Immittance Function Analyzer

Understanding Immittance Functions

Immittance functions are mathematical representations used in electrical engineering to describe the relationship between voltage and current in linear time-invariant systems. The term "immittance" combines the concepts of impedance (Z) and admittance (Y), which are fundamental to circuit analysis and system design.

Key Concepts

• Impedance (Z): The opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.
• Admittance (Y): The reciprocal of impedance, representing how easily a circuit allows current to flow.
• Poles and Zeros: Critical frequencies where the function's value becomes infinite (poles) or zero (zeros).
• Transfer Function: A mathematical representation of the relation between the input and output of a system.

Mathematical Representation

Immittance functions are typically represented as rational polynomial functions in the complex frequency domain (s-domain):

F(s) = N(s)/D(s) = (aₙsⁿ + aₙ₋₁sⁿ⁻¹ + ... + a₁s + a₀) / (bₘsᵐ + bₘ₋₁sᵐ⁻¹ + ... + b₁s + b₀)

Pole-Zero Analysis

The roots of the numerator polynomial (N(s) = 0) are called zeros, while the roots of the denominator polynomial (D(s) = 0) are called poles. The locations of these poles and zeros in the complex plane determine the system's:

• Frequency response
• Stability characteristics
• Transient behavior
• Resonant frequencies

Applications

Immittance function analysis is crucial in:

• Filter design (low-pass, high-pass, band-pass, etc.)
• Control system analysis
• Network synthesis
• Impedance matching
• Antenna design
• Vibration analysis in mechanical systems

Using This Tool

This analyzer helps you visualize the pole-zero plot of immittance functions and understand their characteristics. Enter the coefficients of your numerator (zeros) and denominator (poles) polynomials, select whether you're analyzing an impedance or admittance function, and click "Analyze Function" to see the results.