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Impedance Calculation

Basic Terms

Complex Impedance

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  • Polar Form: \(\displaystyle\vec{Z} = Z \angle\theta\)
  • Cartesian Form: \(Z = R + jX\)

Where:
- \(\vec{Z}\) → Impedance vector in polar form.
- \(Z\) → Magnitude of the impedance.
- \(\theta\) → Phase angle.
- \(R\) → Resistance (real part).
- \(X\) → Reactance (imaginary part).

Conversion Between Forms

Polar → Cartesian Form

  • Resistance: \(R = |Z| \cdot \cos \theta\)
  • Reactance: \(X = |Z| \cdot \sin \theta\)

Cartesian → Polar Form

  • Magnitude: \(Z = \sqrt{R^2 + X^2}\)
  • Phase Angle: \(\angle \theta = \tan^{-1}\left(\frac{X}{R}\right)\)

Typical Equations for LCR Meters

LCR meters measure the impedance by analyzing the current (\(I\)) flowing through the DUT and the voltage (\(V\)) across its terminals. These meters calculate not only the ratio of RMS values of \(I\) and \(V\) but also the phase difference between their waveforms.

Impedance Calculation

\[ \vec{Z} = \frac{|V| \angle \theta_V}{|I| \angle \theta_I} = \frac{|V|}{|I|} \angle (\theta_V - \theta_I) = |Z| \angle\theta \]

Reactance

  • Capacitive Reactance: \(X_C = \frac{1}{2\pi fC}\)
  • Inductive Reactance: \(X_L = 2\pi fL\)

References:
1. LCR Meter Basic Measurement Principles
2. Typical Equations for LCR Meters
3. Complex Impedance (Wikipedia)

Equivalent Circuit Model

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Calibration

Calibration involves performing open and short corrections to ensure accurate measurements.

Details will be added soon...