Digital Systems
EE 245L
Experiment 1
Practical Aspects of Digital Circuits,
Boolean Theorems, and Simplification
1
Objectives
1. Investigate practical voltage levels used by discrete logic components,
2. investigate noise margins,
3. determine the voltage transfer characteristic for a discrete component,
4. measure actual propagation delays, rise times, and fall times,
5. experimentally verify DeMorgan’s two theorems,
6. gain experience with the signal generator and oscilloscope, and
7. use Boolean Theorems in circuit simplification
2
Pre-Laboratory Procedure
The student is required to read this entire lab, and to determine (perhaps with
the help of a lab partner) which portions can be done prior to lab. While future
labs may not have an explicitly stated “pre lab” procedure, the general rule is
that anything that can be done prior to lab, should be done prior to lab.
For example, prior to lab the student should make a list of the equipment that
will be required. This should include a list of IC and discrete components. Since
you have not been in lab yet, the TA will show you how to find the required IC
part numbers so that you can do this for future labs, but you need not do this prior
to this particular lab. The student should perform any algebraic simplification
and circuit design that is required. If a circuit is to be built in lab, the student
should arrive at lab with the a circuit diagram. Any theoretical values required
should be calculated and tabulated (for example, the 5th and 6th columns of
Table 4 should be filled in prior to coming to lab this week).
3
Breadboard Procedure
For this lab you will investigate several electrical properties of the discrete logic
gates that you have will be using this semester.
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3.1
Introduction to Lab Equipment and Data Books
Your TA will demonstrate the safe and proper use of the function generator and
of the oscilloscope. Pay close attention, as you may have to use this equipment
during your lab practical. He will also show you how to use the data books in
order to determine which ICs you need to build your circuit.
3.2
Voltage Characteristic
To begin with, you will try to determine the transfer characteristic of an inverter. The characteristic for a similar component is shown in Figure 3.46 of
your textbook. Your TA will show you how to use our prototype board, and have
you connect the power and ground connections of your chip to the (5 volt and
ground) power supplies. Your TA will show you how to use a separate supply to
vary the input voltage in order to replicate the curve in Figure 3.46 of your text.
After plotting the curve, you should take care to determine the points labeled in
the figure. In particular, you are looking for the two points where the slope of
the characteristic crosses -1 and the (two) points where the curve first begins to
deviate from the maximum and minimum values.
3.3
Noise Margins
Figure 3.1 in your text does not tell the whole story with regard to voltage levels.
In order to ensure correct operation in the presence of noise, a little bit of cushion,
known as a noise margin is included. Thus the voltage level labeled as V1,min is
replaced by two distinct values. The first of these is VIHmin , the minimum input
voltage that the gate will guarantee to count as “high” or “logic 1”. The second
is VOHmin , the minimum voltage that a gate will output for a “high” or “logic
1” output. The difference between these two values is known as the “high state
(DC) noise margin”, which is the amount that a high output can degrade and
still be counted as high by the next gate. A similar relationship exists between
VILmax and VOLmax . The difference between these two is referred to as the “low
state (DC) noise margin”.
Although your textbook does not have a whole lot of detail, there is a decent
discussion of this at:
http://www.allaboutcircuits.com/vol 4/chpt 3/10.html
The entire discussion on the above web page is valid, but we are only interested
in the very first part of the discussion for this week. If you don’t understand my
discussion above and this week’s lecture on this topic, you might refer to the web
page and discussion through the first two figures. In any case you should verify
the advertised numerical values of these parameters using the data books in the
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X
0
0
1
1
Y
0
1
0
1
X’ * Y’
Table 1: Truth Table for X’ * Y’
lab and verify the values using your inverter circuit from the previous portion
of the lab.
3.4
Timing Parameters
Again using your circuit from the first part of the lab, you should next try to
measure the parameters shown in Figure 3.48 of your text. If the propagation
time for one inverter is too small to accurately measure, hook several of them up
in series and measure the propagation delay through the chain of inverters. In
your report you should compare your observed values with the theoretical values.
3.5
Additional Analysis
Your TA may have some additional circuits for you to investigate. You should
include a discussion of any such circuits in your reports.
3.6
Verifying DeMorgan’s theorems: breadboard procedure
1. Set the function generator output to a frequency of 1 kHz and the amplitude
to 5 V.
2. Construct the circuit on the right-hand side of Figure 1(a) Connect the
function generator to the input labeled X and a toggle switch to the input
labeled Y.
3. Monitor the output and the x input using the two channels on the dual
trace oscilloscope
4. Set the Y toggle switch to each input combination in Table 1 and record the
output you observe. Verify that Table 1 is identical to the truth table for a
NOR gate. This part of the experiment shows that an AND with inverted
inputs is equivalent to a NOR gate
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Figure 1: Different Ways of Drawing a NOR gate
X
0
0
1
1
Y
0
1
0
1
X’ + Y’
Table 2: Truth Table for X’ + Y’
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Figure 2: Different Ways of Drawing a NAND gate
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5. Repeat the above procedure for the circuit on the right-hand side of Figure 2(a) and record your results in Table 2. Verify that Table 2 is identical
to the truth table for a NAND gate. This part of the experiment shows
that an OR with inverted inputs is equivalent to a NAND gate
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A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
F(A,B,C)
Table 3: Truth Table for F(A,B,C)
3.7
Simplification using DeMorgan’s Theorems
1. Draw a logic diagram for the expression:
F (A, B, C) = (A0 + B 0 ∗ C)0 ∗ (A0 + B 0 ∗ C 0 )0
2. Construct the circuit using the diagram you drew in step 7 Connect toggle
switches to inputs A, B, and C and an LED monitor to the circuit output.
Set the toggle switches to each input combination listed in Table 3 and
record the output value observed.
3. Based on your results (in Table 3), write an equivalent expression for the
circuit:
4. Use DeMorgan’s theorems to prove that the expression you obtained above
is equivalent to the original expression in Step 7.
3.8
Boolean Simplification
1. Examine the circuit in Figure 3 and write the Boolean expression for the
output, F.
2. Complete the truth table for F, using Table 4
3. Construct the circuit shown in Figure 3 on your circuit board. Connect
toggle switches to the inputs and use an LED to monitor the output
4. Verify the operation of your circuit by checking every possible combination.
Record these values in the appropriate column of Table 4
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Figure 3: Circuit to Simplify: F(w,x,y,z)
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W
X
Y
Z
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Original F
theory-Pre Lab
Original F Simplified F
empirical
empirical
Table 4: Truth Table for various implementation of F(w,x,y,z)
5. Simplify the expression you obtained above, using Boolean theorems. Your
lab report should show each step of this simplification, and you should list
the theorem (or axiom) used in each step.
6. You report should also show the logic diagram for the simplified expression
7. Construct the circuit for the simplified expression, and verify its operation
for each possible input combination. Record these results in the last column
of Table 4
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