Dynamic Phasors

The envelope transform behind the DP domain, derived two ways, and how it relates to EMT.

In the power systems community, dynamic phasors were initially introduced for power electronics analysis Sanders1991 as a more general approach than state-space averaging. They were used to construct efficient models for the dynamics of switching gate phenomena with a high level of detail as shown in Mattavelli1999. A few years later, dynamic phasors were also employed for power system simulation as described in Demiray2008. In Strunz2006 the authors combine the dynamic phasor approach with the Electromagnetic Transients Program (EMTP) simulator concept which includes Modified Nodal Analysis (MNA). Further research topics include fault and stability analysis under unbalanced conditions as presented in Stankovic2000 and also rotating machine models have been developed in dynamic phasors Zhang 2007.

Bandpass Signals and Baseband Representation

Although here, dynamic phasors are presented as a power system modelling tool, it should be noted that the concept is also known in other domains, for example, microwave and communications engineering Maas2003, Suarez2009, Haykin2009, Proakis2001. In these domains, the approach is often denoted as base band representation or complex envelope. Another common term coming from power electrical engineering is shifted frequency analysis (SFA). In the following, the general approach of dynamic phasors for power system simulation is explained starting from the idea of bandpass signals. This is because the 50 Hz or 60 Hz fundamental and small deviations from it can be seen as such a bandpass signal. Futhermore, higher frequencies, for example, generated by power electronics can be modelled in a similar way.

Two Derivations of the Same Transform

The literature reaches the envelope description along two different routes. One starts from the analytic signal and shifts the spectrum; the other averages a sliding Fourier series. They are usually presented independently, under different names, and it is not always stated that they arrive at the same differential operator. The two routes are set out below so that the equivalence is visible.

Route A: shifting the spectrum

A bandpass signal is one whose spectrum is concentrated in a band around some carrier frequency $\omega_s$, narrow compared with $\omega_s$ itself. The network voltages and currents of a power system are of this kind: energy sits around the 50 Hz or 60 Hz fundamental and spreads only a little either side of it during a transient.

Such a signal can be written exactly as

$$x(t) = \operatorname{Re}\{\, X(t)\, e^{j \omega_s t} \,\},$$

where the complex quantity $X(t)$ is the envelope. It is obtained from the analytic signal $x_a(t) = x(t) + j,\mathcal{H}{x}(t)$, with $\mathcal{H}$ the Hilbert transform, by

$$X(t) = x_a(t)\, e^{-j \omega_s t}.$$

Multiplication by $e^{-j\omega_s t}$ translates the spectrum down by $\omega_s$, so the band that sat around the carrier now sits around zero. That translation is what the name shifted frequency analysis refers to. No information is lost, and nothing is approximated: the operation is invertible and $x(t)$ can be recovered from $X(t)$ at any instant.

The consequence that matters for simulation is what happens to the derivative. Differentiating the expression above gives

$$\frac{dx}{dt} = \operatorname{Re}\left\{ \left( \frac{dX}{dt} + j \omega_s X \right) e^{j \omega_s t} \right\},$$

so in envelope coordinates the time derivative becomes

$$\frac{d}{dt} \;\longrightarrow\; \frac{d}{dt} + j \omega_s .$$

An inductor $v = L,\frac{di}{dt}$ therefore satisfies $V = L,\frac{dI}{dt} + j \omega_s L, I$. Two limiting cases are worth noting. When the envelope is constant, the first term vanishes and what remains is $V = j\omega_s L, I$, the classical steady-state phasor relation. When $\omega_s = 0$, the second term vanishes and the envelope is the instantaneous signal itself, which is the electromagnetic transient description. Both the phasor and the instantaneous representation are special cases of the same expression.

Route B: averaging a Fourier series

The dynamic phasor derivation starts elsewhere. Over a window of length $T = 2\pi/\omega_s$ ending at time $t$, the waveform is expanded in a Fourier series

$$x(t + s) = \sum_{k} X_k(t)\, e^{j k \omega_s s}, \qquad s \in (-T, 0],$$

whose coefficients are themselves functions of time,

$$X_k(t) = \frac{1}{T} \int_{t-T}^{t} x(\tau)\, e^{-j k \omega_s \tau} \, d\tau .$$

These sliding-window coefficients are the dynamic phasors. Each one is constant whenever the waveform is periodic and varies only as the waveform’s shape changes, which is why they are also called generalized averages. Integrating by parts gives the property the method is built on,

$$\frac{dX_k}{dt} = \left\langle \frac{dx}{dt} \right\rangle_k - j k \omega_s X_k ,$$

that is, the same shift as Route A, applied to the $k$-th coefficient.

Where the routes meet

Retaining only $k = 1$ makes Route B’s relation identical to Route A’s: the first coefficient obeys $\frac{d}{dt} + j\omega_s$ and the network equations that follow are the same. The two derivations produce one model.

The difference is emphasis rather than substance. Route A treats a single carrier and is exact for any signal, so the accuracy question is only whether the signal is genuinely bandpass around the carrier that was chosen. Route B offers a whole family of coefficients, so it can carry a DC component in $X_0$, the fundamental in $X_1$ and switching harmonics in higher $X_k$ simultaneously, at the cost of one complex state per coefficient retained. Truncating that series is where the approximation enters: energy at frequencies that were not given a coefficient is discarded, and the sliding window acts as a low-pass filter on what remains, so envelope components varying as fast as $\omega_s$ itself are attenuated.

This explains the split in usage. Work concerned with converter switching, harmonic interaction or unbalanced and fault conditions keeps several coefficients and speaks of dynamic phasors. Work concerned with transient simulation of the fundamental keeps one and speaks of shifted frequency analysis, emphasising the enlarged time step rather than the series. The communications and microwave literature calls the same object the complex envelope or the lowpass equivalent. The three names describe one transform, and the choice between them says more about a paper’s lineage than about its mathematics.

Relationship to Electromagnetic Transient Models

An electromagnetic transient (EMT) model solves for the instantaneous quantities directly. Because the waveform it integrates oscillates at the carrier, the step size has to resolve that oscillation, and the accuracy of the result is bounded by how finely the fundamental period is sampled. A shifted frequency model integrates the envelope instead. In sinusoidal steady state the envelope is constant, so the step size is dictated by the bandwidth of the transient rather than by the carrier, and it can be substantially larger for the same accuracy. This is the reason the approach is attractive for transient stability studies and for real-time simulation of large networks.

The saving is not free, and it is worth being precise about where it comes from. It is not that the envelope model is a coarser description of the same system; for a single carrier the transform is exact. It is that the carrier oscillation has been removed from the quantity being integrated and folded into the coefficient $j\omega_s$, which the model handles analytically rather than numerically. What the envelope model cannot represent is content the retained coefficients do not cover, which for a single-coefficient model means anything outside the band around $\omega_s$.

Because $x(t) = \operatorname{Re}{X(t) e^{j\omega_s t}}$ holds at every instant, converting between the two descriptions is an algebraic operation at a given point in time, not a filtering or a fitting problem. That is what makes mixed-domain and co-simulation setups possible: part of a network can be solved in envelope form and part in instantaneous form, with the interface performing the multiplication in one direction and the extraction of the envelope in the other. Two conditions have to be respected at such an interface. The envelope is defined relative to a chosen $\omega_s$ and a chosen time origin, so both sides must agree on the carrier frequency and share a phase reference; and the extraction direction needs the analytic signal, which for a general instantaneous waveform cannot be formed from a single sample. Any delay introduced between the two sides appears as a phase error on the carrier, and that error grows with $\omega_s$ rather than with the envelope’s own rate of change.

The state-space form of the shift, and how the resulting complex system is solved as a real one, are covered in State-Space Nodal.

Last modified 03.08.2026: removeNode: protect the ground node and clean up tear components (9470e46)