同步十进制加法计数器设计

ゞ浴缸里的玫瑰 2023-08-17 17:26 109阅读 0赞

(1)逻辑抽象:

由分析, 共有10个有效状态, 需要![\\left \\lceil \\log(2, 10) \\right \\rceil][left _lceil _log_2_ 10_ _right _rceil] = 4 个触发器.设4个触发器编码变量自高位到低位依次设为![Q\_\{3\}][Q_3]、![Q\_\{2\}][Q_2]、![Q\_\{1\}][Q_1]、![Q\_\{0\}][Q_0].

设加法进位输出变量为Y, 使用上升沿触发的触发器进行设计. 状态转换关系为 从0000开始计数, 经过10个脉冲后, 实现了0000->0001->0010->0011->0100->0101->0110->0111->1000->1001->0000的状态转换. 按照上升沿触发的方式要求, 在1001->0000时, Y由0->1.

(2)绘制状态图, 并转换为状态表:

由于状态转换关系比较明确, 这里直接绘制出状态表即可.

<table style="width:500px;"> 
 <thead> 
  <tr> 
   <th colspan="9" style="text-align:center;">同步十进制加法计数器状态表</th> 
  </tr> 
 </thead> 
 <tbody> 
  <tr> 
   <td><img alt="^{{Q_{3}}^{n}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B3%7D%7D%5E%7Bn%7D%7D"></td> 
   <td><img alt="^{{Q_{2}}^{n}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B2%7D%7D%5E%7Bn%7D%7D"></td> 
   <td><img alt="^{{Q_{1}}^{n}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B1%7D%7D%5E%7Bn%7D%7D"></td> 
   <td><img alt="^{{Q_{0}}^{n}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B0%7D%7D%5E%7Bn%7D%7D"></td> 
   <td><img alt="^{{Q_{3}}^{n+1}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B3%7D%7D%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="^{{Q_{2}}^{n+1}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B2%7D%7D%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="^{{Q_{1}}^{n+1}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B1%7D%7D%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="^{{Q_{0}}^{n+1}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B0%7D%7D%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="Y" src="https://private.codecogs.com/gif.latex?Y"></td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
  </tr> 
  <tr> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">0</td> 
   <td style="text-align:center;">1</td> 
  </tr> 
 </tbody> 
</table>

(3)选择触发器, 绘制状态激励表:

这里选择JK触发器, 对于JK触发器, 有下述关系:

<table style="width:500px;"> 
 <thead> 
  <tr> 
   <th colspan="4" style="text-align:center;">JK触发器状态</th> 
  </tr> 
 </thead> 
 <tbody> 
  <tr> 
   <td><img alt="Q^{^{n}}" src="https://private.codecogs.com/gif.latex?Q%5E%7B%5E%7Bn%7D%7D"></td> 
   <td><img alt="Q^{^{n+1}}" src="https://private.codecogs.com/gif.latex?Q%5E%7B%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="J" src="https://private.codecogs.com/gif.latex?J"></td> 
   <td><img alt="K" src="https://private.codecogs.com/gif.latex?K"></td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>1</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
  </tr> 
 </tbody> 
</table>

<table> 
 <thead> 
  <tr> 
   <th colspan="17" style="text-align:center;">同步十进制加法计数器状态激励表</th> 
  </tr> 
 </thead> 
 <tbody> 
  <tr> 
   <td><img alt="^{{Q_{3}}^{n}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B3%7D%7D%5E%7Bn%7D%7D"></td> 
   <td><img alt="^{{Q_{2}}^{n}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B2%7D%7D%5E%7Bn%7D%7D"></td> 
   <td><img alt="^{{Q_{1}}^{n}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B1%7D%7D%5E%7Bn%7D%7D"></td> 
   <td><img alt="^{{Q_{0}}^{n}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B0%7D%7D%5E%7Bn%7D%7D"></td> 
   <td><img alt="^{{Q_{3}}^{n+1}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B3%7D%7D%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="^{{Q_{2}}^{n+1}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B2%7D%7D%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="^{{Q_{1}}^{n+1}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B1%7D%7D%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="^{{Q_{0}}^{n+1}}" src="https://private.codecogs.com/gif.latex?%5E%7B%7BQ_%7B0%7D%7D%5E%7Bn&amp;plus;1%7D%7D"></td> 
   <td><img alt="Y" src="https://private.codecogs.com/gif.latex?Y"></td> 
   <td><img alt="J_{3}" src="https://private.codecogs.com/gif.latex?J_%7B3%7D"></td> 
   <td><img alt="K_{3}" src="https://private.codecogs.com/gif.latex?K_%7B3%7D"></td> 
   <td><img alt="J_{2}" src="https://private.codecogs.com/gif.latex?J_%7B2%7D"></td> 
   <td><img alt="K_{2}" src="https://private.codecogs.com/gif.latex?K_%7B2%7D"></td> 
   <td><img alt="J_{1}" src="https://private.codecogs.com/gif.latex?J_%7B1%7D"></td> 
   <td><img alt="K_{1}" src="https://private.codecogs.com/gif.latex?K_%7B1%7D"></td> 
   <td><img alt="J_{0}" src="https://private.codecogs.com/gif.latex?J_%7B0%7D"></td> 
   <td><img alt="K_{0}" src="https://private.codecogs.com/gif.latex?K_%7B0%7D"></td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>0</td> 
   <td>1</td> 
   <td>1</td> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>0</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td>1</td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>0</td> 
   <td>1</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>0</td> 
   <td>1</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
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   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
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   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
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   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>1</td> 
   <td>1</td> 
   <td>0</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
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   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
  <tr> 
   <td>1</td> 
   <td>1</td> 
   <td>1</td> 
   <td>1</td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
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   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
   <td><img alt="\times" src="https://private.codecogs.com/gif.latex?%5Ctimes"></td> 
  </tr> 
 </tbody> 
</table>

(4)根据状态激励表求出输出方程和激励方程组:

![J\_\{0\}=1, K\_\{0\}=1][J_0_1_ K_0_1]

(5)检查电路自启动功能:

由于电路在设计时未采用全编码, 所以要考虑无效状态次态的问题, 因此电路必须检查自启动功能.

检查思路:

将激励方程组代入状态特性方程组,

![\{Q\_\{3\}\}^\{n+1\}=J\_\{3\}\\overline\{\{Q\_\{3\}\}^\{n\}\}+\\overline\{K\_\{3\}\}\{Q\_\{3\}\}^\{n\}=\\overline\{\{Q\_\{3\}\}^\{n\}\}\{Q\_\{2\}\}^\{n\}\{Q\_\{1\}\}^\{n\}\{Q\_\{0\}\}^\{n\}+\{Q\_\{3\}\}^\{n\}\\overline\{\{Q\_\{0\}\}^\{n\}\},][Q_3_n_1_J_3_overline_Q_3_n_overline_K_3_Q_3_n_overline_Q_3_n_Q_2_n_Q_1_n_Q_0_n_Q_3_n_overline_Q_0_n]

![\{Q\_\{2\}\}^\{n+1\}=J\_\{2\}\\overline\{\{Q\_\{2\}\}^\{n\}\}+\\overline\{K\_\{2\}\}\{Q\_\{2\}\}^\{n\}=\\overline\{\{Q\_\{2\}\}^\{n\}\}\{Q\_\{1\}\}^\{n\}\{Q\_\{0\}\}^\{n\}+\{Q\_\{2\}\}^\{n\}\\overline\{\{Q\_\{1\}\}^\{n\}\{Q\_\{0\}\}^\{n\}\}=\{Q\_\{2\}\}^\{n\}\\oplus (\{Q\_\{1\}\}^\{n\}\{Q\_\{0\}\}^\{n\}),][Q_2_n_1_J_2_overline_Q_2_n_overline_K_2_Q_2_n_overline_Q_2_n_Q_1_n_Q_0_n_Q_2_n_overline_Q_1_n_Q_0_n_Q_2_n_oplus _Q_1_n_Q_0_n]

![\{Q\_\{1\}\}^\{n+1\}=J\_\{1\}\\overline\{\{Q\_\{1\}\}^\{n\}\}+\\overline\{K\_\{1\}\}\{Q\_\{1\}\}^\{n\}=\\overline\{\{Q\_\{3\}\}^\{n\}\}\\cdot \\overline\{\{Q\_\{1\}\}^\{n\}\}\\cdot \{Q\_\{0\}\}^\{n\}+\{Q\_\{1\}\}^\{n\}\\cdot \\overline\{\{Q\_\{0\}\}^\{n\}\},][Q_1_n_1_J_1_overline_Q_1_n_overline_K_1_Q_1_n_overline_Q_3_n_cdot _overline_Q_1_n_cdot _Q_0_n_Q_1_n_cdot _overline_Q_0_n]

![\{Q\_\{0\}\}^\{n+1\}=J\_\{0\}\\overline\{\{Q\_\{0\}\}^\{n\}\}+\\overline\{K\_\{0\}\}\{Q\_\{0\}\}^\{n\}=\\overline\{\{Q\_\{0\}\}^\{n\}\}+\\overline\{1\} \\cdot \{Q\_\{0\}\}^\{n\}=\\overline\{\{Q\_\{0\}\}^\{n\}\}.][Q_0_n_1_J_0_overline_Q_0_n_overline_K_0_Q_0_n_overline_Q_0_n_overline_1_ _cdot _Q_0_n_overline_Q_0_n_.]

对于设计中的无效状态1010/1011/1100/1101/1110/1111代入状态特性方程组, 得出的次态分别为 1011/0100/1101/0100/1111/0000, 构成 1010->1011->0100, 1100->1101->0100, 1110->1111->0000, 由此可以看出,

所有的无效状态都可进入有效状态的循环, 则电路具备自启动功能.

(6)绘制状态图:

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MjA0ODQ2Mw_size_16_color_FFFFFF_t_70][]

(7)设计心得:

与设计同步八进制可逆加法计数器不同, 在设计同步十进制加法计数器时, 需考虑到无效状态的问题: 电路设计未采用全编码, 故会有一些无效状态.

在判断无效状态能否进入有效状态的循环时, 只需将激励方程组代入触发器的状态特性方程, 求出每一个无效状态对应的次态, 再判断这些无效状态的次态能否进入有效状态的循环即可.

由于同步十进制加法计数器的状态转换关系比较明确, 故在设计时直接给出状态表. 在即将设计的110序列检测器中, 就需要先分析出原始状态表, 再对原始状态表进行等价状态化简, 之后对化简后的原始状态进行编码——最后才能得到110序列检测器的状态表.

从一开始的分析给定同步时序逻辑电路功能, 到后来的设计同步N进制计数器, 再到即将设计的110序列检测器, 我们可以发现分析难度是在递增的. 如果能设计出110序列检测器, 那么对于同步时序逻辑电路的学习就进入了一个新的层次.

虽然设计同步十进制加法计数器难度不大, 但深刻理解该设计用例是十分重要的. 只有把该例子的设计流程捻熟于心, 才能进入同步时序逻辑电路设计下一阶段的学习.

[left _lceil _log_2_ 10_ _right _rceil]: https://private.codecogs.com/gif.latex?%5Cleft%20%5Clceil%20%5Clog%282%2C%2010%29%20%5Cright%20%5Crceil
[Q_3]: https://private.codecogs.com/gif.latex?Q_%7B3%7D
[Q_2]: https://private.codecogs.com/gif.latex?Q_%7B2%7D
[Q_1]: https://private.codecogs.com/gif.latex?Q_%7B1%7D
[Q_0]: https://private.codecogs.com/gif.latex?Q_%7B0%7D
[J_0_1_ K_0_1]: https://private.codecogs.com/gif.latex?J_%7B0%7D%3D1%2C%20K_%7B0%7D%3D1
[Q_3_n_1_J_3_overline_Q_3_n_overline_K_3_Q_3_n_overline_Q_3_n_Q_2_n_Q_1_n_Q_0_n_Q_3_n_overline_Q_0_n]: https://private.codecogs.com/gif.latex?%7BQ_%7B3%7D%7D%5E%7Bn&plus;1%7D%3DJ_%7B3%7D%5Coverline%7B%7BQ_%7B3%7D%7D%5E%7Bn%7D%7D&plus;%5Coverline%7BK_%7B3%7D%7D%7BQ_%7B3%7D%7D%5E%7Bn%7D%3D%5Coverline%7B%7BQ_%7B3%7D%7D%5E%7Bn%7D%7D%7BQ_%7B2%7D%7D%5E%7Bn%7D%7BQ_%7B1%7D%7D%5E%7Bn%7D%7BQ_%7B0%7D%7D%5E%7Bn%7D&plus;%7BQ_%7B3%7D%7D%5E%7Bn%7D%5Coverline%7B%7BQ_%7B0%7D%7D%5E%7Bn%7D%7D%2C
[Q_2_n_1_J_2_overline_Q_2_n_overline_K_2_Q_2_n_overline_Q_2_n_Q_1_n_Q_0_n_Q_2_n_overline_Q_1_n_Q_0_n_Q_2_n_oplus _Q_1_n_Q_0_n]: https://private.codecogs.com/gif.latex?%7BQ_%7B2%7D%7D%5E%7Bn&plus;1%7D%3DJ_%7B2%7D%5Coverline%7B%7BQ_%7B2%7D%7D%5E%7Bn%7D%7D&plus;%5Coverline%7BK_%7B2%7D%7D%7BQ_%7B2%7D%7D%5E%7Bn%7D%3D%5Coverline%7B%7BQ_%7B2%7D%7D%5E%7Bn%7D%7D%7BQ_%7B1%7D%7D%5E%7Bn%7D%7BQ_%7B0%7D%7D%5E%7Bn%7D&plus;%7BQ_%7B2%7D%7D%5E%7Bn%7D%5Coverline%7B%7BQ_%7B1%7D%7D%5E%7Bn%7D%7BQ_%7B0%7D%7D%5E%7Bn%7D%7D%3D%7BQ_%7B2%7D%7D%5E%7Bn%7D%5Coplus%20%28%7BQ_%7B1%7D%7D%5E%7Bn%7D%7BQ_%7B0%7D%7D%5E%7Bn%7D%29%2C
[Q_1_n_1_J_1_overline_Q_1_n_overline_K_1_Q_1_n_overline_Q_3_n_cdot _overline_Q_1_n_cdot _Q_0_n_Q_1_n_cdot _overline_Q_0_n]: https://private.codecogs.com/gif.latex?%7BQ_%7B1%7D%7D%5E%7Bn&plus;1%7D%3DJ_%7B1%7D%5Coverline%7B%7BQ_%7B1%7D%7D%5E%7Bn%7D%7D&plus;%5Coverline%7BK_%7B1%7D%7D%7BQ_%7B1%7D%7D%5E%7Bn%7D%3D%5Coverline%7B%7BQ_%7B3%7D%7D%5E%7Bn%7D%7D%5Ccdot%20%5Coverline%7B%7BQ_%7B1%7D%7D%5E%7Bn%7D%7D%5Ccdot%20%7BQ_%7B0%7D%7D%5E%7Bn%7D&plus;%7BQ_%7B1%7D%7D%5E%7Bn%7D%5Ccdot%20%5Coverline%7B%7BQ_%7B0%7D%7D%5E%7Bn%7D%7D%2C
[Q_0_n_1_J_0_overline_Q_0_n_overline_K_0_Q_0_n_overline_Q_0_n_overline_1_ _cdot _Q_0_n_overline_Q_0_n_.]: https://private.codecogs.com/gif.latex?%7BQ_%7B0%7D%7D%5E%7Bn&plus;1%7D%3DJ_%7B0%7D%5Coverline%7B%7BQ_%7B0%7D%7D%5E%7Bn%7D%7D&plus;%5Coverline%7BK_%7B0%7D%7D%7BQ_%7B0%7D%7D%5E%7Bn%7D%3D%5Coverline%7B%7BQ_%7B0%7D%7D%5E%7Bn%7D%7D&plus;%5Coverline%7B1%7D%20%5Ccdot%20%7BQ_%7B0%7D%7D%5E%7Bn%7D%3D%5Coverline%7B%7BQ_%7B0%7D%7D%5E%7Bn%7D%7D.
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MjA0ODQ2Mw_size_16_color_FFFFFF_t_70]: /images/20230808/b0f1d360c9654ccf981e4b10e447f491.png