Formula Reference

Equations implemented by the simulator (pulse + sine, RL + series RLC + (R+L)||C).

Topologies + Governing Equations

Series RL (state eq.) L · di/dt + R_total · i(t) = v_src(t)

Series RLC (KVL) v_src(t) = v_R(t) + v_L(t) + v_C(t)

(R+L)||C (parallel) i_src(t) = i_RL(t) + i_C(t), v(t) = v_src(t) = v_C(t) = v_RL(t)

Element laws v_R=iR_total, v_L=L·di/dt, i_C=C·dv/dt

Time constant (RL branch) τ = L / R_total

Core definitions

Total resistanceR_total = R_coil + R_load

Resistor drop (series)v_R(t) = i(t) · R_total

Magnetic energyW_L(t) = ½ · L · i²(t)

Capacitor energyW_C(t) = ½ · C · v_C²(t)

Flux linkage changeΔλ = L · Δi

Capacitor chargeq(t) = C · v_C(t)

Capacitor charge changeΔQ_cap = C · Δv_C

Pulse mode

Period / dutyT = 1/f, T_on = T·d, T_off = T − T_on

Ideal pulse sourcev_src(t) = V for 0→T_on, else 0

RL DC limitI∞ = V / R_total

RL ON currenti_on(t) = I∞ + (I₀ − I∞) · e^(−t/τ)

RL OFF currenti_off(t) = I_end · e^(−t/τ)

Steady I₀ (periodic) I₀ = I∞(1−a)b / (1−ab), a=e^(−T_on/τ), b=e^(−T_off/τ)

Pulse RLC / parallel steady state Solved as a periodic state transition over one period T (exact linear system closure)

State vector x = [ i, v_C ]^T

Series RLC state equation ẋ = A x + b·v_src(t), A = [[−R/L, −1/L], [1/C, 0]], b = [1/L, 0]^T

Segment propagation x(t) = Φ(t)x(0) + Γ(t)V, Φ(t)=e^At

Periodic closure (I − Φ_offΦ_on)x₀ = Φ_offΓ_onV

Parallel voltage identity For (R+L)||C: v_C(t) = v_src(t) (same node voltage)

Sine mode (steady-state)

Angular frequencyω = 2πf

ReactancesX_L = ωL, X_C = 1/(ωC)

Series net reactanceX = X_L − X_C

Series impedance magnitude|Z| = √(R² + X²)

Series phaseφ = atan2(X, R)

RMS valuesV_rms = V_pk/√2, I_rms = I_pk/√2

Parallel admittance Y = 1/(R + jX_L) + jωC

Parallel conductance G = R / (R² + (ωL)²)

Parallel susceptance B = −ωL / (R² + (ωL)²) + ωC

Parallel admittance split Y = G + jB

Parallel impedance Z = 1/Y, |Z| = 1/|Y|, φ = arg(Z) = −arg(Y)

Resonance + bandwidth

Series resonancef₀ = 1 / (2π√(LC))

Series quality factorQ₀ = ω₀L / R = 1/(ω₀ C R)

Series bandwidth (approx.)BW ≈ f₀ / Q₀

Detuningδ = (ω − ω₀) / ω₀

Parallel resonance (R+L)||C ω₀² = 1/(LC) − (R/L)² (requires ω₀² > 0)

Parallel resonance behavior At ω≈ω₀: net susceptance ≈ 0 → φ≈0 and |Z| peaks

Power + energy metrics

Mean power split (resistive) P_load = R_load·⟨i²(t)⟩, P_coil = R_coil·⟨i²(t)⟩, P_src = P_load + P_coil

Apparent powerS = V_rms·I_rms

Power factorPF = P/S

Reactive power (sine)Q = V_rms·I_rms·sin(φ)

Reactive split (series RLC) Q_L = I_rms²X_L, Q_C = −I_rms²X_C, Q_net = I_rms²(X_L − X_C)

Instantaneous source powerp(t) = v_src(t) · i(t)

Resistive lossp_R(t) = i²(t) · R

Inductor powerp_L(t) = v_L(t) · i(t) = dW_L/dt

Capacitor powerp_C(t) = v_C(t) · i(t) = dW_C/dt

Energy balance (per period) E_gen = ∫₀ᵀ v_srci dt, E_R = ∫₀ᵀ i²R dt, E_gen ≈ E_R + ΔW_L + ΔW_C

Windowed metrics (0 → τ)

Impulse metric∫₀^τ |v_L(t)| dt

Peak back-EMFmax₀→τ |v_L(t)|

Window charge-transfer metricQ_win = ∫₀^τ i(t) dt

Δi windowΔi_win = i(τ) − i(0)

Δλ checkΔλ_win = L·Δi_win

Peak magnetic energy in windowW_L,pk(0→τ) = ½·L·max₀→τ(i²(t))

Magnetic energy changeΔW_L(0→τ) = ½·L·(i²(τ) − i²(0))

Inductor consistencyΔW_L(0→τ) ?= ∫₀^τ v_L(t)·i(t) dt

Capacitor RMS voltage in windowV_C,rms,win = √[(1/τ)·∫₀^τ v_C²(t) dt]

Peak capacitor voltage in windowV_C,pk(0→τ) = max₀→τ |v_C(t)|

Peak capacitor energy in windowW_C,pk(0→τ) = ½·C·max₀→τ(v_C²(t))

Capacitor charge changeΔQ_cap(0→τ) = C·(v_C(τ) − v_C(0))

Capacitor energy changeΔW_C(0→τ) = ½·C·(v_C²(τ) − v_C²(0))

Capacitor energy integral∫₀^τ v_C(t)·i(t) dt

Capacitor consistency (C>0)ΔW_C(0→τ) ?= ∫₀^τ v_C(t)·i(t) dt

Reactive exchange / tank metrics (0 → τ)

Average inductor powerP_L,avg(0→τ) = (1/τ)·∫₀^τ v_L(t)·i(t) dt

Average capacitor powerP_C,avg(0→τ) = (1/τ)·∫₀^τ v_C(t)·i(t) dt

Net tank average powerP_tank,avg(0→τ) = P_L,avg + P_C,avg

Peak exchange magnitudeP_exchange,pk(0→τ) = max₀→τ |p_L(t)|

Pulse PQ (fundamental extraction)

Fundamental phasorX₁ = (2/T) · ∫₀ᵀ x(t)·e^(−jωt) dt, ω = 2π/T

RMS of fundamentalX_rms,1 = |X₁| / √2

Fundamental powersP₁ = ½·Re{V₁ I₁*}, Q₁ = ½·Im{V₁ I₁*}, S₁ = V_rms,1I_rms,1

PF (fundamental)PF₁ = P₁ / S₁

IEEE-style decompositionS_eq² = P₁² + Q₁² + D²

Symbols follow the code: R = R_total, τ = L/R. Series RLC uses KVL closure v_C = v_src − v_R − v_L. In (R+L)||C the capacitor voltage equals the source node voltage (v_C=v_src). Pulse PQ values are derived from the repetition fundamental.

Boundary Conditions

Valid only for linear inductance (L = const), ideal capacitance (C = const), lumped-element approximation, no saturation, no hysteresis, and no mutual coupling (k = 0).

Any deviation in laboratory measurements implies additional physical phenomena beyond these single-branch reference models.