S-plane

Forming the transfer function incorporating just those poles in the negative half of the s-plane.

Applying the Laplace transform gives the following relation in the s-plane.

This transfer function has no zeros and two poles located in the complex s-plane:

This series converges uniformly on compact subsets of the s-plane to an entire function.

Such diagrams are graphically referred to as s-plane diagrams.

The root locus can also be computed in the z-plane, the discrete counterpart of the s-plane.

However, at least half of these filters are not physically realizable, because the pole is in the right half of the s-plane.

A continuous-time causal filter is stable if the poles of its transfer function fall in the left half of the complex s-plane.

The C' planes are rarely observed except in ultradeformed mylonites, and form nearly perpendicular to the S-plane.

These two constraints imply that both the zeros and the poles of a minimum phase system must be strictly inside the left-half s-plane.

The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane.

So, causality and stability for imply that its poles - the roots of A (s) - must be strictly inside the left-half s-plane.

Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane:

The 'z-plane' is a discrete-time version of the s-plane, where z-transforms are used instead of the Laplace transformation.

Therefore, all poles of the system must be in the strict left half of the s-plane for BIBO stability.

For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero.

Evidently the locations of the poles and zeros in the s-plane determine which frequency ranges are most significant in the spectrum of the signal.

Notice that, since functions do not grow without limit in the real physical world, poles are restricted to the left-hand half of the s-plane in network analysis.

The magnitude condition is a constraint that is satisfied by the locus of points in the s-plane on which closed-loop poles of a system reside.

Closed-loop poles are the positions of the poles (or eigenvalues) of a closed-loop transfer function in the s-plane.

A diagonal line of constant damping in the s-plane maps around a spiral from (1,0) in the z plane as it curves in toward the origin.

The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the z-plane to the s-plane.

Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform).

To appreciate how an s-plane diagram can reveal the Fourier spectrum of a signal, first consider for simplicity the Laplace transform corresponding to an exponentially decaying signal.

When the transfer function method is used, attention is focused on the locations in the s-plane where the transfer function becomes infinite (the poles) or zero (the zeroes).