- Information
- AI Chat
Digital Questions
General Physics I (PH 161)
Montgomery College
Recommended for you
Preview text
warning buzzer using combinational logic. The criterion for the activation of the ing buzzer is as follows: The buzzer activates if the headlights are on and the door is opened or if the key is in the ignition and the door is opened. The logic function for the automobile warning buzzer is illustrated symbolically in Figure The figure illustrates a combination of logic functions that can be ten as a Boolean equation in the form or HandD which is also written as HD This equation can be stated as is HIGH if K and D are HIGH or if H and D are Key in ignition K__ D TE Headlightson H_ 1 D mo aan) K Key in ignition Headlights on An OR gate input to an AND gate requires parentheses. Write the Boolean logic equation, and draw the logic circuit and truth table that represents the following function: A bank burglar alarm (A) is to vate if it is after banking hours and the front door (F) is opened or if it is after banking hours (1) and the vault door is opened (V). Solution: A HF HV. The logic circuit and truth table are shown in Figure After hours a _ Front door open 1 ) (Sar ! . oocosoolr ee OOF e osoln ecececesle Using common reasoning, reduce the logic function described in Example to a simpler form. Solution: The alarm is activated if it is after banking hours and if either the front door is opened or the vault door is opened (see Figure The plified equation is written as V) (Notice the use of parentheses.) F Vault door open Vv Figure shows a gray water reclamation tank having five inputs and three outputs. The inputs are used to monitor levels on the quantities shown, and the outputs are used to illuminate the color lights in the Process Monitoring Station. The system is designed to capture gray water before it goes into a septic system. Gray water is the water drained in the bathroom sink or shower and water drained in a washing machine. This recycled water can then be used in the toilet or for landscape irrigation. In this example, logic gates are connected to the figure to turn on the blue light (B) if the water is at the mid level (WZ) and there is a HIGH pressure (P) or if the water is at the mid level (MM) and there is a HIGH opacity (c). (Opacity is a measure of water clarity.) (a) Reduce that Boolean equation to a simpler form. (b) Write the Boolean equation for the new logic that would turn on the red light (A) if the PH level or the Opacity (C) or the Pressure (P) are HIGH while the water is at the mid level (MW). (The word while indicates an AND function). (c) Write the Boolean equation for the new logic that would turn on the green light (G) if the PH level or the Pressure (P) are HIGH while the water 1s at the mid level (MM) or the full level (F). (d) Write the Boolean equation for the new logic that would turn on the blue light (B) if the Opacity (C) and the pressure (P) are HIGH while the water 1s at the mid level or the full level (F). a) Opacit Solutions: (a) (c) G (H F) (b) (d) B CP(M F) equivalent B to: A Figure Using the commutative law of addition to rearrange an OR gate. A B Is B equivalent c c j to: A Figure Using the commutative law of multiplication to rearrange an AND gate. A A R Is equivalent ta: B c c Figure Using the associative law of addition to rearrange the grouping of OR gates. A J Is X equivalent to: RE Figure Using the associative law of multiplication to rearrange the grouping of AND gates. How many gates are required to implement the following Boolean equations? (a) X B)C (b) AC BC (c) Z (ABC CD)E Which Boolean law is used to transform each of the following equations? (a) (b) CAB BCA (c) (B C(A D) BD CA CD The output of an AND gate with one of its inputs connected to 1 will always output a level equal to the level at the other input. True or false? The output of an OR gate with one of its inputs connected to will always output a level equal to the level at the other input. True or false? If one input to an OR gate is connected to 0, the output will always be 0 regardless of the level on the other input. True or false? Use one of the forms of Rule 10 to transform each of the following equations: (a) (b) BC The logic circuit shown in Figure is used to turn on a warning buzzer at X based on the input conditions at A, B, and C. A simplified equivalent circuit that will perform the same function can be formed using Boolean algebra. Write the equation of the circuit in Figure simplify the tion, and draw the logic circuit of the simplified equation. Figure Logic circuit for Example Solution: The Boolean equation for X is To simplify, first apply Law 3 C) BA BC): Next, factor a C from terms 2 and 3: X BA C(B 1) Apply Rule 4(B 1 1): Apply Rule C): Apply Law J (BA AB): simplified equation The logic circuit of the simplified equation is shown in Figure rt ) c pe Figure Simplified logic circuit for Example EXAMPLE Repeat Example for the logic circuit shown in Figure a4 ot (a) Figure Logic circuit for Example (a) Original circuit and (b) Simplified circuit. Apply Rule 3 (AB 0 AB): X BC Factor a B from terms and 2: X B(A BC Apply Rule 4(A 1 1): Apply Rule 2 1 B): Apply Rule 10(b) (B BC B C): equation The logic circuit of the simplified equation is shown in Figure The logic level at A has no effect on the a output. A Not used B _ Cc (b) EXAMPLE Repeat Example for the logic circuit shown in Figure A ZL . f De B (a) Figure Logic circuit for Example (a) Original circuit and (b) Simplified circuit. The logic circuit of the simplified equation is shown in Figure A B C Not used (a) NOR A A equivalent B B B xX xXx c c Cc Original circuit Inversion Final equivalent circuit bubbles cancel (b) Inverter NAND NOR v4 YY EXAMPLE Write the Boolean equation for the circuit shown in Figure Use De theorem and then Boolean algebra rules to simplify the equation. Draw the simplified circuit. 21 Figure Solution: The Boolean equation at X is X Applying De theorem produces (Notice the use of parentheses to maintain proper grouping. Rule: Whenever you break the bar over a NAND you must use parentheses.) Using Boolean algebra rules produces X AB BB 0 AB simplified equation Bubble inverter. A ( Ao _ _ OR B Figure Simplified logic circuit for Example EXAMPLE Repeat Example for the circuit shown in Figure A AB __ Cc Figure Solution: X D)AB AB AB equation The simplified circuit is shown in Figure The simplified circuit is shown in Figure Dee D Figure Simplified logic circuit for Example EXAMPLE Use De theorem and Boolean algebra on the circuit shown in Figure to develop an equivalent circuit that has inversion bars ing only single variables. a Figure Solution: The Boolean equation at X is X C) Applying De theorem produces X (Notice the use of parentheses to maintain proper grouping.) Using Boolean algebra rules produces X AB AC BB BC BC final equation form) The equivalent circuit is shown in Figure pe Draw the logic circuit for the following equation, simplify the equation, and construct a truth table for the simplified equation Solution: To draw the circuit, we have to reverse our thinking from the previous examples. When we study the equation, we see that we need two NANDs feeding into an OR gate, as shown in Figure Then we have to provide the inputs to the NAND gates, as shown in Figure Figure Partial solution to Example A B A Cc Figure Logic circuit of the equation for Example Next, we use De theorem and Boolean algebra to simplify the equation: : ! ae) a Apply Rule equation This equation can be interpreted as: X is HIGH if A is LOW or C is LOW or B is HIGH. Now, to construct a truth table (Table we need three input columns (A, B, C) and eight entries 8), and we fillina 1 for X when A 0,C Q 1. : 2 : : a ll ml RPreOoOGreer aoc! oc Oo oO C OS ee ee
Digital Questions
Course: General Physics I (PH 161)
University: Montgomery College
- Discover more from:
