NAND, NOR and XOR functions properties

Transcription

1 Laboratory NAND, NOR and XOR functions properties. Laboratory work goals Enumeration of NAND, NOR and XOR functions properties Presentation of NAND, NOR and XOR modules Realisation of circuits with gates in order to practically test some of the properties of NAND, NOR and XOR functions.. Theoretical considerations.. NAND function properties Mathematical symbol: or The truth table is presented in Fig... The graphical symbol of the gate realising the function is depicted in Fig... x x x x x x x x Fig.. Fig.. 0

2 P) x x x n = x x... x n = x + x x n P) Commutativity of NAND: x x =x x P) NAND function is not associative, that is (x x ) x x (x x ) P) NAND function is pseudo-associative, that is x x x = ( x x) x = x ( x x) P) 0 is aggressive with respect to NAND meaning 0 x x x n = P) The pseudo-neutrality of with respect to NAND a) x x x n =x x x n b) x = x P) The pseudo-idempotency of NAND a) x x x y y y m = x y y y m b) x x x= x P) The pseudo-distributivity of NAND with respect to NOR: x (y z)=( x y) ( x z) P) The first absorption law for NAND: x (x y)=x y P0) The second absorption law for NAND: (x y) ( x y)=y P) Any switching function can be represented using only NAND operators (The NAND operator forms a base for switching functions representation).

3 .. NOR function properties Mathematical symbol: The truth table is presented in Fig... The graphical symbol of the gate realising the function is depicted in Fig... x x x x x x x x Fig.. Fig.. P) x x x n = x + x xn = x x... xn P) NOR is commutative: x x =x x P) NOR function is not associative, meaning (x x ) x x (x x ) P) NOR function is pseudo-associative, meaning x x x = (x x) x = x (x x) P) is aggressive with respect to NOR meaning x x x n =0 P) The pseudo-neutrality of 0 with respect to NOR a) 0 x x x n =x x x n b) 0 x = x P) The pseudo-idempotency of NOR a) x x x y y y m = x y y y m b) x x x= x P) The pseudo-distributivity of NOR with respect to NAND:

4 x (y z)=( x y) ( x z) P) The first absorption law for NOR: x (x y)=x y P0) The second absorption law for NOR: (x y) ( x y)=y P) Any switching function can be represented using only NOR operators (the NOR operator forms a base for switching functions representation)... XOR function properties Mathematical symbol: The truth table is presented in Fig... The graphical symbol of the gate realising the function is depicted in Fig... x x x x x x x + x Fig.. Fig.. P) x x = xx + xx (see paragraph..) P) XOR operator is commutative: x x =x x P) XOR operator is associative: x (x x )= (x x ) x P) 0 is neutral with respect to XOR: 0 x=x 0=x

5 P) is pseudo-neutral with respect to XOR: x=x = x Remark! The P and P properties justify the name of commanded inverter, sometimes used to designate the XOR operator, as it can be seen in Fig... Fig.. P) The symmetrical of x with respect to XOR is x itself, meaning x x=0. P) x x... x = n times 0 x if if n = k n = k + P) x y = x y = x y P) Distributivity of AND with respect to XOR: x (y z)=(x y) (x z) P0) Any switching function can be implemented using only XOR, AND operators and the constant (XOR, AND, form a base for switching functions representation).

6 . Presentation of modules comprising NAND, NOR and XOR logic gates The modules used during the lab activities comprise XOR (), NOR (0) and NAND (LS00) logic circuits. Details about each module and pin assignment are presented in Appendix and [0], []. The LS00 module (Fig..) is an integrated circuit that contains NAND gates, implementing the NAND function with inputs. Y=A B= A B = A + B The LS0 module (Fig..) is an integrated circuit that contains NAND gates, implementing the NAND function with inputs. Y=A B C = A B C = A + B + C The LS0 module (Fig..0) is an integrated circuit that contains NAND gates, and implements the NAND function with inputs. Y=A B C D = A B C D = A + B + C + D The LS0 module (Fig..) is an integrated circuit containing independent NOR gates, and implements the NOR function with inputs. Y=A B= A + B = A B The module (Fig..) is an integrated circuit containing independent XOR gates, implementing the XOR function with inputs. Y=A B= A B + AB

7 A A A B A B C B A Y B A Y Y NC B A Y B C A NC Y B 0 A C 0 B B 0 A Y B Y C Y B GND Y GND Y GND Fig.. Fig.. Fig..0 Y Y A A Y B A B A Y B Y B A Y A 0 Y B 0 A B A Y B GND B GND Y Fig.. Fig... Lab activity progress There will be implemented circuits comprising gates in order to verify some of the XOR, NOR and NAND functions properties. Example: Distributivity of conjunction with respect to XOR: x,y,z {0,} x (y z)=(x y) (x z) E= x (y z) E=(x y) (x z)

8 In order to verify these properties a XOR SN module and an AND SNLS0 module will be used. The test diagram is depicted in Fig... Both expressions must be implemented and then for each possible combination of x, y and z the output voltage must be measured for each expression circuit. x y z UA LSA UA LS0 E UB 0 UC LS0 UB LSA E LS0 Fig... Proposed problems. Implement a circuit containing logic gates that verifies property P of NAND function. The NAND function will be implemented using the 0 module.. Implement a circuit containing logic gates that verifies properties P, P, P and P of NAND function.. Implement a circuit containing logic gates that verifies property P of NAND function.. Implement a circuit containing logic gates that verifies properties P and P0 of NAND function.. Implement a circuit containing logic gates that verifies property P of NOR function. The NOR function will be implemented using and 0 modules.

9 . Implement a circuit containing logic gates that verifies properties P, P, P and P of NOR function.. Implement a circuit containing logic gates that verifies property P of NOR function.. Implement a circuit containing logic gates that verifies properties P and P0 of NOR function.. Implement a circuit containing logic gates that verifies properties P, P, P and P of XOR function. 0. Implement a circuit containing logic gates that verifies properties P, P and P of XOR function. Note! The test results will be recorded for each problem in a table as the Fig.. or Fig..