze ≈
u1
T1
R2
R4
u3
RL1
R2
mit Last
V =1
Ze → ∞
− h fe ⋅ Z C
u2
=
u1 hie + (1 + h fe )⋅ Z E
R1
Impedanzwandler
+
Ue
Ua
-
Za → 0
(h fe + 1)⋅ Z E
u
v3 = 3 =
u1 hie + (1 + h fe )⋅ Z E
Z C = R4 R L 3
Z E = R3 R L 2
© Daniel Arnold
www.lanny.ch
Ua
RL2
9.2.2.2
v2 =
Rk
Re2
Ue2
+
ohne Last
− h fe ⋅ R3
u 20
=
u1
hie + (1 + h fe )⋅ R4
8.5.1.2
Re1
Ue1
vUcom
Gegentaktstufe
8.5.1.1
9.2.1.3
vUdiff
8.5
v30 =
REE
Gleichtaktunterdrückung (Common Mode Rejection Ratio)
v CMRR = 20 ⋅ log
v20 =
RE
RE
09.09.2004
© Daniel Arnold
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09.09.2004
41
9.2.3
42
Differenzverstärker
RK
Re1
R
∗
− U e1 K
R
Re1
1 + e2
RQ
1+
U a = U e2
Ue1
Re1
9.3
Rk
Ue2
9.3.1
Re2
Ua
+
Rq
du e
dt
= RK ⋅ ω ⋅ C e = RK ⋅ 2π ⋅ f ⋅ C e
Schmitttrigger
9.2.4.1
Invertierender Schmitttrigger
U eaus = U a max ∗
R1
R1 + R 2
U eein = U a min ∗
R1
R1 + R 2
∆U e = (U a max − U a min )
9.2.4.2
U eaus
Differenzierer
u a = − RK ⋅ C e ⋅
VSinus
9.2.4
Grundschaltungen mit Speicher mit idealem OP
R1
R1 + R 2
nicht-invertierender Schmitttrigger
U
= − a max ∗ R1
R2
U eein = −
U a min
∗ R1
R2
∆U e = (U a max − U a min )
R1
R2
9.3.2
Integrierer
1
du a = −
⋅ u e ⋅ dt
Re ⋅ C K
ua = − ∫
© Daniel Arnold
www.lanny.ch
09.09.2004
1
⋅ u e ⋅ dt
Re ⋅ C K
© Daniel Arnold
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09.09.2004
43
9.3.3
44
9.3.3.6
Schaltungen und ihre Übertragungsfunktionen
Achtung, Invertierende OP-Schaltung (Phase)
9.3.3.1
G=−
P
R2C
R1C + R1 R2 s
R2
R1
-
R2
+
R1
R
G=− 2
R1
9.3.3.2
C
Tiefpass
+
I
C
9.3.3.7
R
1
G=−
RCs
C + R2 s
G=−
R1 s
+
D und P
R2
C
R1
+
9.3.3.3
D
R
G = − RCs
C
+
9.3.3.4
P und I
C
R

G = − 2 + R2Cs 
 R1

R2
R1
+
9.3.3.5
R Cs
G=− 2
1 + R1Cs
© Daniel Arnold
www.lanny.ch
Hochpass
R2
R1
C
+
09.09.2004
© Daniel Arnold
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09.09.2004
45
9.4
Realer OP
9.4.1
GBW (Gain-Bandwith-Product)
V
[dB]
46
20 dB / dek
GBW = v ⋅ f = f T
fT
10
HF-Technik
10.1
KO
10.1.1
log f
9.4.2
Rise Time, Bandbreite
Slew Rate
s r = uˆ ⋅ ω
10.1.1.1
9.4.3
10.1.1.2
Biasstrom
I +I
Ib = + −
2
Biasstromkompensation
Ib =
10.2
Re
+
Ua
Wellenwiderstand ≥10kHz
10.2.1
L
C
Z =
I+ − I−
2
VReal =
HF-Leitungen
Ue
Offsetstrom
9.4.5
1
s
[tr ] = s
[ B] =
Rk
Rb = Re Rk
9.4.4
6GHz
B ⋅ t r = 0.47
Rb
9.4.3.1
<= 1GHz
B ⋅ t r = 0.35
10.2.2
Reale Verstärkung bei Rückkopplung
c=
1 + V0 ⋅ VR
1 V0 ⋅ VR
=
+
V0
V0
V0
k =
V0 → ∞
Ausbreitungsgeschwindigkeit
1
L∗C
c
=
c0
1
εr
⇒ VReal = VR
© Daniel Arnold
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09.09.2004
© Daniel Arnold
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09.09.2004
47
48
11 Oszillatoren
11.1
12 Steuer- und Regelungstechnik
Schwingbedingung im Frequenzbereich
U a (s )
A
=
U e (s ) 1 − A ⋅ B(s )
12.1
1
In1
1
G0
1 − A ⋅ B (s ) = 0
Regeltechnische Glieder
1
Out1
12.1.1
P
B(s)
T
G1
A =1
∠(B(s )) = 0
V
[dB]
ω 0=1/T
T
11.2
log f
Quarze
Cip
11.2.1
fS =
Serieresonanz
1
2π LiS ⋅ CiS
11.2.2
Parallelresonanz
1
CiS + CiP
fP =
= fS
CiP
2π LiS ⋅ C x
Cx =
Cis
Ris
Lis
ϕ
IM(Z )
0.1ω 0
fS
log f
fP
CiS ⋅ CiP
CiS + CiP
-45°
-90°
11.2.3 Güte
2π ⋅ f S ⋅ LS
Q=
10ω 0
12.1.2 P-Regler
RS
V
[dB]
Kp
log f
ϕ
KP =
© Daniel Arnold
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09.09.2004
Ausgang
Eingang
© Daniel Arnold
www.lanny.ch
log f
09.09.2004
49
50
12.1.3 I-Regler
12.1.5
V
[dB]
KI
PI-Regler
Kp
Tn
V
[dB]
KI
TI
log f
log f
Tn
e
ϕ
KI =
ϕ
1
1
=
TI C K ∗ R1
log f
KP =
12.1.6
D-Regler
V
[dB]
KD
RK ∗ RQ
log f
R1 ∗ R2
TN = RK ∗ C K
-90°
12.1.4
PD-Regler
Kp
Tv
V
[dB]
KD
log f
ϕ
KD =
1/Tn
1
1
=
TD C1 ∗ RK
1/Tv
log f
Tv
+90°
log f
KP =
ϕ
RK ∗ RQ
log f
R1 ∗ R2
TV = C1 ∗ R1
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09.09.2004
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09.09.2004
51
12.1.7
52
PID-Regler
Kp
Tn,Tv
12.2
V
[dB]
Zustandsraumdarstellung
12.2.1 Regelungsnormalform
1/Tn
1/Tv
12.2.1.1
log f
G(s) =
Tn
KP =
Tv
Z ( s) b0 + b1s + ... + bn s n
=
N ( s ) a0 + a1s + ... + an s n
an y ( n ) + ... + a1 y& + a0 y = b0u + b1u& + ... + bnu ( n )
Einführen der Zustandsvariablen:
x1 = y
x 2 = y&
ϕ
...
RK ∗ RQ
x n = y ( n−1)
Daraus folgen die Differentialgleichungen:
x&1 = x2
x&2 = x3
log f
R1 ∗ R2
TV = C1 ∗ R1
TN = RK ∗ C K
...
x&n −1 = xn
Durch Einsetzen in die ursprüngliche Differentialgleichung:
an x&n + an −1 xn + ... + a1 x2 + a0 x1 = b0u + b1u& + ... + bnu ( n )
12.1.8 Phasenreserven
Phasenreserve
90 °
65 °
60 °
45 °
25 °
Zusandsgleichung
Überschwingen
0%
5%
10 %
20 %
50 %
b
b
b
an −1
a
a
b
xn − ... − 1 x2 − 0 x1 + 0 u + 1 u& + ... + n −1 u ( n −1) + n u ( n )
an
an
an
an
an
an
an
Zusammenhang von Zustandsgrössen und Ausgang:
y = x1
x&n = −
12.2.1.2
G(s) =
Spezialfall: Eingang wirkt nicht durch seine Ableitungen auf den Ausgang
Z ( s)
b0
=
N ( s ) a0 + a1s + ... + an s n
an y ( n ) + ... + a1 y& + a0 y = b0u
Einführen der Zustandsvariablen:
x1 = y
x 2 = y&
...
x n = y ( n−1)
Daraus folgen die Differentialgleichungen:
x&1 = x2
x&2 = x3
...
x&n −1 = xn
Durch Einsetzen in die ursprüngliche Differentialgleichung:
© Daniel Arnold
www.lanny.ch
09.09.2004
© Daniel Arnold
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09.09.2004
53
54
an x&n + an −1 xn + ... + a1 x2 + a0 x1 = b0u
13
Signale
a
a
a
b
x&n = − n −1 xn − ... − 1 x2 − 0 x1 + 0 u
an
an
an
an
Zusammenhang von Zustandsgrössen und Ausgang:
y = x1
13.1
Qualität
13.1.1
Klirrfaktor
U 12 + U 22 + ... + U n2
U + U 12 + U 22 + ... + U n2
K = 100 [%]
13.2
2
0
Zeitmittelwerte
13.2.1 Gleichwert (arithmetischer Mittelwert)
x=
1
T
t1 +T
∫ x ⋅ dt
t1
13.2.2 Effektivwert (RMS)
X =
t1 +T
1
T
13.3
∫ x(t )
2
⋅ dt
t1
Komplexe Zahlen
13.3.1 Darstellungen
13.3.1.1
Normalform
z = a + bj
13.3.1.2
Polarform
z = r ∗ (cos ϕ + j sin ϕ ) = r ∗ cis ϕ
13.3.1.3
Exponentialform
z = r ∗ e jϕ
13.3.2 Umrechnungen
z = a2 + b2 = z
a = r cos ϕ
b = r sin ϕ
13.4
Korrelation
13.4.1 Autokorrelationsfunktion
13.4.1.1
Periodische Signale
T
ρ 11 (τ ) = x1 (t ) ⋅ x1 (t + τ ) =
1 2
x1 (t ) ⋅ x1 (t + τ ) dt
T −T∫
2
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09.09.2004
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09.09.2004
55
13.4.1.2
56
Stochastische Signale
13.6
T
1 2
x1 (t ) ⋅ x1 (t + τ ) dt
T →∞ T ∫
−T
ρ 11 (τ ) = x1 (t ) ⋅ x1 (t + τ ) = lim
Fouriertransformation
X (ω ) =
2
∞
∫ x(t ) ⋅ e
− jωt
dt
−∞
13.4.1.3
Signale mit begrenzter Signalenergie (zeitbegrenzt)
x(t ) =
∞
ρ 11 (τ ) = x1 (t ) ⋅ x1 (t + τ ) = ∫ x1 (t ) ⋅ x1 (t + τ ) dt
1
2 ⋅π
∞
∫ X (ω ) ⋅ e
jωt
dω
−∞
−∞
13.7
Harmonische Signale in LTI-Systemen
13.4.2 Kreuzkorrelationsfunktion
T
ρ 12 (τ ) = x1 (t ) ⋅ x2 (t + τ ) =
13.7.1 Widerstand
1 2
x1 (t ) ⋅ x2 (t + τ ) dt
T −T∫
2
13.5
u (t ) = R ⋅ i (t )
Faltung
y (t ) = x1 (t ) ∗ x2 (t ) =
R
U = Z ⋅I
Z = R + j0
∞
∫ x (τ )⋅ x (t − τ )dτ
1
2
−∞
13.7.2
Spule
u (t ) = L ⋅
di(t )
dt
U = Z ⋅I
Z = 0 + jωL
i (t ) = C ⋅
du (t )
dt
U = Z ⋅I
L
13.7.3
Kondensator
C
Z =0+
1
j
=0−
jωC
ωC
13.7.4 Spannungsquelle
u (t ) = Uˆ ⋅ cos(ωt + ϕ u )
© Daniel Arnold
www.lanny.ch
09.09.2004
© Daniel Arnold
www.lanny.ch
U=
1 ˆ jϕ
⋅U ⋅ e
2
09.09.2004
57
14
Nachrichtentechnik
14.1
Leitungsgebundene Übertragung
58
14.1.3 Reflexionen und stehende Wellen
14.1.3.1
umin = uˆv − uˆr
14.1.1 Leitungsmodell
14.1.1.1
14.1.3.2
Beläge
r=
L' = ∆L ∆l
R' = ∆R ∆l
C ' = ∆C ∆l
G ' = ∆G ∆l
14.1.1.2
ur
=
uv Z a + Z i
Rückflussdämpfung
u 
a = 20 ⋅ log v 
 ur 
Ausbreitungskoeffizient
gw
∆l
γ = ( jωC '+G ') ⋅ ( jωL'+ R')
14.1.3.4
Anpassungsfaktor / Welligkeitsfaktor
1 u
uˆ − uˆ
m = = min = v r
s umax uˆv + uˆr
uv = uv 0 ⋅ e −γx
ω
β
14.1.4 Glasfasern
ur = uv 0 ⋅ e −γl ⋅ r ⋅ e −γ (l − x )
14.1.1.2.1
14.1.4.1
Dämpfung
14.1.4.2
14.1.1.2.2
( )
14.1.4.3
m
14.1.1.4
( )
( )
Pein = Pges ⋅ sin 2 a g = Pges ⋅ ( NA)
14.1.4.4
Wellendämpfungsmass
gw = l ⋅γ = l ⋅
2
Übertragungsdämpfung
Paus = Pein ⋅10 − a ⋅l
( jωC '+G ') ⋅ ( jωL'+ R')
P 
a ⋅ l = −10 ⋅ log ein 
 Paus 
14.1.2 Anpassung
14.1.2.1
Eingekoppelte Leistung
Pein
= sin 2 a g = ( NA)2
Pges
Wellenwiderstand
jωL'+ R '
jωC '+G '
Zw =
Numerische Apertur (Lichteinsammlungszahl)
NA = n0 ⋅ sin a g = n12 − n22 ≈ n1 ⋅ 2∆
Phasenkoeffizient
[β ] = rad
14.1.1.3
Lichtausbreitung in Glasfasern
n −n
∆= 1 2
n1
a = 8.686 ⋅ α
Za =
Reflexionsfaktor
Z a − Zi
14.1.3.3
γ = α + jβ =
υ=
Überlagerung der vor- und rücklaufenden Signale
umax = uˆv + uˆr
Maximale Wirkleistung
Z i*
Ra ± jX a = Ri m jX i
14.1.2.2
Wellenanpassung
Z a = Zi
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www.lanny.ch
09.09.2004
© Daniel Arnold
www.lanny.ch
09.09.2004
59
14.2
60
14.2.3 Effektiv abgestrahlte Leistung
Drahtlose Übertragung
14.2.3.1
14.2.1 Komponenten einer drahtlosen Übertragung
Kabelverluste
P
Ltc = s
Pt
α = 10 ⋅ log(Ltc )
14.2.3.2
Reflexionsfaktor
u
Z − Z0
rt = r = ant
uv Z ant + Z 0
14.2.3.3
Pant = Pt ⋅ 1 − rt
14.2.2 Übertragungsverluste im freien Raum
14.2.2.1
Sr =
14.2.3.4
isotroper Strahler (Referenz für Gewinne)
Prad
P
= rad
AKugel 4πr 2
(
W 
 m 2 
Empfangene Leistung bei einem isotropen Strahler
Prad
⋅ Ae
4πr 2
Pr = S r ⋅ Ae =
Pr =
14.2.2.5
Strahlungswirkungsgrad
Richtfaktor
S ave
S iso
D '[ dB ] = 10 ⋅ log(D )
14.2.3.6
Gewinn
Gt = η ⋅ D
G '[ dB ] = 10 ⋅ log(G )
Empfangene Leistung allgemein
Prad ⋅ Gr
r

 4π 
 λ
)
D=
[W ]
14.2.2.3
Gewinn
4πAe
Gr = 2
λ
14.2.2.4
2
P
η = rad
Pant
14.2.3.5
14.2.2.2
Verfügbare Leistung an der Antenne
2
=
Prad ⋅ Gr
L fs
14.2.3.7
[W ]
ERP Effective Radiated Power
ERPt = Pant ⋅ Gt
free space path loss
2
r

L fs =  4π 
λ


L fs[ dB ] = 20 ⋅ log( f[ MHz ] ) + 20 ⋅ log(r[ km ] ) + 32.4
14.2.2.6
r≥
Fernfeldbedingung
2⋅d2
λ
14.2.4 Empfangene Leistung
Pr = S ave ⋅ Ae
Pr =
E2
⋅ Ae
Z0
Z 0 ≈ 377Ω
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09.09.2004
© Daniel Arnold
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09.09.2004
61
14.2.4.1
rr =
62
Reflexionsfaktor
Z 0 − Z ant
Z 0 + Z ant
14.3
Codierung
14.3.1 Informationsgehalt
14.2.4.2
(
Leistung am Eingang des Antennenzubringers
Pline = Pr ⋅ 1 − rt
14.2.4.3
2
)
14.3.1.1
Kabelverluste
14.3.1.2
P
Lrc = line
Prec
2 =k =n
I
Leistung am Empfängereingang
(1 − r )⋅ G ⋅ G ⋅ (1 − r )
2
Prec = Ps ⋅
t
Ltc
2
t
r
r
L fs
Lrc
14.3.1.3
14.2.5 Antennengrundlagen
4π
D=
D≈
Informationsgehalt
m
I = m ⋅ ld (n )
I1 = ld (n )
α rc = 10 ⋅ log(Lrc )
14.2.4.4
Information
k = nm
Information des gesamten Ereignisses
n −1
n −1
i =0
i =0
I = −∑ I xi = −m∑ p( xi ) ⋅ ld ( p( xi ))
ΩA
14.3.1.4
4π
Θ − 3dB ⋅ Φ − 3dB
Mittlere Informationsgehalt pro Element (Entropie H)
n −1
I1 = −∑ p ( xi ) ⋅ ld ( p( xi ))
i =0
14.3.2 Kanalkapazität
14.3.2.1
Informationsfluss
I
R=
T
14.3.2.1.1
Rmax = ⋅ ld (n )
14.2.6 Antennenimpedanz
14.2.6.1
τ
Strahlungseffizienz
14.3.2.2
R rad
η=
RΩ + R rad
14.2.6.2
fτ =
Reflexionsgrad
1
14.3.2.3
1
τ0 =
2 BK
14.2.6.3
14.3.2.4
Welligkeitsfaktor
1+ r
s=
1− r
τ0 =
09.09.2004
Symbolrate
τ
Z − Z0
r = ant
Z ant + Z 0
© Daniel Arnold
www.lanny.ch
Maximale Informatonsfluss
1
Maximale Symboldauer
Maximale Auflösung
Emax
=
ER
© Daniel Arnold
www.lanny.ch
S+N
N
09.09.2004
63
14.3.2.5
64
Kanalkapazität
C>R
S

C = BK ⋅ 1 + 
N

15.1
Die Übertragungsfunktion
Mittlere Codewortlänge
15.1.1 Darstellungsarten
m
s = ∑ p( xi ) ⋅ s( xi )
i =1
14.3.3.2
Analoge Filter
Y (s )
G (s ) =
X (s )
s = σ + jω
14.3.3 Codewortlänge
14.3.3.1
15
Redundanz
R = s − I1
G (s ) =
Z (s ) bm s m + bm−1s m−1 + ... + b1s + b0
=
N (s ) an s n + an−1s n−1 + ... + a1s + a0
G (s ) =
Ak
A1
A2
+
+ ... +
s − p1 s − p2
s − pk
G (s ) =
G0 =
(s − z1 )(s − z 2 )...(s − z m )
Z (s )
= G0 ⋅
(s − p1 )(s − p2 )...(s − pm )
N (s )
bm
an
15.1.2 Berechnungen mit TI-89
15.1.2.1
Nullstellen
cZeros(G(s),s) oder cSolve(G(s)=0,s) oder cSolve(Z(s)=0,s)
15.1.2.2
Pole
cFactor(G(s)) oder cSolve(N(s)=0,s)
15.1.2.3
Residuen
Ak = (s − pk ) ⋅ G (s ) s = p
k
expand(G(s))
15.2
Standard Übertragungsfunktionen 2. Ordnung
15.2.1.1
G (s ) =
b2 ⋅ s + b1 ⋅ s + b0
a2 ⋅ s 2 + a1 ⋅ s + a0
15.2.1.2
G (s ) =
© Daniel Arnold
www.lanny.ch
09.09.2004
Allgemeine Übertragungsfunktion eines Filters 2. Ordnung
2
Übertragungsfunktion eines Filters 2. Ordnung mit konjugiert komplexem Polpaar
1
1
1
=
=
a2 ⋅ s 2 + a1 ⋅ s + a0 s 2 − 2 ⋅ Re[ pk ] ⋅ s + Re[ pk ]2 + Im[ pk ]2 s 2 + ω 0 ⋅ s + ω 2
0
Q
© Daniel Arnold
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09.09.2004
65
66
15.2.2 Tiefpassübertragungsfunktion
G ( jω ) = G0TP ⋅ GTP ( jω ) = G0TP ⋅
Q=
G ( jω 0 )
G0TP
15.2.2.1
15.2.5 Bandstoppübertragungsfunktion
1
2
2
ω 
jω 1
1 −   +
⋅
ω0 Q
 ω0 
G ( jω ) = G0 BS ⋅ GBS ( jω ) = G0 BS
ω 
1 −  
 ω0 
⋅
2
ω 
jω 1
1 −   +
⋅
ω0 Q
 ω0 
GBS = GTP + GHP = 1 − GBP
Amplitudenmaximum
ω A _ max
1
= 1−
2 ⋅ Q2
ω0
G ( jω A _ max ) =
Q
1−
1
4 ⋅ Q2
15.2.3 Hochpassübertragungsfunktion
15.2.6 Allpassübertragungsfunktion
2
2
ω 
−  
 ω0 
G ( jω ) = G0 HP ⋅ GHP ( jω ) = G0 HP ⋅
2
ω 
jω 1
1 −   +
⋅
ω0 Q
 ω0 
GHP ( jω ) = GTP ( jω )  ω  =1  ω 
ω 
 0
ω 
1 −   −
ω
G ( jω ) = G0 AP ⋅ G AP ( jω ) = G0 AP ⋅  0 2
ω 
1 −   +
 ω0 
jω 1
⋅
ω0 Q
jω 1
⋅
ω0 Q
ω 
 0
15.2.4 Bandpassübertragungsfunktion
jω 1
⋅
ω0 Q
G ( jω ) = G0 BP ⋅ GBP ( jω ) = G0 BP ⋅
2
ω 
jω 1
1 −   +
⋅
ω
ω
0 Q
 0
ω0
Q=
ωH − ωL
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09.09.2004
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09.09.2004
67
15.3
68
15.3.2.5
Filterberechnungen höherer Ordnung
Phasenoptimiert
Normierte Grenzfrequenz: 1[rad/s]
G ( jω ) =
15.3.1 Vorgehen beim Filterentwurf höherer Ordnung
1.
2.
Wahl der Approximation
Festlegen der Spezifikationswerte
15.3.3.1
-
Butterworth
Möglichst flach im Durchlassbereich
1
G ( jω ) =
1 + ε 2 ⋅ Ω2n
Ω=
ω
ωG
(
)
-
(
)
-
 10
−1

log 0.1⋅ A
max
10
− 1 

n≥
2 log(ω S ωG )
n: Ordnung
ε: Ripple im Durchlassbereiskonstante
Frequenzgang berechnen
H = freqs(Num, Den, w)
Tschebyscheff
15.3.3.6
Möglichst steil im Übergangsbereich
1
G ( jω ) =
1 + ε 2 ⋅ Cn2 (Ω )
(
)
-
(
-
)
Amplitudengang darstellen in [dB]
semilogx(w, 20*log10(abs(H))), grid
15.3.3.7
cos n ⋅ cos −1 (Ω ) für 0 ≤ Ω ≤ 1
Cn (Ω ) = 
−1
 cosh n ⋅ cosh (Ω ) für 1 < Ω
15.3.2.3
Vektor mit logarithmische verteilten Werten
w = logspace(-1, 2, 1000) -> 1000Werte, 10^-1 bis 10^2
15.3.3.5
-
Bestimmung der Zähler- und Nennerpolynome normiert auf 1 rad/s
[Num, Den] = zp2tf(Z, P, K)
15.3.3.4
-
Nullstellen, Pole, Verstärkungsfaktor
[Z,P,N] = buttap(n)
cheb1ap
cheb2ap
ellipap
15.3.3.3
0.1⋅ Amin
Ordnung
n = buttord(Wp, Ws, Rp, Rs, ’s’)
cheb1ord
cheb2ord
ellipord
15.3.3.2
Amax = A(ωG ) = 20 ⋅ log 1 + ε 2 = 10 ⋅ log 1 + ε 2
15.3.2.2
1
1 + ε 2 ⋅ sn 2 (u , k1 )
15.3.3 Matlab Funktonen
15.3.2 Approximationen
ω
Ω=
ωG
15.3.2.1
Bessel
Phasengang darstellen
semilogx(w, 180/Pi*angle(H))
Inverse Tschebyscheff
Möglichst keine Welligkeit im Durchlassbereich und eine Welligkeit im Sperrbereich
1
2
G ( jω ) = 1 −
1
2
1 + ε ⋅ Cn2  
Ω
15.3.2.4
Cauer (elliptische Approximation)
Welligkeit um Durchlass- und Sperrbereich, aber grösste Steilheit im Übergangsbereich
1
G ( jω ) =
1 + ε 2 ⋅ sn 2 (u , k1 )
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09.09.2004
69
70
15.4.1.5
15.4
Aktive Filter
 jω 


ω 0 

G ( jω ) = G 0
 jω 

1 + 
 ω0 
R
G0 = 1 + 2
R1
15.4.1 Aktive RC-Filter 1. Ordnung
15.4.1.1
Differentiator
R
G (s ) = − RCs
jω
G ( jω ) = −
C
-
ω0
15.4.1.2
+
G ( jω ) = −
C
R
-
 jω 
 ω 
 0
+
OP
Tiefpass 1. Ordnung mit invertierendem Verstärker
R2
1
⋅
R1 R2 Cs + 1
G ( jω ) = G 0
R2
+
OP
1
ω0 =
R2 C
G ( jω ) = G 0
1
 jω 

1 + 
 ω0 
R
G0 = 1 + 2
R1
+
OP
Hochpass 1. Ordnung mit nichtinvertierendem Verstärker
 jω 


ω 0 

G ( jω ) = G 0
 jω 

1 + 
 ω0 
R
G0 = − 1
R
1
ω0 =
RC
C
OP
+
-
R2
R
R1
R
OP
+
-
R2
C
R1
R1
R1
-
R
+
OP
C
1
RC
15.4.1.8
Allpass 1. Ordnung 2. Variante
 jω 

1 − 
 ω0 
G ( jω ) = G 0
 jω 

1 + 
 ω0 
G0 = −1
ω0 =
09.09.2004
Allpass 1. Ordnung 1. Variante
 jω 

1 − 
 ω0 
G ( jω ) = G 0
 jω 

1 + 
 ω0 
G0 = 1
ω0 =
1
ω0 =
RC
© Daniel Arnold
www.lanny.ch
R
1
RC
15.4.1.7
R1
R
G0 = − 2
R1
Tiefpass 1. Ordnung mit nichtinvertierendem Verstärker
R1
C
C
1
 jω 

1 + 
 ω0 
15.4.1.4
ω0 =
15.4.1.6
1
15.4.1.3
G (s ) = −
OP
Integrator
1
G (s ) = −
RCs
Hochpass 1. Ordnung mit invertierendem Verstärker
R1
R1
-
C
+
OP
R
1
RC
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71
72
15.4.2 Aktive Filter 2. Ordnung
15.4.2.1.4
15.4.2.1
K =1+
KRC Filter (Sallen-Key Filter)
K & Q sind von Bauteilverhältnissen abhängig
Geeignet für Filter mit Q < 10
15.4.2.1.1
K = 1+
G (s ) =
ω0 =
C1
R1
R2
OP
+
UA
K
=
U E R1C1 R2 C 2 s 2 + [(1 − K )R1C1 + R1C 2 + R2 C 2 ]s + 1
RA
1
1 − R1C1 R2 C 2ω + [(1 − K )R1C1 + R1C 2 + R2 C 2 ] jω
2
15.4.2.1.2
ω0 =
C1
R1
R2
OP
+
1
ω0 =
RC
1
Q=
3− K
R
K = 1+ B
RA
RA
R1C2
+
R2C1
R1C1
R2C2
Bandpass KRC Filter
RB
RA
R3
R1
K
C2
+
R  C R
1 + (1 − K ) 1 + 1 + 1  1
R3  C2  R2
OP
-
RB
C1
R2
R1
R3
RA
R1C1 R2C2

R 
Q =  1 + 1 
R
3 



 1 + (1 − K ) R1  R2C2 + R1C2 + R1C1
 RC

R
R2C1
R2C2
3 
1 1





-
C2
RB
15.4.2.1.6
RA
Q>
Bandpass KRC Filter mit Q >
2
3
R1 = R2 = R3 = R
R3
R1
C2
+
C1 = C 2 = C
K >1
K = 1+
15.4.2.1.3
Tiefpass KRC Filter mit Verstärkung 1
G0TP = 1
ω0 =
RB
R2
R2C2
+
R1C1
15.4.2.1.5
1+
Tiefpass KRC Filter mit gleichen Komponenten
G0TP = K
-
(1 − K )
G0 BP =
R1C1
R1C 2
R2 C 2
+
+
R2 C 2
R2 C1
R1C1
OP
+
1
1
(1 − K )
C2
1
R1C1R2C2
K = 1+
1
R1C1 R2C 2
Q=
R1
C1
RB
C2
G0TP = K
ω0 =
Q=
-
G ( jω ) = K
RB
RA
G0 HP = K
Tiefpass KRC Filter
RB
RA
Hochpass KRC Filter
C1
1
R1
m⋅n
m +1
R2 = R
Q=
K
4− K
2
ω0 =
RC
2
Q=
4−K
RB
C1
R2
RA
G0 BP =
R2
+
m ⋅ n ⋅ RC
RB
RA
OP
-
OP
-
C2
C2 = C
R1 = m ⋅ R
C1 = n ⋅ C
m =1
− > n = 4Q 2
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09.09.2004
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09.09.2004
73
15.4.2.1.7
K =1+
74
Bandstopp KRC Filter
15.4.2.2.3
RB
RA
ω0 =
2C
G0 BS = K
R
R
C
C
+
1
R3
R1 R2 C1C 2
OP
1
RC
1
Q=
4 − 2K
RB
R/2
Q=
RA
R 2 R3
+
R12
G0TP = −
Multiple Feedback Filter
R3
+
R2
C2
R2
R1
C1
C2
-
ω0 =
15.4.2.2
Tiefpass MFB Filter
-
C1
+
OP
R2
R3
R3
R1
Geeignet für Q etwas grösser als 10
15.4.2.2.4
15.4.2.2.1
Bandpass MFB Filter
U
− jωR2 C 2
G ( jω ) = A =
U E 1 − ω 2 R1 R2C1C 2 + jωR1 (C1 + C 2 )
ω0 =
Q=
1
R1 R2 C1C 2
G ( jω ) = G0 BS ⋅ GBS ( jω ) = G0 BS ⋅ (1 − GBP ( jω ))
-
C2
OP
+
+
OP
OP
15.4.3 Universalfilter Filter
Mehrere OP’s für 2. Ordnung
Mehrere Frequenzgänge pro Schaltung
Unempfindlicher auf Bauteiltoleranzen
Einfacher für Einstellungen
Q>100 möglich
15.4.3.1
Zustandvariablen Filter
15.4.3.1.1
Bandpass MFB Filter mit C1 = C2 = C
G0 HP
C
1
R1
C
R2
G0 BP
-
R2
R1
+
-
R2
R1
=
C
1+ 1
C2
Q = 0.5
R3
-
R1
C1
C2
+
C2
C1
R1R2C 2
R2
R1B
R2
C1 = C2 = C
ω0 =
R5
C2
R1A
R2
R1
15.4.2.2.2
C1
C1
−
G0 BP
R4
Bandstopp MFB Filter
+
OP
G0 BP = −2Q 2
Zustandvariablen Filter (state-variable)
R
=− 5
R3
R
1+ 2
R1
=
R3 R3
1+
+
R4 R5
G0TP = −
R4
R3
ω0 =
R5 R4
R4
R3
R5
C1
R6
-
-
-
R1
+
OP
U_TP
U_BP
U_HP
+
C2
R7
OP
+
OP
R2
R6C1R7C2
 R2  R5 R6C1
1 + 
R1  R4 R7C2
Q=
R
R
1+ 5 + 5
R3 R4
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09.09.2004
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09.09.2004
75
15.4.3.1.2
76
Zustandvariablen Filter (state-variable) mit zum Teil gleichen Komponenten
R3 = R4 = R5
15.4.3.2.2
R4 = R5 = R
R5
R6 = R7 = R
R5
R5
C1 = C2 = C
C
R
-
G0 HP = −1
-
+
U_TP
U_BP
+
OP
R1
+
OP
OP
G0TP
+
C2
R4
U_BP
OP
R3
+
U_TP
OP
R3
+
U_TP
OP
Nichtinvertierendes SV Filter (vereinfacht)
R3
G0 BP = −1
R3
G0TP = 1 Q
C
R
-
C
R
-
-
+
+
OP
R2
U_TP
U_BP
U_HP
1
ω0 =
RC
R
Q = 1+ 2
2 R1
+
OP
OP
R2
R1
Biquad Filter
15.4.3.2.1
Biquad Filter
R
=− 2
R1
G0TP =
R5
R1
ω0 =
1
R4 R5C1C2
Q=
R
=
R1
C1
R1
ω0 =
G0 HP = 1 Q
G0 BP
R2
R1
R2
1
RC
R
Q= 2
R
1
RC
1 R 
Q = 1 + 2 
3
R1 
ω0 =
15.4.3.2
G0 BP = −
R2
G0TP = −1
15.4.3.1.3
R5
C1 = C2 = C
C
R
U_HP
G0 BP = Q
Biquad Filter (vereinfacht)
R5
R2
C1
R1
+
C2
R4
U_BP
OP
R3
+
U_TP
OP
R3
+
U_TP
OP
R2 C1
R4 R5C2
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09.09.2004
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09.09.2004
77
15.5
78
15.5.3 Koeffiziententabelle 2
Kaskadenentwurf
0.1 und 1dB Welligkeit im Durchlassbereich
15.5.1 Ent-Normierung
ω 0 _ aktuell = ω 0 _ Tabelle ⋅ ω G
f 0 _ aktuell = f 0 _ Tabelle ⋅ f G
Qaktuell = QTabelle
15.5.2 Koeffiziententabelle 1
15.5.4 Ent-Normierung auf Hochpass
ωG
ω 0 _ aktuell =
ω 0 _ Tabelle
f 0 _ aktuell =
fG
f 0 _ Tabelle
Qaktuell = QTabelle
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09.09.2004
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09.09.2004
79
16
Signalübertragung
16.1
Signalverzerrugen
80
16.3
Rauschen
16.3.1 Rauschquellen
16.3.1.1
16.1.1 Klirrfaktor
k=
u eff Obwerwellen
u eff Gesamtsignal
=
Widerstandsrauschen
U R _ Reff = 4kTRB
A22 + A32 + ... + An2
A12 + A22 + A32 + ... + An2
16.3.1.2
Schrotrauschen (gilt nicht in Metallen)
I R _ eff = 2qI DC B
16.1.2 Verzerrungsfreie Übertragung
y (t ) = K ⋅ x(t − td )
16.3.1.3
Y (s ) = K ⋅ e
∠(H ( jω )) = −ω ⋅ td ± m ⋅ π
Kohlemasse:
0.1...3 µV/V
n-schicht:
0.05..0.3 µV/V
Metallschicht:
0.02...0.2 µV/V
n-film:
0.02...0.2 µV/V
Drahtwiderstände:
0.01...0.2 µV/V
Spektrum ca. 1/f (-10dB/Dek)
16.1.3 Amplitudenverzerrungen
16.3.2 SNR
− jωt d
H ( jω ) = K ⋅ e
⋅ X (x )
− jωt d
H ( jω ) = K
H ( jω ) ≠ K
1/f Rauschen (Flickerrauschen)
U 2
SNR = 10 ⋅ log Signal
 U2
R

16.3.2.1
U R = U + U R2 2
Bandpassähnlicher Kanal mit Laufzeitverzerrungen
(
j −ω t + ϕ
)
(
g
0
H ( jω ) = A ⋅ e
= A⋅e
∠(H ( jω )) = −ω ⋅ t g + ϕ 0
jϕ 0
)⋅ e
− jωt g
ϕ0
ω
t d (ω ) = t g −
16.2
Addition von Rauschen
2
R1
16.1.4 Phasen- oder Laufzeitverzerrungen
∠(H ( jω )) ≠ −ω ⋅ t d ± m ⋅ π
∠(H ( jω ))
t d (ω ) = −
ω
16.1.4.1




Equalization
H ( jω ) = K ⋅ e − jωtd = H Kanal ( jω ) ⋅ H eq ( jω )
H eq ( jω ) =
K ⋅ e − jω t d
H Kanal ( jω )
© Daniel Arnold
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09.09.2004
81
17
Modulation und Demodulation
17.1
Analoge Verfahren
82
17.1.5 Pulsmodulation
TS = t Puls + t Pause
17.1.5.1
Pulsamlitudenmodulation PAM
17.1.1 Trägerschwingung
x(t ) = A ⋅ cos(ω 0t + ϕ )
t Puls = konst
17.1.1.1
A, ϕ
B=
t Pause = konst
Funktionsparameter
1
2τ
17.1.5.1.1
17.1.2 Amplitudenmodulation AM
Flat-Top (Bipolares)
xPAM (t ) = ∑ x(kTS ) ⋅ p (t − kTS )
A(t ) = A ⋅ [1 + m ⋅ s (t )]
k
ϕ = konst
17.1.5.1.2
17.1.2.1
m=
Modulationsgrad
Unipolares Flat-Top
xPAM (t ) = ∑ A0 [1 + m ⋅ x(kTS )]⋅ p (t − kTS )
uˆ i
uˆT
k
1 + m ⋅ x(t ) > 0
17.1.2.2
DSB
dsb(t ) = (K + sin (ω S t )) ⋅ sin (ωT t )
17.1.5.2
1
1
= cos((ω T − ω S )t ) − cos((ω T + ω S )t ) + K sin (ω T t )
2
2
2 ⋅ uˆ S 2 ⋅ u S
m=
=
uˆT
uT
t Pause = konst
17.1.2.3
t Puls (k ) = τ 0 [1 + m ⋅ x(kTS )]
x PDM (t ) = konst
B=
1
2t r
DSBSC
dsbsc(t ) = sin (ω S t ) ⋅ sin (ω T t )
dsbsc(t ) =
Pulsdauermodulation PDM
17.1.5.3
1
1
cos((ω T − ω S )t ) − cos((ωT + ω S )t )
2
2
17.1.2.4
Pulspositionsmodulation (Pulsphasenmodulation) PPM
t Puls = konst
t Beginn (k ) = kTS + t 0 x(kTS )
u PPM (t ) = konst
SSB
B=
1
ssb(t ) = − cos((ω T + ω S )t )
2
1
2t r
17.1.3 Phasenmodulation PM
A = konst
ϕ (t ) = m ⋅ s (t )
17.1.4 Frequenzmodulation FM
A = konst
dϕ (t )
= ∆ω 0 ⋅ s (t )
dt
ϕ (t ) = ∫ ∆ω 0 ⋅ s (t ) dt
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09.09.2004
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09.09.2004
83
17.2
84
17.2.2 Digitale Übertragung analoger Signale
Digitale Verfahren
17.2.2.1
17.2.1 Q-Funktion
B=
Benötigte Bandbreite für einen Puls
1
2τ
t
17.2.2.2
Begriffe
17.2.2.2.1
r=
1/B
Symbolrate [baud]
1
Ts
17.2.2.2.2
Bitrate [bps]
rb = r ⋅ ld(M ) =
1 1
log(M )
= ⋅ ld(M ) =
Tb Ts
Ts ⋅ log(2 )
M = 2n
17.2.2.2.3
Unipolar
ak ∈ {(M − 1)A,....,2 A, A,0}
17.2.2.2.4
Bipolar
5 A 3A A
 (M − 1)A
ak ∈  ±
,....,±
,± ,± 
2
2
2
2

17.2.2.3
17.2.2.3.1
Impulsnebensprechen (Inter-Symbol Interference ISI)
Symbolfolge
ak
17.2.2.3.2
Kanal
b(t )
17.2.2.3.3
Impulsformer ideal
u (t ) = ∑ ak ⋅ δ (t − kT )
k
17.2.2.3.4
Impulsformer real
u (t ) = ∑ ak ⋅ p(t − kT )
k
17.2.2.3.5
linear verzerrter Puls
y(t ) = g (t ) ∗ b(t ) ∗ c(t ) ∗ pk = h(t ) ∗ pk = ~
p (t )
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09.09.2004
85
17.2.2.3.6
86
17.2.2.5
linear verzerrte Pulsfolge
Mittlere Rauschleistung
y(t ) = ∑ ak ⋅ ~
p (t − kT )
N rn = σ 2 = N 0 ⋅ BN ≥ N 0 ⋅
17.2.2.3.7
17.2.2.6
k
Nur Rauschen n(t), ak=0
r
2
Mittlere Symbolfehler-WSK
y(t ) = n(t ) * c(t ) =ˆ r (t )
17.2.2.6.1
17.2.2.3.8
Unipolar a, b gleichhäufig
S ak = a 2 ⋅ P[a] + b 2 ⋅ P[b] = a 2 2
Signal und Rauschen


y(t ) =  ∑ ak ⋅ ~
p (t − kT ) + r (t )
 k

 a 
PS = Q

 2 ⋅σ 


y n = an ⋅ ~
p (t0 ) +  ∑ ak ⋅ ~
p (t0 + nT − kT ) + rn
 k ≠n

17.2.2.6.2
17.2.2.3.9
yn( ISI )
ISI-Anteil
 Sa
a
k
PS = Q  = Q
 Nr
σ 
n



=  ∑ ak ⋅ ~
p (t0 + nT − kT )
 k ≠n

Nyquist Impulsformer (ISI -> 0)
sin(πλ )
sinc(λ ) =
πλ
17.2.2.6.3
17.2.2.4.1
ak2 =
17.2.2.4
17.2.2.4.2
M-ary Fehler-WSK
17.2.2.6.4
t 0 − 3T
t 0 − 2T
t0 − T
t0 + T
t 0 + 2T
t 0 + 3T
Frequenzbereich
M-ary Fehler-WSK-Graycode
 M − 1   Sak
PS = 2
 ⋅ Q
 M ⋅ ld(M )   N rn
für f ≥ B
17.2.2.7
r
B= +β
2
r
für 0 ≤ β ≤
2
u (t ) = ∑ ak ⋅ p(t − kTs )
PAM Signale
17.2.2.8
PCM (Pulse Code Modulation)
p(t ) = pβ (t ) ⋅ sinc(r ⋅ t )
17.2.2.8.1
17.2.2.4.4
σ q2 = ε k2 =
17.2.2.4.5
Bandbreite der Impulsfunktion
Quantisierungsrauschen
1
3q 2
(
für f > β
( )
)
S
n 2
  = 10 ⋅ log 3 ⋅ M ⋅ S D ≤ (4.8 + 6 ⋅ n )[dB]
N
q = Mn
M: Zustände pro Leitung (nach Codierer)
n: Anzahl Leitungen
q: Anzahl Zustände (vor Codierer)
Roll-Off Faktor 0..100%
2β
Rolloff =
r
© Daniel Arnold
www.lanny.ch




k
Funktion
F {pβ (t )} = Pβ ( f ) = 0




M 2 −1 2
⋅A
3
P( f ) = 0
17.2.2.4.3

 = Q a 

σ 

1   Sa k

PS = 21 −  ⋅ Q
M

  N rn
Zeitbereich
0 t = −T ,−2T ,... n ≠ k

~
p (t ) = 1
t =0
n=k
0 t = T ,2T ,... n ≠ k

Bipolar b=-a , gleichhäufig
Sa k = a 2 ⋅ P[a ] + b 2 ⋅ P[b] = a 2
09.09.2004
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09.09.2004
87
88
17.2.2.8.2
PCM mit Rauschen - Gesamte Rauschleistung am Ausgang
17.2.4 Binäre Empfänger
N = σ q2 + σ n2 ≈
1 + 4q ⋅ Pber
1
4
+ Pber =
3q 2 3
3q 2
17.2.4.1
17.2.2.8.3
Threshold-Effekt
2
(
17.2.4.1.1
)
17.2.4.1.2
17.2.3 Digitale Modulation von sinusförmigen Trägern
x(t ) = A[xi (t ) ⋅ cos(ω 0t + ϕ 0 ) − xq (t ) ⋅ sin (ω 0t + ϕ 0 )]
2
(
17.2.3.1
∞
) ∑ P( f )
2
2
⋅δ ( f − n ⋅ r)
]
17.2.4.1.3
Für ein Bitintervall
[
]
x(t ) = A ⋅ I k ⋅ pi (t ) ⋅ cos(ω 0 t ) − Qk ⋅ pq (t ) ⋅ sin (ω 0 t )
ak ∈ [0,1,...(M − 1)]
x(t ) = s m (t − kTb )
k
17.2.4.2
Mittelwert und Varianz des Digitalsignals
Korrelationsempfänger
17.2.4.2.1
M −1
ma = ak =
2
M 2 −1
σ 2 = ak2 − ma2 =
12
mittlere Signalenergie pro Bit
Tb
∫ [s (t ) − s (t )] dt =ˆ E
2
1
0
1
+ E0 + 2 E10
0
Eb =
1
[E1 + E0 ]
2
Spektrum
GTP ( f ) = Gi ( f ) =
17.2.3.2
[
k
x(t ) = A ⋅ xi (t ) ⋅ cos(ω 0t + ϕ 0 )
17.2.3.1.2
N 0 ∈ [1,2,3...]
x(t ) = A ⋅ ∑ I k ⋅ pi (t − kTb ) ⋅ cos(ω 0 (t − kT0 )) − Qk ⋅ pq (t − kTb ) ⋅ sin (ω 0 (t − kT0 ))
n = −∞
xq (t ) = 0
17.2.3.1.1
Vereinfachte Quadraturform
f 0 = N 0 ⋅ rb
ϕ0 = 0
Amplitude Shift Keying ASK
xi (t ) = ∑ ak ⋅ p(t − kT0 )
Allgemeine Quadraturform





x(t ) = A ⋅ ∑ I k ⋅ pi (t − kTb ) cos(ω 0 t + ϕ 0 ) − ∑ Qk ⋅ p q (t − kTb ) sin (ω 0 t + ϕ 0 )

 k

 k

S
= 6 M 2 −1
 
R
 N Threshold
Gi ( f ) = σ a2k ⋅ r ⋅ P( f ) + ma k ⋅ r ⋅
Allgemeines
17.2.4.2.2
2
M 2 −1
 f   M −1
sinc 2   + 
 δ(f )
12
r  4 
ρ=
Phase Shift Keying PSK
x(t ) = A ⋅ ∑ cos(ω 0t + ϕ 0 + Φ k ) ⋅ p(t − kTs )
mit I k = cos(Φ k )
Fehlerwarscheinlichkeit
 Eb − E10
Pber = Q
N0

k
xq (t ) = ∑ Qk ⋅ p(t − kTs )
1
⋅ s1 (t ) ⋅ s0 (t )dt
E1 ⋅ E0 ∫0
17.2.4.2.3
k
xi (t ) = ∑ I k ⋅ p (t − kTs )
Korrelationskoeffizient
Tb
mit Qk = sin (Φ k )



 = Q Eb (1 − ρ ) 



N0



k
Φk =
(2ak + N )π
M
17.2.3.2.1
17.2.4.2.4
N = [0,1]
Optimale Schwelle für Entscheider
K
R = (E1 − E0 )
2
Spektrum
1
f
GTP ( f ) = 2 ⋅ Gi ( f ) = sinc 2  
r
r
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www.lanny.ch
09.09.2004
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www.lanny.ch
09.09.2004
89
Binäre ASK
s1 (t ) = A ⋅ pT (t ) ⋅ cos(ω 0t )
90
17.2.5 Vergleich von digitalen Modulationsverfahren
17.2.4.2.5
b
A ⋅ Tb
Eb =
4
 Eb
Pber = Q
 N0
s0 (t ) = 0
und
Modulationsart
( )




mit :
γb =
17.2.5.1 Binär
2
Eb
N0
Pber = Q γ b
A2 ⋅ Tb
2
 2 Eb
Pber = Q
 N0
s0 (t ) = − s1 (t )
und
Eb =
(
Pber = Q 2γ b
17.2.4.3
)




mit :
γb =
Eb
N0
rb Bk
Pber
γ
ASK, FSK
f d = rd 2
Hüllkurve
1
DPSK
Phasenvergleich
1
BPSK
Kohärent
1
Q 2γ b
MSK, QAM, QPSK
Kohärent, Quadratur
2
2γ b
DPSK (M >= 4)
Phasenvergleich,
Quadratur
ld (M )
2 
 π
Q 4kγ b sin 2 
k 
 2M
 


PSK (M >= 8)
Quadratur, Kohärent
ld (M )
2 
π
Q 2kγ b sin 2 
k 
M
 


QAM (k gerade)
Kohärent, Quadratur
ld (M )
Binäre PSK
s1 (t ) = A ⋅ pTb (t ) ⋅ cos(ω 0t )
17.2.5.2 M-ary
17.2.4.2.6
Detektoren
1 − 2b
e
2
1 −γ b
e
2
(
Q(
)
)

4
1   3M
γb 
1 −
Q
k
M   M − 1 
Nichtkohärente binäre Empfänger
- Wenn ausreichende Signalstärke
- rasch ändernde Kanaleigenschaften (zB Laufzeitänderung bei Mobilfunk)
17.2.4.3.1
Binäre ASK
A2 ⋅ Tb
Eb =
4
γ
 1 −γ b
1  − 2b
Pber =  e + Q γ b  ≈ e 2
2
 2
( )
17.2.4.3.2
Pber =
1
e
2
A2
−
2⋅σ 2
© Daniel Arnold
www.lanny.ch
bei γ b >> 1
DPSK (Differential Coherent PSK)
=
1 −γ b
e
2
09.09.2004
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www.lanny.ch
09.09.2004
91
Erstellt von:
Daniel Arnold, Altdorf, Uri, Schweiz
http://www.lanny.ch
Erstelldatum 29.11.1999 18:20
Letzte Änderung 09.09.2004 00:33
© Daniel Arnold
www.lanny.ch
09.09.2004