Exercise DET-05.07 – The six columns of the BCD clock (Solution)

Prof. Dr. Thorsten Jungmann (last updated 2026-11-02)

Relates to DET-05 Codes, code converters and standard components

Task

A clock in the binary-coded decimal (BCD) code shows the time with 20 light emitting diodes (LEDs) in six columns, for the tens and units of the hours, the minutes and the seconds. Each column shows one decimal digit in the BCD code, with the weights 8, 4, 2 and 1 from top to bottom:

The clock does not drive all 20 LEDs at once. An integrated circuit HEF4051 works as a demultiplexer, and its outputs $C_1$ to $C_6$ switch the six columns on one after another. Its three address inputs select the column: the address 000 selects the tens of the hours, the address 101 the units of the seconds. As long as its enable input $E$ is 0, it works; at $E=1$, no column is on. For each column, the microcontroller applies the BCD code word of its digit.

a) What time does the clock in the first figure show?

b) Fill in the gaps for this time. The table shows what happens in each of the six time slots.

$A_2$	$A_1$	$A_0$	Column that is on	Digit	Weight 8	Weight 4	Weight 2	Weight 1
0	0	0	tens of the hours	1	0	0	0	1
0	0	1	units of the hours
0	1	0	tens of the minutes
0	1	1
1	0	0
1	0	1

c) Fill in the gaps in the truth table of the column selection, and give the logic expression of $C_3$.

$E$	$A_2$	$A_1$	$A_0$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$C_6$
0	0	0	0	1	0	0	0	0	0
0	0	1	0
0	1	0	1
0	1	1	0
1	1	0	1

d) Which LEDs are lit during the time slot with the address 011? Why does the clock not use the addresses 110 and 111?

e) How many electric lines lead from the microcontroller to the demultiplexer and to the LEDs in this way, if $E$ stays at 0? How many would it need with one electric line for each LED?

f) Each column is on for one of six equal time slots. For which part of the time does an LED that belongs to a lit digit shine? Why do you still see all lit LEDs at once?

◀️ Back to the task

---

Solution

Given

Given explicitly:

• the 20 light emitting diodes (LEDs) in six columns with the weights 8, 4, 2 and 1, and the lit LEDs in the first figure
• the address inputs $A_2$, $A_1$ and $A_0$, from 000 for the tens of the hours to 101 for the units of the seconds
• the enable input $E$: at 1, no column is on
• for each column, the microcontroller applies the BCD code word of its digit

Known:

• Binary-coded decimal (BCD) code and reading the clock (DET-05)
• Demultiplexer, enable input (DET-05, DET-05)
• The HEF4051 in the clock and time-division multiplexing (DET-05)

Required

a) the time
b) the complete table of the six time slots
c) the complete truth table and the expression of $C_3$
d) the lit LEDs for the address 011, and why the addresses 110 and 111 are not used
e) the number of electric lines in both cases
f) the part of the time and why you see all LEDs at once

a) Reading the time

Read the figure column by column, and add the weights of the lit LEDs (DET-05, 1.3):

Column	Lit weights	Digit
tens of the hours	1	1
units of the hours	8 + 1	9
tens of the minutes	4	4
units of the minutes	4 + 2 + 1	7
tens of the seconds	2 + 1	3
units of the seconds	4 + 1	5

Check: the tens of the hours do not exceed 2, and those of the minutes and seconds do not exceed 5, as the columns with two and three LEDs require ✓

b) The six time slots

The address selects the column, and the microcontroller applies the code word of its digit (DET-05, 5.3):

$A_2$	$A_1$	$A_0$	Column that is on	Digit	Weight 8	Weight 4	Weight 2	Weight 1
0	0	0	tens of the hours	1	0	0	0	1
0	0	1	units of the hours	9	1	0	0	1
0	1	0	tens of the minutes	4	0	1	0	0
0	1	1	units of the minutes	7	0	1	1	1
1	0	0	tens of the seconds	3	0	0	1	1
1	0	1	units of the seconds	5	0	1	0	1

c) Truth table of the column selection

Each output of the demultiplexer is 1 for exactly one address, as long as $E$ is 0. The input is fixed at 1, so the outputs are the minterms of the address inputs, as in a decoder (DET-05, 5.2), and the enable input is a further condition:

$$\begin{array}{rl}{C}_1& =\overline{E}\cdot \overline{A_2}\cdot \overline{A_1}\cdot \overline{A_0}\\ C_2& =\overline{E}\cdot \overline{A_2}\cdot \overline{A_1}\cdot A_0\\ C_3& =\overline{E}\cdot \overline{A_2}\cdot A_1\cdot \overline{A_0}\\ C_4& =\overline{E}\cdot \overline{A_2}\cdot A_1\cdot A_0\\ C_5& =\overline{E}\cdot A_2\cdot \overline{A_1}\cdot \overline{A_0}\\ C_6& =\overline{E}\cdot A_2\cdot \overline{A_1}\cdot A_0\end{array}$$

$E$	$A_2$	$A_1$	$A_0$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$C_6$
0	0	0	0	1	0	0	0	0	0
0	0	1	0	0	0	1	0	0	0
0	1	0	1	0	0	0	0	0	1
0	1	1	0	0	0	0	0	0	0
1	1	0	1	0	0	0	0	0	0

Check with the address 010: only $C_3$ is 1, and the third column, the tens of the minutes, is the one with the address 2 ✓

d) The time slot with the address 011

The address 011 is the number 3, and this selects column 4, the units of the minutes. The digit 7 has the code word 0111, so the LEDs with the weights 4, 2 and 1 of this column are lit. All other columns are off during this time slot.

The clock has six columns, and three address inputs give eight addresses. The addresses 110 and 111 would select the outputs 6 and 7, which are connected to nothing, so the clock has no use for them.

e) Number of electric lines

In each time slot, the code word needs four electric lines, one for each weight, and the address needs three. With $E$ at 0, this makes $4+3=7$ electric lines from the microcontroller. With one electric line for each LED, it would need 20. This is the saving that time-division multiplexing brings (DET-05, 5.3).

f) Part of the time

Each column gets one of six equal time slots, so an LED of a lit digit shines for one sixth of the time. The clock runs through the columns so fast that your eye cannot follow, and you see all lit LEDs at once (DET-05, 5.3).

The simulation shows the column selection and, as a block, the microcontroller with the code words of the time 19:47:35. Switch the address inputs through the addresses 000 to 101 and compare with the table of b). You do by hand what the clock does much faster. With $E$ at 1, no column is on, whatever the address says.

Open in the simulator

Circuit as text (File → Import From Text…)

<cir f="1" ts="0.000005" ic="1.500424758475255" cb="50" pb="50" vr="5" mts="5e-11">
  <L x="96 64 96 16" f="2"/>
  <L x="144 64 144 16" f="2"/>
  <L x="192 64 192 16" f="2"/>
  <L x="240 64 240 16" f="2"/>
  <I x="288 128 352 128" f="0" sl="0.5" hi="5"/>
  <dmux x="416 128 576 128" f="0" se="3"/>
  <M x="624 128 672 128" f="2"/>
  <M x="624 160 672 160" f="2"/>
  <M x="624 192 672 192" f="2"/>
  <M x="624 224 672 224" f="2"/>
  <M x="624 256 672 256" f="2"/>
  <M x="624 288 672 288" f="2"/>
  <clm nm="bcdclock" f="0" in="A2,A1,A0" o="W8,W4,W2,W1" if="code word of the time">000=0001
001=1001
010=0100
011=0111
100=0011
101=0101
110=0000
111=0000</clm>
  <cl x="416 560 512 560" f="0" mo="bcdclock"/>
  <M x="560 560 608 560" f="2"/>
  <M x="560 592 608 592" f="2"/>
  <M x="560 624 608 624" f="2"/>
  <M x="560 656 608 656" f="2"/>
  <w x="96 128 288 128" f="0"/>
  <w x="96 64 96 128" f="0"/>
  <w x="352 128 416 128" f="0"/>
  <w x="240 448 448 448" f="0"/>
  <w x="448 416 448 448" f="0"/>
  <w x="240 624 416 624" f="0"/>
  <w x="240 64 240 448" f="0"/>
  <w x="240 448 240 624" f="0"/>
  <w x="192 464 480 464" f="0"/>
  <w x="480 416 480 464" f="0"/>
  <w x="192 592 416 592" f="0"/>
  <w x="192 64 192 464" f="0"/>
  <w x="192 464 192 592" f="0"/>
  <w x="144 480 512 480" f="0"/>
  <w x="512 416 512 480" f="0"/>
  <w x="144 560 416 560" f="0"/>
  <w x="144 64 144 480" f="0"/>
  <w x="144 480 144 560" f="0"/>
  <w x="576 128 624 128" f="0"/>
  <w x="576 160 624 160" f="0"/>
  <w x="576 192 624 192" f="0"/>
  <w x="576 224 624 224" f="0"/>
  <w x="576 256 624 256" f="0"/>
  <w x="576 288 624 288" f="0"/>
  <w x="512 560 560 560" f="0"/>
  <w x="512 592 560 592" f="0"/>
  <w x="512 624 560 624" f="0"/>
  <w x="512 656 560 656" f="0"/>
  <x x="89 -12 99 -9" f="0" si="16" te="E"/>
  <x x="137 -12 156 -9" f="0" si="16" te="A2"/>
  <x x="185 -12 204 -9" f="0" si="16" te="A1"/>
  <x x="233 -12 252 -9" f="0" si="16" te="A0"/>
  <x x="404 100 526 103" f="0" si="16" te="like the HEF4051"/>
  <x x="692 134 756 137" f="0" si="16" te="column 1"/>
  <x x="692 166 756 169" f="0" si="16" te="column 2"/>
  <x x="692 198 756 201" f="0" si="16" te="column 3"/>
  <x x="692 230 756 233" f="0" si="16" te="column 4"/>
  <x x="692 262 756 265" f="0" si="16" te="column 5"/>
  <x x="692 294 756 297" f="0" si="16" te="column 6"/>
  <x x="404 536 789 539" f="0" si="16" te="microcontroller: code word of each column at 19:47:35"/>
  <x x="628 566 687 569" f="0" si="16" te="weight 8"/>
  <x x="628 598 687 601" f="0" si="16" te="weight 4"/>
  <x x="628 630 687 633" f="0" si="16" te="weight 2"/>
  <x x="628 662 687 665" f="0" si="16" te="weight 1"/>
</cir>