Pole Zero Plot Laplace

The pole-zero plot is a graphical representation of the transfer function in the complex s-plane. By marking pole locations with crosses and zero locations with circles, an engineer can immediately assess stability, frequency response shape, and transient behavior without computing any integrals. This diagram is essential for GATE and is the foundation of root locus and frequency response methods.

Core Concept Explanation

The s-plane is the complex plane where the horizontal axis represents the real part of s and the vertical axis represents the imaginary part. Every pole of H(s) is marked on this plane with a cross symbol, and every zero is marked with a small circle. The plot instantly communicates whether a system is stable, how quickly it settles, and whether it oscillates.

A real pole at s = -a lies on the negative real axis and contributes a decaying exponential e^(-at) to the response. The farther the pole is from the imaginary axis, the faster the decay. A pair of complex conjugate poles at s = -a plus or minus jb lies symmetrically about the real axis and contributes a damped sinusoidal term e^(-at) sin(bt) or cos(bt). The imaginary part b controls the oscillation frequency and the real part a controls the damping.

The distance of a complex pole pair from the origin equals the natural frequency wn, and the angle the pole makes with the negative real axis determines the damping ratio zeta. Specifically, wn = sqrt(a^2 + b^2) and zeta = a/wn. These two parameters completely describe the standard second-order response.

Mathematical Expression

For a second-order system, the poles are located at s = -zeta*wn plus or minus j*wn*sqrt(1 - zeta^2). When zeta is between 0 and 1, the poles are complex conjugates in the left half plane and the system is underdamped. When zeta equals 1, both poles coincide on the real axis (critically damped). When zeta is greater than 1, both poles are distinct and real (overdamped). All three cases are stable since the poles remain in the left half plane.

The frequency response H(jw) can be visualized geometrically from the pole-zero plot. At any frequency w on the imaginary axis, H(jw) equals K times the product of distances from all zeros to that point, divided by the product of distances from all poles to that point, with appropriate phase angles. This geometric interpretation is the basis of graphical filter design.

Practical Understanding

In filter design, the positions of poles and zeros directly determine the filter's frequency response. A Butterworth low-pass filter has all its poles equally spaced on a semicircle in the left half plane with no zeros. A notch filter places a pair of zeros on the imaginary axis at the frequency to be attenuated, making the response exactly zero at that frequency. Elliptic filters use both poles and zeros for the sharpest possible cutoff.

In control systems, the closed-loop pole positions determine the step response. A dominant pole pair close to the imaginary axis with small damping ratio gives a highly oscillatory step response with large overshoot. Moving poles deeper into the left half plane through feedback increases damping and reduces settling time. Root locus is the systematic method to find how poles move as gain changes.

Given:
H(s) = 10(s+2) / [(s+1)(s^2 + 4s + 8)]
Find poles, zeros, and assess stability.

Why this formula applies:
Factor each polynomial and equate to zero.

Formula:
Zeros: numerator = 0 → s + 2 = 0 → s = -2
Poles: denominator = 0
  s + 1 = 0 → s = -1
  s^2 + 4s + 8 = 0 → s = (-4 ± sqrt(16-32))/2

Substitution:
s = (-4 ± sqrt(-16))/2 = -2 ± j2

Calculation:
Poles: s = -1, s = -2+j2, s = -2-j2
Zero: s = -2
All poles have negative real parts (-1 and -2)

Final Answer with units:
System is STABLE. Natural frequency of complex pair: wn = sqrt(4+4) = 2sqrt(2) rad/s, zeta = 2/(2sqrt(2)) = 0.707

Exam Tip: For GATE, memorize that zeta = 0.707 corresponds to poles at 45 degrees from the negative real axis. This is the maximally flat (Butterworth) condition. Also remember: minimum phase system has all zeros in the LHP; non-minimum phase has at least one zero in the RHP.

Quick Revision

• Poles marked as x, zeros as o in the s-plane. Pole-zero plot fully characterizes H(s) up to a gain constant.
• Stability: all poles strictly in LHP for BIBO stable system.
• Complex pole pair: wn = distance from origin, zeta = cos(angle from negative real axis).
• Real pole at s = -a → time constant tau = 1/a; larger |a| means faster decay.
• Non-minimum phase: zeros in RHP cause initial undershoot in step response.
• For second-order: underdamped (0 < zeta < 1), critically damped (zeta=1), overdamped (zeta>1).
• Dominant poles: poles closest to jw axis dominate transient response; distant poles decay fast and can be neglected.