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The efficiency of the Brayton cycle is given by $1 - \frac{1}{r^{\gamma-1}}$, where $r$ is the pressure ratio across the compressor and turbine, and $\gamma$ is the heat capacity ratio.

The efficiency of a steam turbine is calculated by the ratio of the useful work output divided by the heat input, but specific formulas vary based on the cycle (e.g., Rankine cycle) used.

CHP efficiency cannot be defined by a simple formula, but it's the sum of the electrical and heat efficiency in utilizing the fuel energy, typically much higher than conventional power plants.

The efficiency of an electric motor is typically given by the ratio of the mechanical power output to the electrical power input, often exceeding 90% efficiency.

The COP of a refrigerator is given by $\frac{Q_C}{W}$, where $Q_C$ is the heat removed from the cold space and $W$ is the work input. For ideal refrigerators, COP can also be based on temperatures, $\frac{T_C}{T_H - T_C}$.

Diesel engine efficiency is given by $1 - \left[\frac{1}{r^{\gamma-1}}\left(\frac{\rho^\gamma - 1}{\gamma(\rho - 1)}\right)\right]$, where $r$ is the compression ratio, $\gamma$ is the heat capacity ratio, and $\rho$ is the cut-off ratio (volume before combustion/volume at the end of compression).

The efficiency of an Atkinson cycle engine is increased by a longer expansion stroke compared to its compression stroke, which allows for better thermal efficiency but does not have a simple formula.

The efficiency of magnetic refrigeration can be described by the magnetic Carnot cycle and is dependent on the magneto-caloric effect, but does not have a specific formula for efficiency.

Efficiency for a Stirling engine can be approximated by the same formula as a Carnot engine: $1 - \frac{T_C}{T_H}$, because it operates on a closed regenerative thermodynamic cycle.

Efficiency for a simple Rankine cycle is given by $1 - \frac{T_C}{T_H}$, however, real cycle efficiency is reduced by factors such as pump work and irreversibilities in the actual components.

Otto engine efficiency is given by $1 - \left(\frac{V_1}{V_2}\right)^{\gamma-1}$ where $V_1$ is the volume at the beginning of compression, $V_2$ is the volume at the end of compression, and $\gamma$ is the heat capacity ratio ($c_p/c_v$).