Question 2 of 6: Synchronous up/down counter with JK flip-flops (12 pts)

Question 2 — Synchronous up/down counter with JK flip-flops (12 pts)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Two state bits $AB$ ($A$ = MSB), enable $E$ and direction $X$. The $E=1,X=1$ sequence is the binary up count and $E=1,X=0$ is the binary down count; $E=0$ freezes the state. The JK excitation table is $0\!\to\!0:J{=}0,K{=}\!\times$; $0\!\to\!1:J{=}1,K{=}\!\times$; $1\!\to\!0:J{=}\!\times,K{=}1$; $1\!\to\!1:J{=}\!\times,K{=}0$.

Find. $J_A,K_A,J_B,K_B$ as functions of $A,B,E,X$, and the resulting diagram, table and gate circuit.

Approach. Recognise the specification as an enabled up/down counter ($X$ = direction), tabulate the next state, read the JK excitations, and minimise.

1. (b) State/excitation table. With $s=2A+B$, the next state is $s$ (hold) for $E=0$, $(s{+}1)\bmod4$ for $E{=}1,X{=}1$, and $(s{-}1)\bmod4$ for $E{=}1,X{=}0$. Reading the JK table for each bit gives the excitations below.

State/excitation table ($\times$ = don’t care)

$E$	$X$	$A\,B$	$A^{+}B^{+}$	$J_A$	$K_A$	$J_B$	$K_B$
0	–	any	hold	0	0	0	0
1	1	00	01	0	×	1	×
1	1	01	10	1	×	×	1
1	1	10	11	×	0	1	×
1	1	11	00	×	1	×	1
1	0	00	11	1	×	1	×
1	0	11	10	×	0	×	1
1	0	10	01	×	1	1	×
1	0	01	00	0	×	×	1

2. (c) K-map simplification. Bit $B$ toggles on every enabled clock in both directions, so $J_B=K_B=E$. Bit $A$ toggles only on the carry ($B{=}1$ while counting up) or the borrow ($B{=}0$ while counting down), i.e. when $B\odot X$ (equal); the enable $E$ multiplies every term. Hence$$\boxed{J_A=K_A=E\,(B\odot X)=E\,(BX+\overline B\,\overline X)\qquad J_B=K_B=E}$$The common factor $E$ realises the hold with $J{=}K{=}0$ — there is no need to gate the clock.
3. (d) Circuit. One XNOR forms $B\odot X$; one AND gates it with $E$ to drive $J_A$ and $K_A$ (tied together); $E$ drives $J_B$ and $K_B$ directly. Both flip-flops share the clock.