From Sand to Computers

Transistors -- the basis of computers.

The above image shows an NMOS transistor. When current is applied to its input the gate is pushed into the P substrate and connects the two N areas (source and drain). Thus current flows from the source to the drain. The PMOS transistor is the equivalent but uses a N substrate with P source and drain. It conducts when no current is applied at the gate and shuts off when current is applied. The symbols for NMOS and PMOS transistors are alsow shown in the figure.

From Transistors to Gates

Inverter

Nand and Nor gates

Symbols and Truth Tables for Gates

Truth tables and boolean functions

For every function of boolean variables we can create a truth table. The table simply reports the output for each combination of inputs. From the table we can obtain the circuit that implements the function by using a combination of "and", "or", and "not" gates. We need to and the variables for which the truth table has an output of one and then or the outputs of these and gates. The inputs to the and gates are either the variables or their negation depending on whether the term in the truth table has a 1 or a 0 respectively.

Function Minimization

Many times the truth table representation of a boolean function had redundant terms. It is possible to derive a minimal representation for a Boolean function by using Karnaugh maps. Each entry in such map represents one of the terms derived from the truth table. Adjacent terms can be combined and the variable(s) that appear in both the positive and negative terms can be removed. See the above figure for an example of a Karnaugh map. The labelling on each box follows Gray-code numbering.

Flip-Flops

Flip-flops are boolean circuit whose output depends on both the input and their previous state. The RS flip-flop produces an output of 1 when the RS pair is 01, an output of 0 when the RS pair is 10, maintains its state when the RS pair is 00, and has an undefined output when the RS pair is 11. The JK flip-flop has the same behavior with the RS flip-flop with the exception of the 11 input case. In that case the JK flip-flop toggles its state.

More in digital circuits can be found in "Digital logic and computer design" by Morris Mano which is on reserve at the Carlson library.