# Counter (digital)

In digital logic and computing, a counter is an electronic device which stores the number of times a particular event or process has occurred as indicated by a clock signal. The most common type is a sequential digital logic circuit with an input signal called the clock, and multiple output signals which collectively represent a number in the binary or BCD number system. Each pulse applied to the clock input causes the value stored in the counter to increment or decrement (increase or decrease by one).
A counter circuit is usually constructed of flip-flops, which store the count value, and in many cases, additional logic to effect particular counting sequences, qualify clocks and perform other functions.
Counters are widely used in digital circuits, and are implemented as stand-alone integrated circuits and also commonly incorporated into larger integrated circuits.

## Common characteristics

An electronic counter is a sequential logic circuit that has a clock input signal and a group of output signals that represent an integer "counts" value. Upon each qualified clock edge, the circuit will increment (or decrement, depending on circuit design) the counts. When the counts have reached the end of the counting sequence (maximum counts when incrementing; zero counts when decrementing), the next clock will cause the counts to overflow or underflow, and the counting sequence will start over. Internally, counters use flip-flops to represent the current counts and to retain the counts between clocks. Depending on the type of counter, the output may be a direct representation of the counts (a binary number), or it may be encoded. Examples of the latter include ring counters and counters that output Gray codes.
Many counters provide additional input signals to facilitate dynamic control of the counting sequence, such as:
* Reset – sets counts to zero. Some IC manufacturers name it "clear" or "master reset (MR)".
* Enable – allows or inhibits counting.
* Direction – determines whether counts will increment or decrement.
* Data – parallel input data which represents a particular counts value.
* Load – copies parallel input data to the counts.
Some counters provide a Terminal Count output which indicates that the next clock will cause overflow or underflow. This is commonly used to implement counter cascading (combining two or more counters to create a single, larger counter) by connecting the Terminal Count output of one counter to the Enable input of the next counter.
The modulus of a counter is the number of states in its count sequence. The maximum possible modulus is determined by the number of flip-flops. For example, a four-bit counter can have a modulus of up to 16 (2^4).

## Counter types

Counters are broadly categorized as either synchronous or asynchronous. In synchronous counters, all flip-flops share a common clock and change state at the same time. In asynchronous counters, each flip-flop has a unique clock, and the flip-flop states change at different times.
Counters are further categorized in various ways. For example:
* Modulus counter – counts through a particular number of states.
* Decade counter – modulus ten counter (counts through ten states).
* Up/down counter – counts up and down, as directed by a control input, or by the use of separate "up" and "down" clocks.
* Ring counter – formed by a "circular" shift register.
* Johnson counter – a twisted ring counter.
* Gray-code counter – outputs a sequence of Gray codes.
* Shift register generator counter – based on a shift register with feedback.

### Binary counter

A binary counter is a synchronous or asynchronous digital counter that counts and represents the count in binary. A binary counter is modulus 2$^n$, where n is the number of flip-flops used to store the count. For example, the illustrations below show the behavior of a 5-bit binary counter, which has 32 (2$^5$) states:
* Binary counter sequence
* Voltage changes on the five outputs of the binary counter counting from 00000, left to 11111 (or 31), right (vertically).

### Asynchronous (ripple) counter

An asynchronous counter, or binary ripple counter, is a "chain" of toggle (T) flip-flops in which the least-significant flip-flop (bit 0) is clocked by an external signal (the counter input clock), and all other flip-flops are clocked by the output of the nearest, less significant flip-flop (e.g., bit 0 clocks the bit 1 flip-flop, bit 1 clocks the bit 2 flip-flop, etc.). The first flip-flop is clocked by rising edges; all other flip-flops in the chain are clocked by falling clock edges. Each flip-flop introduces a delay from input clock edge to output toggle, thus causing the counter bits to change at different times and producing a ripple effect as the counter input clock propagates through the chain. When implemented with discrete flip-flops, ripple counters are commonly implemented with JK flip-flops, with each flip-flop configured to toggle when clocked (i.e., J and K are both connected to logic high).
* Schematic diagram of an n-bit ripple counter constructed from JK flip flops
* Functional diagram of the 4020, 14-bit ripple counter IC
* Photograph of a 4020, 14-bit ripple counter IC (RCA CD74HC4020E)
In the simplest case, a one-bit counter consists of a single flip-flop which will increment (by toggling its output) once per clock cycle and count from zero to one before overflowing (starting over at zero). Each output state persists for a full input clock cycle, and consequently the frequency of the flip-flop's output signal is exactly half that of its input clock. If this flip-flop's output is inverted and used as the clock signal for a second flip-flop, the pair of flip-flops will form a two-bit ripple counter with the following state sequence:
| Clock cycle | Q1 | Q0 | (Q1:Q0) decimal |
| 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 2 | 1 | 0 | 2 |
| 3 | 1 | 1 | 3 |
| 4 | 0 | 0 | 0 |
Additional flip-flops may be added to the chain to form counters of any arbitrary word size, with the output frequency of each bit equal to exactly half the frequency of its nearest, less significant bit.
Ripple counters exhibit unstable output states while the input clock propagates through the circuit. The duration of this instability (the output settling time) is proportional to the number of flip-flops. This makes ripple counters unsuitable for use in synchronous circuits that require the counter to have a fast output settling time. Also, it is often impractical to use ripple counter output bits as clocks for external circuits because the ripple effect causes timing skew between the bits. Ripple counters are commonly used as general-purpose counters and clock frequency dividers in applications where the instantaneous count and timing skew is unimportant.

### Synchronous counters

In a synchronous counter, the clock inputs of all flip-flops are connected together and the common clock simultaneously triggers all flip-flops. Consequently, all of the flip-flops change state at the same time (in parallel).

#### Binary counter

The circuit shown below is an ascending (up-counting) four-bit synchronous binary counter implemented with JK flip-flops. Each bit of this counter is allowed to toggle when all of its less significant bits are at a logic high state. For example, upon clock rising edge bit 1 toggles if bit 0 is logic high; bit 2 toggles if bits 0 and 1 are both high; bit 3 toggles if bits 2, 1, and 0 are all high.

#### Decade counter

A decade counter counts in decimal digits, rather than binary. A decade counter may have each (that is, it may count in binary-coded decimal, as the 7490 integrated circuit did) or other binary encodings. A decade counter is a binary counter designed to count to 1001 (decimal 9). An ordinary four-stage counter can be easily modified to a decade counter by adding a NAND gate as in the schematic to the right. Notice that FF2 and FF4 provide the inputs to the NAND gate. The NAND gate outputs are connected to the CLR input of each of the FFs.". It counts from 0 to 9 and then resets to zero. The counter output can be set to zero by pulsing the reset line low. The count then increments on each clock pulse until it reaches 1001 (decimal 9). When it increments to 1010 (decimal 10), both inputs of the NAND gate go high. The result is that the NAND output goes low, and resets the counter to zero. D going low can be a CARRY OUT signal, indicating that there has been a count of ten.

#### Ring counter

A ring counter is a circular shift register that is initiated such that only one of its flip-flops is the state one while others are in their zero states.
A ring counter is a shift register (a cascade connection of flip-flops) with the output of the last one connected to the input of the first, that is, in a ring. Typically, a pattern consisting of a single bit is circulated, so the state repeats every n clock cycles if n flip-flops are used.

#### Johnson counter

A Johnson counter (or switch-tail ring counter, twisted ring counter, walking ring counter, or Möbius counter) is a modified ring counter, where the output from the last stage is inverted and fed back as input to the first stage. The register repeatedly cycles through a sequence of bit-patterns, whose length is equal to twice that of the shift register.
Johnson counters are commonly used in finite state machines and specialized applications such as digital-to-analog conversion, etc. For example, the CMOS 4017 integrated circuit uses a 5-bit Johnson counter to cycle through 10 states, and thus implement a decade counter with encoded outputs:
* Functional diagram of a CMOS 4017 IC, a decade counter which uses a 5-bit Johnson counter to implement 10 states
* Internal logic of a CMOS 4017 IC; one output is active per state

## Implementation

Counters are implemented in a variety of ways, including as dedicated MSI and LSI integrated circuits, as embedded counters within ASICs, as general-purpose counter and timer peripherals in microcontrollers, and as IP blocks in FPGAs. In the latter case, a counter is typically instantiated by synthesizing it from a description written in VHDL, Verilog or some other hardware description language. For example, the following VHDL code describes a 32-bit up/down counter with clock enable and load capability:

## Theory

In computability theory, a counter is considered a type of memory. A counter stores a single natural number (initially zero) and can be arbitrarily long. A counter is usually considered in conjunction with a finite-state machine (FSM), which can perform the following operations on the counter:
* Check whether the counter is zero
* Increment the counter by one
* Decrement the counter by one (if it's already zero, this leaves it unchanged)
The following machines are listed in order of power, with each one being strictly more powerful than the one below it:
* Deterministic or non-deterministic FSM plus two counters
* Non-deterministic FSM plus one stack
* Non-deterministic FSM plus one counter
* Deterministic FSM plus one counter
* Deterministic or non-deterministic FSM
For the first and last, it doesn't matter whether the FSM is a deterministic finite automaton or a nondeterministic finite automaton. They have the same power. The first two and the last one are levels of the Chomsky hierarchy.
The first machine, an FSM plus two counters, is equivalent in power to a Turing machine. See the article on counter machines for a proof.