Working in ladder logic programs is getting further complicated especially for large-scale projects. So, it is very beneficial to know shortcuts for writing ladder logic simpler, more readable, and easier for others to develop. Using this variety of logic gates, enrich the logic and fluency of writing the logic in different situations. For example, using XOR logic is very common for comparing two inputs to decide if they are equal or different. Therefore, we found that it is crucial to go through all logic gates and do simulations for them all in this tutorial. for coherently we get the knowledge to use them fluently in ladder logic programming to solve different logical problems and scenarios. There are seven basic logic gates which are: “AND”, “OR”, “NOT”, “NAND”, “NOR”, “XOR”, and “XNOR”. We have gone through “AND”, “OR”, and “NOT” logic gates including simulation work. So let us go through the other four gates for learning and practicing all basic logic gates.

**The "NAND" Logic Gate**

The “NAND” logic gate is the invert of the “AND” gate like you invert the output of an “AND” logic gate as shown in Fig. 12. Table 5 lists all combinations of the inputs and the output of the “NAND” logic gate.

**Fig. 12: NAND symbol**

**Table 5: the truth table of the “NAND” logic gate**

Input A |
Input B |
Output |

0 | 0 | 1 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

In addition, the timing diagram of a “NAND” logic gate is shown in fig. 13, it shows the output goes low when both of the inputs A and B are true which is the inverse of the “AND” logic gate. Also, Fig. 14 shows a ladder logic of a “NAND” logic.

**Fig. 13: The timing diagram of the “NAND” logic gate**

**Fig. 14: Ladder logic sample of a “NAND” logic**

**NAND Logic in PLC simulator
**

The NAND logic gate can be considered as the invert of the “AND” logic gate. So as listed in table 1, the truth table of AND and NAND logic gates shows how the NAND gate is the reverse of the AND gate. Also, it shows the NAND gate should come out to your mind when you want the output always low except when both inputs A and B are high.

**Table 1: the truth table of the AND versus NAND logic gate**

The NAND gate can be implemented by connecting the negate of the output in series to the inputs A and B. Another way to implement NAND is by inverting both inputs and connecting them in parallel as shown in fig. 1. Rung 1 and rung 2 respectively.

**Fig. 1: The NAND ladder logic in two ways**

Let’s test our ladder logic in both methods. According to the truth table of NAND gate, we have four test cases. figure 2 shows the first test case when both inputs A and B are false. In this case, the output should be true as shown in fig. 1.

**Fig. 2: NAND ladder logic when both inputs are false**

figure 3 shows the second test case when inputs A goes true while B is false. In this case, the output should be true as shown in fig. 3.

**Fig. 3: NAND ladder logic when input A is false and input B is true**

Figure 4 shows the third test case when inputs B goes high while B is low. In this case, the output should be true as shown in fig. 4.

**Fig. 4: NAND ladder logic when input A is true and input B is false**

Figure 5 shows the last test case when both inputs A and B become high so the output goes false as shown in fig. 5.

**Fig. 5: NAND ladder logic when both inputs are true**

Thanks to performing the simulation of the NAND gate, we now can conclude that we should be looking for a NAND logic when we want to shut down an actuator i.e. motor whenever two inputs are in a true logical state simultaneously. For a practical example, when we have three pumps and we want to run them in the mode of two of three. There should be only two of the three pumps to run at any time. In that case scenario, the run condition of any motor can be a NAND of the status of the other two motors.

**The “NOR” Logic Gate**

The “NOR” logic gate receives two inputs and has one output. It is the same as the invert of the “OR” logic gate. Like you follow the output of an “OR” gate by a “NOT” logic gate. Fig. 15 shows the symbol of a “NOR” gate. In addition, the truth table is expressed in table 6. It shows the output becomes false only when one of the input A or input B or both goes high which is the reverse logic of the “OR” logic gate.

**Fig. 15: The symbol of “NOR” gate**

The “NOR” logic gate can be formed by connecting the “OR” logic gate to the inverter “NOT” logic gate. Or by inverting the inputs by using “NOT” logic gates and connecting them to the “AND” logic gate as shown in Fig. 16.

**Fig. 16: structure of “NOR” logic gate**

**Table 6: the truth table of the “NOR” gate**

Input A |
Input B |
Output |

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 0 |

Figure 17 shows an example of a ladder program for implementing the “NOR” logic gate. It shows that the “NOR” gate can be implemented in ladder logic by connecting two contacts of type NC in series.

**Fig. 17: A sample ladder for a “NOR” logic gate**

On the other hand, the timing diagram of the “NOR” logic gate is depicted in Fig. 18. It shows the output is false as long as either input A or input B or both are true.

**Fig. 18: The timing diagram of the “NOR” logic gate**

**NOR logic gate in PLC Simulator
**

This logic gate can be considered as the negate of OR as you can notice in the truth table as listed in table 2. You can now feel when we may need to use the NOR logic? Yes! Exactly you want it when you design for output which is all time off except when both inputs are false.

**Table 2: the truth table of NOR versus OR**

Input A | Input B | OR | NOR |

0 | 0 | 0 | 1 |

0 | 1 | 1 | 0 |

1 | 0 | 1 | 0 |

1 | 1 | 1 | 0 |

Figure 6 shows two ways to implement the NOR logic in ladder logic. To make that happen, we connect the invert of the two inputs in series to the output as shown in the top part of fig. 6. The other way is to connect the two inputs in parallel to form OR and then connect to negate the output as shown in the lower part of fig. 6.

**Fig. 6: The ladder logic of NOR logic gate**

Let’s practice simulation of the NOR gate, in fig. 7, the first test case is when both inputs are false, the output is true as shown in fig. 7.

**Fig. 7: The Simulation of NOR ladder when both inputs are false**

Figure 8 shows the second case when input A is true and input B is false, the output is false as shown in fig. 8.

**Fig. 8: The Simulation of NOR ladder when input A is true and input B is false**

Figure 9 shows the third test case when input B is true and input A is false, the output is false as shown in fig. 9.

**Fig. 9: the Simulation of NOR ladder when input B is true and input A is false**

Figure 10 shows the last test case of NOR ladder logic, it shows the output is false

**Fig. 10: The Simulation of NOR ladder when both input A and B are true**

One practical example of using the NOR logic is that, imagine friends we drive some machine with two motors. And it is required to have at least one of them or both are running all the time otherwise alarm should be energized. The NOR logic is the best to manage that alarm to get energized if and only if both motors are off.

**The Exclusive OR "XOR" Logic Gate**

Despite this logic gate having two inputs and one output like the “AND” and “OR” logic gates, this logic gate is a bit more complicated than the previous logic gates. Table 4 lists the truth table including all combinations of the inputs and the output. By noticing the truth table of the XOR logic gate, you can see the output becomes low when the two inputs are equal like both are high or both are low. But the output goes high when there is a difference in the state of the two inputs. Imagine my friend, how much this logic gate is very beneficial for comparing two signals.

**Table 4: The truth table of XOR logic gate**

Input A |
Input B |
Output |

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

On the other hand, Figure 10 shows the symbol of the XOR logic gate and its schematic. See how the basic logic gates OR, AND, and NOT can be utilized to build the logic of XOR logic gate.

**Fig. 10: The XOR logic gate symbol**

Figure 11 shows a sample of a ladder logic program that implements XOR logic. In addition, it shows the timing diagram of the inputs and output, it shows the output goes high when there is a difference between the two inputs and becomes low when they are equal i.e. both are low or both are high.

**Fig 11: Sample of the ladder logic for XOR logic and the timing diagram**

**XOR Logic Gate in PLC Simulator
**

The XOR is used to compare two signals if they are equal or different. Table 3 lists the truth table of the XOR. It shows that the output comes to true when inputs are different and becomes false when they are equal.

**Table 3: The truth table of XOR logic**

Figure 11 shows the construction of XOR ladder logic. It shows that it is composed of, two parallel branches and each branch is forming AND logic of the two inputs in the opposite logical state.

**Fig. 11: The ladder logic of XOR**

Figure 12, shows the simulation results of XOR when input A and B are false, the output is false.

**Fig 12: Simulation of XOR ladder when both inputs are false**

Figure 13, shows the simulation results of XOR when input A is true and input B is false, the output is true.

**Fig 13: The Simulation of XOR ladder when input A is false and input B is high**

Figure 14, shows the simulation results of XOR when input B is true and input A is false, the output is true.

**Fig 14: The Simulation of the XOR ladder when input B is true and input A is false.**

Figure 15, shows the simulation results of XOR when input A is true and input B is true, the output is false.

**Fig 15: The Simulation of XOR ladder when both inputs are true**

In a conclusion, the XOR logic in simulation shows that the output is low whenever both inputs are equal and goes high when the inputs are different in the logical state. A very good practical example for utilizing the XOR logic is that imagine friends we have a motor that is energized by two different destinations, and it should be requested by only one at a time. So we can get the run signal from the XOR of the two input switches. So, the only case to run the motor is by requesting from one source.

**The “XNOR” Logic Gate**

This logic gate is the invert of the XOR gate. So it is equivalent to applying an inverter to the “XOR” logic gate. Table 7 lists the combination of its two inputs and its output. It shows clearly that, the output becomes true when inputs are equal i.e. both inputs are true or both are false.

**Table 7: the truth table of the “XNOR” logic gate**

Input A |
Input B |
Output |

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

Fig. 20 shows the symbol of the “XNOR” logic gate, it shows clearly how it is the invert of the XOR logic gate. This logic gate is very useful to validate if two signals are equal or not.

**Fig. 20: The symbol of the “XNOR” logic gate**

On the other hand, Fig. 21 shows a sample ladder logic of an “XNOR” logic gate implementation. It shows that there are only two ways to have the output in the TRUE state which are by setting both inputs TRUE or set both FALSE.

**Fig. 21: A sample ladder logic for “XNOR” logic**

Fig. 22 depicts the timing diagram of the inputs and output of the “XNOR” logic gate and clearly shows the output goes high when both inputs have the same state.

**F****ig. 22: the timing diagram of the “XNOR” logic gate**

**XNOR Logic in PLC Simulator **

The XNOR is the invert of the XOR and it is used to compare two input signals. Table 4 lists the cases of the truth table of XNOR logic. It shows the output goes high when both inputs are equal i.e. both are high or both of them are low.

**Table 4: the truth table of XNOR logic**

Figure 16 shows the construction of XNOR ladder logic. It shows that it is composed of, two parallel branches and each branch is forming AND logic of the two inputs in the same logical state.

**Fig 16: XNOR ladder logic**

Figure 17, shows the simulation results of XNOR when input A and B are equal i.e. both are false, the output is high.

**Fig. 17: The Simulation of XNOR ladder when both inputs are false**

Figure 18, shows the simulation results of XNOR when input A is true and input B is false i.e. they are different. So the output goes false.

**Fig. 18: The Simulation of XNOR ladder when input A is false and input B is high**

Figure 19, shows the simulation results of XNOR when input A is false and input B is true i.e. they are different. So the output goes false.

**Fig. 19: The Simulation of the XNOR ladder when input B is true and input A is false.**

In the last case when both inputs A and B are high as shown in Fig. 20, the output becomes true.

**Fig. 20: The Simulation of XNOR ladder when both inputs are true**

We can conclude that the XNOR is marked by the true status of its output whenever both inputs are equal and vice versa. One of the most common scenarios for the best practice of XNOR is the protection of the operator's hands in the cutting machine. In that machine, the command for running the knife driving motor is an XNOR of two switches on the left and right hand of the operator. In that way, it is guaranteed that to run the motor, the operator should use both hands at the same time.

**What’s next **

I am very pleased to see you up to this point of our tutorial, Now you are familiar with all logic gates and practiced their logic on the simulator. In addition, you can feel the importance of mastering all these logic gates to ease your programming and enrich your programming skills. In the next tutorial, we are going to go deeply through the edge signal including rising and falling edge. We are going to introduce the benefits of these edge signals and how they can be utilized in ladder logic programming to solve a lot of problems. So be ready for more learning and practice with simulation in ladder logic series.