January 1969 Electronics World
Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
Electronics World, published May 1959
- December 1971. All copyrights hereby acknowledged.
of IC-based digital circuits was relatively new in 1969 when this story was printed in Electronics
World magazine and was probably considered expert knowledge. Note that it was written by a research
engineer at the Lockheed Missiles and Space Company. Today, introductory digital logic courses begin
with material presented here and advance rapidly into programmable logic, ASICs microprocessors, and
beyond. Even if you have already taken courses on digital logic with counters and dividers, the knowledge
might be in a dusty corner of the gray matter and could stand a refresh. One of my main motivations
for studying for and obtaining my Amateur Radio licenses (recently promoted to Extra level), was to
have an excuse to review and re-learn concepts first studied long ago.
I have not yet acquired the issue of Electronics World in which
Part 1 was published.
IC Frequency Dividers & Counters
By Donald L. Steinbach / Research Engineer
Lockheed Missiles and Space Co.
Complete frequency divider and counter systems. Synchronous dividers for division ratios from two
through ten are given, along with a simple decade counter using inexpensive, readily available integrated
Part 1 of this article discussed in some detail the characteristics of a family of IC flip-flops,
gates, and buffers. In this part we extend this information to complete frequency divider and counter
Logic Elements in Dividers and Counters
The frequency dividers and counters described in this article are made up of one or more JK FF's
(flip-flops) interconnected in such a way that as each CP (clock pulse) arrives at the divider input
one or more of the FF's in the divider change state. This is accomplished by "forcing" the FF's to particular
states by use of the S and C inputs and/or appropriate selection of the source for input T. In some
cases gates are used to derive signals for S and C that do not already exist somewhere in the divider.
Buffers are used as required to increase drive levels and/or provide isolation from external circuitry.
Fig. 1 - These are the before and after flip-flop states that exist for all possible
Fig. 2 - All possible states of divider with up to 4 FF's.
Fig. 3 - Circuit and operation of synchronous n = 3 divider.
Fig. 4 - Synchronous n = 4 divider with an n = 2 output.
Fig. 5 - Circuit and operation of synchronous n = 5 divider.
Fig. 6 - Circuit and operation of synchronous n = 6 divider.
Fig. 7 - Circuit and operation of synchronous n = 7 divider.
Fig. 8 - Synchronous n = 8 divider with synchronous n = 2,4 outputs.
Fig. 9 - Circuit and operation of synchronous n = 9 divider.
Fig. 10 - Arrangement employed for synchronous n = 10 divider with its counting sequence
chosen for easy decoding.
The frequency divider has a division ration if its output waveform passes through one complete cycle
as its input waveform passes through n complete cycles. The number of FF's required in a particular
divider is determined by the desired division ratio. The highest possible division ratio for a given
number of FF's is 2x where x is the number of FF's in the divider. Thus, one FF is needed
to divide by two; two FF's are needed to divide by three or four; three FF's are needed to divide by
five, six, seven, or eight, etc.
The signal input connected to T (pin 2 of the particular C discussed last month in Part 1) of each
FF in the divider will be either the incoming CP or the output of a preceding FF. If T of each FF is
connected to the incoming CP, the divider is a synchronous divider. If the incoming CP is connected
to T of the first FF only, and T of the second FF is connected to the output (Q or
Q) of the first FF, etc., then the divider is called an asynchronous
The FF propagation delays in the asynchronous divider are cumulative and the time between CP's must
be sufficient to allow each FF in the divider string to change state. In general, no more than about
six 9923 JK FF's should be used in anyone asynchronous divider intended to operate at an input frequency
of 2 MHz. All FF's in the synchronous divider are triggered simultaneously and the time delay between
the CP and the resulting change in divider output is equal to the propagation delay time of one FF rather
than the combined propagation delays of a string of FF's. Synchronous dividers should always be used
when the divider input and output waveforms must be in synchronism.
The control inputs 5 and C (pins 1 and 3 respectively) of a particular FF are connected to ground
(0 level), + VCC (1 level), or the outputs of other FF's either directly or via gates. The
level applied to S, the level applied to C, and the "present" FF state determine the state of the FF
after the next CP or 1-to-0 transition of a preceding FF. Fig. 1 is an expanded version of the truth
table in Fig. 7 of Part 1. It lists all possible combinations of S, C, and JK FF states that might exist
before the arrival of a CP and gives the FF state that will then result after the arrival of the CP.
Keep in mind that a CP is simply a 1-to-0 transition at T (pin 2) and that the state of the FF is the
level at Q (pin 7).
If Preset (pin 6) of all the 9923 FF's in a divider are connected together, all of the FF's will
be forced to the 0 state (output Q at the 0 level) when this "preset line" is momentarily connected
to VCC (+3.6 V d.c.). This technique provides a convenient starting point for the dividing
action both in the operating circuit and on paper.
It is customary to define the instantaneous state of the divider as the states of the FF's in the
divider written in some logical order. Thus, if the divider is made up of four FF's labeled FF1, FF2,
FF3, and FF4, and FF states are 1, 0, 1, and 1, respectively, then the state of the divider is 1011.
Since each FF has two states (1 and 0), the number of possible divider states is 2x where x is the number
of FF's in the divider. All possible states of a divider having 1, 2, 3, or 4 FF's are tabulated in
Circuit waveforms for the more complex dividers are determined from a state table. The state table
is a CP-by-CP tabulation of the levels at S, C, and Q of every FF in the divider. It is most convenient
to assume that the divider starts from the Preset state (i.e., all Q's at 0). The levels of each FF
S and C input are then determined from the divider schematic. Knowing S, C, and Q, the FF states after
the first CP may then be determined from Fig. 1. The "new" S and C levels are determined and the FF
states after the second CP are determined. This process is continued until the state table begins to
repeat itself, indicating that one complete division cycle has occurred.
The completed state table should be compared with Fig. 2 to determine which (if any) of the possible
divider states in Fig. 2 do not appear in the state table. Additional state tables are then constructed
using each of these "unused" divider states as the initial starting point in order to determine if the
divider will recover and divide by the desired ratio. If it does not, two courses of action are available:
redesign the divider, or make provisions for Presetting the divider.
There may be more than one circuit that will yield a particular division ratio. The circuit finally
chosen will usually be the one that uses the fewest components or provides the most desirable output
waveform for the particular application. It frequently happens that more than one division ratio can
be obtained from a single divider. For example, a divide-by-ten circuit may be able to simultaneously
deliver a divide-by-two or divide-by-five output from some point in the circuit.
The circuits to follow are drawn using the logic symbols of the IC devices. Refer to Fig. 8 in Part
1 for the actual Fairchild IC pin numbers. Although not shown, pin 4 of each IC is grounded and pin
8 of each IC is connected to VCC (+3.6 V d.c.). If the Preset feature of the JK FF's is employed,
then connect pin 6 of each of the FF's together and connect this to VCC through a normally
open momentary switch.
Either Q or Q of any FF in the divider may be chosen as the divider
output (s). For a given FF, the more lightly loaded of the two output terminals is usually used, although
this is not mandatory as long as the output drive factor of the FF is not exceeded.
In the figures that follow, the input signal is drawn as a square wave only for purposes of illustration.
The input waveform may be of any shape as long as the fall-time is small enough to be accepted by the
FF's as a clock pulse. The area marked "first complete division cycle" is the waveform that will repeat
with every n clock pulse.
Dividing by Two
The simplest possible frequency divider is an n =2 divider made from a single JK FF. If the S and
C inputs are both (permanently) at 0, the FF changes state with each CP. If the FF is initially in the
0 state, it will change to the 1 state when the first CP arrives. When the second CP arrives, the FF
returns to the 0 state. On the third CP, the FF changes back to the 1 state, completing the output cycle.
The FF state alternates with each consecutive CP and the output frequency is one-half the input frequency
- or the output waveform period is twice the input waveform period.
Dividing by Three
Then n = 3 divider in Fig. 3 is a synchronous divider - the CP is applied simultaneously to the T
input of both FF's. The input load factor is 10 since each FF has an input load factor of 5.
The state table for the divider of Fig. 3 is constructed as follows:
a. After Preset, Q1 (output Q of FF1) and Q2 (output Q of FF2) are both 0. These 0's are entered
in the Q1 and Q2 columns on the Preset line of the table.
b. Now that Q1 and Q2 are known, all of the S and C levels may be determined directly from the divider
schematic: S1 = Q2 =1; C1 = 0; S2 = 0; and C2 =
Q1 = 1. These levels are entered in their appropriate columns on the
Preset line of the table.
c. The levels entered on the Preset line of the table are the levels at S, C, and Q that now exist
before the arrival of the first CP. The levels at Q1 and Q2 after the arrival of the first CP are obtained
directly from Fig. 1: Q1 = 1 and Q2 = 0. These levels for Q1 and Q2 are entered on the CP1 line of the
d. Now that Q1 and Q2 after CP1 are known, the S and C levels after CP1 are determined: S1 =
Q2 = 1; C1 = 0; S2 = 0; and C2 = Q1
= 0. These levels are entered in their appropriate columns on the CP1 line of the table.
e. Continuing in this manner, after CP2: Q1 = 1; and Q2 = 1. Also, S1 = 0; C1 = 0; S2 = 0; and C2
f. After CP3: Q1 = 0 and Q2 = 0. Also S1 = 1; C1 = 0; S2 = 0; and C2 = 1. This divider state is identical
to the Preset state; therefore, the cycle will be repetitive.
If a CP4 line were added to the table, it would look exactly the same as the CP1 line; a CP5 line
would be the same as the CP2 line; a CP6 line would be the same as the CP3 line, a CP7 line would be
the same as the CP1 line, etc. The waveforms in Fig. 3 are constructed directly from the information
in the state table.
The three states of the divider are 00, 10, and 11. Comparing these states with Fig. 2 reveals that
the 01 state is missing from the divider operating sequence. A state table using an initial state (U1
) of 01 demonstrates that the divider will perform exactly the same as when the initial state is 00;
the divider state one CP after the initial state is the same in both cases and the waveforms are identical.
Bear in mind that if the Preset function is used as explained earlier, this evaluation of the divider
recovery from an "unused" state is unnecessary. It is explained in this section only to demonstrate
Dividing by Four
The simplest means of dividing by four is to divide by two twice. The output of the first divider
is at one-half the input frequency and is connected to the input of the second divider. The second divider
divides the output frequency of the first divider by two and the resulting output frequency is one-fourth
the input frequency to the first divider. Simultaneous n = 2 and n = 4 outputs may be obtained from
this divider. The n = 2 output is synchronous, but the n = 4 output is asynchronous since it is delayed
from the input CP by the sum of the propagation delays of both flip-flops.
The synchronous divider in Fig. 4 also provides simultaneous n = 2 and n = 4 outputs. The synchronous
n = 4 output is obtained at the expense of slightly increased circuit complexity and a larger input
Dividing by Five, Six, Seven, Eight & Nine
Fig. 5 is a n = 5 divider. All outputs are synchronous and have the same waveshape, but are time-displaced
from one another. The circuit will recover from its unused states so the use of the Preset function
Fig. 6 is a simple synchronous n = 6 divider. In addition to the n = 6 output from FF3, n = 3 ouputs
are available from either FF1 or FF2. The divider has two unused states and will recover from either.
A division ratio of six may also be obtained by dividing by two and then by three (or vice versa).
A synchronous n = 7 divider appears in Fig. 7. The divider has one unused state and will recover
on the next CP. An asynchronous n = 7 divider can be constructed and requires one less gate than the
An asynchronous n = 8 divider is most easily assembled by cascading three n = 2 dividers. A synchronous
divider is shown in Fig. 8. Regardless of the method used, n = 2 and n = 4 outputs will also be available
and there are no unused states.
Fig. 9 is a synchronous n = 9 divider. It has seven unused states and will recover from each. Cascading
two n = 3 dividers will provide an asynchronous n = 9 output and a synchronous n = 3 output.
Dividing by Ten
Many n = 10 dividers have been devised due to their popularity in dividing and counting applications.
An asynchronous n = 10 divider could be built from an n = 2 divider and an n = 5 divider. A synchronous
n = 2 or n = 5 output (depending on which divider is connected to the incoming CP) would also be available.
The synchronous n = 10 divider in Fig. 10 operates in the so-called "shift mode." Although this divider
requires one additional FF and provides only n = 10 outputs, it is particularly useful in counting applications
as we shall see later. The divider has 22 unused states - an ideal opportunity to utilize the Preset
Typical frequency-divider systems consist of one or more divider stages cascaded to provide the desired
division ratios and outputs. A common application is that of cascading several n = 10 dividers to divide
a 1-MHz or 100-kHz signal down to 10 kHz, 1 kHz, etc. Buffers are used between divider stages when an
increase in drive level is required. They should also be provided on the output lines if external circuit
loading is appreciable.
Whatever the ultimate application, the first problem encountered is usually that of converting the
input waveform to a fast-fall-time pulse to act as a clock pulse for the dividers. The circuit of Fig.
11 will accept any input waveform and has been used by the author to drive some of the dividers in this
article at frequencies in excess of 10 MHz.
The circuit functions like a low-hysteresis Schmitt trigger that switches at a threshold voltage
of about 0.9 V d.c. D1 may be any signal diode - its only function is to protect IC1 from negative-going
inputs. Naturally, the voltage at pin 1 of IC1 should not exceed 3.6 volts in the positive direction.
The choice of C1 is based on the input signal amplitude and frequency. For inputs of 100 kHz and over,
a 0.1-μF capacitor is adequate. If R4 is set midway between the two trigger levels, the circuit will
operate reliably on a.c. inputs well under 100 m V peak-to-peak. If R3 and R4 are omitted, then the
minimum a.c. input must be on the order of 2 volts peak-to-peak.
The power supply requires no particular attention other than assuring that its output voltage is
low in ripple and transient-free. A full-wave rectifier followed by a filter capacitance of 25,000 μF
or more will be adequate on both counts. An allowance of about 25 mA d.c. per IC will suffice for estimating
total d.c. current requirements.
Fig. 11. Circuit for generating clock pulses from any input. Output can drive the equivalent of 16
flip-flop "T" inputs.
Fig. 12. Decoder for the n = 10 divider shown in Fig. 10.
Although the term "frequency divider" has been used throughout this article, the frequency dividers
are actually repetitive counters. When the input CP's to the counter are regular and periodic, frequency
division is obtained as a by-product of the repetitive counting operation. In frequency-dividing applications,
one is interested in the time-varying waveforms present at the FF outputs during the counting operation;
in counting applications, one is interested in the states of the individual FF's at a particular instant
In order for the counter to be of any real value, information on the FF states must be presented
in some usable form. Typically a lamp-driver/lamp circuit is used. The lamp driver is designed so that
the lamp illuminates when the lamp driver input is at the 1 level and extinguishes when it is at the
The decade counter is designed to display the number of clock pulses counted in numbers from 0 to
9. On the 10th CP the counter resets to 0 and delivers an output pulse. If this pulse is connected to
the input of a second decade counter, the display of the second counter advances one count for every
ten counts of the first decade counter. Connected in this manner, one decade counter counts from 0 to
9 clock pulses, two decade counters count from 0 to 99 clock pulses, three decade counters can count
up to 999 clock pulses, etc.
A decade counter is designed around an n = 10 divider. Since the divider state is different for each
successive CP, the lamp responses can be related to the divider states through appropriate decoding
techniques. Any n = 10 divider can be decoded, but the divider of Fig. 10 is ideally suited since it
can be completely decoded using only ten two-input gates.
A decoder for the Fig. 10 divider is shown in Fig. 12. The inputs Q1 through Q5 are connected to
the corresponding FF outputs of Fig 10. When the divider/counter is Preset (all FF outputs at the 0
level) only the "0" decoded output is at the 1 level. After the first CP only the ''1'' decoded output
is at the 1 level. After the second CP only the "2" decoded output is at the 1 level. After the ninth
CP only the "9" decoded output and output X are at the 1 level. Coincident with the tenth CP output
X switches to the 0 level providing a CP to drive a second decade counter.
Posted May 23, 2017