Resum de Teoria de Circuits

Lluís Batlle i Rossell 25 de desembre de 2001

1 Amplificadors operacionals

1.1 Amplificador inversor

V_{o} = −\frac{z_{2}}{z_{1}} V_{g}

1.2 Amplificador no inversor

V_{o} = (1 + \frac{z_{2}}{z_{1}} )V_{g}

1.3 Amplificador diferencial

V o = K(V_{2} − V_{1})

1.4 Amplificador amb offset DC

V o = KV_{+} = K \frac{R_{2}}{R_{1} + R_{2}} V 1 + K \frac{R_{1}}{R_{1} + R_{2}} V cc

2 Filtres

2.1 Filtre passa banda de 2on ordre

H(s) = K \frac{\frac{1}{RC} s}{s^{2} + \frac{1}{RC} s + \frac{1}{LC}} = K \frac{BW s}{s^{2} + BW s + ω_{r}^{2}} = K \frac{\frac{ω_{r}}{Q} s}{s^{2} + \frac{ω_{r}}{Q} s + ω_{r}^{2}}

f_{r} = \frac{1}{2π\sqrt{LC}} f_{c1} = \frac{1}{2π} [\sqrt{\frac{1}{LC} + (\frac{1}{2RC} )^{2}} − \frac{1}{2RC} ] f_{c2} = \frac{1}{2π} [\sqrt{\frac{1}{LC} + (\frac{1}{2RC} )^{2}} + \frac{1}{2RC} ] Q = R\sqrt{\frac{C}{L}} BW = \frac{f_{r}}{Q} = \frac{1}{RC} |H(2πfj)| = _{\frac{|K|}{\sqrt{1+[(\frac{f}{fr} − \frac{fr}{f})Q]^{2}}}} φ(H(2πfj)) = φ(k) − arctan ((\frac{f}{f_{r}} − \frac{f_{r}}{f} ) Q)

2.2 Filtre passa banda de 2on ordre sense bobina

H(s) = \frac{R_{2}}{R_{1}} \frac{\frac{1}{R_{2}C} s}{s^{2} + \frac{1}{R_{2}C} s + \frac{1}{(R_{3}C)^{2}}}

A_{vmax} = \frac{R_{2}}{R_{1}} BW = \frac{1}{2πR_{2}C} f_{r} = \frac{1}{2πR_{3}C}

2.3 Filtre pas baix de 2on ordre

H(s) = \frac{\frac{1}{LC}}{s^{2} + \frac{R}{L}s + \frac{1}{LC}}

H(s) = K \frac{ω_{0}^{2}}{s^{2} + \sqrt{2}ω_{0}s + ω_{0}^{2}}

H(jω) = \frac{ω_{0}^{2} − ω^{2}ω_{0}^{2} − \sqrt{2}ωω_{0}j}{ω^{4} + ω_{0}^{4}}

|H(jω)| = \frac{ω_{0}^{2}}{\sqrt{ω_{0}^{4} + ω^{4}}} = \sqrt{1 + (\frac{f_{c}}{f} )^{4}}^{(\frac{f_{c}}{f} )2}

ξ = \frac{1}{\sqrt{2}} f_{c} = \frac{ω_{c}}{2π}

2.4 Filtre trampa de 2on ordre

H(s) = \frac{s^{2} + (\frac{1}{RC} )^{2}}{s^{2} + \frac{4}{RC} s + (\frac{1}{RC} )^{2}}

Deduit de:

[ \begin{matrix}2 (Cs + G) & 0 & −G \\ 0 & 2 (Cs + G) & −Cs \\ −G & −Cs & Cs + G \end{matrix} ] [ \begin{matrix}V_{1}(s) \\ V_{2}(s) \\ V_{o}(s) \end{matrix} ] = [ \begin{matrix}GV g(s) \\ CsV g(s) \\ 0 \end{matrix} ]

2.5 Filtre pas alt de 2on ordre

H(s) = \frac{s^{2}}{s^{2} + \sqrt{2}ω_{0}s + ω_{0}^{2}}

H(jω) = \frac{ω^{2} − ω^{2}ω_{0}^{2} + \sqrt{2}ω_{0}ω^{3}j}{ω^{4} + ω_{0}^{2}}

|H(jω)| = \frac{ω}{\sqrt{ω^{2} + ω_{0}^{2}}} = \sqrt{1 + (\frac{f}{f_{c}} )^{4}}^{(\frac{f}{f_{c}} )2} = \frac{1}{\sqrt{1 + (\frac{f_{c}}{f} )^{4}}}

2.6 Filtre pas alt de 1er ordre

H(s) = \frac{s}{s + \frac{1}{RC}} = \frac{s}{s + ω_{c}}

|H(jω)| = \frac{\frac{f}{f_{c}}}{\sqrt{1 + (\frac{f}{f_{c}} )^{2}}}

ω_{c} = \frac{1}{RC}

2.7 Filtre pas baix de 1er ordre

H(s) = \frac{\frac{1}{RC}}{s + \frac{1}{RC}} = \frac{ω_{c}}{s + ω_{c}}

|H(jω)| = \frac{\frac{f_{c}}{f}}{\sqrt{1 + (\frac{f_{c}}{f} )^{2}}}

ω_{c} = \frac{1}{RC} = \frac{1}{τ}

H(jω) = \frac{ω_{c}^{2} − jωω_{c}}{ω^{2} + ω_{c}^{2}}

3 Oscil·lador

f_{0} = \frac{1}{2π} \frac{1}{\sqrt{LC}} Ideal LC:

H(s) = \frac{1}{C} \frac{s}{s^{2} + \frac{1}{LC}} = \frac{…}{s^{2} + ω_{0}^{2}}

No ideal:

H(s) = \frac{1}{C} \frac{s + \frac{r}{L}}{s^{2} + \frac{r}{L}s + \frac{1}{LC}}

Arreglat amb -R:

H(s) = \frac{1}{C} \frac{s + \frac{r}{L}}{s^{2} + (\frac{r}{L} − \frac{1}{RC} ) s − \frac{R−r}{R} \frac{1}{LC}}

R = \frac{L}{rC}

4 Disseny de circuits a partir de Bode (Baixes freqüències)

z_{1}(s) = \frac{N_{1}}{D_{1}} ;z_{2}(s) = \frac{N_{2}}{D_{2}} H(s) = −\frac{z_{2}(s)}{z_{1}(s)} = −\frac{N_{2}·D_{1}}{N_{1}·D_{2}} = \frac{Z_{2}·P_{1}}{P_{2}·Z_{1}} Per un amplificador no inversor:

H(s) = (1 + \frac{z_{2}}{z_{1}} ) = 1 + \frac{Z_{2}P_{1}}{P_{2}Z_{1}} = \frac{P_{2}Z_{1} + Z_{2}P_{1}}{P_{2}Z_{1}}

4.1 R

z(s) = R Ni pols ni zeros a H(s)

4.2 C

z(s) = \frac{1}{Cs} Pol a l’origen

4.3 RC sèrie

z(s) = R + \frac{1}{Cs} = \frac{RCs+1}{Cs} Zero real a \frac{−1}{RC} i Pol a l’origen

4.4 RC paral·lel

z(s) = \frac{\frac{1}{C}}{s+ \frac{1}{RC}} = R\frac{\frac{1}{RC}}{s+ \frac{1}{RC}}

Pol real a \frac{−1}{RC}