Analog Filters: Biquad Circuits

Analog Filters: Biquad Circuits

Franco Maloberti

Franco Maloberti Analog Filters: Biquad Circuits 2

Introduction

Active filters which realize the biquadratic transfer function

are important building blocks

(biquad)

p

p

0

€

H(s) =a2s

2 +a1s+a0

(s−sp)(s−sp*)

€

ω02 =σ p

2 +ωp2

€

ω0

Q=−2σ p

Franco Maloberti Analog Filters: Sensitivity 3

Introduction

Biquads can build high-order filters

€

H(s) =P(s)Q(s)

=(s−si )1

n∏(s−sj )1

m∏Poles and zeros areReal or complex conjugate

€

H(s) =(s−sz,1)(s−sz,1

* )(s−sp,1)(s−sp,1

* )⋅(s−sz ,2)(s−sz,2

* )(s−sp,2)(s−sp,2

* )⋅(s−sz ,3)(s−sz,3

* )(s−sp,3)(s−sp,3

* )⋅K

s or 1/s

B1 B2 B3

Problem: how to properly pair poles and zeros


Single Amplifier Configurations

RC A+

-

RC A+

-

RR(k-1)

RC A+

-

RR(k-1)

Enhanced positive or negative feedback


Sallen-Key Biquad

R1 R2

C1 C2

E1 E2

€

E1 =E2(1+sR1C1)(1+sR2C2)

Only real poles (or zeros)

C1 C2

E1 E2

The feedback permits us to achieve complex poles


Sallen-Key Biquad (ii)

C1

C2

E1 E2

R1 R2

RbRa

E3E4

€

E2

E1

=

μR1R2C1C2

s2 +(1R1C1

+1

R2C1

+1−μR2C2

)s+μ

R1R2C1C2

€

ω0 =1

R1R2C1C2

Q=

1

R1R2C1C2

1R1C1

+1

R2C1

+1−μR2C2

G=μ


Sallen-Key Biquad (ii)

Five design elements, two properties (G is not important)

€

ω0 =1

R1R2C1C2

Q=

1

R1R2C1C2

1R1C1

+1

R2C1

+1−μR2C2

G=μ

Case 1: C1=C2; R1=R2=R

R=1/ 0 =3-1/QCase 2: C1=C2; Ra=Rb

R1=Q/ 0 R2=1/Q 0 Case 3: R1=R2; =1

C1=2Q/ 0 C2=1/2Q 0

Case 4: C1=31/2Q C2; =4/3

R1=1/Q0 R2=1/31/20


Sallen-Key Biquad (iii)

Sensitivities

€

ω0 =1

R1R2C1C2

Q=

1

R1R2C1C2

1R1C1

+1

R2C1

+1−μR2C2

G=μ

€

SR1

ω0 =SR2

ω0 =SC1

ω0 =SC2

ω0 =−12

SR1

Q =−12

+QR2C2

R1C1

SR2

Q =−12

+QR1C2

R2C1

+(1−μ)R1C1

R2C2

⎛

⎝ ⎜

⎞

⎠ ⎟

SC1

Q =−12

+QR1C2

R2C1

+R1R2C1

R2C2

⎛

⎝ ⎜

⎞

⎠ ⎟

SC2

Q =−12

(1−μ)QR1C1

R2C2

K


Sallen-Key High- and Band-pass

R1 R2

C1 C2

E1 E2

R1 R2C1

C2

E1 E2

R1

R2C1

C2

E1 E2

C1 C2

LP

HP

BP


Generic Sallen-Key

€

E2

E1

=VoutVin

=Z1

'Z2'

(Z1+Z2 +Z2' )Z1

' +Z1Z2


Sallen-Key: finite op-amp gain

The inverting and non-inverting terminals are not virtually shorted

C1

C2

E1 E2

R1 R2

RbRa

E3E4

€

E4 =E2

Ra +RbRa +

E2

A0


Sallen-Key in IC

C1

C2

E1 E2

R1 R2

RbRa

E3E4

C1

C2

E1 E2

R1 R2

E3E4


LP Sallen-Key with real op-amp

€

H(s) =1+as+bs2

α+βs+γs2 +δs3

€

a=2Cgm

b=RC2

gm

α=1+1A0

β=2RC+R0C0 +2R0C+4RC

A0

γ=2R2C2+4RR0C(C+C0)+R2C2

A0

δ=2R2C2

C0

gm


LP Sallen-Key with real op-amp (ii)

The transfer function has two zeros and three poles.If k = Rgm >> 1 the zeros are practically complex conjugates and are located at

The extra-pole is real and is located around the GBW of the op-amp.

€

H(s) =1+as+bs2

α+βs+γs2 +δs3

€

ω0,p =gm

2RC2 =ωpK


LP Sallen-Key with real op-amp (iii)

Possible responses


Sallen-Key IC Implementations

€

Y(s) =sC/R

sinh sRC

Yp (s)=Y(cosh(sRC−1))


Band-reject Biquad

A band-reject response requires zeros on the immaginary axis

It can be obtained with the generic SK implementation

Another option is to use a twin-T network

R1R1R2C1C2C2 R1R1R2C1C2C2


Band-reject Biquad (ii)

Using complementary valuesRRR/22CCCE1E2

€

E2

E1

=μ s2 +

1

R2C2

⎛

⎝ ⎜

⎞

⎠ ⎟

s2 +4(1−μ)

RCs+

1

R2C2

€

Q =1

4(1−μ)


Use of Feed-forward

P(s)Q(s)+k

€

E2

E1

=P(s) + kQ(s)

Q(s)

Assume

€

P(s) = a1s

Q(s) = s2 + b1s+ b0

k = −b1 /a1

€

E2

E1

= −k(s2 + b0)

s2 + b1s+ b0

High-pass Band-pass


Infinite-Gain Feedback Biquad

Sallen-Key architectures require input common mode range.

Input parasitic capacitance of the op-amp can affect the filter response

Keep the inputs of the op-amp at ground or virtual ground

A-

+


Infinite-Gain Multi-Feedback Biquad

A conventional op-amp amplifier is not able to realize complex-conjugate poles

Two or more feedback connections achieve the result

A-

+

Z1Z2

A-

+

Z2Z3Z1Z4


Low-Pass MFB

A-

+

C2R2R1C1R3E2E1

€

E2

E1

=−

1

R1R3C1C2

s2 +1

R1

+1

R2

+1

R3

⎛

⎝ ⎜

⎞

⎠ ⎟s+

1

R2R3C1C2

€

Q =C1

C2

1

R2R3

R1

+R3

R2

+R2

R3

ω0 =1

R2R3C1C2

G = −R2

R1


Design and Sensitivity

Five elements and three equations

“Arbitrarily choose two of them and determine the remaining three parameters

Assess the “quality of design” Sensitivity on relevant design element Spread of components Cost of the implementation

Linearity of components


High-pass and Band-pass

A-

+

C2R2R1C1C3E1A-

+

C2R2R1C1E1E2E2

€

E2

E1

= −

C1

C2

s2

s2 +C1 +C2 +C3

R1C1C2

s+1

R1R2C2C3

€

E2

E1

= −

1

R1C2

s

s2 +1

R2C1

+1

R2C2

⎛

⎝ ⎜

⎞

⎠ ⎟s+

1

R1R2C2C3


Two-Integrators Biquad

Use of state-variable method Derive the block diagram Translate the block diagram into an active

implementation Addition or subtraction Integration Dumped integration (integration plus addition)


Basic BlocksA-

+

V1V2-K1Σ-K2A-

+

V1A-

+

V1Σ-1/s-1/sK1-1/( +s K1)VoutVoutVout

€

−K1V1 −K2V2 =Vout

€

−sVout =V1

€

−(s+K1)Vout1 =V1


State Variables

The state variable are relevant voltages of the network

€

E2

E1

=Ga0

s2 + b1s+ b0

€

E2

a0

s2 + b1s+ b0( ) =GE1

€

E6 s2 + b1s+ b0( ) = E5

H(s)E1 E2

H’(s)E1 E2

E5

G a0

E6

€

E6 s2 + b1s( ) = E4

€

E4 + E6b0 = E5

ΣH”-b0

E5 E6

E4


State Variables (ii)

€

E6 s2 + b1s( ) = E4

€

E3 −s−b1( ) = E4

€

E3 = −sE6

Σ-1/sb1

-1/s

E4E3

E3 E6

Σ-1/sb1-1/sΣ-b0Ga0E1E5E6E2E3E4


State Variables (iii)

€

E6 s2 + b1s( ) = E4

€

E6 −s−b1( ) = E3

€

E4 = −sE3Σ-1/sb1

-1/s

E3E6

E4 E3

Σ-1/sb1-1/sΣ-b0Ga0E1E5E6E3E4


State Variables (iv)

€

E2

E1

=Ga2s

2 + a1s+ a0

s2 + b1s+ b0

€

E2'

E1

=G1

s2 + b1s+ b0

€

E2' a2s

2 + a1s+ a0( ) = E2

€

E3 = −sE2'

E7 = s2E2'

Σ-1/sb1-1/sΣ-b0GE1E5E’2E3E4

+a2

-a1

a0

E2


Implementations

Kervin-Huelsman-Newcomb Tow-Thomson Fleischer-Tow …. Fleischer-Laker


Implementations (ii)Σ-1/sb1-1/sΣ-b0Ga0E1E5E6E3E4

Analog Filters: Biquad Circuits

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