Digital logic-formula-notes-final-1
- 1. Number System and Codes: A number system with base ‘r’, contents ‘r’ different digits and they are from 0 to r – 1. Decimal to other codes conversions: To convert decimal number into other system with base ‘r’, divide integer part by r and multiply fractional part with r. Other codes to Decimal Conversions: Hexadecimal to Binary: Convert each Hexadecimal digit into 4 bit binary. 2 16 (0101 1010 1111) (5AF) 5 A F Binary to Hexadecimal: Grouping of 4 bits into one hex digit. Octal to Binary and Binary to Octal: Same procedure as discussed above but here group of 3 bits is made. Codes: Binary coded decimal (BCD): In BCD code each decimal digit is represented with 4 bit binary format. 10 9 4 9 :(943) 1001 0100 0011 BCD Eg 10 Invalid BCD codes 16 106 These are 1010, 1011, 1100, 1101, 1110, and 1111 Excess-3 code: (BCD + 0011) It can be derived from BCD by adding ‘3’ to each coded number. It is unweighted and self-complementing code. 2 1 0 1 2 10( (A)rx x x . y y ) 2 1 2 2 1 0 1 2A x r x r x y r y r 2 160011 0101.1100(110101.11) (35.C) It is also known as 8421 code Invalid BCD codes Total Number possible 24 16 Valid BCD codes gradeup gradeup gradeup Digital Electronics (Formula Notes)
- 2. Gray Code: It is also called minimum change code or unit distance code or reflected code. Binary code to Gray code: MSB Gray code to Binary code: Alpha Numeric codes: EBCDIC (Extended BCD Interchange code) It is 8 bit code. It can represent 128 possible characters. Parity Method is most widely used schemes for error detection. Hamming code is most useful error correcting code. BCD code is used in calculators, counters. Complements: If base is r then we can have two complements. (i) (r – 1)’s complement. (ii) r’s complement. To determine (r–1)’s complement: First write maximum possible number in the given system and subtract the given number. To determine r’s complement: (r–1)’s complement + 1 First write (r–1)’s complement and then add 1 to LSB Example: Find 7’s and 8’s complement of 2456 7777 5321 2456 1 7's complement 8's complement 5321 5322 Find 2’s complement of 101.110 1’s complement 010.001 For 2’s complement add 1 to the LSB 010.001 1 2'scomplement 010.010 + 1 0 0 1 0 + + + 1 1 0 1 1 MSB Binary Gray gradeup gradeup gradeup
- 3. Data Representation: Unsigned Magritude: Range with n bit Signed Magritude: Range with n bit 1’s complement: Range with n bit 2’ complement: With n bits Range In any representation +ve numbers are represented similar to +ve number in sign magnitude. 0 to 2n 1 5101 5 Not possible (2n 1 1) to (2n 1 1) 60110 1 110 1 sign bit sign bit with 4 bits with 8 bits 6 0000110 (2n 1 1) to (2n 1 1) 60110 1 001 6 sign bit 1's complement of 6 2n 1 to (2n 1 1) 60110 sign bit 2's complement of 6 6 1 010 Boolean Laws: gradeup gradeup gradeup
- 4. Flip Flops : Excitation tables : For ring counter total no.of states = n For twisted Ring counter = “2n” (Johnson counter / switch tail Ring counter ) . To eliminate race around condition t < < t . In Master slave master is level triggered & slave is edge triggered 0 0 00 0 1 x 1 1 0 1 R 1 0 1 S x 0 1 0 00 0 1 0 1 0 1 D 0 1 0 1 1 0 1 01 1 0 1 0 1 0 J K 0 1 x 1 0x x x 1 0 1 T 0 1 1 0 Q(n+1) = S + R Q = D = J + Q = T + Q gradeup gradeup gradeup
- 6. Multiplexer : 2 i/ps ; 1 o/p & ‘n’ select lines. It can be used to implement Boolean function by selecting select lines as Boolean variables For implementing ‘n’ variable Boolean function 2 × 1 MUX is enough . For implementing “n + 1” variable Boolean 2 × 1 MUX + NOT gate is required . For implementing “n + 2” variable Boolean function 2 × 1 MUX + Combinational Ckt is required If you want to design 2 × 1 MUX using 2 × 1 MUX . You need 2 2 × 1 MUXes Decoder : n i/p & 2 o/p’s used to implement the Boolean function . It will generate required min terms @ o/p & those terms should be “OR” ed to get the result . Suppose it consists of more min terms then connect the max terms to NOR gate then it will give the same o/p with less no. of gates . If you want to Design m × 2 Decoder using n × 2 Decoder . Then no. of n × 2 Decoder required = . In Parallel (“n” bit ) total time delay = 2 t . For carry look ahead adder delay = 2 t . PROM , PLA & PAL :- AND OR Fixed Programmable Programmable fixed Programmable Programmable PROM PAL PLA gradeup gradeup gradeup
- 7. Asynchronous Sequential circuits: Asynchronous Sequential circuits do not use a clock and can change their output state as fast as the signal path's propagation delay from the input allows. This means they can be faster than Synchronous Sequential circuits. However, they are considerably more likely to suffer from race conditions (inputs arriving at different times causing different output states) and intermediate output states (as the outputs change from one state to the next final state) than Synchronous Sequential circuits. Synchronous Sequential circuits: Synchronous Sequential circuits use a clock signal to alleviate the two problems mentioned above. The outputs can only change state with the clock and are designed such that all propagation delays are satisfied before the outputs are allowed to change. This however makes them potentially slower (because the whole circuit must run at the speed of the slowest path in it) and consumes significantly more power due to the extra circuitry required by distributing the clock to all flip-flops, and the continual switching. gradeup gradeup gradeup
- 8. • Fan out of a logic gate = IOH IIH IIL or IOL • Noise margin : VOH - VIH or VOL - VIL • Power Dissipation PD = Vcc Icc = Vcc � I +I 2 � I → Ic when o/p low I → Ic when o/p high . • TTL , ECL & CMOS are used for MSI or SSI • Logic swing : VOH - VOL • RTL , DTL , TTL → saturated logic ECL → Un saturated logic • Advantages of Active pullup ; increased speed of operation , less power consumption . • For TTL floating i/p considered as logic “1” & for ECL it is logic “0” . • “MOS” mainly used for LSI & VLSI . fan out is too high • ECL is fastest gate & consumes more power . • CMOS is slowest gate & less power consumption • NMOS is faster than CMOS . • Gates with open collector o/p can be used for wired AND operation (TTL) • Gates with open emitter o/p can be used for wired OR operation (ECL) • ROM is nothing but combination of encoder & decoder . This is non volatile memory . • SRAM : stores binary information interms of voltage uses FF. • DRAM : infor stored in terms of charge on capacitor . Used Transistors & Capacitors . • SRAM consumes more power & faster than DRAM . • CCD , RAM are volatile memories . • 1024 × 8 memory can be obtained by using 1024 × 2 memories • No. of memory ICs of capacity 1k × 4 required to construct memory of capacity 8k × 8 are “16” DAC ADC • 2 FSV = VR �1 − 1 � * LSB = Voltage range / 2n • Resolution = step size FSV = VR /2n VR �1− 2 1 n � = 1 2n −1 × 100% * Resolution = n FSV 2 −1 • Accuracy = ± 1 2 LSB = ± 1 2n+1 * uantisation error = V n R 2 % • Analog o/p = K. digital o/p • Flash Type ADC : 2n−1 → comparators 2n → resistors 2n × n → Encoder Fastest ADC :- • Successive approximation ADC : n clk pulses • Counter type ADC : 2n - 1 clk pulses • Dual slope integrating type : 2n+1 clock pulses . gradeup gradeup gradeup
- 9. A Microprocessor includes ALU, register arrays and control circuits on a single chip. Microcontroller: A device that includes microprocessor, memory and input and output signal lines on a single chip, fabricated using VLSI technology. Architecture of 8085 Microprocessor 8085 MPU: 8 bit general – purpose microprocessor capable of addressing 64 K of memory. Microprocessor It has 40 pins, requires a +5V single power supply and can operate with 3 – MHz single phase clock. Accumulator: Is an 8 bit register that is used to perform arithmetic and logic functions. gradeup gradeup gradeup
- 10. Flag Register: Carry Flag (CY): If an arithmetic operation result in a carry or borrow, the CY flag is set, otherwise it is reset. Parity Flag (P): If the result has au even number of 1s, the flag is set, otherwise the flag is reset. Auxiliary Carry (AC): In an arithmetic operation If carry is generated by and passed to flag is set. Otherwise it is reset. Zero Flag (Z): Zero Flag is set to 1, when the result is zero otherwise it is reset. Sign Flag (S): Sign Flag is set if bit of the result is 1. Otherwise it is reset. Program counter (PC): It is used to store the l6 bit address of the next byte to be fetched from the memory or address of the next instruction to be executed. Stack Pointer (SP): It is 16 bit register used as a memory pointer. It points to memory location in Read/Write memory which is called as stack. 7 6 5 4 3 2 1 0D D D D D D D D S Z AC P CY 3D 4D 7D 8085 Signals: Address lines: and A8 A15 to identify the memoryThere are l6 address lines locations. 0 7AD AD • In memory mapped I/O ; I/O Devices are treated as memory locations . You can connect max of 65536 devices in this technique . • In I/O mapped I/O , I/O devices are identified by separate 8-bit address . same address can be used to identify i/p & o/p device . • Max of 256 i/p & 256 o/p devices can be connected . Programmable Interfacing Devices :- • 8155 → programmable peripheral Interface with 256 bytes RAM & 16-bit counter • 8255 → Programmable Interface adaptor • 8253 → Programmable Interval timer • 8251 → programmable Communication interfacing Device (USART) • 8257 → Programmable DMA controller (4 channel) • 8259 → Programmable Interrupt controller • 8272 → Programmable floppy Disk controller • CRT controller • Key board & Display interfacing Device gradeup gradeup gradeup
- 11. CALL & RET PUSH & POP • * Programmer use PUSH to save the contents rp on stack When CALL executes , µp automatically stores 16 bit address of instruction next to CALL on the Stack • CALL executed , SP decremented by 2 * PUSH executes “SP” decremented by “2” . • RET transfers contents of top 2 of SP to PC * same here but to specific “rp” . • RET executes “SP” incremented by 2 * same here RLC :- Each bit shifted to adjacent left position . D7 becomes D0 . CY flag modified according to D7 RAL :- Each bit shifted to adjacent left position . D7 becomes CY & CY becomes D0 . ROC :-CY flag modified according D0 RAR :- D0 becomes CY & CY becomes D7 Rotate Instructions: CALL and Return Instructions CALL → 18T states SRRWW CC → Call on carry 9 – 18 states CM → Call on minus 9-18 CNC → Call on no carry CZ → Call on Zero ; CNZ call on non zero CP → Call on +ve CPE → Call on even parity CPO → Call on odd parity RET : - 10 T RC : - 6/ 12 ‘T’ states Jump Instructions JMP → 10 T JC → Jump on Carry 7/10 T states JNC → Jump on no carry JZ → Jump on zero JNZ → Jump on non zero JP → Jump on Positive JM → Jump on Minus JPE → Jump on even parity JPO → Jump on odd parity . gradeup gradeup gradeup
- 12. • PCHL : Move HL to PC 6T • PUSH : 12 T ; POP : 10 T • SHLD : address : store HL directly to address 16 T • SPHL : Move HL to SP 6T • STAX : Rp store A in memory 7T • STC : set carry 4T • XCHG : exchange DE with HL “4T” XTHL :- Exchange stack with HL 16 T • For “AND “ operation “AY” flag will be set & “CY” Reset • For “CMP” if A < Reg/mem : CY → 1 & Z → 0 (Nothing but A-B) A > Reg/mem : CY → 0 & Z → 0 A = Reg/mem : Z → 1 & CY → 0 . • “DAD” Add HL + RP (10T) → fetching , busidle , busidle • DCX , INX won’t effect any flags . (6T) • DCR, INR effects all flags except carry flag . “Cy” wont be modified • “LHLD” load “HL” pair directly • “ RST “ → 12T states • SPHL , RZ, RNZ …., PUSH, PCHL, INX , DCX, CALL → fetching has 6T states • PUSH – 12 T ; POP – 10T gradeup gradeup gradeup