Frequency Division and Counting

Frequency Division
In the Sequential Logic tutorials we saw how D-type Flip-Flop´s work and how they can be connected together to form a Data Latch. Another useful feature of the D-type Flip-Flop is as a binary divider, for Frequency Division or as a “divide-by-2″ counter. Here the inverted output terminal Q (NOT-Q) is connected directly back to the Data input terminal D giving the device “feedback” as shown below.

Divide-by-2 Counter

It can be seen from the frequency waveforms above, that by “feeding back” the output from Q to the input terminal D, the output pulses at Q have a frequency that are exactly one half ( f ÷ 2 ) that of the input clock frequency. In other words the circuit produces Frequency Division as it now divides the input frequency by a factor of two (an octave).

This then produces a type of counter called a “ripple counter” and in ripple counters, the clock pulse triggers the first flip-flop whose output triggers the second flip-flop, which in turn triggers the third flip-flop and so on through the chain producing a ripple effect (hence their name) of the timing signal as it passes through the chain.

Of the clock frequency using the appropriate number off FF's, this circuit could devide a frequency by any power of 2.Specifically, using N flip-flops would produce an output frequency from the last FF which is equal to 1/2^N of the input frequency, this application of FF is referred to as frequency division. For frequency division, toggle mode flip-flops are used in a chain as a divide by two counter.

Counting Operation

In addition to functioning as a frequency divider, the circuit of figure 5-45 also operates as a Binary Counter. The clock is actually used for data transfer in these applications. Typically, counters are logic circuits that can increment or decrement a count by one but when used as asynchronous divide-by-n counters they are able to divide these input pulses producing a clock division signal.

Counters are formed by connecting flip-flops together and any number of flip-flops can be connected or “cascaded” together to form a “divide-by-n” binary counter where “n” is the number of counter stages used and which is called the Modulus. The modulus or simply “MOD” of a counter is the number of output states the counter goes through before returning itself back to zero, ie, one complete cycle.

Then a counter with three flip-flops like the circuit above will count from 0 to 7 ie, 2ⁿ-1. It has eight different output states representing the decimal numbers 0 to 7 and is called a Modulo-8 or MOD-8 counter. A counter with four flip-flops will count from 0 to 15 and is therefore called a Modulo-16 counter and so on.

On below, the circuit function as a binary counter in which the states of the FFs represent a binary number equivalent to the number of pulses thet have accured.

State Stansition Diagram

Figure 5-47 show how state of the FFs change with each applied clock pulse. Each circle represent one possible state as indicated by the binary number inside the circle.

The arrows connecting one circle to another show how one state changes to another as a clock pulse is applied.

To help describe, analyze, and design counters and other sequential circuit we will use state transition diagram.

MOD Number

The counter of figure 5-45 referred to as a MOD-8 counter, where the MOD number indicates the number of state in the counting sequence. The sequence of states would count in binary from 0000 to 1111 if fourth FF were added.

MOD-16 consist of toal of 16 states. The MOD number of the counter also indicates the frequency division obtained from the last FF.

Refferences : 

2001,Prentice Hall Digital systems principles and applications 8ed, Tocci

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