Binary Numbers

Binary Numbers

Unlike a linear, or analogue circuit which contains signals that are constantly changing from one value to another, such as amplitude or frequency, digital circuits process signals that contain just two voltage levels or states, labelled logic "0" and logic "1". Generally, a logic "1" represents a higher voltage, which is referred to as a HIGH and a logic "0" is referred to as a LOW. These discrete voltage levels are commonly known as BInary digiTS and are normally referred to as BITS. Because there are only two valid Boolean values for representing either a logic "1" or a logic "0", makes the system of using Binary Numbers ideal for use in digital or electronic circuits and systems. The Binary Numbers system is a Base-2 system which follows the same rules in mathematics as the common decimal system meaning instead of powers of ten, for example 1, 10, 100, 1000 etc, binary numbers use powers of two, doubling the value of each successive bit, 1, 2, 4, 8, 16, 32 etc.

The voltages used to represent a digital circuit are called "logic levels" and ideally one voltage level represents a HIGH and another represents a LOW. Digital waveforms or signals consist of discrete voltage levels that are changing back and forth between these HIGH and LOW levels or states. But what makes a signal or voltage "Digital" and how can we represent these voltage levels. Electronic circuits can be divided into two main categories.

• Analogue Circuits - Analogue or Linear circuits amplify or respond to continuously varying voltage levels over a period of time.
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• Digital Circuits - Digital circuits produce or respond too two distinct voltage levels representing either a Logic level "1" or a Logic level "0".

Analogue Voltage Output

An example of an analogue (or analog) circuit and a digital circuit are shown below:

Analogue Voltage Output Representation

The output from the potentiometer varies as the wiper terminal is rotated producing an infinite number of voltage points between 0 volts and V max. As the voltage output varies either slowly or rapidly there is no sudden change between two voltage levels giving a continuous output voltage. Examples of analogue signals include temperature, pressure, liquid levels and light intensity.

Digital Voltage Output

In this example the potentiometer wiper has been replaced by a rotary switch which is connected to each junction of the resistor chain, forming a potential divider network. As the switch is rotated from one node to the next the output changes quickly in discrete voltage levels of multiples of 1 volt each, as shown in the graph. For example, 2 volts, 3 volts, 5 volts etc but NOT 2.5V, 3.1V or 4.6V. Finer output voltage levels could easily be produced by increasing the number of resistors within the potential divider chain.

Digital Voltage Output Representation

Then the major difference between an analogue signal or quantity and a digital quantity is that an "Analogue" quantity is continuously changing over time while a "Digital" quantity has discrete (step by step) values. LOW to HIGH or HIGH to LOW. Another example of this could be a light dimmer that varies the light intensity up or down between fully-ON and fully-OFF, but with a light switch the light is either "ON", (HIGH) or it is "OFF", (LOW).

Some circuits combine both analogue and digital signals such as an analogue to digital converter (ADC) or a digital to analogue converter (DAC). Either way, the digital input or output signal represents a binary number value of the analogue signal.

Logic Levels

In all electronic circuits, only two logic levels are allowed and these levels are referred to as "logic 1 or logic 0", "high or low", "true or false". Most logic systems use positive logic, in which a logic "0" is represented by zero volts and a logic "1" is represented by a higher voltage, such as +5 volts and the switching from one voltage level, "0" to "1" or "1" to "0" is made as quickly as possible to prevent faulty operation of the logic circuit. In standard TTL (transistor-transistor-logic) IC's there is a defined range of input and output voltage limits for defining what is a logic "1" value and what is a logic "0" value and this is shown below.

TTL Input & Output Voltage Levels

Then, when using a +5 volt supply any voltage input between 2.0v and 5v is recognised as a logic "1" value and any voltage input of below 0.8v is recognised as a logic "0" value. While the output of a logic gate between 2.7v and 5v represents a logic "1" value and a voltage output below 0.4v represents a logic "0" value. This is called "positive logic" and is used in these tutorials.

Noise

However, between these defined HIGH and LOW values lies what is generally called a "no-man's land" (the blue area's above) and if we apply a signal voltage of a value within this no-man's land area we do not know whether the logic gate will respond to it as a level "0" or as a level "1", and the output will become unpredictable. Noise is the name given to a random and unwanted voltage that is induced into electronic circuits by external interference, such as from nearby switches, power supply fluctuations or from wires and other conductors that pick-up stray electromagnetic radiation. Then in order for a logic gate not to be influence by noise in must have a certain amount of noise margin or noise immunity.

Noise Immunity

In the example above, the noise signal is superimposed onto the Vcc supply voltage and as long as it stays above the min level (Von-min) the input an corresponding output of the logic gate are unaffected. But when the noise level becomes large enough and a noise spike causes the HIGH voltage level to drop below this minimum level, the logic gate may interpret this spike as a LOW level input and switch the output accordingly producing a false output switching. Then in order for the logic gate not to be affected by noise it must be able to tolerate a certain amount of unwanted noise on its input without changing the state of its output.

Then binary numbers are represented by either a logic "0" or a logic "1" and in the next tutorial about Binary Logic we will look at converting decimal numbers into binary numbers and vice versa and introduce the concept of the Byte and the Word to represent parts of a binary number.