Mathematical Foundations

Analytic equations, state definitions, assumptions, and validation criteria

This page documents the analytic model implemented by the RLC Analyzer. The intent is reproducibility and auditability: for identical parameter sets the computed waveforms and metrics are deterministic.

Model Assumptions

• Linear, time-invariant (LTI) components only: R, L, C constants.
• Ideal voltage source unless explicitly modeled otherwise.
• No parasitic ESR/ESL unless explicitly included in R or L parameters.
• No magnetic saturation or hysteresis model (no B–H nonlinearity).
• No temperature dependence of component values.
• No semiconductor device dynamics (dead-time, diode recovery, switching losses) unless represented via equivalent parameters.
• Waveforms represent circuit-level quantities; radiation/EMI and spatial field coupling are out of scope.

Numerical Method Statement

The simulator uses closed-form analytic solutions for the supported LTI networks. In steady-periodic pulse mode, solutions are computed to satisfy periodic boundary conditions. There is no explicit time-step ODE integration loop used to reach steady state; therefore, there is no integration drift or solver tolerance dependence in the steady-periodic result.

Symbols and Conventions

• \(R\) denotes total series resistance in a branch (e.g., \(R = R_{coil} + R_{load}\)).
• \(L, C\) are inductance and capacitance.
• \(\omega = 2\pi f\) is angular frequency.
• Pulse repetition period \(T = 1/f\), duty ratio \(D \in [0,1]\), \(T_{on} = D T\), \(T_{off} = T - T_{on}\).
• Time-domain sign convention: passive sign; \(p(t)=v(t)i(t)\) positive indicates power delivered to the element.

State-Space Definitions

Topology	Excitation	True dynamic state vector	Notes
Series RLC	Pulse	\(x = \begin{bmatrix} i \\ v_C \end{bmatrix}\)	Two independent energy stores (L and C). Periodicity check uses both states.
Series RLC	Sine	Phasor steady state	Amplitude–phase form; no transient state tracking required for steady metrics.
RL	Pulse	\(x = i\)	Single energy store (L). Time constant \(\tau=L/R\).
\(C \parallel (R+L)\)	Pulse	\(x = i_{RL}\)	Capacitor voltage is source-clamped in this topology: \(v_C(t)=v_{src}(t)\). Only RL branch current is a true state.
\(C \parallel (R+L)\)	Sine	Phasor steady state	Admittance formulation; resonance defined by \(\operatorname{Im}(Y)=0\).

Topology A: RL

Time Domain

\[ L \frac{di}{dt} + R i = v(t) \] Time constant: \[ \tau = \frac{L}{R} \] Step response for constant \(v(t)=V\), initial current \(i(0)=i_0\): \[ i(t)=\frac{V}{R} + \left(i_0-\frac{V}{R}\right)e^{-t/\tau} \]

Sinusoidal Steady State

\[ Z_{RL}=R+j\omega L,\quad |Z|=\sqrt{R^2+(\omega L)^2},\quad \phi=\tan^{-1}\!\left(\frac{\omega L}{R}\right) \] With source \(v(t)=V_{pk}\sin(\omega t)\): \[ i(t)=I_{pk}\sin(\omega t-\phi),\quad I_{pk}=\frac{V_{pk}}{|Z|} \]

Topology B: Series RLC

Differential Form

With series current \(i(t)\) and capacitor voltage \(v_C(t)\): \[ v(t)=Ri(t)+L\frac{di}{dt}+v_C(t),\qquad i(t)=C\frac{dv_C}{dt} \] Equivalent second-order ODE: \[ L\frac{d^2 i}{dt^2}+R\frac{di}{dt}+\frac{1}{C}i=\frac{dv}{dt} \] Natural frequency and damping: \[ \omega_0=\frac{1}{\sqrt{LC}},\qquad \zeta=\frac{R}{2}\sqrt{\frac{C}{L}} \]

State-Space Form

Let \(x=\begin{bmatrix} i \\ v_C \end{bmatrix}\). Then: \[ \dot{x}= \begin{bmatrix} -\frac{R}{L} & -\frac{1}{L}\\[4pt] \frac{1}{C} & 0 \end{bmatrix}x + \begin{bmatrix} \frac{1}{L}\\[4pt] 0 \end{bmatrix}v(t) \]

Sinusoidal Steady State

\[ Z=R+j\left(\omega L-\frac{1}{\omega C}\right),\quad |Z|=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2} \] \[ \phi=\tan^{-1}\!\left(\frac{\omega L-\frac{1}{\omega C}}{R}\right),\qquad I_{pk}=\frac{V_{pk}}{|Z|} \] Resonance (ideal series): \[ \omega=\omega_0=\frac{1}{\sqrt{LC}} \]

Topology C: Parallel \(C \parallel (R+L)\)

Sinusoidal Steady State (Admittance)

RL branch impedance: \[ Z_{RL}=R+j\omega L \] RL branch admittance: \[ Y_{RL}=\frac{1}{R+j\omega L}=\frac{R-j\omega L}{R^2+(\omega L)^2} \] Capacitor admittance: \[ Y_C=j\omega C \] Total admittance: \[ Y=Y_{RL}+Y_C \] Parallel resonance condition (susceptance cancels): \[ \operatorname{Im}(Y)=0 \] Explicitly: \[ -\frac{\omega L}{R^2+(\omega L)^2}+\omega C=0 \] This can be rearranged to: \[ C=\frac{L}{R^2+(\omega L)^2} \]

Pulse Mode (Source-Clamped Capacitor)

In the implemented topology, capacitor voltage equals the source node voltage: \[ v_C(t)=v_{src}(t) \] The only true dynamic state is the RL-branch current \(i_{RL}\): \[ L\frac{di_{RL}}{dt}+R\,i_{RL}=v_{src}(t) \] Total source current is the sum: \[ i(t)=i_{RL}(t)+i_C(t),\qquad i_C(t)=C\frac{dv_{src}}{dt} \]

Pulse Train Definitions and Periodicity

\[ T=\frac{1}{f},\qquad T_{on}=DT,\qquad T_{off}=T-T_{on} \]

Periodic Closure Conditions (Steady-Periodic Pulse)

Series RLC: \[ x(T^-)=x(0)\quad\text{with}\quad x=\begin{bmatrix} i \\ v_C \end{bmatrix} \] Parallel \(C \parallel (R+L)\) pulse: \[ i_{RL}(T^-)=i_{RL}(0) \] RL pulse (single-state): \[ i(T^-)=i(0) \]

Power and Power-Quality Definitions

Given time-domain voltage and current \(v(t)\), \(i(t)\) over an integer number of cycles: \[ V_{rms}=\sqrt{\langle v^2\rangle},\qquad I_{rms}=\sqrt{\langle i^2\rangle} \] Real power (average): \[ P=\langle v(t)\,i(t)\rangle \] Apparent power: \[ S=V_{rms}I_{rms} \] Power factor: \[ PF=\frac{P}{S} \] For sinusoidal fundamental (phasor-based): \[ P=V_{rms}I_{rms}\cos\phi,\qquad Q=V_{rms}I_{rms}\sin\phi \] Reactive decomposition (conceptual): \[ Q_{net}=Q_L+Q_C,\qquad Q_L>0,\quad Q_C<0 \]

Energy and Flux Relations

Inductor (magnetic) energy: \[ W_L=\frac{1}{2}Li^2 \] Capacitor (electric) energy: \[ W_C=\frac{1}{2}Cv^2 \] Flux linkage: \[ \lambda=Li \] Energy consistency (element-level): \[ \Delta W = \int_{t_0}^{t_1} v(t)\,i(t)\,dt \] Windowed checks often used in analysis: \[ \Delta W_L = \frac{1}{2}L\left(i^2(t_1)-i^2(t_0)\right),\qquad \Delta W_C = \frac{1}{2}C\left(v^2(t_1)-v^2(t_0)\right) \]

Resonance and Detuning Metrics

Natural frequency: \[ \omega_0=\frac{1}{\sqrt{LC}} \] Detuning: \[ \delta=\frac{\omega-\omega_0}{\omega_0} \] Series RLC quality factor (ideal small damping): \[ Q_s=\frac{\omega_0 L}{R} \] Bandwidth approximation: \[ \Delta\omega\approx\frac{\omega_0}{Q_s} \]

Validation Criteria

These checks are designed to be topology-aware and to validate true state closure and consistency:

• State periodicity closure (pulse): \(x(T^-)=x(0)\) for the topology’s true state vector.
• Energy consistency: compare \(\Delta W\) from state energy (\(W_L, W_C\)) against \(\int v i\,dt\) over defined windows or periods.
• Resonance conditions: series: \(\operatorname{Im}(Z)=0\); parallel: \(\operatorname{Im}(Y)=0\) in sine mode.
• Dimensional consistency: units and scaling sanity checks (V, A, W, J, var).
• Determinism: identical inputs produce identical results (no randomization, no tolerance-driven branching).

Known Limitations

• No nonlinear magnetics (saturation, hysteresis, core loss models).
• No dielectric absorption, leakage, or frequency-dependent C unless explicitly modeled.
• No switch/device dynamics (dead-time, diode recovery, switching loss) unless mapped into equivalent parameters.
• No stray coupling between conductors, mutual inductance, or radiative effects.
• No temperature drift, tolerance spread, or measurement chain modeling (probe bandwidth, phase delay) unless explicitly added as a measurement layer.

Reproducibility

All computed outputs are deterministic for a given set of inputs and topology selection. For lab documentation, it is recommended to record the full parameter set (including units) and the exact software revision.