Question $5(20\backslash \%)$

You are asked to design a PLC program for a heat$-$treatment furnace temperature control. The operations of the automated furnace temperature control are shown as follows:

$1.$ The whole system will be ON and stay on when a start push$-$button START is pressed $($and released later$).$ When the system is ON$,$ a green light GREEN is turned on indicating the system is ready. The following operations are performed during the system has been turned ON $.$

$2.$ When the system is ON $,$ the temperature of a furnace is to be maintained between the temperatures of a lower temperature setting sensed by a lower temperature sensor LT and an upper temperature setting sensed by an upper temperature sensor UT$.$

$3.$ Given the system has been ON $,$ a heater $\backslash (\backslash$mathbf$\{$H E$\}\backslash )$ will be turned on when the furnace temperature drops below the lower temperature stetting by triggering the sensor LT$.$ The heater HE will remain ON until the furnace temperature reaches the upper temperature setting by triggering the sensor UT and the heater HE will then be turned off.

$4.$ When the system is on but the heater $\backslash (\backslash$mathbf$\{$H E$\}\backslash )$ is turned OFF $($furnace is cooling$),$ the heater $\backslash (\backslash$mathbf$\{$H E$\}\backslash )$ stays OFF until the furnace temperature drops below the lower temperature setting sensed by the sensor LT$.$

$5.$ Two indicator lights $($GREEN and RED$)$ are used to show the status of the furnace and heating element. GREEN light indicates that the furnace is turned on and the furnace system is ready for heat treatment. The RED indicates that the heater HE is on or heating.

$6.$ For operations safety, a door limit switch DR is used to detect whether the furnace door is closed or not during the furnace operations. Anytime the door is opened during furnace normal operations, the furnace system$-$ready status and the heater HE will be turned OFF and an alarm ALM will be sounded. The alarm will be OFF by closing the door. The green light GREEN will also be turned OFF whenever the door is opened during the normal operations. The furnace system should not be turned ON unless the door is closed, sensed by the normally closed limit switch DR$.$

Construct a ladder logic diagram for the control of the automatic heat$-$treatment furnace temperature. Clearly define all the inputs and outputs used for the furnace temperature control system. Clearly specify the correspondent Boolean logic expressions of each of the outputs: HE$,$ ALM, GREEN, and RED for the furnace temperature control system.