半加器
如果不考慮來自低位的進位將兩個1二進制數相加，稱為半加。
實現半加運算的邏輯電路稱為半加器。
真值表
>> 邏輯表達式和 \begin{align}\notag s = a{b}' + {a}'b \end{align} >> 邏輯表達式進位輸出 \begin{align}\notag co = ab \end{align}
verilog code
module halfadder(
output s, //sum
output co, //carry
input a,
input b
);
assign s = a ^ b;
assign co = a & b;
//assign {co,s} = a + b;
endmodule
testbench
module halfadder_tb;
wire s;
wire co;
reg a;
reg b;
initial
begin
a = 0;
b = 0;
#10 a = 0 ;b = 0;
#10 a = 0 ;b = 1;
#10 a = 1 ;b = 0;
#10 a = 1 ;b = 1;
#10 $finish;
end
initial begin
$fsdbDumpfile("test.fsdb");
$fsdbDumpvars();
end
halfadder u_halfadder(
.s(s),
.co(co),
.a(a),
.b(b)
);
endmodule
全加器
在將兩位多位二進制數相加時，除了最低位以外，每位都應該考慮來自低位的進位，即將兩個對應位的加數和來自低位的進位3個數相加。這種運算稱為全加，所用的電路稱為全加器。
真值表
邏輯表達式和
\begin{align}\notag s = {({a}'{b}'{ci}' + a{b}'ci + {a}'bci + ab{ci}'})' \end{align} >> 邏輯表達式進位輸出 \begin{align}\notag co = {({a}'{b}' + {b}'{ci}' + {a}'{ci}')}' \end{align}
verilog code
module fulladder(
output s, //sum
output co, //carry to high bit
input a,
input b,
input ci //carry from low bit
);
//RTL level
assign s = ~((~a&~b&~ci)||(a&~b&ci)||(~a&b&ci)||(a&b&~ci));
assign co = ~((~a&~b)||(~b&~ci)||(~a&~ci));
//assign {co,s} = a + b + ci;
endmodule
testbench
module fulladder_tb;
wire s;
wire co;
reg a;
reg b;
reg ci;
initial
begin
ci = 0; a = 0 ;b = 0;
#10 ci = 0; a = 0 ;b = 1;
#10 ci = 0; a = 1 ;b = 0;
#10 ci = 0; a = 1 ;b = 1;
#10 ci = 1; a = 0 ;b = 0;
#10 ci = 1; a = 0 ;b = 1;
#10 ci = 1; a = 1 ;b = 0;
#10 ci = 1; a = 1 ;b = 1;
#10 $finish;
end
initial begin
$fsdbDumpfile("test.fsdb");
$fsdbDumpvars();
end
fulladder u_fulladder(
.s(s),
.co(co),
.a(a),
.b(b),
.ci(ci)
);
endmodule
多位加法器
串行進位加法器
依次將低位全加器的進位輸出co接到全加器的進位輸入端ci，就可以構成多位加法器。
顯然，每一位的相加結果都必須等到低一位的進位產生才能建立起來，因此，這種結構的電路稱為串行進位加法器（或稱為行波進位加法器）。
verilog code (fulladder為上面所述的全加器)
module serialadder(
output [3:0] s,
output co,
input [3:0] a,
input [3:0] b,
input ci
);
wire [3:0] co_tmp;
fulladder u_add0(
.s(s[0]), //sum
.co(co_tmp[0]), //carry to high bit
.a(a[0]),
.b(b[0]),
.ci(ci) //carry from low bit
);
fulladder u_add1(
.s(s[1]), //sum
.co(co_tmp[1]), //carry to high bit
.a(a[1]),
.b(b[1]),
.ci(co_tmp[0]) //carry from low bit
);
fulladder u_add2(
.s(s[2]), //sum
.co(co_tmp[2]), //carry to high bit
.a(a[2]),
.b(b[2]),
.ci(co_tmp[1]) //carry from low bit
);
fulladder u_add3(
.s(s[3]), //sum
.co(co_tmp[3]), //carry to high bit
.a(a[3]),
.b(b[3]),
.ci(co_tmp[2]) //carry from low bit
);
assign co = co_tmp[3];
endmodule
testbench
module serialadder_tb;
wire [3:0] s;
wire co;
reg [3:0] a;
reg [3:0] b;
reg ci;
initial
begin
a = 4'b0000; b = 4'b0000; ci = 0;
#10 a = 4'b1111; b = 4'b1111; ci = 0;
#10 a = 4'b1100; b = 4'b1001; ci = 0;
#10 a = 4'b0111; b = 4'b0110; ci = 0;
#10 a = 4'b0101; b = 4'b0101; ci = 1;
#10 a = 4'b1110; b = 4'b1001; ci = 1;
#10 a = 4'b0010; b = 4'b0110; ci = 1;
#10 a = 4'b0110; b = 4'b1100; ci = 1;
#10 $finish;
end
initial begin
$fsdbDumpfile("test.fsdb");
$fsdbDumpvars();
end
serialadder u_serialadder(
.s(s),
.co(co),
.a(a),
.b(b),
.ci(ci)
);
endmodule
超前進位加法器
超前進位信號的產生原理
- ab = 1 --> co = 1
- a + b = 1,且ci = 1 --> co =1
兩位多位數中第i位相加產生的進位輸出co(i)可以表示位
\begin{align}\notag co_{i} = a_{i}b_{i} + (a_{i} + b_{i})(ci_{i}) \end{align}
>> 從全加器的真值表寫出第i位和s(i)的邏輯式： \begin{align}\notag s_{i} = a_{i}{b_{i}}'{ci_{i}}' + {a_{i}}'b_{i}{ci_{i}}' + {a_{i}}'{b_{i}}'ci_{i} + (a_{i} + b_{i})ci_{i} \end{align} >> 上式變換位異或函數位： \begin{align}\notag s_{i} = a_{i} \oplus b_{i} \oplus ci_{i} \end{align}
verilog code
module carry_look_aheadadder(
output [3:0] s,
output co,
input [3:0] a,
input [3:0] b,
input ci
);
wire [3:0] co_tmp;
wire [3:0] cin;
assign cin[3:0] = {co_tmp[2:0],ci};
//計算中間進位
assign co_tmp[0] = a[0]&b[0] || (a[0] || b[0])&(cin[0]);
assign co_tmp[1] = a[1]&b[1] || (a[1] || b[1])&(cin[1]);
assign co_tmp[2] = a[2]&b[2] || (a[2] || b[2])&(cin[2]);
assign co_tmp[3] = a[3]&b[3] || (a[3] || b[3])&(cin[3]);
//計算和
assign s[0] = a[0] ^ b[0] ^ cin[0];
assign s[1] = a[1] ^ b[1] ^ cin[1];
assign s[2] = a[2] ^ b[2] ^ cin[2];
assign s[3] = a[3] ^ b[3] ^ cin[3];
assign co = co_tmp[3];
endmodule
testbench
module carry_look_aheadadder_tb;
wire [3:0] s;
wire co;
reg [3:0] a;
reg [3:0] b;
reg ci;
initial
begin
a = 4'b0000; b = 4'b0000; ci = 0;
#10 a = 4'b1111; b = 4'b1111; ci = 0;
#10 a = 4'b1100; b = 4'b1001; ci = 0;
#10 a = 4'b0111; b = 4'b0110; ci = 0;
#10 a = 4'b0101; b = 4'b0101; ci = 1;
#10 a = 4'b1110; b = 4'b1001; ci = 1;
#10 a = 4'b0010; b = 4'b0110; ci = 1;
#10 a = 4'b0110; b = 4'b1100; ci = 1;
#10 $finish;
end
initial begin
$fsdbDumpfile("test.fsdb");
$fsdbDumpvars();
end
carry_look_aheadadder u_carry_look_aheadadder(
.s(s),
.co(co),
.a(a),
.b(b),
.ci(ci)
);
endmodule
參考資料
[1] 數字電子技術基礎(第五版) 閻石主編