Introduction
When trying to understand the physical meaning of poles and zeros and how they affect a system, common textbook statements like "H(s)=A0/(s+a), pole at s=-a, s=jw so at w=a gain drops by 20 dB/decade" can be confusing. The variable s equals σ+jw, so how does it become jw, and how can jw become w or change sign? This article summarizes related concepts to clarify these points.
Fourier and Laplace Transforms
Fourier transform: converts a time-domain signal into a sum of sinusoidal components at different frequencies.
Laplace transform: because the Fourier transform requires the signal f(t) to converge, for nonconvergent functions a damping factor e^{-σt} is applied so that f(t)e^{-σt} tends to zero as t→∞, and then the Fourier transform is taken.
Meaning of a Pole
For H(s)=H0/(s-a), the pole at s=a corresponds to the time-domain function h(t)=H0·e^{at}.
From the above, H(s) is the Fourier transform of h(t) multiplied by the damping factor. If g(t)=H0·e^{at}·e^{-σt}=H0·e^{-(σ-a)t}, then for g(t) to converge σ must be greater than a.
Therefore the pole s=a means s=σ+jw equals a, not jw=a. In other words, σ=a.
The pole indicates the region of convergence (ROC) for which the Laplace transform of the corresponding time-domain function converges: σ must lie to the right of the pole. (See the figure below; a simple diagram made in Excel.)
For a complex pole s=a+bj, σ=a represents the decay rate and b represents the oscillation frequency. Complex poles will be analyzed in more detail later.
Poles and System Stability
The shaded area in the figure is the region of convergence. If a1 is a pole in the left half-plane, the ROC is to the right of a1. If a2 is a pole in the right half-plane, the ROC is still to the right of a2.
From pole location one can judge system stability:
- Left-half-plane poles: the ROC includes the origin, so at σ=0 the time-domain function f(t)·e^{-σt} converges, which means f(t) itself converges.
- Right-half-plane poles: the ROC is to the right of a positive σ, so f(t)·e^{-σt} only converges for σ greater than zero, which means f(t) does not converge.
s-Domain vs Frequency Response
Why does s=σ+jw sometimes reduce to s=jw? This reflects the relationship between the s-domain representation and the circuit frequency response.
Using the Laplace transform differentiation properties, the s-domain impedance of a capacitor is 1/(sC).
In AC analysis, the capacitive reactance Zc equals 1/(jωC).
Thus setting s=jω gives the circuit frequency response H(ω) from the transfer function H(s). Note that this H(ω) is the frequency response evaluated on the jω axis, not the Fourier transform H(ω).
Once the frequency response is known, amplitude-frequency and phase-frequency characteristics are straightforward to compute.
