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.

If you are looking for intuitive, simulator-replicable circuit behavior explanations before diving into formal derivations, see the Applied Circuit Behavior Guides below. Those pages translate the mathematics into practical verification steps inside the tool.

Reading guide. Equations are organized by topology and excitation mode. For lab work, the most important sections are Model Assumptions, State-Space Definitions, Validation Criteria, and Known Limitations.

Applied Circuit Behavior Guides

The following pages translate the formal mathematics into physically intuitive, instrument-style explanations with fully worked, simulator-replicable examples. These are recommended reading before exploring advanced derivations.

RL Circuit (Series R–L)

Time constant \(\tau=L/R\), phase lag under sine excitation, pulse response, energy in magnetic field \(\tfrac12Li^2\), and step-by-step worked examples.

Open RL Guide →

Series RLC (R–L–C in Series)

Resonance \(\omega_0=1/\sqrt{LC}\), damping ratio, ringing under pulses, reactive power flow, and resonance verification inside the simulator.

Open Series RLC Guide →

Parallel Tank C ∥ (R+L)

Admittance view, anti-resonance \(\operatorname{Im}(Y)=0\), impedance peak, reactive current cancellation, and pulse behavior in voltage-driven parallel networks.

Open Parallel Tank Guide →
Reading order recommendation: Start with RL → then Series RLC → then Parallel Tank. Each page builds intuition for the next topology.

Model Assumptions

The excitation source is modeled as an ideal voltage source with zero internal impedance (Zs = 0). Therefore, no grid impedance, back-coupled network stiffness, or bidirectional source dynamics are included.

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

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 \(\operatorname{Im}(Y)=0\).
Important. Algebraic derivatives such as \(v_L = L\,di/dt\) are not treated as independent state variables. They can be discontinuous at switching edges even when the true state is perfectly periodic.

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: \[ i(t)=i_{RL}(t)+i_C(t),\qquad i_C(t)=C\frac{dv_{src}}{dt} \]
Edge handling. In pulse mode, discontinuous \(dv/dt\) would create idealized current impulses in \(i_C\). The simulator uses an edge-aware source representation to avoid non-physical infinite impulses while preserving analytic determinism.

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): \[ i(T^-)=i(0) \]
Metric note. For parallel pulse topology, periodicity must be evaluated on \(i_{RL}\) only. Including derivative-based quantities (e.g., \(v_L=L\,di/dt\)) can falsely report large errors due to edge discontinuities.

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): \[ Q_{net}=Q_L+Q_C,\qquad Q_L>0,\quad Q_C<0 \]

Fundamental, Equivalent, and Distortion Power Decomposition

For purely sinusoidal steady-state operation, active power, reactive power, apparent power, and power factor are fully described by the fundamental voltage–current phase relationship:

\[ P = V_{rms} I_{rms}\cos\phi,\qquad Q = V_{rms} I_{rms}\sin\phi,\qquad S = V_{rms} I_{rms},\qquad PF = \frac{P}{S} \]

In the analyzer, these values are reported both in their fundamental form and in equivalent total form where appropriate. This distinction is useful because non-sinusoidal excitation can contain harmonic content, for which no single physical phase angle fully characterizes the voltage–current relationship.

Accordingly, the simulator distinguishes between:

A common decomposition is:

\[ S_{eq}^2 = P_{eq}^2 + Q_{eq}^2 + D^2 \]

where \(D\) is distortion power. In the special case of a purely sinusoidal waveform, \(D=0\) and the equivalent quantities reduce to the familiar phasor-domain relations.

Interpretation note. For sine excitation, \(\phi\) and \(Q\) are directly physical phasor quantities. For non-sinusoidal excitation, the displayed equivalent phase angle and reactive power should be interpreted as RMS-derived summary quantities unless a fundamental-only decomposition is used.

Reactive Split into Inductive and Capacitive Components

For topologies containing both inductive and capacitive storage, the analyzer further separates reactive exchange into inductive and capacitive parts:

\[ Q_{net} = Q_L + Q_C \]

with the sign convention

\[ Q_L > 0,\qquad Q_C < 0 \]

and, in sinusoidal steady state,

\[ X_L = \omega L,\qquad X_C = \frac{1}{\omega C},\qquad X = X_L - X_C \]

This decomposition is useful for identifying whether the system is operating in a net inductive regime, a net capacitive regime, or near resonance. At resonance in the ideal series case, \(X_L = X_C\) and the net reactive term tends toward zero.

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:

Known Limitations

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.