Logic Gate and Ladder Logic Diagram

We can construct simply logic functions ( AND, OR, NOT, NOR, XOR etc ) for our hypothetical lamp circuit, using multiple contacts, and document these circuits quite easily and understandably with additional rungs to our original "ladder." If we use standard binary notation for the status of the switches and lamp (0 for unactuated or de-energized; 1 for actuated or energized), a truth table can be made to show how the logic works:

OR Gate

Boolean Equation

Now, the lamp will come on if either contact A or contact B is actuated, because all it takes for the lamp to be energized is to have at least one path for current from wire L1 to wire 1. What we have is a simple OR logic function, implemented with nothing more than contacts and a lamp.

AND Gate
We can mimic the AND logic function by wiring the two contacts in series instead of parallel:

Boolean Equation

Now, the lamp energizes only if contact A and contact B are simultaneously actuated. A path exists for current from wire L1 to the lamp (wire 2) if and only if both switch contacts are closed.

NOT Gate
The logical inversion, or NOT, function can be performed on a contact input simply by using a normally-closed contact instead of a normally-open contact:
Boolean Equation 
Now, the lamp energizes if the contact is not actuated, and de-energizes when the contact is actuated.

If we take our OR function and invert each "input" through the use of normally-closed contacts, we will end up with a NAND function. In a special branch of mathematics known as Boolean algebra, this effect of gate function identity changing with the inversion of input signals is described by DeMorgan's Theorem, a subject to be explored in more detail in a later chapter.

Boolean Equation 

The lamp will be energized if either contact is unactuated. It will go out only if both contacts are actuated simultaneously.

NOR Gate
Likewise, if we take our AND function and invert each "input" through the use of normally-closed contacts, we will end up with a NOR function:

Boolean Equation 

A pattern quickly reveals itself when ladder circuits are compared with their logic gate counterparts:

  • · Parallel contacts are equivalent to an OR gate. 
  • · Series contacts are equivalent to an AND gate. 
  • · Normally-closed contacts are equivalent to a NOT gate (inverter). 
XOR Gate
We can build combinational logic functions by grouping contacts in series-parallel arrangements, as well. In the following example, we have an Exclusive-OR function built from a combination of AND, OR, and inverter (NOT) gates:

Boolean Equation 

The top rung (NC contact A in series with NO contact B) is the equivalent of the top NOT/AND gate combination. The bottom rung (NO contact A in series with NC contact B) is the equivalent of the bottom NOT/AND gate combination. The parallel connection between the two rungs at wire number 2 forms the equivalent of the OR gate, in allowing either rung 1 or rung 2 to energize the lamp.

To make the Exclusive-OR function, we had to use two contacts per input: one for direct input and the other for "inverted" input. The two "A" contacts are physically actuated by the same mechanism, as are the two "B" contacts. The common association between contacts is denoted by the label of the contact. There is no limit to how many contacts per switch can be represented in a ladderdiagram, as each new contact on any switch or relay (either normally-open or normally-closed) used in the diagram is simply marked with the same label.

Sometimes, multiple contacts on a single switch (or relay) are designated by a compound labels, such as "A-1" and "A-2" instead of two "A" labels. This may be especially useful if you want to specifically designate which set of contacts on each switch or relay is being used for which part of a circuit. For simplicity's sake, I'll refrain from such elaborate labeling in this lesson. If you see a common label for multiple contacts, you know those contacts are all actuated by the same mechanism.

If we wish to invert the output of any switch-generated logic function, we must use a relay with a normally-closed contact. For instance, if we want to energize a load based on the inverse, or NOT, of a normally-open contact, we could do this:

We will call the relay, "control relay 1," or CR1. When the coil of CR1 (symbolized with the pair of parentheses on the first rung) is energized, the contact on the second rung opens, thus de-energizing the lamp. From switch A to the coil of CR1, the logic function is noninverted. The normally-closed contact actuated by relay coil CR1 provides a logical inverter function to drive the lamp opposite that of the switch's actuation status.

Applying this inversion strategy to one of our inverted-input functions created earlier, such as the OR-to-NAND, we can invert the output with a relay to create a noninverted function:

From the switches to the coil of CR1, the logical function is that of a NAND gate. CR1's normally-closed contact provides one final inversion to turn the NAND function into an AND function.


  • · Parallel contacts are logically equivalent to an OR gate. 
  • · Series contacts are logically equivalent to an AND gate. 
  • · Normally closed (N.C.) contacts are logically equivalent to a NOT gate. 
  • · A relay must be used to invert the output of a logic gate function, while simple normally-closed switch contacts are sufficient to represent inverted gate inputs. 

Permissive and interlock circuits

A practical application of switch and relay logic is in control systems where several process conditions have to be met before a piece of equipment is allowed to start. A good example of this is burner control for large combustion furnaces. In order for the burners in a large furnace to be started safely, the control system requests "permission" from several process switches, including high and low fuel pressure, air fan flow check, exhaust stack damper position, access door position, etc. Each process condition is called a permissive, and each permissive switch contact is wired in series, so that if any one of them detects an unsafe condition, the circuit will be opened:

If all permissive conditions are met, CR1 will energize and the green lamp will be lit. In real life, more than just a green lamp would be energized: usually a control relay or fuel valve solenoid would be placed in that rung of the circuit to be energized when all the permissive contacts were "good:" that is, all closed. If any one of the permissive conditions are not met, the series string of switch contacts will be broken, CR2 will de-energize, and the red lamp will light.

Note that the high fuel pressure contact is normally-closed. This is because we want the switch contact to open if the fuel pressure gets too high. Since the "normal" condition of any pressure switch is when zero (low) pressure is being applied to it, and we want this switch to open with excessive (high) pressure, we must choose a switch that is closed in its normal state.

Another practical application of relay logic is in control systems where we want to ensure two incompatible events cannot occur at the same time. An example of this is in reversible motor control, where two motor contactors are wired to switch polarity (or phase sequence) to an electric motor, and we don't want the forward and reverse contactors energized simultaneously:

When contactor M1 is energized, the 3 phases (A, B, and C) are connected directly to terminals 1, 2, and 3 of the motor, respectively. However, when contactor M2 is energized, phases A and B are reversed, A going to motor terminal 2 and B going to motor terminal 1. This reversal of phase wires results in the motor spinning the opposite direction. Let's examine the control circuit for these two contactors:

Take note of the normally-closed "OL" contact, which is the thermal overload contact activated by the "heater" elements wired in series with each phase of the AC motor. If the heaters get too hot, the contact will change from its normal (closed) state to being open, which will prevent either contactor from energizing.

This control system will work fine, so long as no one pushes both buttons at the same time. If someone were to do that, phases A and B would be short-circuited together by virtue of the fact that contactor M1 sends phases A and B straight to the motor and contactor M2 reverses them; phase A would be shorted to phase B and vice versa. Obviously, this is a bad control system design!

To prevent this occurrence from happening, we can design the circuit so that the energization of one contactor prevents the energization of the other. This is called interlocking, and it is accomplished through the use of auxiliary contacts on each contactor, as such:

Now, when M1 is energized, the normally-closed auxiliary contact on the second rung will be open, thus preventing M2 from being energized, even if the "Reverse" pushbutton is actuated. Likewise, M1's energization is prevented when M2 is energized. Note, as well, how additional wire numbers (4 and 5) were added to reflect the wiring changes.

It should be noted that this is not the only way to interlock contactors to prevent a short-circuit condition. Some contactors come equipped with the option of a mechanical interlock: a lever joining the armatures of two contactors together so that they are physically prevented from simultaneous closure. For additional safety, electrical interlocks may still be used, and due to the simplicity of the circuit there is no good reason not to employ them in addition to mechanical interlocks.


  • Switch contacts installed in a rung of ladder logic designed to interrupt a circuit if certain physical conditions are not met are called permissive contacts, because the system requires permission from these inputs to activate. 
  • Switch contacts designed to prevent a control system from taking two incompatible actions at once (such as powering an electric motor forward and backward simultaneously) are called interlocks.

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