Measurement Model

Definitions, windows, sign conventions, and reproducibility notes

This page describes how the RLC Analyzer interprets waveforms into reported metrics such as \(V_{rms}\), \(I_{rms}\), \(P\), \(Q\), \(S\), \(PF\), detuning, and periodicity. The intent is lab-style clarity: every number should have an explicit definition.

1. Sign Conventions

The simulator follows the passive sign convention at the element level. Instantaneous power is defined as:

\[ p(t) = v(t)\,i(t) \]

A positive \(p(t)\) means power is delivered into the element or load. Average real power is:

\[ P = \langle p(t) \rangle \]

2. Measurement Window

All time-domain statistics are computed over an explicit measurement window. For periodic excitations, the preferred window is an integer number of periods \(N T\).

\[ \langle x \rangle = \frac{1}{T_w}\int_{t_0}^{t_0+T_w} x(t)\,dt \]

with window length \(T_w\). For ideal steady-periodic pulse mode, \(T_w\) is typically one period unless the UI selects a longer window.

Lab note: RMS and power computed over non-integer cycle windows can show bias, especially with pulse trains and ringing. For reproducible comparisons, use a fixed integer-period window.

3. RMS Definitions

Root-mean-square values are defined as:

\[ V_{rms} = \sqrt{\langle v^2(t)\rangle}, \qquad I_{rms} = \sqrt{\langle i^2(t)\rangle} \]

These definitions are time-domain and remain valid for non-sinusoidal waveforms.

4. Real Power

Average (real) power is computed directly from the time domain:

\[ P = \langle v(t)\,i(t)\rangle \]

This remains valid for pulse trains, transient-rich waveforms, and mixed harmonic content.

5. Apparent Power and Power Factor

\[ S = V_{rms} I_{rms} \] \[ PF = \frac{P}{S} \]
For non-sinusoidal waveforms, \(PF\) is not equivalent to \(\cos\phi\). It includes distortion effects because it is defined from true RMS and true average power.

6. Reactive Power

Reactive power may be presented in two common forms:

For sinusoidal steady state:

\[ P = V_{rms} I_{rms}\cos\phi, \qquad Q = V_{rms} I_{rms}\sin\phi \]
Lab note: In pulse mode, a single phase angle \(\phi\) does not describe the waveform. In that case, \(Q\) should be interpreted carefully and preferably via element energy exchange metrics.

7. Harmonics and Distortion

Non-sinusoidal waveforms can be described by harmonic content. A typical distortion metric is total harmonic distortion (THD):

\[ THD = \frac{\sqrt{\sum_{k=2}^{\infty} X_k^2}}{X_1} \]

where \(X_1\) is the fundamental RMS magnitude and \(X_k\) are harmonic RMS magnitudes.

8. Element Energy Accounting

Energy stored in the inductor and capacitor:

\[ W_L(t) = \frac{1}{2} L i^2(t), \qquad W_C(t) = \frac{1}{2} C v^2(t) \]

Energy consistency check over a window:

\[ \Delta W = W(t_1)-W(t_0) \approx \int_{t_0}^{t_1} v(t)\,i(t)\,dt \]
Lab note: For pulse edges, derivative-based quantities can spike. Energy consistency should be checked over well-defined windows (e.g., excluding idealized discontinuities).

9. Periodicity Closure Metric

In steady-periodic pulse mode, the simulator validates state closure using topology-aware state vectors. The periodicity condition is:

\[ x(T^-) = x(0) \]

State definition depends on topology:

Important: quantities such as \(v_L = L\,di/dt\) are not independent states. They may be discontinuous at switching boundaries even when the true state is perfectly periodic.

10. Reproducibility Checklist

If a metric comparison to an oscilloscope or power analyzer disagrees, first confirm: window length, window alignment, RMS definition, and whether the instrument reports fundamental-only or true-RMS quantities.