What is the equation for the complex frequency variable?

The complex frequency variable is expressed as ( s = \sigma + j\omega ). In this equation, ( s ) denotes the complex frequency, where ( \sigma ) represents the real part, often associated with exponential growth or decay, and ( j\omega ) represents the imaginary part, which corresponds to the oscillatory behavior of a system.

The real part ( \sigma ) can indicate whether the system is stable (when ( \sigma < 0 )) or unstable (when ( \sigma > 0 )). The imaginary part ( \omega ) gives the frequency of oscillation in radians per second. This relationship is particularly useful in the analysis of linear time-invariant systems in control theory and circuit analysis, allowing engineers to study system behavior in the frequency domain using tools like Laplace transforms.

This formulation highlights how complex frequency is vital for understanding behavior in the time and frequency domains. It provides insights into system stability and response characteristics, which are crucial for designing and analyzing electrical and control systems. The alternative equations do not represent the complete definition of the complex frequency variable and fail to include both the real and imaginary components appropriately.