Trasarea diagramelor Bode si a diagramei polare

Partea I: Functii de transfer elementare

Universitatea Tehnica "Gh. Asachi", Iasi, Facultatea de Electronica si Telecomunicatii

Laboratorul de Semnale, Circuite si Sisteme

Breviar teoretic

Scopul lucrarii:

Partea I : Functii de transfer elementare - determinarea regulilor de trasare rapida a diagramelor Bode

Partea II : Functii de transfer de ordinul I si II - diagramele Bode si polara pentru functiile de transfer corespunzatoare unor filtre de tip FTJ, FTB, FTS, FTT.

Partea III : Functii de transfer ale unor circuite reale - utilizarea in exemple complexe a regulilor de trasare rapida

Rezumat teoretic:

Diagrame Bode liniarizate pe portiuni

Mod de lucru

In afara de functiile incluse in libraria standard Maple, in aceasta lucrare vor fi utilizate cateva functii din libraria aditionala SCSlib.

Pentru trasarea diagramelor Bode de castig si faza, precum si a diagramelor polare:

Functii de transfer elementare

Zerou simplu in origine

> H:=s:

> PZ[numeric](H,s);

> Bode[castig](H);

> Bode[faza](H);

> Bode[polara](H);

Pol simplu in origine

> H:=1/s:

> PZ[numeric](H,s);

> Bode[castig](H);

> Bode[faza](H);

> Bode[polara](H);

Zerou simplu pe axa reala

Functia de transfer in acest caz este de forma:

> H:=s-alpha0;

> eval(H,alpha0=-1);

> PZ[numeric](eval(H,alpha0=-1),s);

> Bode[castig](eval(H,alpha0=-1));

> Bode[faza](eval(H,alpha0=-1));

> Bode[polara](eval(H,alpha0=-1),compresie=[4,1]);

>

> eval(H,alpha0=1);

> PZ[numeric](eval(H,alpha0=1),s);

> Bode[castig](eval(H,alpha0=1));

> Bode[faza](eval(H,alpha0=1));

> Bode[polara](eval(H,alpha0=1),compresie=[4,1]);

Concluzie. Diagramele de castig sunt identice in cele doua cazuri anterioare, fiind distincte diagramele de faza.

Pol simplu pe axa reala

Functia de transfer in acest caz este de forma:

> H:=1/(s-alpha0);

> eval(H,alpha0=-1);

> PZ[numeric](eval(H,alpha0=-1),s);

> Bode[castig](eval(H,alpha0=-1));

> Bode[faza](eval(H,alpha0=-1));

> Bode[polara](eval(H,alpha0=-1));

> 0

> eval(H,alpha0=1);

> PZ[numeric](eval(H,alpha0=1),s);

> Bode[castig](eval(H,alpha0=1));

> Bode[faza](eval(H,alpha0=1));

> Bode[polara](eval(H,alpha0=1));

Concluzie. Diagramele de castig sunt identice in cele doua cazuri anterioare, fiind distincte diagramele de faza.

Zerouri simple complex conjugate

Functia de transfer in acest caz este de forma:

> H:=s^2-2*alpha0*s+omega0^2;

> eval(H,[alpha0=-0.05,omega0=1]);

> PZ[numeric](eval(H,[alpha0=-0.05,omega0=1]),s);

> PZ[grafic](eval(H,[alpha0=-0.05,omega0=1]),s);

> Bode[castig](eval(H,[alpha0=-0.05,omega0=1]));

> Bode[faza](eval(H,[alpha0=-0.05,omega0=1]));

> Bode[polara](eval(H,[alpha0=-0.05,omega0=1]),compresie=[8,1]);

> 0

> eval(H,[alpha0=0.05,omega0=1]);

> PZ[numeric](eval(H,[alpha0=0.05,omega0=1]),s);

> Bode[castig](eval(H,[alpha0=0.05,omega0=1]));

> Bode[faza](eval(H,[alpha0=0.05,omega0=1]));

> Bode[polara](eval(H,[alpha0=0.05,omega0=1]),compresie=[8,1]);

Zerouri simple complex conjugate pe axa imaginara

> H:=s^2+omega0^2;

> eval(H,omega0=10);

> PZ[numeric](eval(H,omega0=10),s);

> Bode[castig](eval(H,omega0=10));

> Bode[faza](eval(H,omega0=10));

> Bode[polara](eval(H,omega0=10),compresie=[5,1]);

Poli simpli complex conjugati

In acest caz functia de transfer este:

> H:=1/(s^2-2*alpha0*s+omega0^2);

> eval(H,[alpha0=-0.05,omega0=1]);

> PZ[numeric](eval(H,[alpha0=-0.05,omega0=1]),s);

> PZ[grafic](eval(H,[alpha0=-0.05,omega0=1]),s);

> Bode[castig](eval(H,[alpha0=-0.05,omega0=1]));

> Bode[faza](eval(H,[alpha0=-0.05,omega0=1]));

> Bode[polara](eval(H,[alpha0=-0.05,omega0=1]),compresie=[4,1]);

> 0

> eval(H,[alpha0=0.05,omega0=1]);

> PZ[numeric](eval(H,[alpha0=0.05,omega0=1]),s);

> Bode[castig](eval(H,[alpha0=0.05,omega0=1]));

> Bode[faza](eval(H,[alpha0=0.05,omega0=1]));

> Bode[polara](eval(H,[alpha0=0.05,omega0=1]),compresie=[4,1]);

Poli simpli complex conjugati pe axa imaginara

> H:=1/(s^2+omega0^2);

> eval(H,omega0=10);

> PZ[numeric](eval(H,omega0=10),s);

> Bode[castig](eval(H,omega0=10));

> Bode[faza](eval(H,omega0=10));

> Bode[polara](eval(H,omega0=10),compresie=[8,1]);

Probleme. Intrebari

Sa se reprezinte manual si sa se verifice cu ajutorul calculatorului diagramele Bode de castig si faza, precum si diagrama polara pentru functiile de transfer:

> H:=-s;

> H:=-1/s;

> H:=-(s+100);

> H:=-1/(s+100);

> H:=s^2+50*s+10000;

> H:=s^2+100*s+10000;

> H:=s^2+200*s+10000;

> H:=1/(s^2+50*s+10000);

> H:=1/(s^2+100*s+10000);

> H:=1/(s^2+200*s+10000);

> H:=-(s^2+10000);

> H:=-1/(s^2+10000);