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DESIGN OF SINGLE FREQUENCY FILTERS

BY FORREST F. FULTON , JR .

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical <7iete

No. 23

August 20, 1959

Design of Single Frequency Filters

by

Forrest F. Fulton, Jr.

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Price $ .50

Design of Single Frequency Filters

Forrest F. Fulton, Jr.

ABSTRACT

Efficient procedures are shown for designing filters formed by

a number of identical resonant circuits loosely coupled together, and

which are required to accept one narrow band of frequencies and reject

another narrow band somewhat removed in frequency, without any

special requirements on the shape of the attenuation curve in between

these regions. The design is based on using a large number of

sections.

INTRODUCTION

A convenient form of filter for some applications is that formed

by a number of resonant circuits loosely coupled together, having all

circuits tuned to the same frequency. One application is in the

suppression of the spurious responses of a superheterodyne receiver

having a narrow i-f system, where the necessary filtering of the r-f

signal is merely to reject by some minimum attenuation all frequencies

separated from the signal by a fixed amount, without any special

requirements on the shape of the attenuation curve between the signal

frequency and the frequencies to be rejected. In general, any application where the desired signal is much narrower than the separation of

the signal and the frequencies to be rejected can conveniently use this

type of filter. Some typical response curves obtained for this type of

filter with different values of a coupling parameter, n, are shown in

Fig. 1.

The advantages of this type filter are that it can be properly

adjusted while it is in the equipment simply by maximizing the signal

transmission through it, that it is built of identical sections, and that

the characteristics are expressed simply in terms of the number of

sections, the circuit Q, and a coupling parameter.

It is the purpose of this paper to show two methods of

efficiently designing these filters to meet accurately different kinds of

specifications; a design for minimum insertion loss, and a design for

the minimum number of sections. The designs are based on the

assumption that a large enough number of sections will be used that the

performance can be calculated from the per-section characteristics of

an infinite string of sections.
_ 2 - BASIC RELATIONSHIPS

There are many configurations that can be used for this type of

filter with only minor changes in the calculations. Fig. 2 shows the

configuration used throughout this discussion; Fig. 3 shows some of

the alternative configurations and the changes in parameters to be used

with them.

The propagation through one section of a network that consists

of an infinite string of identical sections may be described by a

complex propagation constant P, which is the principal natural . logarithm of the ratio of the input and output currents of the section . The real and imaginary parts of P describe the attenuation and phase

characteristics respectively. For sections of the type shown in Fig. 2,

we find that

cosh P = C +2C, m 1 2C

1

C,C

1 m

C + 20. m 1

2

p + R

C,C

1 m

C + 2C. m 1 p + 1 (1)

where p is the complex frequency variable. Normalizing to put the

signal frequency at one radian per second, and making some definitions

to bring in useful parameters, this can be written for real frequencies

as:

/-\ -> (2)

where

cosh P = cosh (a + j|3) = —

n

2 . to 1 - to + jJ —

Q _

too

2 1 C

l Cm QJ — , f — ...... o LC' C +20. m 1

2 2C co L

— O , fr»f /"* —

1 OR Q

C

f ° r Dm mL ~" R *

To obtain the insertion loss at the signal frequency this

expression is evaluated at to = 1, giving

cosh P = cosh (a + jS ) = j —

o o o n

(3)

and

sinh aon= — for a nepers

o

(4)

Shea, T. E. , Transmission Networks and Wave Filters,

pp. 76-79, D. Van Nostrand Company, Inc. , New York.
- 3 -

The off-band loss can be approximated by

a = cosh — j 1 - co j in nepers (5)

the approximation being satisfactory for off-band losses greater than

three times the signal frequency loss.

DESIGN FOR MINIMUM INSERTION LOSS

In applications where the loss at the signal frequency must be

kept as low as possible, the optimum design is that which gives the

maximum for the ratio of loss at the frequency to be rejected to loss

at the signal frequency. It is shown in Appendix 1 that the condition for

this maximum is:

a coth a = a tanh a (6) o o

This equation is awkward to use, but the information it provides is given in

Fig. 4, which shows the value of a at the rejected frequency that results

from the optimum design, plotted against the Q of the coils used and the

relative frequency to be rejected. The graph shows that the off-band

loss resulting from the optimum design is very nearly constant at

1. 20 nepers per section, departing only a few percent from this value

toward the limit of the approximation, which occurs only for low Q

coils and very narrow filters. It is apparent that no significant error

will be incurred by assuming that the most efficient filter is obtained by

designing for a loss of 1. 20 nepers per section at the frequency to be

rejected.

With this parameter determined, the coupling parameter, n, can

be determined from equation (5) and the signal frequency loss from (4).

The difference between the midband loss, a , and the off-band loss, a, gives the rejection obtained per section, ana from this and the specified

total rejection required at the off-band frequency, the number of

sections can be determined. For these calculations, it is convenient to

rearrange (5) and put in the value of cosh 1. 20, obtaining:

n = 0.552 Q| 1-co

2

| (7)
- 4 -

Example - A filter for 2. 5 Mc, to reject an image at 2.4 Mc by 126 db,

built with coils having Q5 s of 300. For

2.4 Q = 300 and go = 275 '

(7) gives an n of 12.99. From (4) the signal frequency loss is

0. 077 nepers, giving a rejection of 1. 123 nepers or 9. 75 decibels per

section. For 126 decibels rejection, 13 sections will be required,

having a total signal frequency loss of 1. 00 nepers or 8. 686 decibels.

DESIGN FOR MINIMUM NUMBER OF SECTIONS

If the signal frequency loss that can be tolerated is somewhat

greater than the loss given by the minimum loss filter, then the

number of sections can be reduced, decreasing the cost and size of the

unit. For a filter having the minimum number of sections, the ratio of

required loss at the undesired frequency to the allowable loss at the

signal frequency is to be precisely equal to the ratio of the per section

loss at the undesired frequency to the per section loss at the signal

frequency. For this condition

j ££ i. j t cosh — 1 - w

required off-band loss n

J ' tnx . i.i. = r = i (8) allowable on-bandloss .,-11

sinn —

n

in which the coupling parameter n required for these sections is the

only unknown quantity. With the coupling parameter determined, the

number of sections necessary can be determined from the given allowable loss and the loss per section, obtained from (4).

An explicit solution of (8) for the value of the coupling parameter required for these sections is not practical, but fortunately it is

also not necessary; in general, the design as outlined above does not

come out with an integral number of sections, so the exact solution

seldom gives the final design. A more useful procedure is to determine the proper number of sections by using an approximate solution

of (8) and round off the number of sections to the next highest integer.

The number of sections and the given allowable loss then determine

the a for the section, o
- 5 - A suitable approximation can be quickly obtained from an

iterative process by solving for aQ rather than n, using the relationship (4). Designating the first assumed value of aQ by Gq1» a correc " tion, Ac , can be obtained from:

a oi(r

i " r)

Aa = (9) o 1

where ri is the ratio obtained for the assumed value, oni, and may be

approximated by:

r

l = 7- cosh1 (a

()1

Qll/-«o

2

l) (10)

Example: A filter for 2. 5 Mc, to reject an image at 2.4 Mc by 126 db,

having a signal loss of 1 2 db, using coils having Q8 s of 300.

The ratio of required loss at the image to allowable signal loss

is

iz

y2

iz

= n.so

For an assumed signal loss per section, Oqi, of 0. 2 nepers, the ratio

obtained would be approximately 11.15. The first correction, AaQ

, to

the assumed value cq1» * s -0.011, which is small enough that further

iteration would be profitless. For aQ

= 0.189 nepers or 1.64 db per

section, 1 2 db loss would be obtained with 7. 3 sections. Using eight

sections with a loss of 1. 50 db or 0.173 nepers per section, (4) gives a

coupling ratio of 5. 76. From (5), the loss at the image frequency is

2. 09 nepers, or 1 8. 2 db per section; the total loss at the image is

145 db, giving 133 db rejection. Thus, 8 sections give more rejection

than is required, but a smaller number would give too little, and 8 is

the minimum that can be used.
- 6 -

APPENDIX: CONDITION FOR MINIMUM INSERTION LOSS

The necessary condition for the minimum insertion loss design

is that the ratio of loss at the frequency to be rejected to loss at the

signal frequency be a maximum, which means that the derivative of

this ratio be zero. The derivative is conveniently taken with respect

to the coupling parameter n. This is:

/ a \ da da

d( J a a —

\a / odn dn

o dn ao

and the condition for zero is

ao

since

da da

o dn dn

a = smh —

o n

da a sinh a o o dn n cosh ao

-1

n coth ao

and

a - cosh — [ 1 - co n ' '

da -1

dn n tanh a

the condition for minimum insertion loss becomes:

a coth a = a tanh a o o
- 7 -

For low values of a Q

, the quantity aQ cotha departs slowly

from the limiting value of 1. 00, reaching 1. 05 at a = 0. 4 nepers per

section. On the other hand, the quantity a tanh a has the value 1. 00 at

a = 1. 20 nepers, and varies relatively rapidly, reaching 1. 05 at

a = 1. 24 nepers.
TYPICAL PER- SECTION FREQUENCY RESPONSE CURVES

FOR SECTIONS OF THE TYPE SHOWN IN FIGURE 2,

USING A Q OF 300

cu, Normalized Frequency

Figure
FILTER SECTIONS USED FOR CALCULATIONS

L R c i o * iRm>—vv\/—|F

Cm Cm

2

Figure 2

ALTERNATIVE CONFIGURATIONS FOR FILTER SECTIONS

Lm Lm

2

-\ , 2

* 0000 *-

L, 1 G<

O < o

i

> 1c

T

o . 1 —

o

L =

Q =

n = Q

L| Lm

2L, + L m w C

Lm

2L

2Cm 2Cm

1

J

C= C,+ 2Cm

Q= —COqC

G

n = Q

2Cm

Note: frequency scale inverted; —— = ja;

2Lm

-^WOTT1 W\/ 1| 1—

o

a

s 2Lm

L= L, + 2Lm

Q =

cu L

n = Q

2Lm

L

Note: frequency scale

inverted; —— = jcu

Figure 3
Q

LU < h-o

CO<

LU

UJ

LT CO

(/-) LU o LU 1 o 2 h- o ?! hcr >

UJ

O CO -z. 2 LU "— 3 2 Ohi

_j IT ^ U_ 2 Q 2 LUN

cr 1 o < Li- S

en LT CO O Q<

GOO

I-O <O

u.oo

I-o

-z. Z>

u_

^~. sj- CM3 1_ 1 3

CP o u_

SJ9d9N ' D
U.S. DEPARTMENT OF COMMERCE

Frederick H. Mueller, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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