# Digital Logic Design

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Lesson No. 44
THE LOGIC BLOCK
Each Logic Block consists of a several Logic elements. The details of the Logic
Element are shown in figure 44.1.
Programmable
Carry In
To
Select
Interconnect
Data from
programmable
In
interconnects
Look
Up
Logic
Table
Flip-Flop
Clock/Clear/
Preset
Select
Logic
Carry
Out
Out
Figure 44.1
Block diagram of a Logic Element
The Logic Element
The Look-Up Table LUT has 4-inputs and can be programmed as a logic function
generator. The LUT can be programmed to produce SOP functions or logic functions such as
adders and comparators. When the LUT is configured as an adder the Carry In and Carry Out
allows one LUT to be cascaded with another LUT in other logic units. There are two
Programmable selects, the first Programmable selects allows selection of either the
combinational functions from the LUT output or a direct input to be connected to the input of a
flip-flop. The second Programmable select allows selection of the combinational function from
the LUT output or the registered functions from the flip-flop output. The clock/Clear/Preset
Select Logic controls the operation of the flip-flop through the Clear and Select Asynchronous
signals and the Clock Synchronous signal.
The Look-Up Table
The Look-Up Table shown in the Logic Element is implemented using a memory
element that can be programmed to implement different logic functions. Two examples of LUT
programmed to generate a logic function and to implement a full-adder are shown. Table 44.1
In the first example illustrated by Table 44.1a a three variable SOP function
F = ABC + ABC + ABC is implemented. A memory having 8 locations and storing a single bit
value at each location is required to implement the three variable function. The three bit
address lines which are used to select one out of eight memory locations represents the three
function variables A, B and C respectively. The product terms that are included in the function
439 CS302 - Digital Logic & Design
are represented by memory locations that store logic 1 binary value. The product terms or the
Min-terms that are missing from the function are represented by memory locations that store
logic 0 value. In the function shown, memory locations 2, 5 and 7 which represent the product
terms ABC, ABC and ABC respectively have logic 1 values stored. Thus when ever
addresses 2, 5 or 7 are issued, the data output is a 1 for all other addresses the data output is
a 0.
Data
Output
A
B
C
F
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
0
1
0
0
0
1
0
1
1
1
1
0
0
1
1
1
1
F = ABC + ABC + ABC
Table 44.1a
LUT programmed to generate a function
Data Output
A
B
Cin
Sum
Cout
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
1
1
0
0
1
0
1
0
1
0
1
1
1
0
0
1
1
1
1
1
1
Sum = ABC + ABC + ABC + ABC
C  out = ABC + ABC + ABC + ABC
Table 44.1b
A Single bit Full-Adder can be similarly implemented by using a memory which has
three storage locations, each location storing two bits. A single bit full-adder has three input
variables A, B and Cin and two output variables Sum and Cout. The three address lines which
select one out of eight memory locations are connected to the three input variables A, B and
Cin. The eight two bit memory locations are programmed to represent the Sum and Cout
functions. An address 110 represents the variables A=1 and B=1 and Cin = 0. The Sum output
440 CS302 - Digital Logic & Design
should be 0 and the Cout should be 1 which is represented by the data output 01 corresponding
Analogue to Digital Conversion
Digital systems process digital information. The input and the output to the digital
systems is in digital binary format. Real world quantities are in analogue form, which are
converted into digital format for processing by the digital system. The processed digital output
has to be converted back into analogue format. Mobile phones convert the analogue speech
signal into a digital signal which is processed digitally. The digital signal which is received is
converted back to an analogue form which one hears. Digital thermometers measure
temperature which is in analogue form. The analogue signal is converted into digital format
which is displayed in the form of numbers representing the temperature value. Measuring
instruments such as digital voltmeters also sample an electrical signal in its analogue form.
The analogue samples are digitized and displayed in the form of numbers representing voltage
values. CDs which store digital audio and video files have the original audio and video
analogue signals converted into digital format for storage on CDs. To replay the audio or video
file the digital information is converted back into analogue form. Industrial controller system
sample analogue values, digitize the sampled values, process the digitized data and convert
the digitized processed information into corresponding analogue outputs.
Analogue signals are converted into Digital signals by Analogue to Digital (A/D)
converters. The conversion of the analogue signal into a corresponding digital signals is done
by first sampling the analogue signal and holding it stable for the A/D converter to convert the
signal into a digital value.
Sample and Hold Operation
A sample and hold circuit performs two important operations. The analogue signal is
sampled at regular sampling intervals to take sufficient number of discreet values at points on
the analogue waveform. The more samples are taken the more accurate is the representation
of the original analogue signal. The sampling frequency according to the Nyquist Criteria
should be twice the maximum frequency of the highest analogue frequency. A sampled value
has to be held stable for a certain minimum time to allow the A/D converter to convert the
sample values into equivalent digital values. Figure 44.2a shows an input analogue signal
representing temperature varying with time. To convert the analogue signal it is sampled at
regular intervals. The analogue signal which is sampled at 15 equal intervals is shown in figure
44.2b. The number of samples determine the accuracy of the digitized signal. If the 15
samples are plotted the resulting signal represents reconstructed analogue signal based on
the 15 samples. Figure 44.2c. The reconstructed signal is not an exact replica of the original
analogue signal but is similar. The exactness of the reconstructed signal depends upon the
number of samples. If the number of samples are few then the reconstructed signal losses its
resemblance to the original signal. Figure 44.2d. The signal with fewer samples then the
desired number of samples that are required to accurately reconstruct the signal is known as
an under sampled signal. The under sampled signal is shown to lose the information in the
original signal when it is reconstructed. If the original signal is over sampled by increasing the
number of samples beyond 15, the reconstructed signal will be a very accurate representation
of the original signal however processing too many samples may require too much processing
time.
441 CS302 - Digital Logic & Design
45
40
35
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
time
Figure 44.2a
Continuous signal showing temperature varying with time
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42
41
40
37
35
35
34
30
29
25
25
25
23
22
20
18
15
10
7
5
4
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
time
Figure 44.2b Sampling the Continuous Signal at 15 equal intervals
442 CS302 - Digital Logic & Design
45
42
41
40
37
35
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34
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29
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25
25
23
22
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7
5
4
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
samples
Figure 44.2c Reconstructed Signal by plotting 15 sampled values
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40
35
30
25
20
15
10
5
0
1
3
5
7
9
11
13
15
samples
Figure 44.2d Reconstructed Signal by plotting 7 sampled values
443 CS302 - Digital Logic & Design
The number of samples that are essential to accurately represent the original signal is
determined by the Nyquist Criteria which requires that the sampling frequency should be twice
the frequency of the sampled signal. Assuming the original signal to have a frequency of 50
Hz, the sampling which allows accurate reconstruction of the signal should be carried out at
100 Hz.
The sampled signal at Nyquist frequency have to be held stable for a minimum time
period to allow the A/D converter to convert the analogue sample into a digital value. If the
sampled signal is not held stable, the A/D converted would not have enough time to accurately
convert the signal into a digital value. The samples that are held stable for converting into
digital values by the sample and hold circuit are depicted by a staircase signal shown in figure
44.3.
30
28
26
24
25
20
20
20
16
15
15
12
12
10
10
10
9
10
7
6
5
5
4
5
3
1
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Figure 44.3a Sample and Hold signal
Figure 44.3b Sample and Hold Circuit
Quantization
444 CS302 - Digital Logic & Design
The process of converting the analogue signal into a digital representation (code) is
known as quantization. The number of bits that are used to represent the digital code
determine the accuracy of the digitized signal. An analogue 220 volt signal can be represented
in digital terms by a 2-bit binary number. The four possible digital values 00, 01, 10 and 11
represent four levels of the input 220 volt analogue signal. The binary value 00 represents 0
volts, 01 represents 73 volts, 10 represents 146 volts and 11 represents 220 volts. Since four
values can be represented, therefore analogue voltages in the ranges 37 to 109 are
represented by binary 01. Similarly voltage ranges between 110 to 183 are represented by
binary 10. If a three bit representation is used then the range of analogue signals represented
by the eight, 3-bit values is reduced thereby increasing the accuracy.
30
30
28
30
30
28
23
21
23
21
16 15
15
16
15
15
11
10
9
11
8
10
9
8
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11 12 13
Figure 44.4a Analogue Signal
Figure 44.4b
Sample & Hold Signal
3
3
3
3
2
2
2
2
1
1
1
1
0
1
2
3
4
5
6
7
8
9
10
11 12 13
Figure 44.4c Digitized Signal
In figure 44.4 the representation of an analogue signal using 2 bits or four quantization
levels is shown. The original signal has analogue value range from 0 to a peak value of 31.
Figure 44.4a. The analogue signal is sampled, the output of the Sample and Hold circuit is
shown in figure 44.4b. The sampled values range between 0 and 30. The sampled signal is
digitized using four quantization levels or 2-bits. Figure 44.4c. The original signal having
values in the range 0 to 7.5 are represented by a digital representation 00. Analogue values in
the range 8 to 14.5 are represented by a digital representation 01. Analogue values in the
range 15 to 22.5 are represented by digital representation 10 and the values ranging between
23 and 30 are represented by digits 11. If the quantization level is quadrupled to 16 levels the
digitized representation of the analogue signal becomes more accurate. Figure 44.5.
445 CS302 - Digital Logic & Design
30
30
30
30
28
28
23
23
21
21
16
15
15
16
15
15
11
10
11
9
10
8
9
8
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11 12 13
Figure 44.5a Analogue Signal
Figure 44.5b
Sample & Hold Signal
15
15
14
12
11
8
8
8
6
5
5
4
0
1
2
3
4
5
6
7
8
9
10
11 12 13
Figure 44.5c Digitized Signal
In figure 44.5, the analogue signal is shown to be digitized using a 16 level
quantization. The digitized signal shown in figure 44.5c closely resembles the analogue signal
as compared to the 4 level quantized signal shown in figure 44.4c.
Operational Amplifier (Op-Amp)
Operational Amplifier is a linear amplifier which has two inputs (inverting and non-
inverting) and one output. It has a high voltage gain, high input impedance and low output
impedance. The Op-Amp amplifies the difference signal between its inverted and non-inverted
inputs. Figure 44.6a
The Op-Amp is used as an inverting amplifier and as a comparator. When the Op-Amp
is used as an inverting amplifier, the input signal is applied at its Inverted input through a
series resistance Ri. The output of the Op-Amp is connected to the inverted input through a
feedback resistance Rf. Figure 44.6b. The voltage gain of the Inverting Amplifier is given by
the relation
Vout/Vin = - Rf/Ri
When the Op-Amp is used as a comparator two voltages are applied at the inputs,
when these voltages differ by a very small amount the output of the Op-Amp is driven into one
of its two saturated output sates logic high or low depending upon which of the two input
voltages is higher. Figure 44.6c
446 CS302 - Digital Logic & Design
Figure 44.6a Op-Amp
Figure 44.6b Op-Amp as an Inverting Amplifier
Figure 44.6c Op-Amp as a Comparator
Analogue to Digital converters use Op-Amps as an Integrator and Comparator. An
Integrator integrates the input voltage. An Integrator is implemented by replacing the feedback
resistance Rf by a Capacitor.
Flash Analogue-to Digital Converter
The Flash A/D converter is based on a resistor potential divider, where multiple
resistors of identical value form a voltage divider. A reference voltage is applied at one end of
the potential divider which divides the voltage equally across all the resistors. The input
analogue voltage is applied at the non-inverting inputs of a set of Op-Amp based comparators.
The inverting input of each comparator is connected to the resistive voltage divider which
provides reference voltages for all the comparators. If the input voltage is larger than the
reference voltage the output of the comparator is logic high otherwise it is logic low. The
outputs of all the comparators are connected to the input of a priority encoder which converts
the comparator outputs to a binary coded equivalent value. Figure 44.7a.
447 CS302 - Digital Logic & Design
Figure 44.7a Flash A/D Converter
The analogue input sampled signal is applied at the input of the seven comparators.
The inverted input of each of the seven comparators is connected to voltage divider circuit
made up of eight resistors having the same value R. A reference voltage +VREF is connected at
the top end of the voltage divider circuit and the lower end of the voltage divider is connected
to the ground. The voltage drops across the eight resistors starting from the top most resistor
are VREF, 7/8VREF, 6/8VREF, 5/8VREF, 4/8VREF, 3/8VREF, 2/8VREF and 1/8VREF respectively. If the
input sampled voltage input is greater than the reference input voltage for any comparator, the
comparator output is logic 1, otherwise the output is logic 0. The inputs of an eight-to-three
Priority Encoder are connected to the comparator outputs. The lowest priority input of the
Encoder is grounded. The priority encoder is enabled at each sampled input and a 3-bit code
representing the value of the input sample appears at the output. Consider an example, the
input sample is 4.2 volts. The reference voltage VREF is equal to 8 volts, the seven reference
voltages applied at the inverted inputs of the seven comparators starting from the first
comparator are 7, 6, 5, 4, 3, 2 and 1 volts respectively. With an input of 4.2 volts the outputs of
the first three comparators are set to logic 0 and the outputs of the lower four comparators are
set to logic 1 which sets the encoders first three inputs to inactive-low and the next four inputs
to active high. The highest priority input the encoder outputs 100 which is binary equivalent of
4. Figure 44.7b.
448 CS302 - Digital Logic & Design
Figure 44.7b 3-bit FLASH A/D Converter
The 3-bit Flash converter requires seven comparators, a 4-bit Flash converter requires
fifteen converters. A large number of comparators are required to implement a reasonable-
sized converter. The advantage is that the conversion is done in parallel and the binary
equivalent value is available at the output of the converter almost instantaneously. Flash
converters are used for high speed conversion applications such as conversion of analogue
video signals into digital signals. For accurate reproduction of the digital signal, Flash A/D
converters are based on high number of Quantization levels which requires the use of many
Op-Amp based Comparators which makes the Flash converters expensive and power hungry
Figure 44.8 shows a set of sampled analogue voltage inputs applied at the input of the
Flash converter shown in the figure 44.7. The reference voltage of the Flash converter is set to
8 volts. At each sampling interval an enable pulse allows the Flash converter to convert the
corresponding analogue input voltage sample to be converted and represented in its binary
form. Figure 44.9.
449 CS302 - Digital Logic & Design
9
7.8
7.5
8
6.7
6.7
6.5
7
5.6
6
4.6
4.5
5
3.6
4
3
2
1.7
2
1
0.4
0
1
2
3
4
5
6
7
8
9
10
11
12
time
Figure 44.8
Input analogue voltage samples
Figure 44.9
Binary output representing input analogue voltage samples
Dual-Slope Analogue to Digital Converter
The Dual-Slope A/D converter is used in digital voltmeters and other types of
measuring instruments. A Dual-Slope A/D converter is slower than the Flash Converter. The
circuit diagram of the converter is shown. Figure 44.10. The converter consists of a switch.
Initially, the switch connects the circuit to the input analogue voltage which is to be converted
into its corresponding binary representation. During the conversion process the switch
connects the circuit to a negative reference voltage. The switching between the input voltage
and the reference voltage allows a capacitor connected between the Op-Amp output and input
to be charged and discharged. An Op-Amp based Integrator integrates the analogue input
voltage over a fixed period of time. An Op-Amp based Comparator compares the output of the
Integrator with the ground voltage to enable or disable a counter circuit. A counter and latch
counts the binary output corresponding to the analogue input value. A logic control circuit is
used to switch between analogue voltage input and the reference voltage. It also
enables/disables the latch.
450 CS302 - Digital Logic & Design
C
Analogue
Input (Vin)
CLK
R
Switch
A1
A2
-VREF
Counter
R
n
Integrator
CLEAR
(ramp generator)
Comparator
Control
Latches
Logic
En
Switch Control
D0
D7
Binary or BCD
Output
Figure 44.10 Dual-Slope A/D Converter
The first Op-Amp is connected as an Integrator. Initially, the counter is reset and has a
zero count. The Input switch is connected to the Analogue input which is to be converted into
equivalent binary value. The counter is reset to count zero by the Control Logic circuit. It also
sets the switch to the Analogue input voltage. The Input analogue voltage is assumed to be
constant for the duration of the conversion process. Due to the high input impedance of the
Integrator, the current from the Analogue Input source flows through the Resistor R and the
Capacitor C. The Capacitor will charge and there will be a negative-going linear voltage ramp
at the output of A1. The non-inverted input of the Comparator is connected to the ground,
therefore as the inverted input of the comparator becomes negative, the output changes to
logic 1. The Logic 1 output triggers the Control Logic which in turn resets the counter. The
logic 1 output enables the AND gate which allows the clock signal to be applied at the counter
clock input which increments the counter at each clock pulse. The Integrator output remains at
negative voltage as the negative-going linear ramp continues the integration process. As the
counter count reaches its maximum count value (terminal count), it rolls over and sends a
signal to the control Logic circuit which switches the switch to ­VREF. The Capacitor which is
charged to a positive input voltage discharges resulting in a positive going slope at the output
of integrator. When the voltage at the inverted input of the comparator reaches zero volt, the
comparator output become logic 0 disabling the AND gate and therefore inhibiting the counter
from counting. The Control Logic circuit sends a pulse which loads the latch with the count
value.
Table 44.2 depicts the working of the Dual-Slope A/D converter. At interval t = 0 the
converter is switched to the Vin input which is assumed to remain constant during the
conversion operation. The capacitor starts charging at a constant rate. The output of the
Integrator (voltage output) decreases at constant slope (-V). The output of the comparator is
set to 1 enabling the clock signal and incrementing the counter. The converter remains
connected to Vin for a fixed duration determined by time interval t = n. The duration of the
interval is determined by the maximum count value of the counter. During this interval the
capacitor has charged to a maximum value determined by the input voltage. At time interval t =
n, the counter reaches its terminal count and rolls over. The logic control circuit switches to ­
451 CS302 - Digital Logic & Design
Vref. At interval t=n+1 the capacitor begins to discharge as now it is connected to a negative
voltage -Vref . The Integrator output starts increasing towards a 0 voltage at a constant rate.
The output of the Comparator is logic high allowing the counter to count. At interval t = n+m
the capacitor has completely discharged and the comparator inputs become equal setting its
output to 0. The clock signal is disabled, disabling the counter from counting. The count value
represents the input voltage. Interval m is determined by the magnitude of the charge stored
on the capacitor. Higher the voltage stored on the capacitor longer it will take to discharge to 0
volts, thereby allowing the counter to a larger count value. If the input analogue voltage is
small, the capacitor will be charged to a smaller voltage. It will therefore discharge in a shorter
interval of time allowing the counter to count to a small value.
Time
Input
Output of
Output
of
Clock
Counter
interval t
signal
Integrator
Comparator
Input
0
Vin
-V
1
enabled
Counting
1
Vin
-V
1
enabled
Counting
n
Vin
-V
1
enabled
Terminal count reached.
Counter reset.
Switched to ­Vref
n+1
-Vref
-V
1
enabled
Counting
n+2
-Vref
-V
1
enabled
Counting
n+m
-Vref
0
0
disabled
Binary value representing
Vanalogue
Table 44.2
Operation of Dual-Slope A/D Converter
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