# What does the real axis of a transfer function mean?

Let's say I have a system that has a transfer function $$\ H(s) \$$. Now $$\ s = \sigma + j \omega \$$, so $$\ H(s) = H(\sigma + j \omega) \$$. If I am not mistaken, the imaginary axis $$\ j \omega \$$ describes the oscillation frequency of an input, and $$\ H( j \omega) \$$ describes how the system responds to that frequency. What I don't understand is what the real axis $$\ \sigma \$$ describes, nor why is it in units of Hertz.

Welcome to EE.SE.

As you noted, the imaginary axis shows the frequency of oscillation of a system.

The real axis shows how the amplitude of that oscillation varies with time. For σ = 0 (i.e. all along the imaginary jw axis) the amplitude is constant.

For negative σ the amplitude will decay exponentially. As the system response moves further away to the left from the imaginary axis (becoming more negative) the faster it will decay.

For positive σ the amplitude will grow exponentially. As the system response moves further away to the right from the imaginary axis (becoming more positive) the faster it will grow.

A system that is operating in the right-half of the s plane, with positive σ, will have a response that keeps growing (until it hits the physical limits of the system). It is said to be unstable.

In control system theory considerable effort goes into determining if a system is stable, and making it stable if it is found to be unstable.

why is σ in units of Hertz?

Hertz have units of per second. σ is a time constant, telling us how much the amplitude of the system response grows or shrinks per second so units of Hz are still dimensionally appropriate.

Lets assume that we have the differential equation in the time domain for a frequency-dependent circuit. For solving this diff.equation (finding the "characteristic equation") we are using an ansatz Ke^st - without knowing the meaning of the factor "s".

However, within this process we come to the conclusion that "s" has the unit 1/time and very often consists of a real as well as imag. part (Example: RLC circuits).

This leads to the well-known expression "s=σ+jω".

When we introduce this complexe frequency term into the exponential function e^st and after applying Euler´s formula we find that the solution in the time domain has the form:

K(e^σt)*(sin(ωt)).

For a passive system, the factor σ will be always negative and the solution, therefore, is a damped sinusoidal signal.

This is the background for introducing the concept of "complexe frequencies". For some graphical analyses (pole location, root locus, Nyquist curve) we use the so-called s-plane with the damping coefficient "σ" on the real axis and "jω" on the vertical imag. axis.

This allows us to analyze magnitude and phase of a complexe function in one single diagram. Such a diagram contains the same information as the Bode diagram, which however, consists of two separate diagrams (magnitude and phase).