BlockWerk Documentation
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ZeroPole

Continuous

The ZeroPole block represents a transfer function using its poles and zeros:

H(s) = K · (s - z₁)(s - z₂)...(s - zₘ) / (s - p₁)(s - p₂)...(s - pₙ)

This form is intuitive for control system design, stability analysis, and frequency response visualization.

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Mathematical Model

H(s)=K(s−z1)(s−z2)⋯(s−zm)(s−p1)(s−p2)⋯(s−pn)H(s) = K \frac{(s - z_1)(s - z_2)\cdots(s - z_m)}{(s - p_1)(s - p_2)\cdots(s - p_n)}

where:

  • z₁, z₂, ..., zₘ: zeros (roots of numerator)
  • p₁, p₂, ..., pₙ: poles (roots of denominator)
  • K: static gain

Inputs & Outputs

Direction ID Label Type Status
→ In in Input number Required
→ In reset Reset number Required
← Out out Output number Output

Parameters

Parameters Label Type Default Description
zeros Zeros (space-separated real parts) polynomial -1
poles Poles (space-separated real parts) polynomial -2
gain Gain (K) number 1
initialCondition Initial Condition number 0

Usage Examples

Example 1: Simple Low-Pass Filter

Zeros: (empty)
Poles: "-1"
Gain: 1

Represents the transfer function H(s) = 1/(s+1), a first-order low-pass filter with cutoff frequency 1 rad/s.

Example 2: Lead Compensator (Phase Boost)

Zeros: "-1"
Poles: "-5"
Gain: 5

Represents H(s) = 5(s+1)/(s+5), useful for phase margin improvement in classical control design.

Example 3: Complex Conjugate Poles (Underdamped System)

Zeros: (none)
Poles: "-1+1.732j, -1-1.732j"
Gain: 1

This represents an underdamped second-order system with damping ratio ζ ≈ 0.5 and natural frequency ωₙ ≈ 2 rad/s.

Remarks & Best Practices

  • Conjugate pairs: Complex poles and zeros must be specified in conjugate pairs for real-coefficient systems
  • Stability: All poles must have negative real parts for a stable system; poles on the imaginary axis are marginally stable
  • Gain K: Adjusts overall magnitude without affecting phase or pole/zero locations
  • Empty zeros: Omitted zeros are treated as no zeros (all-pole system); an empty numerator defaults to gain K only
  • High-order systems: Above ~20 poles, numerical issues may arise; consider model reduction or state-space form

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