Question 7 of 7: Combinational Logic Design — HVAC Controller
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams December 2015 — 04-BS-4 Electric Circuits and Power, 3 hours, closed book (one aid sheet permitted, Casio/Sharp approved calculators). The paper instructs "any five questions constitute a complete paper" and marks only the first five answered; every question is solved so the set stands as a complete study resource.
Reference texts: Sadiku, Fundamentals of Electric Circuits (6th ed.) — DC/AC circuit analysis, Thevenin/Norton, first-order transients, phasors, magnetic circuits, diode rectifiers; Mano & Ciletti, Digital Design — combinational logic design (Question 7).
Check: several figure dimensions/polarities in this paper are read directly from the printed figure (vision-caption topology is a known failure mode on this subject); the specific reads are flagged inline where they affect a result (Q2 Thevenin distractor branch, Q5 mean magnetic path lengths, Q7 "desired temperature reached" logic).
Given. Seven binary sensor signals A–G as defined above (1=condition true).
Sensor definitions
Signal
Meaning (1 = true)
A
Compressor rest time (tREST) has been exceeded — OK to restart
B
Stage 1 fan time (tStage1) has been exceeded
C
Ambient temperature > tHI (over-temperature)
D
Ambient temperature < tLO (under-temperature)
E
Heating switch ON
F
Cooling switch ON
G
Furnace temperature > tFurnace
Find. Boolean expressions and AND/OR/NOT gate logic circuits for: Furnace ON, Compressor ON, Fan-low-speed, Fan-high-speed.
Check: the sensor list has no direct signal for "desired temperature reached." The only sensors that describe the ambient temperature relative to a setpoint are C (over-temperature) and D (under-temperature), so "desired temperature reached" is taken here as the deadband condition NOT(C) AND NOT(D) — ambient is neither above tHI nor below tLO. This reading is used consistently in parts (c) and (d) below.
Approach. Translate each control-English sentence directly into an AND/OR/NOT expression over A–G, keep a running "Compressor" and "Fan ON" signal since both are reused by later parts, then draw each as a small gate network.
(a) Furnace control. "Heating switch ON AND ambient lower than tLO": $$\boxed{\text{Furnace} = E\cdot D}$$
Furnace = E AND D.
(b) Compressor control. "Cooling switch ON AND ambient higher than tHI", gated by the minimum-rest-time interlock (A=1 means the rest time HAS been exceeded, i.e. restart is allowed): $$\boxed{\text{Compressor} = F\cdot C\cdot A}$$
Compressor = F AND C AND A (two 2-input AND gates in cascade).
Fan-ON helper signal. Used by both (c) and (d): the fan runs whenever the compressor is on, or the furnace is overheating: $$\text{FanOn} = \text{Compressor} + G$$and the "desired temperature reached" condition adopted above: $$\text{DesiredReached} = \overline{C}\cdot\overline{D}$$
(c) Fan low-speed. Low-speed runs whenever the fan is on AND neither the Stage 1 timer has expired NOR the desired temperature has been reached: $$\boxed{\text{FanLow} = \text{FanOn}\cdot\overline{B}\cdot(C+D)}$$(using $\overline{\text{DesiredReached}}=\overline{\overline{C}\cdot\overline{D}}=C+D$ by De Morgan's theorem).
Fan LOW-speed = [ (Compressor OR G) AND NOT(B) ] AND (C OR D).
(d) Fan high-speed. High-speed takes over exactly when low-speed's "still within Stage 1" condition fails, i.e. B has become true while the fan is still needed and the target has not been reached: $$\boxed{\text{FanHigh} = \text{FanOn}\cdot B\cdot(C+D)}$$FanLow and FanHigh share the same FanOn·(C+D) term but are gated by complementary B/̄B, so they are mutually exclusive whenever the fan is on and the target is not yet reached.
Fan HIGH-speed = [ (Compressor OR G) AND B ] AND (C OR D).
