1. Combinational Logic

2. Multi-input gates - (x+(y*z))’
Out
Vdd z x y z
out 1
Very easy with inverting gates
University of Texas at Austin CS Computer Organization Spring Don Fussell

3. Half Adder 1-bit addition (x + y) outputs: sum s and carry c
s = xy’ + x’y (x XOR y or x  y)
c = xy x y s c 1 x y s c
University of Texas at Austin CS Computer Organization Spring Don Fussell

4. Full 1-bit adder Need a carry in ci x y ci s co 11
s = x’yci’+ xy’ci’ + x’y’ci + xyci
= ci’(x’y+xy’) + ci (xy+x’y’)
= ci’(x’y+xy’) + ci (x’y+xy’)’
= ci  (x  y)
co = x’yci + xy’ci + xyci’ + xyci
= ci (x’y+xy’) + xy(ci’+ci )
= ci (x  y) + xy x y s ci co
University of Texas at Austin CS Computer Organization Spring Don Fussell

5. CMOS adder For the carry, we can first use De Morgan’s lawsx y s ci co co x y s ci
For the xor, recall that a xor b = ab’ + a’b and (a xor b)’ = ab + a’b’
Vdd x’ y
Out x y’
University of Texas at Austin CS Computer Organization Spring Don Fussell

6. Multiplexor
A multiplexor (selector or mux) selects one of n inputs to be output
Let’s make it easy and start with a 2-input mux
when s = 0, f = a
when s = 1, f = b a b c d f s0 s1 a b f s
University of Texas at Austin CS Computer Organization Spring Don Fussell

7. 2 input Multiplexor designa b f 1
f = s’ab’ + s’ab + sa’b + sab
= s’a(b’+b) + sb(a’+a)
= s’a + sb b f a s
University of Texas at Austin CS Computer Organization Spring Don Fussell

8. Making bigger muxes
You can design them directly, or just glue together smaller muxes: c d f s1 a b s0
University of Texas at Austin CS Computer Organization Spring Don Fussell

9. Decoders Decode an n-bit input to set output line 2n-1 a b 00 01 10 111 11 a 10 01 b 00 a 11 10 b 01 00
University of Texas at Austin CS Computer Organization Spring Don Fussell

10. Programmable Logic Array (PLA)
Since all Boolean functions can be expressed in sum of products form, we can implement a set of n input functions systematically in hardware using a PLA b a
Connections f2 f1 f0
University of Texas at Austin CS Computer Organization Spring Don Fussell