State-Space Nodal Across Domains

A component’s state-space model does not change between simulation domains. The same $\boldsymbol{A}$, $\boldsymbol{B}$, $\boldsymbol{C}$ and $\boldsymbol{D}$ describe the same physics whichever domain solves them. What changes is the operator that is discretised, and that difference propagates all the way to whether the resulting nodal stamp is real or complex.

The construction of the stamp itself is unchanged and is derived under state-space nodal components.

The instantaneous case

Here the state equation is integrated as written,

the trapezoidal rule gives a real discrete pair, and the equivalent admittance $\boldsymbol{W} = \boldsymbol{C}\boldsymbol{B}_d + \boldsymbol{D}$ is real.

The envelope case

In an envelope domain the state is a complex envelope $\tilde{\boldsymbol{x}}$ carrying an implicit $e^{j\omega_s t}$. Differentiating that product contributes the carrier term derived under dynamic phasors, so the operator seen by the envelope is

and the system that must be discretised is not $\boldsymbol{A}$ but $\boldsymbol{A} - j\omega_s \boldsymbol{I}$.

The real-augmented form

Rather than integrate a complex system, split the envelope into real and imaginary parts. The shifted operator becomes a real system of twice the size,

The off-diagonal $\pm\omega_s \boldsymbol{I}$ blocks are the carrier rotation, and the block structure $\begin{bmatrix} P & -Q \ Q & P \end{bmatrix}$ is the real representation of the complex number $P + jQ$. Discretising this real system with the same trapezoidal rule and recombining the blocks recovers the complex discrete pair, from which the equivalent admittance and the history term follow exactly as in the instantaneous case.

Two things follow. The equivalent admittance is complex in an envelope domain and real in an instantaneous one, so the same component stamps differently. And setting $\omega_s = 0$ collapses the augmented system back to the instantaneous one, which is the general statement about the envelope transform applied here: the instantaneous formulation is the zero-carrier special case, not a separate method.

Why this matters for accuracy

The carrier rotation is handled analytically, inside $\boldsymbol{A}$, rather than numerically by resolving the oscillation with small steps. For a component whose envelope varies slowly against a fast carrier, the step size is then set by the envelope’s own bandwidth rather than by the carrier frequency. That is the entire accuracy argument for using an envelope domain here, and it fails for exactly the reason it succeeds: content outside the retained band around $\omega_s$ has no representation at all.