# LiveSimulators

> LiveSimulators (https://livesimulators.com) is an open-access engineering simulation and virtual laboratory platform. It executes first-principles physical and mathematical models directly in the user browser at 60 FPS using Float64 numerical integrators (including 4th-Order Runge-Kutta). Built for university engineering syllabi, laboratory exploration, and industrial plant troubleshooting.

## Academic & Industrial Disciplines
- [Electrical & Electronic Systems (EE-200)](https://livesimulators.com/department/electrical): Analog circuits, switching power topologies, RF transmission lines, 3-phase grid power, and semiconductor dynamics. Core equation: `v(t) = L\frac{di}{dt} + Ri(t) + \frac{1}{C}\int i(t)dt`
- [Control Systems & Robotics (CS-300)](https://livesimulators.com/department/control): Closed-loop feedback stability, PID gain tuning, root locus analysis, state-space representations, and inverted pendulum dynamics. Core equation: `u(t) = K_p e(t) + K_i \int_0^t e(\tau)d\tau + K_d \frac{de(t)}{dt}`
- [Mechanical & Thermal Dynamics (ME-400)](https://livesimulators.com/department/mechanical): Damped harmonic oscillators, multi-degree-of-freedom vibrations, thermodynamic Carnot cycles, and heat transfer convection. Core equation: `m\ddot{x} + c\dot{x} + kx = F_0 \cos(\omega t)`
- [Chemical & Process Engineering (CH-250)](https://livesimulators.com/department/chemical): Continuous Stirred Tank Reactors (CSTR), McCabe-Thiele distillation columns, TEMA shell-and-tube heat exchangers, and packed gas absorbers. Core equation: `\frac{dC_A}{dt} = \frac{F}{V}(C_{A0} - C_A) - k_0 e^{-E_a/RT} C_A^2`
- [Civil & Structural Mechanics (CE-320)](https://livesimulators.com/department/civil): Truss axial stress distribution, Euler-Bernoulli beam deflection under uniform distributed loads, and seismic response spectra. Core equation: `EI \frac{d^4 w}{dx^4} = q(x)`
- [Quantum & Semiconductor Physics (PH-500)](https://livesimulators.com/department/physics): P-N junction energy band bending, SiC MOSFET switching dynamics, IGBT multi-layer thermal Foster networks, and BSIM4 inversion channels. Core equation: `i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r},t)`

## Interactive Engineering Simulators
- [RLC Resonant Circuit & Damping Transient](https://livesimulators.com/simulator/rlc-resonance): Observe underdamped ringing, critical damping, and impedance resonance in real time. Physical law: Kirchhoff's Voltage Law & Faraday Induction. Governing formulation: `\omega_0 = \frac{1}{\sqrt{LC}}, \quad \zeta = \frac{R}{2}\sqrt{\frac{C}{L}}`. Level: Fundamentals.
- [3-Phase AC Synchronous Motor & Rotating Field](https://livesimulators.com/simulator/three-phase-ac): Track how three balanced 120° sinusoidal currents generate a continuous rotating stator flux vector. Physical law: Faraday-Lenz Law & Lorentz Force Torque. Governing formulation: `\vec{B}_{net}(t) = \frac{3}{2} B_m \left[ \cos(\omega t)\hat{i} + \sin(\omega t)\hat{j} \right]`. Level: Intermediate.
- [Buck-Boost DC-DC Switching Regulator](https://livesimulators.com/simulator/buck-boost-converter): Explore inductor energy storage, Continuous vs Discontinuous mode, and voltage inversion. Physical law: Conservation of Energy & Volt-Second Balance. Governing formulation: `V_{out} = -V_{in}\frac{D}{1 - D}, \quad \Delta I_L = \frac{V_{in} D}{L f_{sw}}`. Level: Intermediate.
- [2nd-Order Sallen-Key Active Low-Pass Filter](https://livesimulators.com/simulator/sallen-key-filter): Tweak component values to shape Butterworth, Chebyshev, or Bessel frequency responses. Physical law: Negative Feedback & Complex Pole Placement. Governing formulation: `H(s) = \frac{K \omega_c^2}{s^2 + \frac{\omega_c}{Q} s + \omega_c^2}, \quad f_c = \frac{1}{2\pi \sqrt{R_1 R_2 C_1 C_2}}`. Level: Fundamentals.
- [RF Transmission Line & Reflection Coefficient](https://livesimulators.com/simulator/transmission-line): See incident and reflected TEM voltage waves collide to form standing wave envelopes. Physical law: Telegrapher's Equations & Boundary Wave Reflection. Governing formulation: `\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}, \quad \text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}`. Level: Advanced.
- [Fourier Series Harmonic Decomposition & THD](https://livesimulators.com/simulator/fourier-synthesis): Sum discrete sinusoidal harmonics to assemble square, triangle, and pulse waveforms. Physical law: Orthogonality of Sinusoidal Basis Functions. Governing formulation: `f(t) = \frac{4}{\pi} \sum_{n=1,3,5,\dots}^{N} \frac{1}{n} \sin(n \omega_0 t)`. Level: Fundamentals.
- [Planar 4-Bar Grashof Kinematics & Coupler Curves](https://livesimulators.com/simulator/four-bar-mechanism): Synthesize crank-rocker vs double-crank inversions and trace continuous coupler point trajectories. Physical law: Grashof Mechanism Inversion & Loop-Closure Constraint. Governing formulation: `s + l \le p + q, \quad K_1 \cos\theta_4 - K_2 \cos\theta_2 + K_3 = \cos(\theta_2 - \theta_4)`. Level: Intermediate.
- [Thermodynamic Rankine Steam Power Cycle](https://livesimulators.com/simulator/rankine-cycle): Track steam expansion through high-pressure boilers, turbines, condensers, and feed pumps on a T-s diagram. Physical law: First & Second Laws of Thermodynamics (ASME PTC 4.4). Governing formulation: `\eta_{th} = \frac{w_{net}}{q_{in}} = \frac{(h_1 - h_2) - (h_4 - h_3)}{h_1 - h_4}`. Level: Advanced.
- [Damped Forced Harmonic Oscillator & Resonance Peak](https://livesimulators.com/simulator/harmonic-oscillator): Visualize mass-spring-damper resonance, phase lag φ(ω), and displacement amplification M(ω). Physical law: Newton’s 2nd Law & Hookean Elastic Restoration. Governing formulation: `m\ddot{x} + c\dot{x} + kx = F_0 \cos(\omega t), \quad M(\omega) = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}`. Level: Fundamentals.
- [Involute Spur Gear Mesh Dynamics & Backlash](https://livesimulators.com/simulator/spur-gear-mesh): Track tooth contact along the line of action, verify contact ratio CR ≥ 1.4, and check sliding velocity. Physical law: Fundamental Law of Conjugate Gearing & AGMA 2001. Governing formulation: `r_b = r \cos \alpha, \quad CR = \frac{\sqrt{r_{a1}^2 - r_{b1}^2} + \sqrt{r_{a2}^2 - r_{b2}^2} - C\sin\alpha}{p_b}`. Level: Intermediate.
- [Industrial Closed-Loop PID Tuning & Anti-Windup](https://livesimulators.com/simulator/pid-tuning): Tune proportional, integral, and derivative gains with integrator anti-windup clamping. Physical law: Nyquist Stability & Classical Negative Feedback. Governing formulation: `u(t) = K_p e(t) + \frac{K_p}{T_i} \int_0^t e(\tau)d\tau + K_p T_d \frac{de(t)}{dt}`. Level: Intermediate.
- [Pneumatic Control Valve Flow Cv & Cavitation](https://livesimulators.com/simulator/control-valve): Compare Linear, Equal-Percentage, and Quick-Opening trims under varying differential pressures. Physical law: Bernoulli Fluid Continuity & Cavitation Choking. Governing formulation: `Q = C_v N_1 F_p \sqrt{\frac{\Delta P}{SG}}, \quad \sigma_c = \frac{P_1 - P_v}{\Delta P}`. Level: Intermediate.
- [4-20mA Current Loop Transmitter & Line Resistance](https://livesimulators.com/simulator/current-loop): Simulate 2-wire smart transmitters, cable loop resistance drops, and NAMUR fault alarms. Physical law: Ohm’s Law & Kirchhoff’s Voltage Loop. Governing formulation: `I_{loop} = 4\text{mA} + 16\text{mA}\left(\frac{PV - LRV}{URV - LRV}\right), \quad V_{term} = V_s - I_{loop}(R_{wire} + R_{load})`. Level: Fundamentals.
- [Differential Pressure Orifice Plate Flowmeter](https://livesimulators.com/simulator/orifice-meter): Calculate volumetric flow through concentric thin-plate square-edge orifices and tap dP. Physical law: Bernoulli Conservation of Fluid Energy & Continuity. Governing formulation: `q_m = \frac{C_d}{\sqrt{1 - \beta^4}} \epsilon \frac{\pi}{4} d^2 \sqrt{2 \rho \Delta P}, \quad \beta = \frac{d}{D}`. Level: Intermediate.
- [Euler-Bernoulli Beam Deflection, Shear & Moment](https://livesimulators.com/simulator/beam-bending): Calculate Shear Force Diagrams (SFD), Bending Moments (BMD), and elastic deflection profiles. Physical law: Euler-Bernoulli Beam Theory & Hooke’s Law. Governing formulation: `EI \frac{d^4 w}{dx^4} = q(x), \quad \sigma_{max} = \frac{M_{max} y}{I}`. Level: Fundamentals.
- [Warren & Pratt Truss Bridge Analysis](https://livesimulators.com/simulator/truss-bridge): Solve Method of Joints 2D nodal equilibrium under moving deck truck live loads and check Euler buckling. Physical law: Newton’s Static Equilibrium & Euler Column Buckling. Governing formulation: `\sum \vec{F}_{node} = 0, \quad P_{cr} = \frac{\pi^2 E I}{(K L)^2}`. Level: Intermediate.
- [Seismic Base Isolation & Dynamic Multi-Story Response](https://livesimulators.com/simulator/seismic-base-isolation): Compare conventional fixed-base vs elastomeric Lead-Rubber Bearing (LRB) building lateral drift. Physical law: Newton’s Second Law & Elastomeric Hysteresis. Governing formulation: `M \ddot{u} + C \dot{u} + K u = -M \ddot{u}_g(t), \quad T_n = 2\pi \sqrt{\frac{m}{k}}`. Level: Advanced.
- [Mohr’s Circle of Stress & Soil Shear Slip Failure](https://livesimulators.com/simulator/mohr-circle-soil): Transform 2D normal and shear stresses to principal axes (σ₁, σ₂) and evaluate Coulomb failure envelopes. Physical law: Stress Tensor Coordinate Transformation & Coulomb Friction. Governing formulation: `\tau_f = c + \sigma_n \tan \phi, \quad R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}`. Level: Intermediate.
- [Non-Isothermal CSTR & Thermal Runaway Dynamics](https://livesimulators.com/simulator/cstr-kinetics): Simulate continuous reaction kinetics, cooling jacket heat transfer, and thermal runaway thresholds. Physical law: Conservation of Mass, Arrhenius Law & First Law of Thermodynamics. Governing formulation: `V \frac{dC_A}{dt} = F(C_{A0} - C_A) - V k_0 e^{-E_a/RT} C_A, \quad V \rho C_p \frac{dT}{dt} = F \rho C_p(T_0 - T) + (-\Delta H) V r_A - U A (T - T_c)`. Level: Advanced.
- [Binary Distillation Column & McCabe-Thiele Stages](https://livesimulators.com/simulator/distillation-column): Step through McCabe-Thiele theoretical equilibrium trays, adjust reflux ratio, and evaluate minimum reflux R_min. Physical law: Raoult's Law & Conservation of Mass and Enthalpy (AIChE). Governing formulation: `y = \frac{\alpha x}{1 + (\alpha - 1)x}, \quad y = \frac{R}{R+1}x + \frac{x_D}{R+1}, \quad y = \frac{q}{q-1}x - \frac{z_F}{q-1}`. Level: Advanced.
- [Shell-and-Tube Heat Exchanger & ε-NTU Rating](https://livesimulators.com/simulator/heat-exchanger): Evaluate countercurrent thermal gradients, logarithmic mean temperature difference (LMTD), and ε-NTU effectiveness. Physical law: Fourier's Law of Conduction, Newton's Law of Cooling & Conservation of Energy (TEMA). Governing formulation: `Q = U A F \cdot \text{LMTD}, \quad \varepsilon = \frac{1 - \exp(-\text{NTU}(1 - C_r))}{1 - C_r \exp(-\text{NTU}(1 - C_r))}, \quad \text{NTU} = \frac{UA}{C_{min}}`. Level: Intermediate.
- [Packed Gas Absorption Tower & Flooding Hydraulics](https://livesimulators.com/simulator/gas-absorption): Simulate countercurrent solvent scrubbing, calculate packing height Z via HTU-NTU, and detect Sherwood flooding limits. Physical law: Whitman Two-Film Theory & Sherwood-Eckert Flooding Hydrodynamics. Governing formulation: `N_A = K_y(y - y^*), \quad Z = \text{HTU}_{OG} \cdot \text{NTU}_{OG}, \quad \text{NTU}_{OG} = \int_{y_2}^{y_1} \frac{dy}{y - y^*}`. Level: Advanced.
- [P-N Junction Diode Band Bending & Carrier Transport](https://livesimulators.com/simulator/pn-junction): Visualize energy band bending (Ec, Ev, Ef), space-charge depletion width, and carrier drift-diffusion. Physical law: Poisson's Electrostatic Equation & Fermi-Dirac Carrier Distribution. Governing formulation: `W = \sqrt{\frac{2\epsilon_s}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right)(V_{bi} - V_a)}, \quad I = I_s\left(e^{qV_a / k_B T} - 1\right)`. Level: Intermediate.
- [Silicon Carbide (SiC) MOSFET Double-Pulse Switching & Slew Rates](https://livesimulators.com/simulator/sic-switching): Capture ultra-fast dv/dt (up to 100 V/ns), Miller plateau dynamics, inductive ringing, and switching energy losses E_on / E_off. Physical law: Maxwell's Displacement Current, Carrier Drift-Diffusion & Parasitic RLC Resonant Tank. Governing formulation: `E_{sw} = \int_0^{t_{sw}} v_{ds}(t) \cdot i_d(t) dt, \quad \left(\frac{dv}{dt}\right)_{max} = \frac{V_{gs} - V_{plat}}{R_{g,tot} C_{gd}(V_{ds})}`. Level: Advanced.
- [IGBT Module Multi-Layer Thermal Foster Network & SOA](https://livesimulators.com/simulator/igbt-thermal): Model multi-layer conduction from Silicon Die through DBC ceramic and baseplate to heatsink, tracking T_j ripple. Physical law: Fourier Heat Conduction & Electro-Thermal Network Analogy (JEDEC JESD51). Governing formulation: `Z_{th}(t) = \sum_{i=1}^4 R_i \left(1 - e^{-t / \tau_i}\right), \quad T_j(t) = T_a + P_{tot} R_{th(j-a)} + \Delta T_{j,ripple}`. Level: Intermediate.
- [MOSFET Inversion Channel, Band Bending & 2D Electron Gas](https://livesimulators.com/simulator/mosfet-channel): Solve surface potential band bending, visualize 2D electron sheet accumulation, and observe channel pinch-off. Physical law: Poisson-Boltzmann Electrostatics & Fermi-Dirac Statistics (BSIM4). Governing formulation: `\phi_s = \frac{q N_A W_{dep}^2}{2\epsilon_s}, \quad Q_{inv} = -C_{ox}(V_{gs} - V_{th}), \quad I_D = \mu C_{ox} \frac{W}{L}\left[(V_{gs} - V_{th})V_{ds} - \frac{V_{ds}^2}{2}\right]`. Level: Intermediate.

## Industrial Engineering Labs Workbench
- [Power Electronics Lab](https://livesimulators.com/lab/power-electronics-lab): High-frequency switching converters, resonant inverters, and PWM gate driver dynamics.. Discipline: electrical | 16 Interactive Modules | Reference: IEEE 1547 / IEC 62040-3.
- [Power Systems Lab](https://livesimulators.com/lab/power-systems-lab): Transmission line power flow, transient swing stability, and symmetrical fault analysis.. Discipline: electrical | 14 Interactive Modules | Reference: IEEE 141 / NERC PRC-002.
- [SafeOps UPS](https://livesimulators.com/lab/safeops-ups): Double-conversion topology, static bypass transitions, and battery string impedance.. Discipline: instrumentation | 12 Interactive Modules | Reference: IEC 62040-3 / Uptime Tier IV.
- [ElectroLive Electrical Safety](https://livesimulators.com/lab/electrolive-electrical-safety): Arc flash incident energy, NFPA 70E boundary calculations, and touch/step potential.. Discipline: electrical | 10 Interactive Modules | Reference: NFPA 70E / IEEE 1584 / OSHA.

## Companion Engineering Portals
- [DesignCalculators.co.in](https://designcalculators.co.in): Free multi-disciplinary engineering calculators (Electrical, Mechanical, Instrumentation) referencing published standards (IEEE, IEC, ASME, API, ISA).
- [ReliabilityTools.co.in](https://reliabilitytools.co.in): Free plant reliability, asset uptime analytics, 2P/3P Weibull failure modeling, MTBF/MTTR, RCA, and IEC 61508/61511 SIL verification.

## Standards Reference & Non-Affiliation Notice
All standard designations and acronyms (IEEE, IEC, ASME, API, ISO, ISA, NFPA, ASTM, ANSI) belong to their respective owners and are referenced solely for technical identification and academic study. LiveSimulators is an independent educational platform and makes no claim of endorsement or official affiliation.

## Full Technical Documentation
- [Complete Equations, Mathematical Proofs & Benchmark Tests](https://livesimulators.com/llms-full.txt): Detailed mathematical derivations, analytical validation benchmarks, and complete parameter tables for all simulators.