Distortion Contribution Analysis with the Best Linear Approximation

Model-Free Closed-Loop Stability Analysis:
A Linear Functional Approach

Figure 2.1 A Small-signal current source is connected to a well-chosen node in the circuit under test to perform the local stability analysis.

In this paper, the (trans)impedance presented by the circuit to a small-signal current source will be used as FRF (Figure 2.1). In the remainder of this paper, it will be assumed that the unstable poles are observable in the FRF. To reduce the chance of missing an instability in the circuit due to a pole-zero cancellation, many different FRFs can be analysed one-by-one. Having a fast method to determine stability of a single FRF is therefore critical to a robust stability analysis.

The FRF of the linearised circuit is obtained by first placing the circuit in the required orbit, using either a DC or HB analysis and running a small-signal simulation around this orbit.

In a small-signal stability analysis, the stability of the DC solution of the circuit is investigated, so the FRF of the linearised circuit is obtained with an AC simulation. The impedance of the circuit is then obtained as:

Z_{mn}\left(j\omega\right)=\frac{V_{m}\left(j\omega\right)}{I_{n}\left(j\omega\right)}

where $I_{n}\left(j\omega\right)$ is the small-signal current injected into the selected node $n$ and $V_{m}\left(j\omega\right)$ is the voltage response of the circuit measured at node $m$ in the circuit.

In a large-signal stability analysis, the stability of a large-signal solution of the circuit is investigated. The circuit is driven by a periodic continuous-wave excitation at a pulsation $\omega_{\mathrm{ex}}$ and the circuit solution is obtained with a HB simulation.

The FRF of the linearised system around the HB orbit is obtained with a mixer-like simulation¹. As the small-signal will mix with the large signal, several transfer impedances with a different frequency translation are obtained:

Z_{mn}^{\left[ b \right]}\left(j\omega\right)=\frac{V_{m}\!\left(j\omega+bj\omega_{\mathrm{ex}}\right)}{I_{n}\!\left(j\omega\right)}\quad b\in\mathbb{Z}

The stability analysis now needs to determine whether the obtained impedances have poles in the right half-plane. The stability analysis of a large-signal orbit doesn’t differ much from the analysis of a DC solution [7]. The small-signal stability analysis can be considered a special case where only $Z_{mn}^{\left[ 0 \right]}\left(j\omega\right)$ is analysed.

In Keysight’s ADS, this mixer-like simulation is called a Large-Signal Small-Signal (LSSS) analysis. ↩

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2. Determining the frequency responses

Footnotes