Theoretical cycles are approximations to the real cycles. They serve to study the performance of an engine. In a real cycle there are so many variables that affect the efficiency of the engine that it would be impossible to perform exact calculations. However, theoretical cycles allow us to make very reliable approximations.

For the cycles of the endothermic motors, the approximations most used in order of approximation to the real conditions are three:

- Actual cycle or indicated cycle, measured empirically.
- Air cycle
- Air-fuel cycle

In these theoretical cycles, real cycles are compared in practice. Actual cycles are obtained experimentally through the indicators. The use of these indicators is the reason why real cycles are also called indicated cycles.

The theoretical cycles are not identical to real cycles, but they are a very useful tool for thermodynamically studying internal combustion engines. Especially to understand how much the operating conditions influence its use and to compare different types of motors with each other.

## Assumptions of the theoretical cycles

Theoretical cycles can also be called ideal cycles since it is calculated assuming that the operating conditions are ideal as detailed below.

In theoretical cycles (or ideal cycles) it is assumed that the operating fluid is constituted by air and that it behaves like a perfect gas. Therefore, the values of specific heats are considered constant and equal to that of air at conditions such as temperature 15º and 1 atmosphere of pressure:

Cp = 0.241 Ca / kg ºC;

Cv = 0.172 Ca / kg ºC;

Where it turns out:

We also assume that in the theoretical cycle (or ideal cycle) the phases of introduction and subtraction of heat have a well-determined duration, depending on the type of cycle (Otto cycle, diesel cycle, sabathé cycle), and that in the other phases there are no heat losses.

With these hypotheses the maximum values of temperature and pressure, as well as, the work and the thermal efficiency calculated for the theoretical cycle (or ideal cycle), are higher than those corresponding to the other types of cycles.

The theoretical cycle (or ideal cycle) represents the maximum limit that the engine can theoretically reach and allows an easy mathematical study based on the laws of perfect gases.

## Air cycle

In the air cycle, the operating fluid is also air, but specific heats are assumed to vary throughout the temperature range in which it is operated.

The conditions of introduction and subtraction of heat are equal to those of the ideal cycle and there is also no heat loss. As the calculation of the thermodynamic variables of the average specific heats is complicated, tables that directly give the values of heat and work are used. These energy values are in terms of internal energy and enthalpy for the various points of isentropic air transformations. Taking into account the variations in specific heats, values are obtained, for maximum temperatures and pressures, lower than those calculated for the ideal cycle; therefore, the work and the thermal efficiency also lower. However, they are even larger than those corresponding to a real cycle.

## Air-fuel cycle

Among all the cycles that are calculated, the air-fuel cycle is the closest to the real cycle. In the spark ignition engine (Otto cycle), the fluid is composed, during the aspiration phase, by the mixture and residual gases from the previous combustion. In the compression ignition engine it is formed by air and waste gases. After combustion, the fluid is constituted by products thereof, that is, a mixture of CO _{2} , CO, H _{2} O, N _{2} .

These gases have an average specific heat still higher than that of air; but in addition, there is a subsequent increase in specific heats. This increase is due to the chemical dissociation or decomposition of the lighter molecules subjected to the action of high temperatures. The increase in specific heats, as well as the dissociation that, as an endothermic reaction, absorbs a part of the heat of combustion, produces a subsequent decrease in temperature and maximum pressure compared to those calculated for the air cycle.

For the calculation of the air-fuel cycle, tables containing data obtained experimentally are used. Even for this cycle it is admitted not only that heat is introduced and subtracted instantaneously, as in the ideal cycle, but that no heat losses occur.

## Actual cycle of endothermic motors

The real cycle is obtained experimentally. The actual cycle is carried out by means of various indicating devices, capable of recording the pressure diagram according to the volumes, in a working engine cylinder. The indicated diagram reflects the actual conditions of the cycle and, therefore, also takes into account - in addition to the variations already stated for the air cycle and for the air-fuel cycle in the comparison of ideal cycles - heat losses, duration of combustion, the losses caused by friction of the fluid, the duration of the opening time of the valves, the ignition time, as well as injection and the losses of the exhaust.