Noty an27af Linear Technology

Application Note 27A

June 1988

A Simple Method of Designing Multiple Order All Pole
Bandpass Filters by Cascading 2nd Order Sections

Nello Sevastopoulos
Richard Markell

INTRODUCTION

Filter design, be it active, passive, or switched capacitor,
is traditionally a mathematically intensive pursuit. There
are many architectures and design methods to choose
from. Two methods of high order bandpass filter design
are discussed herein. These methods allow the filter
designer to simplify the mathematical design process

®

and allow LTC’s switched capacitor filters (LTC

1059,
LTC1060, LTC1061,LTC1064) to be utilized as high quality
bandpass filters.

The first method consists of the traditional cascading of
non-identical 2nd order bandpass sections to form the
familiar Butterworth and Chebyshev bandpass filters.
The second method consists of cascading identical 2nd
order bandpass sections. This approach, although “non­textbook,” enables the hardware to be simple and the
mathematics to be straightforward. Both methods will
be described here.

AN27A is the first of a series of application notes from
LTC concerning our universal filter family. Additional notes
in the series will discuss notch, lowpass and highpass
filters implemented with the universal switched capacitor
filter. An addition to this note will extend the treatment of
bandpass filters to the elliptic or Cauer forms.

This note will first present a finished design example and
proceed to present the design methodology, which relies on
tabular simplification of traditional filter design techniques.

DESIGNING BANDPASS FILTERS

Table 1 was developed to enable anyone to design But­terworth bandpass filters. We will discuss the tables in
more detail later in this paper, but let’s first design a filter.

EXAMPLE 1—DESIGN

A 4th order 2kHz Butterworth bandpass filter with a –3dB
bandwidth equal to 200Hz is required as shown in Figure 1.

Noting that (f

/BW) = 10/1 we can go directly to Table1

oBP

for our normalized center frequencies. From Table 1
under 4th order Butterworth bandpass filters, we go to

/BW) = 10.

(f

oBP

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Technology Corporation. All other trademarks are the property of their respective owners.

0

–3

–10

–20

–30

GAIN (dB)

–40

–50

–60

0.5k 0.7k 1k 2k 3k 4k 5k

Figure 1. 4th Order Butterworth BP Filter, f

FREQUENCY (Hz)

200Hz

AN27A F01

oPB

10k

= 2kHz

an28f

AN27A-1

Application Note 27A

We find fo1 = 0.965 and fo2 = 1.036 (both normalized to

= 1). To find our desired actual center frequencies,

f

oBP

we must multiply by f
and f

The Qs are Q

= 2.072kHz.

o2

1

= Q2 = 14.2 which is read directly from

= 2kHz to obtain fo1 = 1.930kHz

oBP

Table 1. Also available from the table is K, which is the
product of each individual bandpass gain H

. To put it

oBP

another way, the value of K is the gain required to make
the gain, H, of the overall filter equal to 1 at f

. Our filter

oBP

parameters are highlighted in the following table:

f

oBP

2kHz 1.93kHz 2.072kHz Q1 = Q2 = 14.2 2.03

f

o1

f

o2

Qs K

HARDWARE IMPLEMENTATION

Universal switched capacitor filters are simple to imple­ment. A bandpass filter can be built from the traditional
state-variable filter topology. Figure 2 shows this topology
for both switched capacitor and active operational amplifier
implementations. Our example requires four resistors for
each 2nd order section. So eight resistors are required
to build our filter.

We start with two 2nd order sections (1 LTC1060,
2/3 LTC1061 or 1/2 LTC1064), Figure 3.

We associate resistors as belonging to 2nd order sections,
so R1x belongs to the x section. Thus R12, R22, R33 and

R42 all belong to the second of two 2nd order sections
in our example.

Our requirements are shown in the following table:

SECTION 1 SECTION 2

= 1.93kHz

f

o1

Q1 = 14.2
H

= 1

oBP1

Note that H
choosing H

oBP1

oBP2

× H

oBP2

= 2.03.

= K and this is the reason for

For this example we choose the fo=

= 2.072kHz

f

o2

Q2 = 14.2

H

= 2.03

oBP2

f

CLK

50

R2
R4

mode,

so we will tie the 50/100/Hold pin on the SCF chip to V+,
generally (5V to 7V). We choose 100kHz as our clock and
calculate resistor values. Choosing the nearest 1% resistor
values we can implement the filter using Figure3’s topol­ogy and the resistor values listed below.

R11 = 147k R12 = 71.5k
R21 = 10k R22 = 10.7k
R31 = 147k R32 = 147k
R41 = 10.7k R42 = 10k

Our design is now complete. We have only to generate a
TTL or CMOS compatible clock at 100kHz, which we feed
to the clock pin of the switched capacitor filter, and we
should be “on the air.”

STATE VARIABLE SCF ACTIVE (OP-AMP) STATE VARIABLE

R4

R3

R2 R2

HP

R1

V

IN

–

+

AGND

f

R2

R3

fo=

CLK

100(50)

|

R4

Q =

R2

|

H

oHP

R2

R4

S BP LP

–

+

1/2 LTC1060
1/3 LTC1061
1/4 LTC1064

= –R2 /R1 H

MODE 3

∫ ∫

= –R3 /R1 H

oBP

oLP

= –R4 /R1

R1

V

IN

–

R5

+

1

fO=

2πRC

Figure 2. Switched Capacitor vs Active RC State Variable Topology

AN27A-2

R4

C

C

R

–

R

–

+

R6

3/4 LTC1014

R2
R4

+

AN27A F02

an28f

Application Note 27A

R41

R31

R21

R11

V

IN

–

+

fo=

f

CLK

50

R3

R2

; Q=

R2

R4

Figure 3. Two 2nd Order Sections Cascaded to Form 4th Order BP Filiter

HP

–

+

–

1/2 LTC1060

R2

H

oBP

R4

S1

A

∫ ∫

= –R3 /R1

BP

LP

DESIGNING BANDPASS FILTERS—THEORY BEHIND
THE DESIGN

Traditionally, bandpass filters have been designed by labori­ous calculations requiring some time to complete. At the
present time programs for various personal or laboratory
computers are often used. In either case, no small amount
of time and/or money is involved to evaluate, and later
test, a filter design.

R42

R32

R12

R22

–

+

HP

–

+

–

1/2 LTC1060

S1

A

∫ ∫

BP

LP

BP
OUTPUT

AN27A F03

with the required characteristics (generally the Qs are too
high). We wish to explore here the use of cascaded identical
2nd order sections for building high Q bandpass filters.

For a 2nd order bandpass filter

2

1–G

Q =

G

×

1– f /f

f/f

o

( )

o

2

(1)

Many designers have inquired as to the feasibility of cas­cading 2nd order bandpass sections of relatively low Q to
obtain more selective, higher Q, filters. This approach is
ideally suited to the LTC family of switched capacitor filters
(LTC1059, LTC1060, LTC1061 and LTC1064). The clock
to center frequency ratio accuracy of a typical “Mode1”
design with non “A” parts is better than 1% in a design
that simply requires three resistors of 1% tolerance or
better. Also, no expensive high precision film capacitors
are required as in the active op amp state variable design.

We present here an approach for designing bandpass filters
using the LTC1059, LTC1060, LTC1061 or the LTC1064
which many designers have “on the air” in days instead
of weeks.

CASCADING IDENTICAL 2ND ORDER BANDPASS SECTIONS

When we want to detect single frequency tones and si­multaneously reject signals in close proximity, simple 2nd
order bandpass filters often do the job. However, there are
cases where a 2nd order section cannot be implemented

Where Q is the required filter quality factor

f is the frequency where the filter should have gain, G,
expressed in Volts/V.

is at the filter center frequency. Unity gain is assumed

f

o

.

at f

o

EXAMPLE 2—DESIGN

We wish to design a 2nd order BP filter to pass 150Hz
and to attenuate 60Hz by 50dB. The required Q may be
calculated from Equation (1):

2

So,Q =

1– 3.162× 10

( )

3.162× 10

–3

–3

60 /150

×

1– 60/ 150

( )

= 150.7

2

This very high Q dictates a –3dB bandwidth of 1Hz.

Although the universal switched capacitor filters can
realize such high Qs, their guaranteed center frequency
accuracy of ±0.3%, although impressive, is not enough to
pass the 150Hz signal without gain error. According to the

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AN27A-3

Application Note 27A

f

–R3

R3

previous equation, the gain at 150Hz will be 1 ±26%; the
rejection, however, at 60Hz will remain at –50dB. The gain
inaccuracy can be corrected by tuning resistor R4 when
mode 3, Figure 2, is used. Also, if only detection of the
signal is sought, the gain inaccuracy could be acceptable.

This high Q problem can be solved by cascading two identi­cal 2nd order bandpass sections. To achieve a gain, G, at
frequency f the required Q of each 2nd order section is:

1–G

Q =

G

×

1– f / f

f / f

o

( )

o

2

(2)

The gain at each bandpass section is assumed unity.

In order to obtain 50dB attenuation at 60Hz, and still pass
150Hz, we will use two identical 2nd order sections.

We can calculate the required Q for each of two 2nd order
sections from Equation (2):

So,Q =

1–3.162× 10

3.162× 10

–3

–3

×

60 /150

1– 60/ 150

( )

= 8.5!!

2

With two identical 2nd order sections each with a poten­tial error in center frequency, f

, of ±0.3% the gain error

o

at 150Hz is 1 ±0.26%. If lower cost (non “A” versions
of LTC1060 and LTC1064) 2nd order bandpass sections
are used with an fo tolerance of ±0.8%, the gain error at
150Hz is 1 ±1.8%! The benefits of lower Q sections are
therefore obvious.

HARDWARE IMPLEMENTATION

Mode 1 Operation of LTC1060, LTC1061, LTC1064

As previously discussed, we associate resistors with each
2nd order section, so R1x belongs to x section. Thus R12,
R22 and R23 belong to the second of the two 2nd order
sections, Figure 4.

Each section has the same requirements as shown:

= fo2 = 150Hz

f

o1

Q1 = Q = 8.5
H

oBP1

= H

oBP2

= 1

Note that we could get gain out of our BP filter structure
by letting the product of the H

terms be >1 (within the

oBP

performance limits of the filter itself).

For our example using the LTC1060 we will use f

/100. So we input a 15kHz clock and tie the 50/100/

= f

CLK

= fo2

o1

Hold pin to mid-supplies (ground for ±5V supplies).

We can implement this filter using the two sections of an
LTC1060 filter operated in mode 1. Mode 1 is the fastest
operating mode of the switched capacitor filters. It provides
Lowpass, Bandpass and Notch outputs.

Each 2nd order section will perform approximately as
shown in Figure 5, curve (a).

Implementation in mode 1 is simple as only three resistors
are required per section. Since we are cascading identical
sections, the calculations are also simple.

R31

R21

R11

V

IN

AN27A-4

–

+

AGND AGND

CLK

fo=

H

=

oBP

100

R1

S

N

–

+

1/2 LTC1060 1/2 LTC1060

Q =

∫ ∫

–

R2

Figure 4. LTC1060 as BP Filter Operating in Mode 1

BP

LP

R12

R32

R22

–

+

S

N

–

+

∫ ∫

–

BP

AN27A F04

LP

BP
OUTPUT

an28f

Application Note 27A

( )

We can calculate the resistor values from the indicated
formulas and then choose 1% values. (Note that we let
our minimum value be 20k.) The required values are:

R11 = R12 = 169k
R21 = R22 = 20k
R31 = R32 = 169k

Our design is complete. The performance of two 2nd order
sections cascaded versus one 2nd order section is shown
in Figure 5, curve (b). We must, however, generate a TTL
or CMOS clock at 15kHz to operate the filter.

Mode 2 Operation of LTC1060 Family

Suppose that we have no 15kHz clock source readily avail­able. We can use what is referred to as mode 2, which
allows the input clock frequency to be less than 50:1 or

0

–10

–20

–30

–40

BANDPASS GAIN (dB)

–50

–60

30 50 100 150 300 500

FREQUENCY (Hz)

Figure 5. Cascading Two 2nd Order BP Sections for Higher
Q Response

(a): ONE SECTION

(b): TWO SECTIONS

1k

AN27A F05

100:1 [f

= 50 or 100]. This still depends on the con-

CLK/fo

nection of the 50/100/Hold pin.

If we wish to operate our previous filter from a television
crystal at 14.318MHz we could divide this frequency by
1000 to give us a clock of 14.318kHz. We could then set
up our mode 2 filter as shown in Figure 6.

We can calculate the resistor values from the formulas
shown and then choose 1% values. The required values are:

R11, R12 = 162k
R21, R22 = 20k
R31, R32 = 162k
R41, R42 = 205k

CASCADING MORE THAN TWO IDENTICAL 2ND ORDER BP SECTIONS

If more than two identical bandpass sections (2nd order)
are cascaded, the required Q of each section may be
shown to be:

2/n

1–G

Q =

1/n

G

where Q, G, f and f

f / f

1– f / f

( )

o

2

o

×

are as previously defined and n = the

o

(3)

number of cascaded 2nd order sections.

R41

R31

R21

R11

V

IN

–

+

f

R2

CLK

fo=

1+

Q =

R4

100

H

=R3/ R1

oBP

N

–

+

–

1/2 LTC1060

R3

R2

1+

R2

R4

S

∫ ∫

BP

LP

R12

R42

R32

R22

S

N

BP

LP

–

–

+

+

1/2 LTC1060

∫ ∫

–

BP
OUTPUT

AN27A F06

Figure 6. LTC1060 as BP Filter Operating in Mode 2

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AN27A-5