4. Examples

The stability analysis will now be applied to four different examples. The first is an artificial example generated in Matlab on which we can demonstrate that the unstable poles in the circuit are recovered perfectly. In the second example, an unstable balanced amplifier is analysed to show that the method works for RF circuits. The third example is a two-stage GaN Power Amplifier (PA). In the final example, a large-signal stability analysis is performed to verify the stability of a circuit orbit obtained in a HB simulation.

All simulations were performed in Keysight’s Advanced Design System (ADS) and the post-processing was performed in Matlab.

As a first example, the stability analysis is applied to a random system of order $202$ generated with the rss function from Matlab¹. The test system has an unstable pole pair at $1 \mathrm{GHz}$, as can be seen on its pole-zero map (Figure E1.1). A zero is placed close to the unstable poles. This makes that the unstable poles are difficult to observe in the FRF. To introduce delay in the test system, a time delay of $2 \mathrm{ns}$ is added to the system. The frequency response of the system is calculated on 5000 linearly spaced frequency points between $0 \mathrm{Hz}$ and $5 \mathrm{GHz}$ and is shown in green in Figure E1.2. The obtained stable and unstable parts after projection are shown in red and blue on the same figure. The maximum interpolation error is very low in this example ($-120 \mathrm{dB}$) . We will focus on the effect of the interpolation error in more detail in example 2.

The obtained unstable part peaks at $1 \mathrm{GHz}$, which matches the location of the unstable pole pair of the system. Note also that the obtained $Z_{\mathrm{unstable}}$ is very simple: it is clearly a second-order system. Most of the complexity of the frequency response, including the delay, is projected onto the stable part. This observation supports the proposed approach of estimating a rational model only after projection. A good fit was obtained with a rational model that consisted of two unstable poles and a single zero. The two poles obtained with a rational approximation of $Z_{\mathrm{unstable}}$ coincide exactly with the unstable poles in the circuit as is shown in Figure E1.1.

Figure E1.1 Pole-zero map of the test system. There are 202 poles (black +) and 200 zeroes (black o). The two poles placed in the right half-plane are easily recovered after the projection by fitting a low-order model on the unstable part (red square).

Figure E1.2 The stable/unstable projection splits the original frequency response (green -) in a stable part indicated with (blue -) and an unstable part indicated with (red -)