# Take-Home Exercise. z y x. Erik Jonsson School of Engineering and Computer Science. The University of Texas at Dallas

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1 Take-Home Exercise Assume you want the counter below to count mod-6 backward. That is, it would count , etc. Assume it is reset on startup, and design the wiring to make the counter count properly. Major hint that was supplied: Remember the K-map method will work for ANY counter. Note: x is the most significant bit, z the least significant. z y x 1

2 Designing Sequential Logic Last lecture demonstrated the design of two simple counters (a third counter was a homework problem). Today s exercise: Three additional designs: A modulo-10 binary counter A timer or signal generator A sequential multiplexer Note that all the designs utilize counters. 5

3 Design 1: A Decimal Counter A decimal counter is one that counts modulo ten. This is a realistic design, since in many electronic devices or appliances, it is often necessary to count in tens. We will design only a single 0-9 counter, recognizing that we can count to 99 or 999 by simply adding more counter digits. We will need a 4-bit counter (counting up to requires 4 bits). The counter must: Count from zero to nine and reset on the tenth clock pulse. Count synchronously (as usual), that is, in parallel, not ripple. Have the four counter bits available as outputs, so that they might be decoded to indicate various counts, from 0 to 9. 6

4 First Step: A Four-Bit (Modulo 16) Counter In general, the best way to design any modulo-m counter (m not a power of 2) is to start with the smallest 2 n counter that will count up to m. In our case, we need a 4-bit counter (as mentioned on the previous page), as it takes 4 bits to count to 9. Thus, we will start with a four-bit, modulo-16 counter. Once we have it, we can modify it to count modulo 10. So as our first step, we will design that modulo-16 counter. It turns out that this design will also help us in the other two design exercises in this lecture. 7

5 Modulo-16 Counter 1 Reset Clock LSB 2 nd LSB 2 nd MSB MSB We are building a parallel or synchronous counter. Therefore, the clock is connected to all four stages of the counter as shown above. We remember that for any 2 n counter, the lowest stage is always connected to logic 1. Now we only have to consider the other three stages. We will use the truth table method to find T for the last three stages of the counter. 8

6 Truth-Table/K-Map Approach to Counter Design Using this approach, we will make a truth table for the T inputs to each counter flip-flop versus the count. We remember that for a T-FF, if the T input is 1, the counter stage will change states on the clock pulse. However, if the T input is 0, when the input clock ticks, the flip-flop will not change states. We normally make a composite truth table for all necessary stages (in this case, three), but then consider the counter stages, or flip-flops, one at a time to define the T for each count. Then we transfer this information to the Karnaugh map. 9

7 10 The University of Texas at Dallas Truth Table for Bit y T-Input For a 4-bit counter, the truth tables will have 16 possibilities (i.e., the 16 counts of the counter). To eliminate calling the counter bits LSB, second LSB, etc., we call the counter outputs w, x, y, and z (where w = MSB). Omitting z as noted, we examine the Q output of y (shaded green), along with its next output state (brackets). These are then listed as y and y NEXT as before. As in the previous lecture, comparing the y values tell us what the T-input should be. If y = y NEXT, T = 0, but if y y NEXT, T = 1. Count T-Input y y NEXT wxyz Etc

8 11 Karnaugh Map for Bit y T-Input As usual, we plot the 1 s from the truth table on a Karnaugh map, just as we did for the modulo-8 and modulo-6 counters in the last lecture. From the K-Map, a simplifying prime implicant is identified. Clearly, the implicant does not depend on w, x, or y. Therefore we can define the Boolean expression or term for the T-input to the counter bit y as: T y = z Note that this is just the T input we get using our rule. wx yz

9 Bit y T Input Connected 1 z Ty = z y x w Reset Clock Our timer counter circuit now looks as shown above, with the new w-x-y-z labels, and with the y-input connected. 12

10 Truth Table for Bit x T-Input The truth table is made up for bit x as we did for bit y, comparing the value of x for each count to the value of x in the next count. These are then plotted in the right two columns. As before, When x and x NEXT are equal, T is 0. When they are different, T is 1. Note that there are fewer 1 s on the table in the T-input column because the 2nd MSB, x, will toggle fewer times. Count T-Input x x NEXT w x y z Etc

11 Karnaugh Map for Bit x T-Input The bit x T-input K-map is shown. As before, there is an obvious simplifying Boolean expression for this T-input. 00 yz We can see that outputs w and x do not affect the switching of 2nd MSB. Given the Karnaugh map shown, what is the Boolean expression for T x. wx T x = yz 14

12 Bit x T-Input Connected 1 z Ty = z Tx = yz y x w Reset Clock Note that the inputs for x, y, and z are exactly the same result that we got for our three T inputs on our modulo-8 counter, just as we would expect. We now follow our longer procedure to define the bit w T-input. 15

13 Bit w T-Input Truth Table 16 We proceed in our analysis as before. By now, we know that bit w only toggles after counts 7 (0111) and 15 (1111). Thus the truth table is very simple (only two 1-states for the T-input). We plot these relatively simple results on the K-Map following. Count T-Input w w NEXT wxyz Etc s 1 0 s 1

14 Bit w Expression This K-Map confirms what we could by now, hopefully, tell by inspection: whenever the three lower bits of the counter are 1 (xyz=1), then w toggles to the opposite state, which is exactly what our short cut rule states. Then, bit w toggles to 1 on count 8, and back to 0 on count 16/0. Given the above, what is the expression for T w? wx yz T w =xyz

15 Completing the Counter 1 z Ty z y T = yz x w = x Tw = xyz Reset Clock The input to counter bit w is shown above, so that the counter is now completed. We also learned a short cut method to design 2 n counters. We will go through the short cut solution later in this lecture. 18

16 Decimal Counter Design We now have the starting point for our mod-10 counter a basic mod- 16 counter. This case is similar to the mod-6 counter discussed in the last lecture. In that case, the counter counted to 5, then reset to 0 clock pulse 6. Here, the counter, will reset on count 10, not count 16. Reset Clock 1 z Ty y T = yz x w = z x Tw = xyz 19

17 Modulo-m Counter Design, Where m = 10 As we saw in the last lecture, we obtain a nonmod-2 n counter by starting with a 2 n counter and then modifying it to count in the desired cycle. To do this, we make a truth table of the new count cycle, then examine how each counter stage must behave to count mod-m rather than mod-2 n (in this case, mod-10 rather than mod- 16). 20

18 21 The University of Texas at Dallas Stage-By-Stage Analysis Counter bit z needs to transition to 1 from 0. It will do this anyway. Bit z toggles every clock cycle. It resets on its own, without any change. Bit y is zero on count 9, but since z (LSB) is one, the Ty input is 1, and it will therefore transition to 1 as z goes to 0. We need to prevent this. Bit x is also 0, but since y is 0, the AND gate driving the T x input is 0, so we do not need to do anything bit x will stay 0 since T x is 0. MSB (bit w) is 1 for binary 9. But its T input is also 0 so it will not toggle on count 10, but will stay 1, unless we arrange for it to reset. In summary, bits x and z take care of themselves. However: Bit y Prevent from toggling to 1 Bit w Cause reset to 0 w x y z Mod-10 w x y z Etc. Mod-16

19 Stage y Analysis 22 The Ty input truth table is: Count T-Input y y NEXT And the K-Map is: wx yz X X X X X X w x y z We see that there are four 1 s to be plotted on the K-map. For the counter outputs wxyz, the simplified Boolean expression is T. y = wz Note that counts (10-15) are don t cares. ANDing the outputs of counter states w and z ensures that bit y toggles correctly for a decimal counter.

20 Counter Stage W Now consider the most significant bit of the counter (MSB, bit w): Bit w must toggle back to zero on count 10, (not the normal behavior). 23 The T w truth table is: Count T-Input w w NEXT wx And the K-Map is: yz w x y z This Boolean expression does not appear amenable to simplification.

21 Counter Stage W (Continued) As noted previously, counts 10 through 15 ( ) are all don t cares, since they will never occur. With the don t cares plotted as before, (remembering that a don t care may be treated as a 1) we use them as in Exercise 1, today: wx yz X X X X 1 X X 24 The simplified expression for the Tw input is T. This w = wz + xyz is a considerable simplification using don t cares.

22 Final Decimal Counter Design Tx = yz 1 Reset Clock z Ty = wz Tw = wz + xyz y x w 25 Note that we did not explicitly have to decode the count 9, because the term wz will only be 1 for one count (the count 9 ). This eliminates the x and y in the Boolean expression. We will decode 9 in the short-cut method of design below. The completed counter is shown above.

23 Using the Short-Cut Approach (m 2 n ) We previously learned a second short cut method to design a mod-m counter, m not a power of 2. The short cut rules are: To count any number m, m not a power of 2, determine the smallest mod 2 n counter such that 2 n >m. Build the parallel 2 n counter as we did previously. Determine stages that will 1 on count m, and those that stay at 1 on count m (some stages may not need any help ). Using the outputs of the 2 n counter, decode (m-1). For each stage of the 2 n counter that remains at 1 on count m, (has a 0 T-input on count m-1), OR the current T-input with decoded (m-1). For each stage that will 1 on count m (T input of 1), AND inverted (m-1) with that current T-input. 26

24 Exercise 1 Using the four T FFs below, design a mod-10 (0-9) counter using the short-cut method from the previous slide. 1 z y x w Reset Clock 27

25 Final Decimal Counter Timing Decoded decimal count 9 Count 8 Count 4 Count 2 Count 1 Clock

26 Design 2: A Signal Generator The desired design is a signal generator based on a counter. Specification: 4-bit binary synchronous (parallel) counter, driven by a 50/50 clock. Counter runs continually, reset only at startup by a Reset- signal. Since 2 4 = 16, counter counts modulo-16. Counter generates sync signals on counts 3, 7, and 14, which: Occur immediately after the counts are valid (i.e., immediately after the counter output stages change). Occur for 1/2 clock cycle only. Per spec, we need a 4-bit counter. We can use the standard T FF that we used to build the 3-bit counter in the last lesson. We will need four of these. 30

27 Modulo-16 Counter: Short Cut Method Earlier in the lecture, we designed a mod-16, 4- bit counter using the K-map method. We now need another 16-bit counter, not as the starting point for a non-2 n counter, but to actually count mod-16. Instead of using the counter developed earlier, lets review the short-cut method for a 2 n counter first. Then we can use the version of the counter that makes the most sense. 31

28 Shortcut to Build 2 n Counter For ANY mod-2 n counter (2, 4, 8, 16, 32, etc.): Use n toggle flip-flops. Connect the clock to the clock inputs of all T FFs Connect counter LSB to 1 Connect LSB Q-output to the T input of 2 nd LSB. For all subsequent counter stages, connect the AND of all previous stage outputs to the T-input of the next stage. This makes use of the principle that in a mod- 2 n counter, a stage only toggles when all upstream stages = 1. 32

29 Exercise 2 Complete the mod-16 counter below using the shortcut method. 1 z y x w Reset Clock 33

30 Completing the Timer We now have the mod-16 counter. Our specification calls for three 1/2-clock-cycle signals that denote when counts 3, 7 and 14 have occurred. Consider signal 3: It occurs when LSB and second LSB are both one. Could we just AND y and z together to get the signal 3? NO! Bits y and z are both one on counts 3, 7, 11 and 15. Thus we must AND together all four counter outputs. ONLY for y and z = 1, and w and x = 0 is the count really 3 (0011). In Boolean terms, then, we need: "3"=0011=w x yz The signal is 1/2-clock-cycle in duration, occurring immediately after the count 3 (0011), or on the downward transition of the clock. To do this, we invert clock and AND it with count 3. 35

31 Completing the Timer (2) 3 1 z Ty z y T yz x T = xyz w = x = w Reset Clock We already have the AND of the yz outputs available. Now, simply AND them with the two inverted higher clock stage outputs, w and x, and inverted clock to get a signal 3 that satisfies our specification. Note that in this approach, we are using the K-map mod-16 counter rather than the short-cut counter. 36

32 Completing the Timer (3) We can now create the other two signals similarly: wxyz 7 = The Boolean expression is. Remember that we have to use all four bits, inverting MSB (w), to get the right count, since the bits x, y, and z are all ones for both counts 7 and 15. The input to the MSB T input is the AND of the three lowest bits outputs, so we can use that directly, plus the inverted MSB and clock (using inverted clock to get a 1/2-cycle signal, once again). Likewise, 14 is Thus the Boolean expression for the signal 14 is: "14"=1110 =wxyz. Once again, we AND this expression with inverted clock to get an immediate 1/2-cycle signal. We now create the last two signals, also connecting reset to the T FF reset inputs, remembering that we do not have to time this signal, since it occurs only at startup. 37

33 Completing the Timer (4) 3 1 z Ty y T yz x T = xyz w = z x = w 7 Reset 14 Clock Clock not The completed timer circuit is shown above. 38

34 Completing the Timer (5) We wish to plot the timing of our timer circuit, to ensure that it performs properly. It can be done as follows: All timing is done with respect to the clock. We first build the basic timing diagram for counter outputs (we ignore Reset, since it occurs only at the start-up). The timing of clock and the four counter outputs is shown on the following slide. On the subsequent slide, we add the timing for the three signals that we have created, 3, 7, and 14: 3 is true for 1/2 cycle on counter output 0011, 7 is true on count 0111, and 14 is true on count 1110 (all signals are ½ clock cycle). 39

35 Counter Outputs Count 8 (w) Count 4 (x) Count 2 (y) Count 1 (z) Clock Timing for clock and counter bits. 40

36 Timing Diagram for Completed Timer w x y z 41 Clock

37 Design 3: A Sequential Multiplexer Our last design project is a sequential multiplexer. A multiplexer (or MUX) is a combinational logic circuit that can switch (or gate ) one of several inputs onto a common output. The input which is gated to the output line depends on an address bus. Our sequential MUX is specified as follows: Switches four inputs, a-d, onto a common output, each on a specific count. A four-bit address sets the count for each input to be gated to the output. The address consists of the outputs of a 4-bit synchronous counter. The counter address goes from 0000 to 1111 (0x 0 to f, or modulo-16). Input a is switched on count 3, input b on count 7, input c on count 11, and input d is switched to the output on count 15. Each input stays connected to the output for the duration of that count. 42

38 Design of a MUX We can use the 4-bit counter we previously designed. An illustration of a simple MUX from Lecture # 4 is shown at right. The MUX consists of a decoder section, which decodes the address assigned to each input. Each address is then AND-ed with an input to gate it into the OR gate (selector section). Since only one AND is active at a time, only one output goes through the OR gate at a time. Decoder Selector Output 43

39 Design of a MUX (2) We need to design a MUX with addresses 3, 7, 11, and 15 as gating signals for inputs a, b, c, and d, respectively. The basic AND-OR part of the MUX (the selector) can be as shown below: 44

40 Design of a MUX (3) We now need to provide our basic combinational logic MUX with the four switching addresses 3, 7, 11, and 15. This can be the decoded outputs of our already-designed 4-bit counter. However, we need to decode two different outputs from the ones that we put together for our first timer. Also, our decoded signals now need to last a full clock cycle, NOT 1/2 cycle. Since each of the 16 counts of the counter lasts exactly one clock period, we no longer need the clock input to be AND-ed with the count. We therefore need only to decode the clock counts 3, 7, 11, and

41 Exercise 3 MUX Address Decodes From our first exercise, we know that all four counter bits must be used to decode each output count. Assuming w = MSB, z = LSB, compose the Boolean expressions for the counter states 3, 7, 11, and 15 on which the MUX outputs are made. Count 3 (0011): Count 7 (0111): Count 11 (1011): Count 15 (1111): 46

42 Completed Sequential MUX Ty = z Tx = yz Tw = xyz 3 a 1 z Reset Clock y x w b c d MUX Out Note that only a, b, c, or d will be passed through the OR gate, depending on the counter value. 47

43 MUX Timing The following slide shows the timing for the Sequential MUX. Note that each signal, a-d, is gated out for an entire clock cycle of the given number (3, 7, 11, 15). When we look at the output of the MUX, we see that there are four cycles of activity: clock count 3, count 7, count 11, and count 15. The MUX output is shown as a box. That is, the active cycles of the MUX show both a high and low level. That is because we do not know whether the level of each input is high or low; it could be either (it could even change during the gating cycle although that would probably represent a bad design!). 48

44 MUX out a b c d Count 8 Count 4 Count 2 Count 1 49 Clock

45 Summary This series of lessons in design shows how we can easily design and use binary counters to perform many necessary logic functions. By paying attention to the specification and by modifying circuits we had already designed, we were able to quickly come up with other circuits that met our requirements (engineers never, EVER start completely over to solve a problem if there is ANY possible shortcut). We achieved our designs using K-maps and the logic principles that we have learned. We also reviewed two techniques that provided quicker answers, even though the circuit was not the simplest possible. This is a good illustration that sometimes a less optimal solution to an engineering problem, achieved rapidly, is the preferred solution! 50

46 Puzzler Quiz Reset T C R Q T Q T Q C R C R z y x Z y X 51 Clock This counter counts VERY oddly. By writing down the expression for each T input, then starting at count 000 (since the counter is reset at startup), you can deduce the count sequence. First five to visit me during office hours and analyze it gets +3 points on test 2. Note: You can t just say It counts like this You have to explain how the counter works, stage-by-stage. If you get this right, the hardware part of test 2 will be a breeze!

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