EECS 465: Digital Systems Lecture Notes # 7 (A) Introduction to Sequential Circuits (B) Latches and Flip-Flops (C) Counter Design SHANTANU DUTT Department of Electrical and Computer Engineering University of Illinois, Chicago Phone: (312) 355-1314: e-mail: dutt@eecs.uic.edu URL: http://www.eecs.uic.edu/~dutt

(A) Introduction to Sequential Circuits • Current o/p depends on the current i/p and past history of all i/ps seen by the circuit. Where the relevant past history should be representable by a finite number of classes or states Light = Green O/P = 1 Light = Not red O/P = 0 Reset No Red Red Light State Encode as state = 1 Light = Red O/P = 0 Light = Not green O/P = 0 Encode as state=0 State Transition Diagram Design Problem: Output of the circuit is 1 only if it has seen a red light in the past and currently light is green.

Circuit-Level Model of a Sequential Circuit. I/p from external point x0 Z0 Going to external world Combinational Circuit xn-1 Zm-1 State bits of seq. ckt. yk-1 y’k-1 y0 y’0 Memory Unit Next State Current State

(B) Latches and Flip-Flops Components to store bits ( latches or flip flops ) 1) Problem: can’t store new data 1 1 1 Cascade of inverter LD 2) I/P 1/0 O/P LD A Will conduct when A=1, and open when A=0

Another storage element: NOR gates ( R-S latch ) 3) Cross coupled NOR gates ( R-S latch ) (Reset) R Q S Q Q 1 R=0 S=1 (Set) B Property of a NOR gate A=0 When one I/P of NOR is 0, it acts like an inverter. When one I/P is 1, then O/P=0. Different I/P conditions for R-S latch: i) R=S=0, current I/P is stored indefinitely ( becomes cascade of inverters) Hold

ii) R=1, S=0, when we want to store a 0 in the R-S latch. Q=0, iii) R=0, S=1, when we want to store a 1 in the latch. Q=1, iv) R=1, S=1; Forbidden inputs! Both Q = 0, : Q and its complement have the same value ! Will play havoc in the rest of the logic circuit. Transit to: R=0, S=0. R=1, S=1, both Q and and 0. 0 0 O/P oscillates. Q R=10 S=10 1 01 0  Oscillates between 1 and 0 when we transit from R=S=1 to R=S=0.

From R, S = 1, 1 transit to R=0, S=1 then Q, transit to 1, 0 ( correctly ) From R, S = 1, 1 transit to R=1, S=0 then Q, correctly transit to 0, 1 Two implementations for R-S latch: Cross-coupled NOR Cross-coupled NAND R Q S R S Q Hold State R=S=1 Hold State R=S=0 R=0, S=0 R=1, S=1 Q Forbidden I/Ps

(level-sensitive clock latch) — see terminology defined later. 4) The D-Latch R R1 Q R-S Latch Clocked Latch (level-sensitive clock latch) — see terminology defined later. enb D S1 S D=1, S=1, R=0 Q=1, D=0, S=0, R=1 Q=0, enb=0 R1,S1=0 (hold state) enb=1 enb=1

— Proposed to get rid of the forbidden I/P problem of R-S 5) The J-K Latch: — Proposed to get rid of the forbidden I/P problem of R-S i) J=1, K=0: (a) Let Q=1,  R=0,S=0  Hold state of R-S  Q=1, (b) Let Q=0, , R=0, S=1  Q=1, ii) J=0, K=1  Q=0, using a similar analysis iii) J=K=0  Hold state iv) J=K=1, suppose Q=1, =0  R=1, S=0  Q=0, =1  S=1, R=0  Q=1, =0 This type of toggling continues as long as J=K=1, and the latch is enabled ( CLK=1 below ) Q 10 10 R K R CLK R-S J 1 S Q 01 01

Latch classification with respect to response to “control signal” Terminology: Note that the terminology below applies to all types of latches: R-S, D, J-K, T, etc., though the examples are given for the R-S latch. i) Transparent Latch: O/P responds to latch I/Ps without any enable or clock signal. R Q S Symbol: R S Q Clock: Fixed frequency alternating 1 and 0 signal ii) Clocked or Level-Sensitive Latch: R O/P responds to I/Ps only when enb or clock is at a pre-determined level (high or low — In this example, it is High) Q Clock or enb S Symbol: R S Q R S Q or CLK (low enable) (high enable) CLK

iii) Edge-Triggered Flip-Flop (FF) or simply Flip-Flop O/P will respond to I/Ps only at either: (a) the positive or rising edge of the enb/clock signal (positive edge-triggered FF), or (b) the negative or falling edge of the enb/clock signal (negative edge-triggered FF). Symbol: R Q S CLK Symbol: R S Q CLK Clock: O/P resp. period for a low-enable/clock level sensitive latch O/P response period for a positive edge-triggered FF. O/P response period for a HIGH-enable/clock level-sensitive latch O/P response period for a negative edge-triggered FF

Setup Times and Hold Time of FFs and Latches • Assume, positive edge-triggered D-FF THold relates to propagation delay of another part of circuit. D CLK TSetup relates to propagation delays of various gates in the FF. The high point of the CLK determines the positive edge’s arrival. • If negative edge-triggered TSetup THold D CLK Negative edge arrival • If D-Latch is high-level sensitive: Tsetup and Thold have to be around the negative edge of clock (more specifically, when the clock begins to go low), similar to negative edge-triggered. •If D-Latch is low-level sensitive: Tsetup and Thold have to be around the positive edge of clock, similar to positive edge-triggered.

Solutions to Race Condition Problem with Level Sensitive Latches Solution 1: Master-Slave FF: Q 1 R R-S Latch J R 1 1 1 R-S P Qm K Qs S S Q CLK Master R-S is level sensitive. Slave R-S is level sensitive. Master-Slave J-K is a solution to race-condition problem: Any change in Q, during CLK=0 is not propagated to P, and hence back to Q, during the same CLK=0. Any change to Q, will occur in next CLK=0 period. J Q J-K M-S Master-Slave J-K works similar to a J-K latch: E.g. Let J=1, K=1, CLK=1 Q=1, =0  =1 , P=0 When CLK=0 Q=0, =1 K (O/P responds when CLK goes From 1 to 0)

Solution 2: Edge-Triggered FF: Assume D=1 D=1=S Q=1, Solution 2: Edge-Triggered FF: D Holds when clock goes low R Q =D Clk=0 R Q S Clk=1 D Q responds to internal S signal; responds to internal R signal. S D Holds D when clock goes low D CLK D When CLK is 1 D I/P is internally sampled but does not appear at the O/P. O/P appears (Q=D) R Q O/P is held (changing D does not cause any change in internal signals in the FF or in its output) Clk=0 S D D

Characteristic Equations of Latches/FFs The next O/P Q+ defined in terms of the current O/P Q and the I/P. (FF/Latch is the simplest possible sequential ckt.) 1) R-S Latch— Truth Table: Values at time t S(t) R(t) Q(t) Q+ = Q( t+ ) 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 x 1 1 1 x Hold Reset Set Forbidden Q+= S+ Q (Characteristic equation) Q(t)\SR 00 01 11 10 0 0 0 x 1 1 1 0 x 1

Similarly: Characteristic Equations of 2) J-K, Q+ = Q + J. 3) D-FF, Q+ = D 4) Toggle FF/Latch Q+ = T + Q or T-FF / Latch Symbol: Q T Whenever I/P T is high, the FF will toggle, i.e., Q+ = . When T=0, Q+=Q. Of course, these characteristic equations come into play only when the FF/Latch is enabled.

Excitation Table — Reversed Truth Table — What the inputs to FFs should be for given output transitions (Q  Q+) Q Q+ R S J K T D 0 0 x 0 0 x 0 0 0 1 0 1 1 x 1 1 1 0 1 0 x 1 1 0 1 1 0 x x 0 0 1

— Conversion between FFs Example: J-K to D This should behave like a D-FF. J K Q D CLK Logic D D 0 1 x x 0 1 1 Q D Q O/P function = J J=D Function = K K= x x 1 0 Map the D,Q input combination to a QQ+ transition and then map this to J-K excitation required. Thus, when D=1, Q=0, Q+=1  J,K = 1,x D=0, Q=0, Q+=0  J,K = 0,x D=1, Q=1, Q+=1  J,K = x,0 D=0, Q=1, Q+=0  J,K = x,1. D-FF J K Q D CLK

Example 2: D  J-K Excitation Table for D should work like a J-K Q Q+ D 0  0 0 0  1 1 1  0 0 1  1 1 J K Q Q+ 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 Q J D Logic K CLK JK 00 01 11 10 Q TT for J-K 1 0 0 1 1 1 0 0 1 Function is J-K FF/Latch Q D Q K J CLK

• A counter is a special case of an FSM that cycles through its states (C) Counter Design • A counter is a special case of an FSM that cycles through its states on receiving triggering clock pluses. • It does not have any external data I/Ps. 100 No external I/Ps Reset E A Counter O/P 000 011 Logic D Next State bits 001 B C 010 FFs n n CLK • The states need to be encoded by binary bits.

State Transition Diagram and Table for a 3-bit Binary Up-Counter Synthesis (3-Bit Up Counter) Reset Output Next State Toggle Flip-Flop Inputs Input Present State 000 111 001 110 010 101 011 100 C B A C+ B+ A+ TC TB TA 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 1 0 1 0 0 1 1 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 (a) State Transition Diagram (b) State Transition Table (What next state will be given the current state.) FF Excitation Table Revisited Q Q+ R S J K T D 0 0 x 0 0 x 0 0 0 1 0 1 1 x 1 1 1 0 1 0 x 1 1 0 1 1 0 x x 0 0 1 Excitation table for R-S, J-K, T, and D Flip-Flops

From excitation table for FF inputs, get K-map for the FF inputs. CB CB 00 01 11 10 A 00 01 11 10 A 1 0 0 0 0 0 1 1 0 1 1 1 1 1 TA=1 TC=AB CB A 00 01 11 10 K-maps for Up-Counter Using Toggle Flip-Flops. 1 0 0 0 0 1 1 1 1 TB=A Obtain logic expr. for FF I/Ps (as functions of current state bits A, B, C, --- A=QA, B=QB, C=QC) and realize the counter

Implementation Using J-K FFs: Counters with More Complex Sequencing (Non-Consecutive Binary Outputs) Present State Next State 000 110 010 101 011 C B A C+ B+ A+ 0 0 0 0 1 0 0 0 1 x x x 0 1 0 0 1 1 0 1 1 1 0 1 1 0 0 x x x 1 0 1 1 1 0 1 1 0 0 0 0 1 1 1 x x x State Transition Diagram Implementation Using J-K FFs: State Transition Table Present State Next State Remapped Next State C B A C+ B+ A+ JC KC JB KB JA KA 0 0 0 0 1 0 0 x 1 x 0 x 0 0 1 x x x x x x x x x 0 1 0 0 1 1 0 x x 0 1 x 0 1 1 1 0 1 1 x x 1 x 0 1 0 0 x x x x x x x x x 1 0 1 1 1 0 x 0 1 x x 1 1 1 0 0 0 0 x 1 x 1 0 x 1 1 1 x x x x x x x x x Q Q+ J K 0 0 0 x 0 1 1 x 1 0 x 1 1 1 x 0 J-K Flip-Flop Excitation Table State Transition Table and Remapped Next-State Functions

Next State Functions x x 1 x 0 0 x x x x x 0 x 1 x x 1 x x x x 0 1 x CB CB A 00 01 11 10 00 01 11 10 A x x 1 x x x x 0 0 0 x x x 1 x x JC KC 1 1 CB CB A 00 01 11 10 A 00 01 11 10 1 x x x x x x 1 x 0 1 x x 1 x x JB KB 1 1 CB CB 00 01 11 10 A 00 01 11 10 A 0 1 0 x x x x x x x x x x 0 x 1 JA KA 1 1 Remapped K-Maps for J-K Implementation.

Actual Implementation ( Using J-K) + C A B JA J Q CLK K Q J Q CLK K Q J Q CLK K Q A KB C Count signal A B JA KB C J-K Flip-Flop Implementation of 3 Bit Counter.