Filtro pasa bajo bass boost

Red de refuerzo de graves
“BASS BOOST”
(Compensación de atraso).

Diagrama del circuito:

Diagrama de red escalera:

De lo anterior tenemos entonces:
Z1s=R1; Z2(s)= R2+1Cs
Función de transferencia:

VosVss=Z2Z1+Z2=R2+1CsR1+R2+1Cs=R2Cs+1R1+R2Cs+1
Fs=R2Cs+1R1+R2Cs+1=PsQs
Ps=R1+R2Cs+1
Qs=R2Cs+1
POLOS:

Polos: Ps=R2Cs+1=∞ cuando s→∞. (Se cancela).
Ceros: Qs=1+(R1+R2)Cs=0cuando s→-1R1+R2C.

CEROS:

Ceros: Ps=R2Cs+1=0 cuando s→-1R2C.
Polos: Qs=1+(R1+R2)Cs=0 cuando s→∞. (Se cancela).

Graficando los Polos en el plano S:

Como lo vimos anteriormente:
Sp: Qs=0 si 1+(R1+R2)Cs=0 ∴ Sp=-1R1+R2C
∴ Sp=-110*103+2.2*103*0.22*10-6=-372.58
Tenemos un polo real y negativo.

Sz: Ps=0 si 1+R2Cs=0 ∴ Sz=-1R2C
∴ Sz=-12.2*103*0.22*10-6=-2066.11Tenemos un polo real y negativo.
Finalmente:
ωL=Sp=-372.58=372.58
ωH=Sz=-2066.11=2066.11
ωC=ωH*ωL=372.58*2066.11=877.38

Simplificando la función de transferencia:
Dado que:
A frecuencias bajas; XC→ ∞

F(s)=1

A frecuencias altas; XC→0

Fs=R2R1+R2
Entonces K=R2R1+R2=22002200+10000=0.18033[adimensional].

De la función obtenemos entonces:Fs=R2Cs+1R1+R2Cs+1=R2C(R1+R2)C*1R2C+s1(R1+R2)C+s=R2(R1+R2)*ωH+sωL+s=K*ωH+sωL+s

De donde:
ωLωH=1(R1+R2)C1R2C=R2C(R1+R2)C=R2(R1+R2)=K
∴ ωL=K*ωH
Fs=K*s+ωHs+K*ωH

Función F(s)|s=jω simplificada:

Fjω=Kjω+ωHjω+ωH*K=KjωωH*K+ωHωH*KjωωH*K+ωH*KωH*K=jωωH+1jωωH*K+1=ja+1jaK+1
donde a=ωωH
De esta forma:
Fjω=jaK+Kja+K=jaK+Kja+K-ja+K-ja+K=a2k+jak2-jak+k2a2+k2=a2k+k2+ak2-akja2+k2
Magnitud de Fjω:Fjω=a2k+k22+ak2-ak2a2+k22=a4k2+2a2k3+k4+a2k4-2a2k3+a2k2a2+k22
=a4k2+k4+a2k4+a2k2a2+k22=k2a4+k2+a2k2+a2a2+k22=ka4+k2+a2k2+a2a4+2a2k2+k4
=ka4+2a2k2+k4-k4+k2-a2k2+a2a4+2a2k2+k4=k1+-k4+k2-a2k2+a2a4+2a2k2+k4
=k1+-k4+k2+a2(1-k2)a2+k22=k1+-k4+k2+(ωωH)2(1-k2)(ωωH)2+k22
Fase de Fjω:

< Fjω=arctanak2-aka2+k2a2k+k2a2+k2=arctan ak2-aka2k+k2arctan ak2-ka2k+k2
=arctan ωωHk2-kωωH2k+k2
De forma simplificada obtenemos:

= arctanωωH*k (k-1)k*ωωH2+k

<F(jω)= arctan ωωH* (k-1)ωωH2+k

TABULACION
w [rad/s] | W/WH | |F(jw)| [#] | 20 log(|F(jw)|) [dB] | Fase de F(jw) [rad] | Fase de F(jw) [º] |
1 | 0.0005 | 1.0000 | 0.0000 | -0.0022 | 0.1260 |
1.25 | 0.0006 | 1.0000 | 0.0000 | -0.0027 | 0.1576 |
1.6 | 0.0008 | 1.0000 | -0.0001 | -0.0035 | 0.2017 |
2 | 0.0010 | 1.0000 | -0.0001 | -0.0044 | 0.2521 |
2.5 | 0.0012 |1.0000 | -0.0002 | -0.0055 | 0.3151 |
3.2 | 0.0015 | 1.0000 | -0.0003 | -0.0070 | 0.4033 |
4 | 0.0019 | 0.9999 | -0.0005 | -0.0088 | 0.5042 |
5 | 0.0024 | 0.9999 | -0.0008 | -0.0110 | 0.6302 |
6.4 | 0.0031 | 0.9999 | -0.0012 | -0.0141 | 0.8066 |
8 | 0.0039 | 0.9998 | -0.0019 | -0.0176 | 1.0082 |
10 | 0.0048 | 0.9997 | -0.0030 | -0.0220 | 1.2601 |
12.5 | 0.0061 | 0.9995 | -0.0047 |-0.0275 | 1.5749 |
16 | 0.0077 | 0.9991 | -0.0077 | -0.0352 | 2.0153 |
20 | 0.0097 | 0.9986 | -0.0121 | -0.0439 | 2.5180 |
25 | 0.0121 | 0.9978 | -0.0189 | -0.0549 | 3.1455 |
32 | 0.0155 | 0.9965 | -0.0309 | -0.0702 | 4.0216 |
40 | 0.0194 | 0.9945 | -0.0481 | -0.0876 | 5.0186 |
50 | 0.0242 | 0.9914 | -0.0750 | -0.1092 | 6.2571 |
64 | 0.0310 | 0.9860 | -0.1221 | -0.1391 | 7.9726 |
80| 0.0387 | 0.9784 | -0.1892 | -0.1728 | 9.9010 |
100 | 0.0484 | 0.9669 | -0.2919 | -0.2139 | 12.2530 |
125 | 0.0605 | 0.9498 | -0.4474 | -0.2633 | 15.0842 |
160 | 0.0774 | 0.9216 | -0.7091 | -0.3283 | 18.8122 |
200 | 0.0968 | 0.8852 | -1.0592 | -0.3961 | 22.6976 |
250 | 0.1210 | 0.8364 | -1.5513 | -0.4706 | 26.9620 |
ω L
320 | 0.1549 | 0.7677 | -2.2967 | -0.5560 | 31.8542 |
372.58| 0.1803 | 0.7185 | -2.8713 | -0.6070 | 34.7776 |
400 | 0.1936 | 0.6942 | -3.1698 | -0.6296 | 36.0755 |
500 | 0.2420 | 0.6148 | -4.2259 | -0.6930 | 39.7037 |
640 | 0.3098 | 0.5267 | -5.5688 | -0.7432 | 42.5828 |
ω C
800 | 0.3872 | 0.4527 | -6.8832 | -0.7655 | 43.8608 |
877.38 | 0.4247 | 0.4247 | -7.4393 | -0.7676 | 43.9826 |
1000 | 0.4840 | 0.3879 | -8.2261 | -0.7634 | 43.7384 |...