THE BASIC RS NOR LATCH
Feedback Sequential Circuit Introduction
Digital logic systems have been divided into two main categories: combinational circuits, and feedback sequential circuits. A combinational circuit is a circuit whose output can be determined from its inputs. Examples from this group are Half-Adders, Decoders, Multiplexers, and the basic gates. If I tell you the inputs of an AND gate, for instance, you can readily tell me the output. Feedback sequential circuits, on the other hand, are a bit fancier than that. The output of a feedback sequential circuit depends not only on the inputs but also on the given state (or condition) of the circuit. Before we continue, however, let’s take a short break here for a story.
While playing with your friends on campus, you stood on the edge of a four foot bench and from there jumped to the ground. At the exact same time, my very emotional friend Francois stood on the edge of a forty story building and from there jumped to the ground.
Function and Application
The most basic useful feedback sequential circuit is the Set-Reset Latch with NOR gates. The Set-Reset Latch, as you might guess from the name, is a finite state machine with two states: a Set state and a Reset state. If you have ever used the cruise control¹ feature of your car, you have essentially made use of a Set-Reset latch. When you select a setting/speed, you set the control circuit; when you cancel your selection, you reset the circuit. It’s that simple — at least from the driver’s perspective.
But clearly, if we use the cruise control example just a little bit more, you and I both know that at any given time a car is either on cruise control or it is not on cruise control — it’s never on neither or on both at the same simultaneous time. As a result, a designer usually don’t allow the funny conditions where set and reset are both equal (RS = 00 and RS = 11) to affect the safety feature of the system where the circuit is used — in this case the cruise control system.
ANALYSIS AND VERIFICATION
We analyze circuits to verify certain details about their operation. For combinational circuits, this means deriving a truth table or a Boolean function or both, depending on what your professor is in the mood for. For feedback sequential circuits, however, the list can be a bit longer. Case in point, our analysis of the set-reset latch needs to answer three questions:
–1. Cruise control systems are actually complex electromechanical circuits. However, the illustration is conceptionally intuitive.
The Boolean Function
Whenever someone asks you to analyze an object, they will give you the object; otherwise, you couldn’t verify anything. I can’t verify that your pizza is too small to share if you won’t show it to me. Hence, your professor will give you the circuit you need to analyze. As such, that is since we already have the circuit, the easiest thing to do is get the Boolean function.
From figure 1 we can readily get two equations:
Q = (R + not-q)
if you are taking an exam and you are running out of time, the two expressions will suffice to show your professor that you get it. Nonetheless, it is customary to present the characteristic equation of any latch or flipflop as a solution of Q in terms of q. so let’s follow the tradition:
Voila! The solution of Q in as a function of q!
That was question three on the list. Now we can answer question two: what are the possible transitions.
The State Transition Table
A state transition table is a table that shows how a circuit moves between states. From the characteristic equation we see that we have one state variable, q, and two input variables, R and S. Therefore, since each variable in Boolean logic can have two possible values, we can say with certitude that the set-reset latch has two possible states (q = 0; q = 1) and four possible input conditions (RS = 00; RS = 01; RS = 10; RS = 11). On that account, we can now create a table that shows the relationship between the two states and the four input conditions. Accordingly we label the rows of the table by states and the columns by input conditions. And we call the table a state transition table. Table 2 below shows what I am talking about.
So far our table is empty. We will fill the table by evaluating Q for each combination of given state and input condition. For example, the value of Q when q = 0 and RS = 00 is
Accordingly we put 0 in the cell where q =0 intersects with RS = 00, as shown in Table 3.
Without showing all the evaluations, I fill the state transition table for you in Table 4. Feel free to check
that my calculations are correct.
Table 4 effectively answers question two, which asked about the relationship between the input conditions and the states of the circuit. Because tables are not always intuitive, however, an easy to understand diagram, called a state diagram, is usually submitted as part of an analysis. The diagram, figure 2, has the same information as the transition table. But as you can see, the state diagram is the simplest method to show how a circuit behaves — after all it is a picture.