Cascade Active Filters

Cascade Active FiltersWe’ve seen that odd-order filters may be built by cascading second-order stages or sections (with appropriately chosen poles, zeros and gain) and one first-order section, and even order filters can be built by cascading second order sections. For example:

n=1 n=2 n=2

Cascade Active Filters

The transfer function of a second-order section may be written as:

22

2

kkk

kk

ss

AsH

And that of a first order stage can be written as:

0

00

s

AsH

The values of and are given in Cartinhour’s tables

Cascade Active Filters

Active first- and second-order filter sections may be realized as active networks. One opamp is sufficient for a first-order section, second order sections may be built using one, two, three or four opamps. We’ll use the single-amplifier realization.

A first-order section is shown in the next slide:

Cascade Active Filters

R

C

-

+

Ra

Rf

X(s)Y(s)

RCs

RCR

RsH

a

f

1

1

1)(

Cascade Active FiltersHere’s a Sallen and Key second-order section:

R

C

-

+

Ra

Rf

X(s)Y(s)

R

C

Cascade Active Filters

22

2

1113

11

)(

RCs

RCR

Rs

RCR

R

sH

a

f

a

f

This filter has the following transfer function:

22

2

kkk

kk

ss

AsH

And we want to set it’s coefficients equal to:

Cascade Active Filters

So we’ll pick resistors and capacitors such that

kka

f

k

ka

f

AR

R

RC

AR

R

313

1

Cascade Active FiltersNote that Ak is determined by k and k,, so we can’t freely determine the section gain – it’s determined by the parameters of the quadratic equation in the denominator of the transfer function.

We do need to be able to set the overall filter gain, which can be done by selecting the gain for the first-order section (if the filter order is odd) or by an additional gain stage (if the filter order is even.

Cascade Active FiltersThere are other procedures for selecting the component values for the Sallen and Key filter, some of which may allow more freedom in selction of the section gain. Some procedures are optimized for a particular quality, such as insensitivity to variations in component value. There are also other second-order filter topologies (circuits) using from 1 to 4 opamps, each of which has its own advantages and disadvantages. This isn’t an active filter course, so we won’t consider the others.

Cascade Active Filters

One more thing: Cartinhour states on p. 95 that active filters are not suitable for frequencies above the audio range, because of opamp frequency response limitations. This is no longer true, because opamps have been vastly improved over the last 15-20 years in every respect, and some are now suitable for use in the RF range. At high frequencies, selection of the component type, and circuit board layout become critical, however.

Highpass Filters

We can take a normalized lowpass filter (c = 1, in Cartinhour’s nomenclature) and transform it to a denormalized highpass filter. Recall that to denormalize a lowpass prototype, we substituted s/c

for s in H(s). To denormalize and transform to highpass, we substitute c/s for s.

Highpass Filters If we take a second order section of the form:

22

2

kkk

kk

ss

AsH

Now substitute c/s for s:

22

2

kc

kkc

kkc

ss

A

sH

Highpass Filters Multiplying by s2/s2:

222

22

ss

sA

sH

kckkc

kkc

and then dividing the numerator and denominator by k

2:

2

22

2

k

c

k

ck

kc

ss

sA

sH

where cc f2

Highpass Filters This way of deriving the transformation is somewhat different than Cartinhour’s, but the results are equivalent. Let’s try a second order Butterworth highpass filter, with fc = 10 Hz. From the tables:

7071.0Re

1

414.1

k

k

k

p

p

Highpass Filters Using Cartinhour’s procedure,

22

2

22

2

)3944()86.88()8.62()414.1)(8.62(

414.1Re2

zed)(denormali 62.8

ss

sA

ss

sAsH

p

p

p

kk

k

kk

k

ck

Highpass Filters

22

2

22

2

)3944()86.88()8.62()414.1)(8.62(

414.1

8.62102

d)(normalize 1

ss

sA

ss

sAsH kk

k

c

k

Using my procedure,

So the results are the same. Ya pays your dollar, ya takes yer choice.

Highpass Filters Either procedure may be used to transform a normalized Butterworth or Chebyshev lowpass prototype to a denormalized highpass filter of the same type.

As Cartinhour points out, the amplitude response of an Nth order Butterworth filter is:

N

c

a

f

f

GfH

2

1

)(

Highpass Filters

And that of a Chebyshev highpass filter is:

N

cN

a

f

fC

GfH

2

221

)(

These are similar to the lowpass amplitude responses, but (f/fc) has been replaced by (fc/f). Examining the plots on pages 97 and 98 of the book, the amplitude response of the highpass filter is simply that of the lowpass prototype, but reversed in frequency.

Active Highpass Filters

These are similar to active lowpass filters, and may be constructed by cascading first- and second-order sections. Here’s a first-order section:

R

C-

+

Ra

Rf

X(s)Y(s)

RCs

s

R

RsH

a

f

11)(

Active Highpass Filters

R

-

+

Ra

Rf

X(s)Y(s)

R

C C

Here’s a Sallen and Key highpass section:

Active Highpass Filters

22

2

1113

1

)(

RCs

RCR

Rs

sR

R

sH

a

f

a

f

22

2

kkk

k

ss

sAsH

And we want to set it’s coefficients equal to:

This filter has the transfer function:

Required Filter Order

Typically, we will have a requirement for a filter with a particular passband, stopband, minimum/maximum passband gain, and minimum stopband attenuation. The first thing we need to do is choose the filter type and order. We’ll assume the type has already been chosen, so we need to determine the order.

Required Filter Order

From Cartinhour for a Butterworth lowpass filter:

nattenuatio stopband minimum

frequency cutoff log

11

log21

frequency edge stopband 10

2

20

A

fx

aN

ff

fxa

c

uc

u

A

Required Filter Order

For other filter types, refer to Cartinhour, filter design software, or a book of filter tables.

Problems: