Exercise DET-04.07 – Hot-water charging in comfort and eco times

Prof. Dr. Thorsten Jungmann (last updated 2026-10-25)

Relates to DET-04 Algorithmic minimization of functions with many inputs and hazard analysis

Task

In our example system, the controller of the heat pump starts a hot-water charging when the temperature of the hot water falls 5 K below its set point. The hot-water programme switches between comfort times with a set point of 52.5 °C and eco times with a set point of 47.5 °C, so a charging starts below 47.5 °C in the comfort time and below 42.5 °C in the eco time. From Monday to Thursday, the comfort time ends at 23:00. During a block by the energy supplier, the heat pump must not start. In simplified form, the inputs and the output are:

• $A$: the comfort time of the hot-water programme is active
• $B$: the hot water is below 47.5 °C
• $C$: the hot water is below 42.5 °C
• $D$: the energy supplier blocks the heat pump
• $Y$: a hot-water charging may start

a) Set up the truth table of $Y$ with $A$ as the input with the highest weight. In the rows that never occur, mark the output as a don’t care.

b) Minimize $Y$ with the Quine–McCluskey algorithm, using the don’t cares.

c) A colleague builds the circuit directly from the description, as a two-level circuit with inverters for $A$ and $D$: $Y_{\mathrm{direct}}=A\cdot B\cdot \overline{D}+\overline{A}\cdot C\cdot \overline{D}$. On a Thursday at 23:00, the comfort time ends, while the hot water is below 42.5 °C and there is no block. Show that the output of this circuit can drop to 0 for a moment.

d) Suppose a counter counts every rise of the output from 0 to 1 as the start of a hot-water charging. What does the dip of c) do to the counter? Is the counter a combinational logic circuit or a sequential circuit?

e) Can the output of your minimal circuit from b) drop to 0 at 23:00 as well? Give a reason.

Solution

a) a 1 in rows 6, 12 and 14; X in rows 2, 3, 10 and 11, where the water would be below 42.5 °C but not below 47.5 °C; a 0 in all other rows
b) $Y=C\cdot \overline{D}+A\cdot B\cdot \overline{D}$, from the two essential prime implicants; the prime implicant $\overline{B}\cdot C$ contains only don’t cares
c) for $B=1$, $C=1$ and $D=0$: $Y_{\mathrm{direct}}=A+\overline{A}$; when $A$ falls, $A\cdot B\cdot \overline{D}$ falls before $\overline{A}\cdot C\cdot \overline{D}$ rises, and the output drops to 0 for a moment, a static 1 hazard
d) the counter counts a hot-water charging that never happened and shows one start too many from then on; it is a sequential circuit
e) no: in the minimal DNF, $A$ appears only as it is; at 23:00, the term $C\cdot \overline{D}$ stays 1 and holds the output at 1

▶️ Full worked solution