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EMT simulation ​

Electromagnetic transient (EMT) simulation computes the instantaneous voltages and currents in an electrical network as functions of time, reproducing the full waveform — switching ripple, travelling-wave reflections, inrush, fault transients — rather than just a steady-state phasor. This page introduces what EMT simulation is, where it fits among the other studies you might run, and the numerical idea that makes it work.

Where EMT fits among network studies ​

Power engineers run several different kinds of study, each answering a different question over a different time scale:

  • Short-circuit / fault analysis — how large is the fault current?
  • Power (load) flow — what are the steady-state bus voltages and line flows? (See Power flow.)
  • Stability analysis — does the system stay synchronized after a disturbance, over seconds?
  • Electromagnetic transients (EMT) — what does the actual voltage/current waveform look like, microsecond by microsecond?

EMT is generally the most detailed and the most computationally demanding of these. A phasor study assumes a single sinusoidal frequency and solves for magnitudes and angles; an EMT study makes no such assumption and instead integrates the network's differential equations directly in the time domain, so it captures harmonics, switching events, and high-frequency transients that a phasor model cannot represent.

The core idea: discretize, then solve a resistive network ​

A network with inductors and capacitors is described by differential equations — the inductor relates voltage to the rate of change of current (v = L · di/dt), the capacitor relates current to the rate of change of voltage (i = C · dv/dt). Solving differential equations directly at every node would be expensive.

The standard EMT approach, introduced by Dommel and now used by every major EMT program, avoids that. It does two things:

  1. Pick a fixed time step h (often microseconds to tens of microseconds) and advance the simulation one step at a time: t = 0, h, 2h, 3h, ….

  2. Replace every energy-storage element with a "companion model" — a simple resistor (conductance) in parallel with a current source whose value is known from the previous time step. This is done by applying an implicit numerical integration rule (the trapezoidal rule) to the element's differential equation.

After this substitution, the network at each individual time step contains only conductances and current sources. That is a purely resistive network, and a resistive network is solved with a single set of linear equations — nodal analysis. So an EMT run becomes: at every step, build the known history sources, then solve one linear system for the node voltages.

Companion models in one line each ​

Applying the trapezoidal rule over one step of size h turns each element into a conductance G_eq in parallel with a history current source I_hist that depends only on past (already known) quantities:

Resistor   R   ->  G_eq = 1/R              (no history term)
Inductor   L   ->  G_eq = h / (2L)         I_hist from previous v, i
Capacitor  C   ->  G_eq = 2C / h           I_hist from previous v, i

The history source I_hist is what carries the element's stored energy forward from one step to the next. Because it is computed from the previous step's solution, it is a known constant by the time the current step's linear system is assembled.

The simulation loop ​

Once everything is a conductance plus a known current source, each time step is the same three-stage cycle:

  1. Update history sources. Using the node voltages and branch currents from the previous step, compute each reactive element's I_hist for the upcoming step.
  2. Solve the network. Assemble the linear system and solve for all node voltages at the new time t (this is the nodal / modified nodal analysis step).
  3. Recover branch quantities. From the new node voltages, compute each element's branch current, ready to feed the next step's history update.

Repeat for every time step until the run ends. The matrix that has to be solved is the same from step to step as long as the network's topology doesn't change, so it can be factorized once and reused — this is what makes EMT fast enough to be practical.

Why the trapezoidal rule ​

The trapezoidal rule is the workhorse integration method for EMT because it is A-stable (it will not blow up at large step sizes the way an explicit method can) and second-order accurate (error shrinks quadratically as you reduce h). It is also symmetric, which keeps the companion-model conductances constant for a fixed step size.

Its one well-known drawback is a tendency to produce sustained numerical oscillations after an abrupt change (for example, a switch opening an inductive current). EMT programs damp these with techniques such as critical-damping adjustment or a brief switch to backward Euler around discontinuities; the references below cover these in detail.

Switches and nonlinear elements ​

Diodes, IGBTs, breakers, and other switching elements change the network's conductance pattern when they turn on or off. EMT handles this by re-solving (iterating) within a step when a switch changes state, so the solution is consistent with the switch positions at the end of the step. Smoothly nonlinear elements (saturating inductors, surge arresters) are linearized around the operating point and solved the same way.

Choosing the time step ​

The time step h is the single most important accuracy/speed trade-off in an EMT study:

  • Smaller h resolves faster transients and reduces integration error, but costs more solves per second of simulated time.
  • Larger h runs faster but can miss or distort fast events and, for travelling-wave line models, must stay below the line's travel time.

A common rule of thumb is to choose h at least an order of magnitude smaller than the fastest time constant or switching period you care about.

See also ​

References ​

The treatment here is a high-level summary. For full derivations, companion-model details, and the numerical theory, see:

  • H. W. Dommel, Electromagnetic Transients Program (EMTP) Theory Book, Bonneville Power Administration. The original and definitive reference for the companion-model / nodal solution method.
  • J. Arrillaga and N. R. Watson, Power Systems Electromagnetic Transients Simulation, IET Power and Energy Series 39. A modern, textbook-style treatment of EMT methods.