Digital Electronics Part I : Combinational Circuits

Introduction

So that it may serve to later clarify the important design concepts of synchronous versus asynchronous circuit design, I follow the classical approach and introduce two broad sub-divisions of digital circuits:

Combinational: Circuits whose output depends only on the current state of the inputs.
Sequential: Circuits whose output depends on the history as well as the current state of the inputs.

We shall begin with the simpler combinational configurations before advancing to sequential circuits. In general, one would have to look hard to find a complete circuit with only combinational elements, however, in pratice, large and complex sub-units of a complete design hierarchy can be entirely combinational in nature and moreover, as will be shown later, sequential circuits are themselves composed of logic gates with feedback. So, let's begin with an example of a circuit which might be of interest to an experimental physicist.

Example Implementation: Muon Cosmic Ray Experiment Trigger

Figure 1: Muon decay cosmic ray experiment

The first is a very specific example of a cosmic ray experiment where one wants to capture cosmic ray muon decay events. The apparatus consists of 3 planes of scintillator and two boxes of high density material: sand or rock or something even heavier. Muons are very penetrating particles which will pass the absorber easily and ionize the scintillators producing light which is picked up by photomultipliers (PMTs) coupled to each scintillator plane. The photomultipliers produce large electrical pulses (10's to 100's of mV) which can be fed through a discriminator to produce logic level pulses.

Now, consider the case of a cosmic ray (CR) muon entering the apparatus as shown in Figure 1. A possible schematic of the electronic signals this experiment might produce are shown in Figure 2. Scintillator plane S1 is hit first and then S2 slightly later due to time-of-flight of the muon (3 ns per meter traveled). The hits somehow produce the logic signals shown. Were the muon to continue instead of decaying between planes 2 and 3, one would also observe a 3rd hit in S3. However, in this example the muon comes to rest between planes S2 and S3, decays some 100's of nanoseconds or µs later, and emits an electron which travels back through plane S2 and produces a second pulse in S2. The electron could also travel downward and strike plane S3 at a later time and be distinguishable from a non-decaying muon. In order to capture events of this nature one would like to design a circuit that triggers when:

There is a coincidence of hits within a small time window, several ns depending on the spacing, in planes S1 and S2,
There is no signal in plane S3 coincident with S1 and S2,
There is a late hit in either S2 or S3. The decay is a random process with a mean of 2.2 µs, therefore there is an entire range of possible times following the signals in S1 and S2 which should be accepted (the green bar region in the Figure, for example).

Figure 2: Digital signals produced in the muon decay experiment

The 3rd condition will be the subject of a later discussion when we arrive at sequential logic. The first two conditions state that we want signals in S1 and S2 and not S3. This is typically written as a logic expression

S1·S2·S3'	(1)

Logic Signals

It is evident that the signals being processed by this example trigger system are not the raw PMT signals which would have a complex form dependent on the amount of energy deposition in the scintillator, rather they are square pulses of fixed width and fixed height that only serve to mark the presence (or absence) of a hit in the photomultiplier. This ideal can be realized by using a pulse discriminator which emits pulses of constant height and a monostable to set the width in time. Because there are in principle only two possible voltage levels, the signals can be treated as being in one of two logic states: logic HIGH and logic LOW. By interpreting these states as TRUE and FALSE we can use logic gates to act on the digital input signals S₁, S₂, and S₃, and produce a digital output which corresponds to the logic expression (1).

Logic Gates

These logic gates are available as integrated circuit packages, often with multiple gates per package and with a. A popular, industry-wide standard set of IC's known as the 7400-series logic dates back to the 1960's and still is actively being produced by many manufacturers. The following table lists some of the basic logical expressions, their truth tables, circuit symbols, and the corresponding 7400-series ICs:

Gate Name

Expression

Standard Symbol

IEC Symbol

Truth Table

7400-series IC

NOT

X = A
X = A'

A	X
0	1
1	0

7404	Hex inverter

AND

X = A · B
X = AB

A	B	X
0	0	0
0	1	0
1	0	0
1	1	1

7408	Quad 2-input
7411	Triple 3-input
7421	Dual 4-input

X = A + B

A	B	X
0	0	0
0	1	1
1	0	1
1	1	1

7432	Quad 2-input

XOR

X = A ⊕ B

A	B	X
0	0	0
0	1	1
1	0	1
1	1	0

7486	Quad 2-input

NAND

X = A · B

A	B	X
0	0	1
0	1	1
1	0	1
1	1	0

7400	Quad 2-input
7401	Quad 2-input, open collector outputs
7410	Triple 3-input
7420	Dual 4-input
7430	Single 8-input

NOR

X = A + B

A	B	X
0	0	1
0	1	0
1	0	0
1	1	0

7402	Quad 2-input
7427	Triple 3-input
7425	Dual 4-input

XNOR

X = A ⊕ B

A	B	X
0	0	1
0	1	0
1	0	0
1	1	1

74266

Quad 2-input

Logic Identities

Boolean logic identities are summarized below:

1' = 0
A · A = A
A + A = A
A · 1 = A
A + 1 = 1
A · 0 = 0
A + 0 = A
A · A' = 0
A + A' = 1
(A · A)' = A'
(A + A)' = A'
A · (B + C) = A · B + A · C
(A + B) · (A + C) = A + B · C
(A)' = A
A + A'B = A + B
AB = BA
A + B = B + A
A(BC) = (AB)C
A ⊕ B = A'B + B'A

And finally there are the laws of DeMorgan which can switch AND / OR operations:

(AB)' = A' + B'
(A + B)' = A'B'

DeMorgan's laws have the consequence that all combinational logic circuits could be constructed out of NAND or NOR gates (but not AND or OR gates). Take as example the muon trigger system; one may desire to construct the logic completely from 7400 NAND gates since there are 4 to a package. The logic expression could be re-written as the following sequence of logical identities:

Y = (S₁S₂)'
X = ((YY)'(S₃S₃)')'

(2)

This requires 5 NAND gates to implement which would appear as the circuit:

Logic families and logic thresholds

Each logic family defines thresholds for valid high and low levels:

V_IL: The voltage region in which the logic gate considers the input to be LOW or logic FALSE.
V_IH: The voltage region in which the logic gate considers the input to be HIGH or logic TRUE.
V_OL: The maximum voltage at the output for a logic LOW output.
V_OH: The minimum voltage at the output for a logic HIGH output.

Table: Logic thresholds for various logic families.
Family	V_CC		V_IL		V_IH		V_OL	V_OH	Comments
Family	Min	Max	Min	Max	Min	Max	V_OL	V_OH	Comments
TTL	4.75	5.25	-	0.8	2	-	0.4	2.4
LVTTL	2	5.5	-	0.3×V_CC	0.7×V_CC	-	0.4	2	At V_CC = 2.3 V to 2.7 V
			-	0.3×V_CC	0.7×V_CC	-	0.44	2	At V_CC = 3 V to 3.6 V
			-	0.3×V_CC	0.7×V_CC	-	0.55	2	At V_CC = 4.5 V to 5.5 V
HCMOS	2	6	GND	0.5	1.5	V_CC	0.1	1.9	At V_CC = 2 V
			GND	1.35	3.15	V_CC	0.1	4.4	At V_CC = 4.5 V
			GND	1.8	4.2	V_CC	0.1	5.9	At V_CC = 6 V
LVCMOS	2	3.6	GND	0.7	1.7	V_CC	0.4	2.2	At V_CC = 2.7 V
LVCMOS	2	3.6	GND	0.8	2	V_CC	0.55	2.4	At V_CC = 3 V

This is illustrated in the following graphs which were taken from the Texas Instruments™ Logic Guide.

Inverters and Boolean Gates : Transistor Implementation

Demonstration with an ideal switch

Figure 3: Simple switch inverter

Transistors switches internally compose the circuits of the logic gates discussed previously. Let's consider the trivial example of the circuit shown at right to understand the basic principle behind an inverter. The switch will be in the open state (non-conducting) if V_CTRL is below some threshold voltage, V_T, nominally V_CC/2, and will be closed (conducting) if it is above V_T.

If V_CTRL is in a HIGH state then the switch is closed. The output is grounded (V_OUT in LOW logic state).
If V_CTRL is in a LOW state the switch is open. The output is pulled up through the output resistance R.

Therefore, the output follows the logic inverse of the input: a NOT gate.

Transistor inverter

Figure 4: BJT inverter

A BJT inverter is shown in the graphic at left. It works as follows: assuming a V_CC of 5 V, then if point A is above about 0.8 V, the transistor will enter saturation as the collector current increases to the point where the voltage drop across resistor R2 is too great to keep the base-collector junction reverse-biased. Q1 will pull the collector, and thus the output, X, to approximately 0.05 V. Conversely, if point A is below about 0.6 V the transistor is cut-off, no current flows, and point X is pulled up through 480 Ω to logic HIGH.

Were the BJT replaced by a MOSFET, one could get rid of the base resistor as the FET gate draws no current. The MOSFET drain-source channel will act as a low-impedance (a few ohms are typical) path if V_GS, the potential difference between the gate and the source is above V_T which is typically in the range of several volts, it is a characteristic of each MOSFET.

Figure 5: CMOS inverter

CMOS Inverter

The problem with such inverter circuits is that, as you may have noticed, for outputs at logic LOW, quite a bit of power can be dissipated through the pull-up resistor. The resistor can be made arbitrarily large but with a penalty in switching speed. A better solution is to have a pair of transistors of which only one is conducting at any time far enough away from an output state transition. This is shown in Figure 5. The CMOS (Complementary MOS) basic structure shown here forms the foundation of much of modern digital circuits. It is made from a p-channel MOSFET (Q2) and an n-channel MOSFET (Q3). When A is HIGH Q2 is OFF and Q3 is ON, thus X is LOW. When A is LOW, Q2 is ON and Q3 is OFF, thus X is HI. Note that in each case only 1 output transistor is ON at any point, thus there is no quiescent power dissipation.

CMOS NAND gate

TBD

Open Collector and Three-State Outputs

TBD

Numeric Computing

Binary Numbers: Introduction to Number Bases

A single output of a digital gate can be in one of two states: 0 or 1. A pair of independent ouputs can be in one of four states: (0, 0), (0, 1), (1, 0), (1, 1). In general, the outputs of N gates, Q₀, Q₁, ..., Q_N, can hold 2^N states. These states may represent numbers, a representation which has powerful consequences when circuit elements which perform numerical operations such as addition and subtraction are introduced. This will be done shortly. For the present, we occupy ourselves with understanding the representation of numbers using sequences of HIGH/LOW logic states: this is the system of binary numbers.

Table C: Enumeration of 4-bit binary (base 2) numbers in decimal (base 10), octal (base 8), and hexadecimal (base 16).
Q₃Q₂Q₁Q₀	Decimal	Octal	Hexadecimal
0000	0	0	0
0001	1	1	1
0010	2	2	2
0011	3	3	3
0100	4	4	4
0101	5	5	5
0110	6	6	6
0111	7	7	7
1000	8	10	8
1001	9	11	9
1010	10	12	A
1011	11	13	B
1100	12	14	C
1101	13	15	D
1110	14	16	E
1111	15	17	F

Binary numbers are written as strings of '0' or '1' bits, usually written most significant bit (MSB) first and least significant bit (LSB) last, in analogy with decimal numbers. The corresponding decimal number may be computed using a simple algorithm:

Starting with the LSB, set b = 0
Accumulate (add) 2^b if the bit is '1'
Move to the next bit, b = b + 1

Transformation from decimal into binary may be accomplished with the following algorithm:

Divide the number by 2 and place the remainder (0 or 1) in LSB
Repeat this step, appending the bits in the next-most-significant-bit position (forming the binary string from right-to-left) until the quotient is zero.

Hexadecimal numbers offer a compact format for expressing long bit strings. Because the number base is 16, each quartet of bits orthogonally separates into a hex digit so that the transformation between binary and hex representations is trivial: for example the rather long binary string 10 111 010 001 011 110 010 011 111 can be written as 2E8BC9F by consulting Table C. Octal numbers offer an another orthogonal number format, base 8, with triplets of bits mapping onto a single octal digit. Thus, the binary string above in octal would be 272136237. When it is not clear which number base a sequence of digits is expressed in different conventions are used. Sometimes a number will be tagged with 'bin', 'dec', or 'hex' to resolve the ambiguity:

10010110_bin = 226_oct = 150_dec = 96_hex

In the C programming language, and also languages with similar syntax (C++, Java, Python to name a few), octal literals (constants in the code) may be denoted as such by a leading '0' which is redundant in decimal. Hexadecimal literals are expressed with a leading '0x', thus the following C function would return 1:

int test() {
    int a = 150;
    int b = 0226;
    int c = 0x96;
    return (a == b && b == c);
}

Addition of binary numbers

Despite it being rather more tedious to add binary numbers, there is nothing difficult at all about the mechanics behind addition in base 2. It is completely analogous to addition of decimal numbers so let's jump right in with an example, say the addition using 4-bit adders of 11 + 7:

1	1	1	1		Cy
	1	0	1	1	A
+	0	1	1	1	B
1	0	0	1	0

The process starts with the bits in the right hand column: 1 + 1 = 2 but in binary that must then be 0 with a carry into the next column as signified by the 'Cy' row on top. In the next row the three 1's add to 1 with a carry, and so on ... Finally there is a carry out of the last bit. The resulting number is 18 which agrees well with the decimal version of this operation. Note that it is often the case in binary additions that registers are of fixed width and thus there is no obvious place to put the carry bit in the result. Rather, it would be truncated to 4-bits giving the odd answer of 0010 = 2 instead of 18.

Table D: Truth table for binary full-adder
Cin	A	B	X	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

But how is this operation to be undertaken by logic circuits? The solution is obtained by examination of the truth table that represents the addition of a single pair of bits - one column in the example. Notice that there are 3 inputs: the bit of A, the bit of B, and the carry bit, and two outputs: the sum bit and the carry out to the next column bit. We will call the input carry the carry in or Cin, and the carry out, Cout in the truth table, Table D.

Notice that the sum output, X, is true when the input lines A, B, and Cin have an odd number of bits. This may be implemented as the following logical expression:

X = A ⊕ B ⊕ Cin

while the carry out, Cout, is TRUE when A and B are TRUE or when Cin is TRUE and A ⊕ B is TRUE:

Cout = AB + (A ⊕ B) · Cin

Thus a 1-bit full adder can be realized by the following circuit diagram:

Wider (i.e. more bits) adders are constructed from the single bit basic building block by chaining the carry out of one stage to the carry in of the next stage:

Signed and unsigned

The binary representation explained above covers only numbers from 0 to 2^N - 1 where N is the number of bits in the binary string. Of course, it is entirely your business how you enumerate the logic states: the enumeration listed in Table C is merely one of many possible enumerations and circumstances particular to an application may favor one of these other possibilities. Very often, numeric problems involve negative numbers. Numbers that can take on both negative and positive values are called signed numbers, as opposed to unsigned numbers which are only positive or zero. An enumeration of states known as 2's complement allows one to represent signed numbers and has the important property that the adder developed above for unsigned numbers all work without modification for signed numbers. As was seen in the section on addition, adding two N-bit numbers in general requires N+1 bits to hold the result.

Magnitude Comparators

TBD

Multiplexers

A multiplexer (also muxer or even mux) takes M signal inputs and N selector inputs (usually 2^N = M) and places the input addressed by the selector lines onto the output.

Circuit Symbol

Truth Table

Description

A	B	S	X
0	X	0	0
1	X	0	1
X	0	1	0
X	1	1	1

2-input multiplexer

A	B	C	D	S₁	S₀	X
0	X	X	X	0	0	0
1	X	X	X	0	0	1
X	0	X	X	0	1	0
X	1	X	X	0	1	1
X	X	0	X	1	0	0
X	X	1	X	1	0	1
X	X	X	1	1	1	0
X	X	X	1	1	1	1

4-input multiplexer

Figure 8: Internal diagram for the NXP 74LVC1G157.

Demultiplexers and decoders

The inverse circuit is called a demultiplexer or a decoder. It takes N selector inputs, usually called address inputs, and activates one of the 2^N outputs. This circuit is often used to activate circuitry interconnected using a shared data bus. Another application of decoders is as a driver for a 7-segment display.

Logic symbol and truth table for '138 3-to-8 line decoder (from NXP Semiconductor datasheet). Note that the outputs are ACTIVE LOW.

E1'	E2'	E3	A0	A1	A2	Y0'	Y1'	Y2'	Y3'	Y5'	Y5'	Y6'	Y7'
1	X	X	X	X	X	1	1	1	1	1	1	1	1
X	1	X	X	X	X	1	1	1	1	1	1	1	1
X	X	0	X	X	X	1	1	1	1	1	1	1	1
0	0	1	0	0	0	0	1	1	1	1	1	1	1
0	0	1	0	0	1	1	0	1	1	1	1	1	1
0	0	1	0	1	0	1	1	0	1	1	1	1	1
0	0	1	0	1	1	1	1	1	0	1	1	1	1
0	0	1	1	0	0	1	1	1	1	0	1	1	1
0	0	1	1	0	1	1	1	1	1	1	0	1	1
0	0	1	1	1	0	1	1	1	1	1	1	0	1
0	0	1	1	1	1	1	1	1	1	1	1	1	0

Priority Encoders

A priority encoder takes N inputs and outputs the address of the highest set bit. The '148 offers this function with ACTIVE LOW inputs and outputs, and is cascadable using EI' and EO':

EI'	I0'	I1'	I2'	I3'	I4'	I5'	I6'	I7'	A2'	A1'	A0'	GS'	EO'
1	X	X	X	X	X	X	X	X	1	1	1	1	1
0	1	1	1	1	1	1	1	1	1	1	1	0	1
X	0	1	1	1	1	1	1	1	1	1	0	0	1
X	X	0	1	1	1	1	1	1	1	0	1	0	1
X	X	X	0	1	1	1	1	1	1	0	0	0	1
X	X	X	X	X	1	1	1	1	0	1	1	0	1
X	X	X	X	X	X	0	1	1	0	1	0	0	1
X	X	X	X	X	X	X	0	1	0	0	1	0	1
X	X	X	X	X	X	X	X	0	0	0	0	0	1