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Analysis and Application of Analog Electronic Circuits to Biomedical Instrumentation - Northrop.pdf
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Analysis and Application of Analog Electronic Circuits

 

Q = 1 2 ξ =

R1R2R3 (R3

+ R1)

(7.22)

 

 

 

 

R2 (2R3 + R1)

 

 

 

The S & K BPF’s peak gain and Q are independent of C; C can be used to set ωn, once the three resistors are chosen to find the desired Q and peak gain. This is not as easy to design as the S & K LPF and HPF filter.

Before closing this section on Sallen and Key controlled (voltage)-source quadratic filters, it should be noted that the designer must simulate the filter’s behavior with an accurate model of the op amp to be used in order to predict the filter’s behavior at high frequencies (near the op amp’s fT), as well as how the filter behaves with transient inputs.

7.2.3Biquad Active Filters

For many filter applications, the biquad architecture offers ease in design when two-pole filters are desired; independent setting of ωn, ξ and mid-band gain, Kvo, are possible with few components (Rs and Cs). There are two biquad AF architectures: one-loop and two-loop. One cost of the biquad AFs is that they use more than one op amp (three or four) to realize that which an S & K AF can do with one op amp. The benefit of the biquad architecture is ease in design. Figure 7.5 shows the oneand two-loop configurations.

The two-loop biquad’s transfer functions will be analyzed at the V3, V4, and V5 output nodes. The output at the V5 node is low pass; this can be shown by writing the transfer function using Mason’s rule (Northrop, 1990, Appendix A):

V

(s) =

(R R1)(R R2 )(−1 sR3C)(−1 sR3C)

 

5

1− [(−1)(R R2 )(−1 sR3C)+ (−1)(−1 sR3C)2 ]

(7.23)

VS

This LPF transfer function can be simplified algebraically to time-constant form:

 

V5

(s) =

 

 

 

R2 R1R2

 

 

 

(7.24)

 

 

2

2

C

2

]

+ 1

 

V

 

s

R

+ s RR C R

 

 

S

 

 

3

 

[

3 2

 

 

 

By comparison with the standard quadratic polynomial form,

 

ωn = 1/R3C r/s (ωn set by R3 and C)

 

 

 

(7.25A)

ξ = R/2R2

(damping factor set by R2)

(7.25B)

Kvo = R2/R1R2

(dc gain set by R1, once R2 is chosen)

(7.25C)

© 2004 by CRC Press LLC

Analog Active Filters

 

 

289

 

R3

 

 

 

C

 

 

 

 

C

R

 

 

 

R4

R1 V2 R2

V3

R

VS

 

 

 

 

 

 

V4

 

IOA

IOA

IOA

A

 

R

 

 

 

 

R

 

R

C

 

 

 

 

R1

V2

R2

V3

R3

VS

 

 

 

V4

 

 

 

 

 

IOA

 

IOA

IOA

 

 

R

 

 

 

B

 

C

 

 

 

R3

 

 

 

 

 

 

V5

 

IOA

 

 

 

 

 

FIGURE 7.5

(A) The three-op amp, one-loop biquad active filter. (B) The four-op amp, two-loop biquad active filter.

Next, consider the transfer function for V4. Note that the denominator is the same as for the V5 transfer function:

V

(s) =

(−R R1)(−R R2 )(1 sR3C)

 

4

1[(1)(−R R2 )(1 sR3C)+ (1)(1 sR3C)2 ]

(7.26)

VS

Multiplying the numerator and denominator of Equation 7.26 by

(–1/sR3C)2 yields:

 

 

 

 

 

 

 

 

 

V4

 

−s R2R C

R R

 

 

 

 

(s) =

 

3

(

1 2 )

 

(7.27)

 

S

2 2 2

[

3

2

]

 

 

 

3

+ 1

 

V

 

s R C

+ s RR C R

 

which is a quadratic BPF. The filter parameters are:

© 2004 by CRC Press LLC

290

 

 

Analysis and Application of Analog Electronic Circuits

 

 

ωn = 1/R3C r/s (ωn set by R3 and C)

(7.28A)

 

 

Q = R2/R

(Q set by R2: Q = 1/2 ξ)

(7.28B)

 

 

Kvo = R/R1

(peak gain at ωn set by R1)

(7.28C)

Finally, examine the transfer function to the V3 node. By Mason’s rule,

 

V3

(s) =

 

(R R1)(R R2 )

 

 

 

1− [(−1)(R R2 )(−1 sR3C)+ (−1)(−1 sR3C)2 ]

(7.29)

 

VS

Again, divide numerator and denominator of Equation 7.29 by (−1/sR3C)2 and obtain the transfer function of a quadratic high-pass filter:

 

V4

 

s2 R2R 2C2

R R

 

 

 

 

(s) =

 

3

(

1

2 )

 

(7.30)

 

S

2 2 2

[

3

 

2

]

 

 

 

3

 

+ 1

 

V

 

s R C

+ s RR C R

 

As before:

 

 

 

 

 

 

 

 

 

 

 

 

ωn = 1/R3C r/s

 

 

 

 

(7.31A)

 

 

 

ξ = R/2R2

 

 

 

 

 

(7.31B)

 

 

 

Kvhi = R2/(R1R2)

 

 

 

 

(7.31C)

An interesting benefit from the biquad AF design is the ability to realize quadratic notch and all-pass configurations with the aid of another op amp adder. An ideal notch filter has a pair of conjugate zeros on the jω axis in the s-plane; its transfer function is of the form:

V

(s) =

 

s2 ω 2

+ 1

(7.32)

n

 

n

 

 

s2

ωn2 + s (Qωn )+ 1

VS

Clearly, when s = jωn, Vn/VS (jωn) = 0. Figure 7.6 illustrates how a biquad AF and an op amp adder can be used to realize a notch filter of the form modeled by Equation 7.32.

The biquad + op amp summer architecture can also be used to realize a quadratic all-pass filter in which a pair of complex-conjugate zeros is found in the right-half s-plane with positive real parts the same magnitude as the C-C poles in the left-half s-plane. The general form of this filter is:

Vap

(s) =

s2

ωn2 s2ξ ωn + 1

(7.33)

V

s2

ω

2

+ s2ξ ω

n

+ 1

 

S

 

 

 

n

 

 

 

 

© 2004 by CRC Press LLC

Analog Active Filters

291

 

 

s ωn /Q

 

 

+

s2

+ s ωn /Q + ωn2

 

 

 

 

 

 

VS

 

 

 

 

 

 

Vn

 

 

 

 

 

 

A

 

 

 

 

 

 

 

R

 

LPF

V5

 

 

Basic

 

R

biquad

BPF

V4

Vn

 

 

 

HPF

V3

 

 

VS

 

R

B

FIGURE 7.6

(A) Block diagram showing how a notch filter can be formed from a quadratic BF. (B) A practical circuit showing how a two-loop biquad’s V4 BPF output can be added to Vs to make the notch filter.

 

 

V5

R6

 

LPF

 

 

 

 

Basic

 

V4

R4

biquad

BPF

 

 

Vap

 

 

V3

 

HPF

IOA

 

 

 

VS

 

 

R5

FIGURE 7.7

A practical circuit showing how a two-loop biquad’s V4 BPF output can be added to Vs to make a quadratic all-pass filter.

Figure 7.7 illustrates the circuit architecture used to realize a symmetrical APF. The APF output is given by:

Vap = R6[VS R5 + V4 R4 ]

(7.34)

Now substitute the transfer function for the BPF into Equation 7.34 to get the APF transfer function:

Vap

 

 

 

(R6 R4 )(−s R2 R3C R1R2 )

˘

 

 

 

 

 

 

 

 

(s) = − R6 R5

+

2 2

 

2

[

3

 

 

 

2

]

+

 

˙

 

 

 

 

 

(7.35)

 

S

 

 

 

 

 

 

 

 

 

 

 

 

 

s R

3

C

 

C R

1

˙

 

 

 

 

 

 

 

V

 

 

 

 

 

+ s RR

 

 

˚

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(R6 R5 )s2 R32C2 + (R6 R5 )s[RR3C R2 ]+ (R6

R5 ) (R6

R4 )(s R2 R3C R1R2 )˘

 

= −

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

˙

 

 

 

 

 

 

 

 

 

2

 

 

2

 

 

2

 

[

3

 

2

]

+ 1

 

 

 

 

 

 

 

 

 

 

 

s

R

3

C

 

 

C R

 

˙

 

 

 

 

 

 

 

 

 

 

 

 

 

+ s RR

 

 

 

˚

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

© 2004 by CRC Press LLC