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Question

# Provide a CMOS logic gate circuit that realizes the function $Y=\overline{A} B C+A \overline{B} C+A B \overline{C}.$ How many transistors are required? Explore the possibility of reducing the number of the transistors required.

Solution

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We need to provide a CMOS circuit that will realize the Boolean expression:

$Y=\overline{A}BC+A\overline{B}C+AB\overline{C}$

Considering that we have Y (and not $\overline{Y}$), this expression will be realized by the $\textbf{pull-up network}$. The pull-up network will have 9 transistors (for each of the variables). If we group two of the summands we can lower the number of transistors by 1.

$Y=\overline{A}BC+A\left(\overline{B}C+B\overline{C}\right)$

Now, we will complement the given expression Y:

\begin{align*} \overline{Y}&=\overline{\overline{A}BC+A\left(\overline{B}C+B\overline{C}\right)}\\ &=\left(A+\overline{B}+\overline{C}\right)\cdot \left(\overline{A}+\overline{\overline{B}C+B\overline{C}}\right)\\ &=\left(A+\overline{B}+\overline{C}\right)\cdot \left(\overline{A}+\overline{\overline{B}C}\cdot \overline{B\overline{C}}\right)\\ &=\left(A+\overline{B}+\overline{C}\right)\cdot \left(\overline{A}+\left(B+\overline{C}\right)\cdot \left(\overline{B}+{C}\right)\right)\\ &=\left(A+\overline{B}+\overline{C}\right)\cdot \left(\overline{A}+\overline{B}\cdot \overline{C}+BC\right)\\ &=A\overline{A}+\overline{A}\cdot\overline{B} + \overline{A}\cdot\overline{C}+ A\overline{B}\cdot\overline{C}+ \overline{B}\cdot\overline{B}\cdot\overline{C}+\overline{C}\cdot\overline{B}\cdot\overline{C}+ABC+\overline{B}BC+\overline{C}BC\\ &=\overline{A}\cdot\overline{B} + \overline{A}\cdot\overline{C}+ A\overline{B}\cdot\overline{C}+ \overline{B}\cdot\overline{C}+\overline{B}\cdot\overline{C}+ABC\\ &=\overline{A}\cdot\overline{B} + \overline{A}\cdot\overline{C}+ A\overline{B}\cdot\overline{C}+ \overline{B}\cdot\overline{C}+ABC\\ &A\overline{B}\cdot\overline{C}+ \overline{B}\cdot\overline{C}=\overline{B}\cdot\overline{C}\\ \Rightarrow \overline{Y}&=\overline{A}\cdot\overline{B} + \overline{A}\cdot\overline{C}+\overline{B}\cdot\overline{C}+ABC\\ &=\overline{A}\cdot\overline{B} +\overline{C} \left(\overline{A}+\overline{B}\right)+ABC \end{align*}

This expression will be realized by the $\textbf{pull-down network}$. For that we need 8 transistors. Now we can synthetize the complete CMOS circuit, for which we will need 16 transistors in total.

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