Topic 3 Counters

At the end of this section you should be able to:

Describe the use of feedback to make a D-type flop-flip divide by two.
Convert a four-bit binary number into decimal or HEX notation.
Design 4-bit up or down counters based on rising edge triggered D-type flip-flops.
Design 4-bit modulo-N counters and draw timing diagrams.
Describe the use off a BCD or HEX decoder with a seven segment display.

Counters

Latches can act as a memory for binary numbers that have been put into them. Eight flip-flops can act as a memory for an eight-bit word, or a single byte. In computers, which work with bytes, each eight-bit number stands for something, be it a letter, a number, or a character, determined by the ASCII code, universal for all computers.

Counters are special memories that store a word that represents the number of pulses that have passed into the circuit. The D-type flip-flop is the simplest counter for 1 bit.

The timing diagram shows the behaviour of the circuit:

Question 1. What do you notice about the output compared to the input? ANSWER

This circuit can count two bits:

When the reset is made high, the outputs Q go to give a two bit word BA 00. The circuit then changes state on the falling edge of the clock pulse, according to the following table:

*Pulse*	B	A
0	0	0
1	0	1
2	1	0
3	1	1
4	0	0

The word BA tells us the number of pulses that have arrived. B is the most significant bit worth 2¹ (= 2), while A is the least significant bit worth 2⁰ (= 1). So BA = 11 represents 2 x 1 + 1 x 1 = 3. The timing diagram is like this:

Question 2 What do you notice about the output trace B? ANSWER

Divide by 2 Circuit

As well as acting as a single bit counter, the circuit above, which is a D-type flip-flop with feedback, acts as a divide by two circuit. If we look at the timing diagram, we can see that the number of output pulses is half the number of clock pulses.

Question 3 When does the output trace change state? ANSWER

The circuit works like this:

The output Q-bar is 1 when Q is 0.
On the first rising edge, Q changes to 1.
And stays there as the clock pulse falls to 0.
Then on the next clock pulse, the rising edge causes the output to change to 0.
The output changes every other clock pulse.

Binary and Hexadecimal Counting

Computers work on binary numbers, which mean numbers to the base 2. We normally count in tens, because we have ten digits on our front and hind paws. This is base 10 or decimal. We can count in any base we like. In the UK the currency was run on a duodecimal system, counting in base twelve. 1 shilling = 12 old pence; 1 pound = 20 shillings.

When we express a number, we start off with the most significant bit at the left hand side, and the least significant at the right. Consider the number 245:

	*Hundreds*	*Tens*	*Units*
*Powers of Ten*	10²	10¹	10⁰
*Number*	2	4	5
	2 x 100	4 x 10	5 x 1

So 245 is the sum of 200 + 40 + 5

We do a similar thing in binary. The least significant bit is 2⁰(= 1), followed by 2¹ (= 2), followed by 2² (= 4), etc. We will look at a four bit number:

	*Eight*	*Four*	*Two*	*Units*
Powers of Two	2³	2²	2¹	2⁰
Number	1	0	1	1
	1 x 8	0 x 4	1 x 2	1 x 1

So 1011 = 8 + 0 + 2 + 1 = 11.

Try Question 4

Computer memories are designed to act rather like a set of pigeonholes, or lockers, in which data is posted. Each location has a unique address, which is given a number in base 16 (= 2⁴), called a hexadecimal or HEX code.

The first 9 hexadecimal numbers are like the first 9 decimal numbers. The character 10 represents decimal 16. So there have to be alternative characters for decimal 10, 11, 12, 13, 14, and 15. These are A, B, C, D, E, and F respectively. The table shows decimal numbers 0 to 16 with their four bit binary and hexadecimal codes:

*Decimal*	*Four-bit binary*	*Hexa-decimal*
0	0000	0
1	0001	1
2	0010	2
3	0011	3
4	0100	4
5	0101	5
6	0110	6
7	0111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F
16	0000	10

The binary code for decimal 16 or hexadecimal 10 is 10000, which is a five bit number.

Address codes of four figures give 16⁴ (= 65536) combinations. In real computers the addresses can be 16digit codes or even 32 digit, which give many more combinations.

Try Question 5

4-bit Counters

We can cascade the flip-flops so that we can have as many bits as we want. The next diagram shows a four bit counter:

This circuit is rising edge triggered, and each flip-flop has its Q-bar output fed back to the data input. The timing diagram shows the idea:

Notice that the unit counter goes through a change every two clock pulses, and the two’s counter every four pulses. The four’s would be every eight pulses, and the eight’s every sixteen pulses. If we look at the output of the counter, we would see it increase by 1 every two clock pulses. This is an up-counter.

Question 6 What is the maximum decimal number that this counter can count to? ANSWER

To make a down-counter, we connect the Q-bar output to the CK input of the next flip-flop, while the Q output is connected to the data input.

Modulo N Counters

The 4-bit counters we have seen above count from 0 to 15 (decimal) before resetting to zero. We say that it is a modulo 16 counter. The modulo refers to the number of states that a counter goes through until it resets to zero.

A counter with n flip-flops will go through 2ⁿ states before it resets to zero.

If we want to reset to zero before that 2ⁿth state is reached, we need to add an AND gate to the circuit and feed the output of the AND gate to the reset line. The diagram shows the four bit counter with the AND gate feeding the reset line:

The counter counts up to binary 1010 (decimal 10). Since Q₁ and Q₃ are 1, the output of the AND gate is 1 and that makes the reset line 1, knocking the counter back to zero. This circuit is called a binary coded decimal (BCD) counter. We could chose any of the lines. The AND gate between Q2 and Q3 would give us a modulo 12 counter.

Question 7 Why is the circuit above a Modulo-12 counter? ANSWER

Click here to move on to counter displays.

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