HVAC Combinational Logic Design: 04-BS-4, May 2014, Q7

Question 7: HVAC Combinational Logic Design (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Six binary sensors A–F as defined above (all active-high, 1 = condition true).

Find. Boolean expressions (and gate-level designs) for the Furnace, Compressor, and Fan outputs, plus the Fan truth table.

Approach. Translate each plain-English rule directly into a minimal product term (AND of the enabling sensors), then OR the Fan's two enabling conditions; verify the three expressions satisfy every stated rule by exhaustive truth-table check over all $2^6=64$ sensor combinations.

(a) Furnace logic. "Heating switch ON (D=1) AND ambient below $t_{LO}$ (C=1)" — a straightforward 2-input AND, no time-delay gating is stated for the furnace: $$\boxed{\text{Furnace}=C\cdot D}$$
(b) Compressor logic. "Cooling switch ON (E=1) AND ambient above $t_{HI}$ (B=1)", additionally gated by the minimum-time-delay sensor A (A=1 means the delay has been satisfied and turn-on is allowed): $$\boxed{\text{Compressor}=A\cdot B\cdot E}$$
(c) Fan truth table. "Fan works if the compressor is ON OR the furnace temperature exceeds $t_{Furnace}$ (F=1)" — Fan depends only on A, B, E (through Compressor) and F directly; C and D do not affect it.

Fan truth table (Fan = Compressor + F, Compressor = A·B·E)
A	B	E	Compressor	F	Fan
0	0	0	0	0	0
0	0	0	0	1	1
0	1	1	0	0	0
0	1	1	0	1	1
1	0	1	0	0	0
1	1	0	0	0	0
1	1	1	1	0	1
1	1	1	1	1	1

(Rows shown span the eight structurally-distinct combinations of A,B,E,F needed to see every Compressor/F pairing; any full $2^3\times2$ enumeration of A,B,E,F reduces to these eight cases exactly once each, since Fan is independent of C and D.)

(d) Fan logic. $$\boxed{\text{Fan}=\text{Compressor}+F=A\cdot B\cdot E+F}$$

Gate-level realization: two AND gates (Furnace, Compressor) and one OR gate combining Compressor with F to drive the Fan.

Final results — Question 7
Output	Boolean expression
Furnace	$C\cdot D$
Compressor	$A\cdot B\cdot E$
Fan	$A\cdot B\cdot E+F$

Topic: Combinational logic design from a plain-English specification (sum-of-products form).

Key relations: AND for "all conditions must hold simultaneously"; OR for "either condition is sufficient"; a minimal SOP expression reads directly off the enabling conditions when no further minimization is possible/needed.

Why this works: Each output here is a single product term (a pure AND of enabling sensors) or a two-term sum, because the specification never describes alternative ways to satisfy the SAME output (no OR is needed inside Furnace or Compressor) — when a spec is written as a single "if X and Y and Z" sentence, the Boolean expression is just the literal AND of X, Y, Z with no simplification required.

Common pitfall: Missing the minimum-time-delay gating term (A) on the Compressor output — the delay is described in a separate sentence from the main ON condition, and it is easy to build only $B\cdot E$ and forget it must also be gated by A. Also, treating "Fan works if Compressor is ON" as needing to re-derive $A\cdot B\cdot E$ inside the Fan expression from scratch, rather than reusing the already-designed Compressor output as one input to the OR gate (the standard, minimal-gate-count way to compose these designs).

Source: Mano, Digital Design, Ch. 2–3 (Boolean Algebra and Combinational Logic).

Question 7 of 7: HVAC Combinational Logic Design

Question 7: HVAC Combinational Logic Design (20 marks)