3 A motor driving a water pump has a safety system which uses three
inputs to a logic circuit. A certain combination of conditions results
in an output, X, from the logic circuit being equal to 1. When the
value of X- 1 then the motor and pump are shut down.
The following table shows which parameters are being monitored
and form the three inputs to the logic circuit.
Parameter description
motor speed
bearing temperature
water velocity
The output, X, will have
of conditions occur:
Parameter
S
T
Binary value Description of condition
motor speed <-2000 rpm
motor speed > 2000 rpm
0
1
0
1
0
1
bearing temperature <-90°C
bearing temperature > 90°C
water velocity <- 5 m/s
wind velocity > 5 m/s
value of 1 if any of the following combination
»> either motor speed > 2000 rpm and bearing temperature > 90°C
>> or motor speed <-2000 rpm and water velocity <- 5 m/s
>>
or bearing temperature > 90°C and water velocity <- 5 m/s
a Design a logic circuit for the above scenario.
b Complete a truth table for the above scenario.
Problem
(we need to make it clear, first)
A motor driving a water pump has a safety system which uses three inputs to a logic circuit. A certain combination of conditions results in an output, X, from the logic circuit being equal to 1. When the value of X = 1 then the motor and pump are shut down.
The following description shows which parameters are being monitored and form the three inputs to the logic circuit. (converted from table)
Parameter description:
- S = motor speed
- T = bearing temperature
- W = water velocity
[tex]\begin{aligned}S&=\begin{cases}0,&\!\!\!\rm if\ motor\ speed\ \le \ 2000\ rpm\\1,&\!\!\!\rm if\ motor\ speed\ > \ 2000\ rpm\end{cases}\\T&=\begin{cases}0,&\!\!\!\rm if\ bearing\ temp.\ \le \ 90^{\circ}\:C\\1,&\!\!\!\rm if\ bearing\ temp.\ > \ 90^{\circ}\:C\end{cases}\\W&=\begin{cases}0,&\!\!\!\rm if\ water\ velocity\ \le \ 5\ m/s\\1,&\!\!\!\rm if\ water\ velocity\ > \ 5\ m/s\end{cases}\end{aligned}[/tex]
The output, X, will have a value of 1 if any of the following combination of conditions occur:
- either motor speed > 2000 rpm and bearing temperature > 90°C
- or motor speed <= 2000 rpm and water velocity <= 5 m/s
- or bearing temperature > 90°C and water velocity <= 5 m/s
a. Design a logic circuit for the above scenario.
b. Complete a truth table for the above scenario.
Solution
a. Logic Circuit
First, we convert each of the three cases above to logic statements, and then convert each one of them to equivalent logic circuit.
Condition 1
Description: motor speed > 2000 rpm and bearing temperature > 90°C
Logic statement:
[tex]\begin{aligned}C_1&=(S=1)\ {\rm AND}\ (T=1)\\&\left({\rm assume}\ S=1,\,T=1\right)\\\Rightarrow C_1&=S\ {\rm AND}\ T\end{aligned}[/tex]
Logic circuit: see Figure 1.
Condition 2
Description: motor speed <= 2000 rpm and water velocity <= 5 m/s
Logic statement:
[tex]\begin{aligned}C_2&=(S=0)\ {\rm AND}\ (W=0)\\&\left({\rm assume}\ S=1,\,W=1\right)\\\Rightarrow C_2&={\rm NOT}\ S\ {\rm AND}\ {\rm NOT}\ W\end{aligned}[/tex]
Logic circuit: see Figure 2.
Condition 3
Description: bearing temperature > 90°C and water velocity <= 5 m/s
Logic statement:
[tex]\begin{aligned}C_3&=(T=1)\ {\rm AND}\ (W=0)\\&\left({\rm assume}\ T=1,\,W=1\right)\\\Rightarrow C_3&=T\ {\rm AND}\ {\rm NOT}\ W\end{aligned}[/tex]
Logic circuit: see Figure 3.
Join all subcircuits
Each of the three conditions are joined together by the word OR. Thus, we need to join all subcircuits by the OR gate.
[tex]C_1\ {\rm OR}\ C_2[/tex] : see Figure 4.
[tex]C_1\ {\rm OR}\ C_2\ {\rm OR}\ C_3 = X[/tex] : see Figure 5.
CONCLUSION:
∴ Logic circuit in Figure 5 is the answer to question a.
(However, it can be further simplified.)
______________
b. Truth Table
To generate the truth table, we will use the intermediate values given by [tex]C_1[/tex], [tex]C_2[/tex], and [tex]C_3[/tex].
[tex]\begin{array}{|c|c|c||c|c|c||c|}S&T&W&C_1&C_2&C_3&X\\&&&&&&\footnotesize\text{$(C_1\ {\rm OR}\ C_2\ {\rm OR}\ C_3)$}\\0&0&0&0&1&0&\bf1\\0&0&1&0&0&0&\bf0\\0&1&0&0&1&1&\bf1\\0&1&1&0&0&0&\bf0\\1&0&0&0&0&0&\bf0\\1&0&1&0&0&0&\bf0\\1&1&0&0&0&1&\bf1\\1&1&1&1&0&0&\bf1\\\end{array}[/tex]
[answer.2.content]
