Analog and digital
, and , in fact, most other fields of endeavor, we are constantly dealing with quantities. quantities are measured, monitored , manipulated arithmetically , observed , or in some other way utilized in most physical systems . it is important when dealing with various quantities that we be able to represent their values efficiency and accurately . there are basically two ways of representing the numerical value of quantities :analog and digital
analog representation: in analog representation a quantity is represented by a voltage, current, or meter movement that is proportional to the value of that quantity b. analog quantities such as those cited above have an important characteristic, they can vary over a continuous range of values.
Digital representation:here the quantities are represented not by proportional quantities but by symbols called digits. As an example, consider the digital watch, which provides the time of day in the form of decimal digits which present hours and minutes (and sometimes seconds). As we know, the time of day changes continuously , but the digital watch reading does not change continuously; rather, it changes in steps of one per minute (or per second) . in other words this digital representation of the time of day changes in discrete step, as compared with the representation of time provided by an analog watch, where the dial reading changes continuously step, as compared with the representation
the major difference between analog and digital quantities, then can be simple stated as follows :
analog=continuous
Digital = discrete (step by step)
Units in electricity
the three most basic units in electricity are voltage (V), current (I) and resistance (r) . voltage is measured in volts, current is measured in amps and resistance is measured in ohms. there is a basic equation in electrical engineering that states how the three terms relate. it say that the equal to the voltage divided by the resistance.
A neat analogy to help understand these terms is a system of plumbing pipes. the voltage is equivalent to the water pressure, the current is equivalent to the flow rate , and the resistance is like the pipe size.
I =V/r
Let's see how this relation applies to the plumbing system. Let's say you have a tank of pressurized water connected to a hose that this are using to water the garden. what happens if you increase the pressure in the tank? you probably can guess that this makes more water come out of the hose. the same is true of an electrical system:increasing the voltage will make more current flow. Let's say you increase the diameter of the hose and all of the fittings to the tank. you probably guessed that this also makes more water come out of the hose. this is like decreasing the resistance in an electrical system, which increases the current flow. Electrical power is measured in watts. In an electrical system power (p) is equal to the voltage multiplied by the current
P =VI
The water analogy still applies. take hose and point it at a waterwheel like the ones that were used top turn grinding stones in watermills. you can increase the power generated by the waterwheel in two ways. if you increase the pressure of the water coming out of the hose, it hits the waterwheel with a lot more force and the wheel turns faster, generating more power. if you increase the flow rate , the waterwheel turns faster because of the weight of the extra water hitting it.
The figure to right shows the basic type of electrical circuit , in the form of a block diagram . it consists of a source of electrical energy , some sort of load to make use of that energy , and electrical conductors connecting the source and the load . the electrical source has to terminals, this designated positive (+) and negative (-) . as long as there is an unbroken connection from source to load and back again as shown here , electrons will be pushed from the negative terminal of the source, through the load, and then back to the positive terminal of the source . the arrows show the direction of electron current flow through this circuit . because the electrons are always moving in the same direction through the circuit. their motion is known as a direct current (DC) .
This in turn means that so long as the circuit is operating within its bounds (output voltage within the range of ± 10 volts),the junction of these components will be a virtual ground.Knowing this,we can use Ohm's Law to calculate the currents through the two resistors.Furthermore, since the two currents must be exactly the same,we can set them equal to each other and use that relationship to determine the output voltage of the amplifier:
V out = - V in
R f = - R in
Using a little algebra , we can solve this equation for volt ( out) , or we can solve it for the ratio of volt -(out) / volt-(in) to get the voltage gain of the overall circuit :
from these equations, we can see that the voltage gain of the overall circuit is set entirely by the ratio
Rf /
Rin
. this is the secret of the operational amplifier; it use extremely high gain combined with a lot of negative feedback in order to achieve accurate and predictable result. if we precision resistor, we can obtain precise and measurable result.
The figure to right shows the basic type of electrical circuit , in the form of a block diagram . it consists of a source of electrical energy , some sort of load to make use of that energy , and electrical conductors connecting the source and the load . the electrical source has to terminals, this designated positive (+) and negative (-) . as long as there is an unbroken connection from source to load and back again as shown here , electrons will be pushed from the negative terminal of the source, through the load, and then back to the positive terminal of the source . the arrows show the direction of electron current flow through this circuit . because the electrons are always moving in the same direction through the circuit. their motion is known as a direct current (DC) .
the source can be any source of electrical energy. in practice , there are three general possibilities : it can be a battery , an electrical generator , or some sort of electronic power supply. the load is any device or circuit powered by electricity . it can be as simple as light bulb or as complex as modern high-speed computer. the electricity provided by the source has two basic characteristics, called voltage and current. these are defined as follows:
Voltage
the electrical "pressure" that causes free electron to travel through an electrical circuit. also known as electromotive force (emf) . it is measured in volts .
current
the amount of electrical charge (the number of free electrons ) moving past a given point in an electrical circuit per unit of time . current is measured in amperes . the load, in turn , has a characteristics called resistance .
resistance
that characteristic of a medium which opposes the flow of electrical current through itself . resistance is measured in ohms . the relationship between voltage, current, and resistance in an electrical circuit is fundamental to the operation of any circuit or device .verbally , the amount of current flowing through a circuit is directly proportional to the applied voltage and inversely proportional to the circuit resistance. by explicit definition , one volt of electrical pressure can push one ampere of current through one ohm of resistance . two volts can either push one ampere through a resistance of two ohms , or can push two amperes through one ohm. mathematically,
E =*R
where
E =Applied voltage, or EMF
I = circuit current
R = resistance in the circuit
electrical power is the rate at which electrical energy is converted to another form , such as motion , heat , or an electromagnetic field . the common symbol for power is the uppercase letter P. The standard unit is the watt, symbolized by W. in utility circuits , the kilowatt (kW) is often specified instead ; 1 kW =1000 W.
one watt is the power resulting from an energy dissipation, conversion , or storage process equivalent to one joule per second. when expressed in watt , power is sometimes called wattage . the wattage in a direct current(DC) Circuit is equal to the product of the voltage in volts and the current in amperes. this rule also holds for low-frequency alternating current (AC) circuits in which energy is neither stored not released . at high AC frequencies, in which energy is stored and released (as well as dissipated or converted ), the expression for power is more complex.
Less technically , regular household appliances use DC and the power outlet in the wall is AC. the computer's power supply converts AC from the wall to DC for the computer. the modern analog computer is based on an electronic circuit known as an operational amplifier. Early operational amplifiers ("op amps" for short ) usEDVACuum tubes, since that was the only available technology. modern op amps are constructed as semiconductor integrated circuits. either way, the general theory is the same.
We will discuss the internal working of op amps later in the course. for the overall discussion of analog computer circuits and op amp behavior in search applications , we will make three assumptions about op amps :
the figure shows the basic circuit used in analog computers . the triangle represents our amplifier . per our discussions here, we will assume standard IC amplifiers permitting a typical signal voltage range of ± 10 volts . associated with the amplifier are two resistor: an input resistor (Rm) and a feedback resistor (Rf).in addition,we will state that the amplifier inverts the signal. that is, a positive input signal will result in a negative output signal, and vice-versa. with this combination of characteristics, we can use precision resistor and other components of to accurately determine how the circuit will behave.
Now let's consider what will happen if some input voltage is applied to the V in
there will be no voltage drop across this resistor, and the applied input voltage will appear at the input to the amplifier itself . this will be amplified and inverted by the amplifier , which will try to produce an infinite but opposite output voltage(remember # 1 above)
obviously , this cannot happen. that inverted output voltage will produced a voltage drop across both R and R causing current to flow through both resistors. none of this current will be accepted or used by the amplifier itself ( # 2 above), so the current flowing through RF most be the same as the current flowing through R. TO determine the current through the two resistors , we must determine the output in voltage and the voltage at the amplifier input, where both resistors are connected. once again, refer to # 1 above. with an infinite voltage gain, any voltage at the amplifier's input will cause an excessive output voltage. therefore, the voltage at the junction must always be zero. the amplifier output will provide whatever voltage is required to maintain that condition, and keep the currents through
- they have infinite voltage again.
- they have infinite input resistance (or zero input current)
- they have zero output resistance (infinite output current capability .
the figure shows the basic circuit used in analog computers . the triangle represents our amplifier . per our discussions here, we will assume standard IC amplifiers permitting a typical signal voltage range of ± 10 volts . associated with the amplifier are two resistor: an input resistor (Rm) and a feedback resistor (Rf).in addition,we will state that the amplifier inverts the signal. that is, a positive input signal will result in a negative output signal, and vice-versa. with this combination of characteristics, we can use precision resistor and other components of to accurately determine how the circuit will behave.
Now let's consider what will happen if some input voltage is applied to the V in
there will be no voltage drop across this resistor, and the applied input voltage will appear at the input to the amplifier itself . this will be amplified and inverted by the amplifier , which will try to produce an infinite but opposite output voltage(remember # 1 above)
obviously , this cannot happen. that inverted output voltage will produced a voltage drop across both R and R causing current to flow through both resistors. none of this current will be accepted or used by the amplifier itself ( # 2 above), so the current flowing through RF most be the same as the current flowing through R. TO determine the current through the two resistors , we must determine the output in voltage and the voltage at the amplifier input, where both resistors are connected. once again, refer to # 1 above. with an infinite voltage gain, any voltage at the amplifier's input will cause an excessive output voltage. therefore, the voltage at the junction must always be zero. the amplifier output will provide whatever voltage is required to maintain that condition, and keep the currents through
R in and R f the same.
V out = - V in
R f = - R in
Using a little algebra , we can solve this equation for volt ( out) , or we can solve it for the ratio of volt -(out) / volt-(in) to get the voltage gain of the overall circuit :
from these equations, we can see that the voltage gain of the overall circuit is set entirely by the ratio
Rf /
Rin
. this is the secret of the operational amplifier; it use extremely high gain combined with a lot of negative feedback in order to achieve accurate and predictable result. if we precision resistor, we can obtain precise and measurable result.
Digital to analog conversion
One common requirement in electronics is to convert signals back and forth between analog and digital forms. most such conversions are ultimately based on a digital-to-analog converter circuit. therefore, it is worth exploring just how we can convert a digital number that represents a voltage value into an actual analog voltage.
The circuit to the above is a basic digital -to-analog (D to A) converter It assumes a 4-bit binary number in binary -coded decimal (BCD) format , using +5 volts as a logic 1 and 0 volts as a logic 0. it will convert to applied BCD number to a matching (inverted) output voltage the digits 1,2,4, and 8 refer to the relative weights assigned to each input thus , 1 is the least significant bit 9 LSB
of the input binary number, and 8 is the most significant bit (MSB) .
if the input voltages are accurately 0 and +5 volts, then, the "1" input will cause an outputvoltage of-5*(4k/20k) =-5*(1/5) = -1 volt whenever it is a logic 1. voltage will take on one of 10 specific voltages, in accordance with the input bcd CODE
Unfortunately , there are several practical problems with this circuit. first most digital logic gates do not accurately produce 0 and +5 volts as their outputs therefore, the resulting analog voltages will be close , but not really accurate . in addition, the different input resistors will load the digital circuit outputs differently which will almost certainly result in different voltages being applied to the summer inputs.
The circuit above performs D to A conversion a little differently. Typically the inputs are driven by CMOS gate, which have low but equal resistance for both logic 0 and logic 1. also, if we use the same logic levels, CMOS gates really do provide +5 and 0 volts for their logic levels. the maximum output voltage from this circuit will be one step of the least significant bit below 10 volts. thus, an 8-bit ladder can produce output voltages up to 9.961 volts (255/256*10 volts). this is fine for many applications. if you have an application that requires a 0-9 volt output from a BCD input , you can easily scale the output upwards using an amplifier with a gain of 1.6 (8/5)
if you want an inverting D to A conveter, the circuit shoen above will work well. you may need to scal the output voltage, depending on your requirements. also it is possible to have a bipolar D to A converter . if you apply the most significant bit R-2R ladder, the binary number applied to the ladder will be handled as a two's- complement number, going both positive and negative.
if the input voltages are accurately 0 and +5 volts, then, the "1" input will cause an outputvoltage of-5*(4k/20k) =-5*(1/5) = -1 volt whenever it is a logic 1. voltage will take on one of 10 specific voltages, in accordance with the input bcd CODE
Unfortunately , there are several practical problems with this circuit. first most digital logic gates do not accurately produce 0 and +5 volts as their outputs therefore, the resulting analog voltages will be close , but not really accurate . in addition, the different input resistors will load the digital circuit outputs differently which will almost certainly result in different voltages being applied to the summer inputs.
The circuit above performs D to A conversion a little differently. Typically the inputs are driven by CMOS gate, which have low but equal resistance for both logic 0 and logic 1. also, if we use the same logic levels, CMOS gates really do provide +5 and 0 volts for their logic levels. the maximum output voltage from this circuit will be one step of the least significant bit below 10 volts. thus, an 8-bit ladder can produce output voltages up to 9.961 volts (255/256*10 volts). this is fine for many applications. if you have an application that requires a 0-9 volt output from a BCD input , you can easily scale the output upwards using an amplifier with a gain of 1.6 (8/5)
if you want an inverting D to A conveter, the circuit shoen above will work well. you may need to scal the output voltage, depending on your requirements. also it is possible to have a bipolar D to A converter . if you apply the most significant bit R-2R ladder, the binary number applied to the ladder will be handled as a two's- complement number, going both positive and negative.
Analog to digital conversion
the convert a digital code to an analog voltage, we only had to find a way to effectively assign an appropriate voltage to each bit, and then combine them. but is there some equally easy way of finding the digital code that corresponds to a given analog voltage?
consider the very simple requirement to determine whether an analog voltage was closet to 0,1,2, or 3 volts. the result would be stored as a two-bit binary number. the first step in making this determination might be a set of three comparators, connected as shown to the right. as the analog voltage increases, the comparators will, one by one from the bottom up, change state from false to true. of course, additional digital circuitry will be determine directly which code will be closest to the actual analog voltage.
this approach will work, and can be expanded to any number of steps for finer resolution of the analog voltage. however, as you have probably already perceived, there is a problem with this approach, in that the number of comparators required increase exponentially with the number of binary bits used to store the code. thus, using this approach to convert a 0 to 9-volt range to a BCD number will require nine comparators. a 4-bit binary number, (counting from 0 to 15, requires 15 comparators. and a typical 8-bit circuit will require 255 comparators! clearly this approach becomes repidly too expensive for ordinary use, although it is approach involves the use of a D to A converter and a single comparators. the output voltage of the D to A converter is compared with the unknown analog voltage , then changed and compared again. we won't go into much detail about digital circuit here , but suppose a digital counter were to have its outputs applied to the D to A converter inputs. the counter would start at zero, A converter to exceed the unknown analog voltage, thus causing the comparators output to change. the count that causes this to happen is taken as the digital code that most accurately represents the unknown analog voltage.
Advantages and Limitations of digital techniques
advantage
1 Easier to design . exact values of voltage or current are not important , only the range (HIGH ) or ( low ) in which they fall.
2 information storage is easy .
3 Accuracy and precision are greater.
4 Operation can be programmed. analog systems can also be programmed, but the variety and complexity of the available operations is severely limited.
5 Digital circuits are less affected by noise. as long as the noise is not large enough to prevent us from distinguishing a HIGH from a LOW .
6 more digital circuitry can be fabricated on IC chips.
Limitations
there is really only one major drawback when using digital techniques:
the real world is mainly analog
most physical quantities are analog in nature, and it is these quantities that are often the inputs and outputs that are being monitored, operated on , and controlled by a system. to take advantage of digital techniques when dealing with analog inputs and output, three steps must be followed:
1 convert the real-world analog inputs to digital form ( ADC)
2 process (operate on) the digital information.
3 convert the digital outputs back to real-world analog form (DAC) etc.
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