Ch 4. Combinational Logic Design Principles

• Combinational Logic Circuit– Outputs depend only on its current inputs– No feedback loop

• Sequential Logic Circuit– Outputs depend on its current inputs and present states– Feedback loop

4.1 Switching Algebra

‘+’ : ‘OR’

‘’ : ‘AND’

& are obvious (by perfect induction)

is identical to the distributive law for integers and reals

Switching-algebra theorems

NAND

DeMorgan’s Th.

NOR

DeMorgan’s Th.

Black box Representation with Truth table

Active Low

Active Low

Black box Representation with Truth table

DualityPositive Negative 𝑭 (𝑿𝟏 ,𝑿𝟐 ,… 𝑿𝒏 )⇒ 𝑭 𝑫 (𝑿𝟏

′ , 𝑿𝟐′ ,…𝑿𝒏′ )

𝑭 (𝑿𝟏 ,𝑿𝟐 ,… 𝑿𝒏 )=[𝑭 𝑫 (𝑿𝟏′ ,𝑿𝟐

′ ,…𝑿𝒏′ ) ]′

[𝑭 (𝑿𝟏 ,𝑿𝟐 ,…𝑿𝒏 ) ]′=[𝑭 𝑫 (𝑿𝟏′ , 𝑿𝟐

′ ,… 𝑿𝒏′ ) ]❑

→𝑮𝒆𝒏𝒆𝒓𝒂𝒍𝒊𝒛𝒆𝒅 𝑫𝒆𝑴𝒐𝒓𝒈𝒂𝒏′ 𝒔𝑻𝒉 .

𝑫𝒖𝒂𝒍𝒊𝒕𝒚

Function with 3 inputX, Y, Z

𝑭 (𝑿 ,𝒀 ,𝒁 )

𝑪𝒂𝒏𝒐𝒏𝒊𝒄𝒂𝒍 𝑺𝒖𝒎−𝑨𝒔𝒖𝒎𝒐𝒇 𝒎𝒊𝒏𝒕𝒆𝒓𝒎𝒔 𝒕𝒐𝒑𝒓𝒐𝒅𝒖𝒄𝒆𝒐𝒖𝒕𝒑𝒖𝒕𝒔 ′ 𝟏 ′

DeMorgan’s Th.

4.2 Combinational-Circuit Analysis

F(X, Y, Z)

F(X, Y, Z)

¿ 𝑿𝒁+𝒀 ′ 𝒁+𝑿 ′𝒀𝒁 ′

Three-level without considering inverter

(𝑽 ∙𝑾 ∙𝐗 )+ (𝐘 ∙𝒁 )=(𝑽 +𝒀 ) ∙ (𝑽 +𝒁 ) ∙ (𝑾 +𝒀 ) ∙(𝑾 +𝒁 )∙(𝑿+𝒀 ) ∙(𝑿+𝒁 )

“Add-out” to obtain a product of sums by T8’

T8’

Sum of products

Product of sums

(Figure 4-12)

(Figure 4-13)

Same

4.3 Combinational-Circuit Synthesis

Prime number detector

¿𝑷𝒂𝒏𝒊𝒄+𝑬𝒏𝒂𝒃𝒍𝒆 ∙𝑬𝒙𝒊𝒕𝒊𝒏𝒈 ′ ∙𝑺𝒆𝒄𝒖𝒓𝒆 ′

𝒏𝒐𝒏𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅𝒈𝒂𝒕𝒆

𝒆𝒍𝒎𝒊𝒏𝒂𝒕𝒆𝒔 𝒏𝒐𝒏𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅𝒈𝒂𝒕𝒆

NAND

NOR

𝐹= ∑𝑁3𝑁2𝑁1𝑁0

❑

(1,3,5,7,2,11,13 )

𝑮𝒓𝒂𝒚 𝑺𝒆𝒒𝒖𝒆𝒏𝒄𝒆

𝑴𝒊𝒏𝒕𝒆𝒓𝒎𝑮𝒓𝒂𝒚 𝑺𝒆𝒒𝒖𝒆𝒏𝒄𝒆

𝒅𝒊𝒔𝒕𝒊𝒏𝒈𝒖𝒊𝒔𝒉𝒆𝒅𝟏−𝒄𝒆𝒍𝒍𝒔

𝑬𝑷𝑰

𝑬𝑷𝑰

4.4 Timing Hazards

𝑿𝒁𝑷=𝟏𝑿=𝟏

𝐘 𝒁=𝟎

Unwanted Output

Static-1 Hazard : Momentary output ‘0’ to occur during a transition in the differing input variable

Static-0 Hazard : Momentary output ‘1’ to occur during a transition in the differing input variable

Unwanted Output

𝑻𝒐𝒆𝒍𝒊𝒎𝒊𝒏𝒂𝒕𝒆 𝒔𝒕𝒂𝒕𝒊𝒄𝟏−𝒉𝒂𝒛𝒂𝒓𝒅