Example 4: R-L-diode circuit

The final example in this paper shows that the stability analysis can also be used to determine the stability of HB simulations of the R-L-diode circuit shown in Figure E4.1. The circuit is based on [3], but a realistic diode model was used to represent the diode in the circuit instead of the three equations provided in the original paper.

The circuit is excited by a single-tone voltage source with an amplitude $V_{\mathrm{in}}$ and a frequency of $100 \mathrm{kHz}$. Because the diode has a transit-time of $4 \mathrm{\mu s}$, the circuit generates period-doubling solutions starting from sufficiently high amplitudes $V_{\mathrm{in}}$. For even higher $V_{\mathrm{in}}$, the circuit will create chaotic solutions.

To visualise this behaviour, a bifurcation diagram is constructed using time-domain simulations in the same way as is described in [3]: For every value of $V_{\mathrm{in}}$, 1030 periods of $100 \mathrm{kHz}$ are simulated and the final 30 periods are sampled every $\frac{1}{100 \mathrm{kHz}}$. If the circuit solution is periodic with the same period as the input source, all 30 sampled points will fall on top of each-other. If a period-doubling occurs in the circuit, two different values will be obtained.

The obtained bifurcation diagram for our R-L-diode example is shown in Figure E4.2. It is clear that a period-doubling occurs for $V_{\mathrm{in}}$ higher than $0.8 \mathrm{V}$. Starting from $1.8 \mathrm{V}$ the period quadruples. For the highest input amplitudes, a chaotic solution is obtained.

If this R-L-diode circuit is simulated with HB, the circuit solution is constrained to harmonics of $100 \mathrm{kHz}$. For input amplitudes higher than $0.8 \mathrm{V}$, where the circuit wants to go to a period-doubling solution, the constrained HB solution will be locally unstable.

We run two HB simulations on this circuit. Both HB simulations have a base frequency of $100 \mathrm{kHz}$ and an order of 10. In the first simulation, $V_{\mathrm{in}}$ is set to $0.5 \mathrm{V}$, which will result in a stable orbit. The second simulation has a $V_{\mathrm{in}}$ of $1.5 \mathrm{V}$, which will cause the orbit to be unstable.

The frequency response of the circuit around the HB solution is obtained with a mixer-like simulation, as explained in the introduction of this paper. The small-signal excitation was swept in both cases on a linear frequency grid starting from $(1 \mathrm{kHz}+1 \mathrm{Hz})$ up to $(2 \mathrm{MHz}+1 \mathrm{Hz})$ in $1 \mathrm{kHz}$ steps. The $1 \mathrm{Hz}$ was added to the start and stop values of the sweep to avoid overlap with the tones of the HB simulation.

The mixer-like simulation in ADS uses Single-Sideband (SSB) current excitations $i\left(t\right)=e^{j\omega t}$, which causes the obtained frequency responses $Z_{mn}^{\prime\left[b\right]}\left(j\omega\right)$ with $b\neq0$ to be non-Hermitian:

An alternative representation can make $Z_{mn}^{\prime\left[b\right]}\left(j\omega\right)$ Hermitian by transferring to a sine and cosine basis from the exponential basis [17] [26]

$Z_{mn}^{\left[ -1 \right]}\left(j\omega\right)$ $Z_{mn}^{\left[ 0 \right]}\left(j\omega\right)$ and $Z_{mn}^{\left[ +1 \right]}\left(j\omega\right)$ are then analysed with the stable/unstable projection method. The results are shown in Figure E4.3. The HB solution obtained for $V_{\mathrm{in}}=0.5 \mathrm{V}$ is clearly stable: its unstable part is more than $70 \mathrm{dB}$ smaller than its stable part.

In the case for $V_{\mathrm{in}}=1.5 \mathrm{V}$, the solution is clearly unstable as the unstable part lies far above the stable part of the frequency response. Note that the lowest-frequency peak in the unstable part is located around $50 \mathrm{kHz}$ and that copies of the resonance are found at $150 \mathrm{kHz}$, $250 \mathrm{kHz}$,… This behaviour is to be expected and indicates that the circuit wants to go to a period-doubling solution.

During the stability analysis of a periodic orbit, the unstable part will contain both the unstable base pole and all its higher-order copies. The unstable part will be simple, just like in the small-signal case, and it will be possible to approximate it by a finite set of base poles. Due to the infinite amount of higher-order copies however, it will not be possible to approximate it by a low-order rational approximation as is the case in the stability analysis of a DC solution.

Figure E4.1 The circuit R-L-diode circuit is excited with a small-signal current source at the diode.

Figure E4.2 The bifurcation diagram of the R-L-diode circuit shows that a period-doubling occurs for input amplitudes higher than $0.8 \mathrm{V}$.

Figure E4.3 The results of the stability analysis of $Z_{mn}^{\left[ -1 \right]}\left(j\omega\right)$ $Z_{mn}^{\left[ 0 \right]}\left(j\omega\right)$ and $Z_{mn}^{\left[ +1 \right]}\left(j\omega\right)$ in the two HB simulations show that, for $V_{\mathrm{in}}=0.5 \mathrm{V}$, the orbit is stable, but for $V_{\mathrm{in}}=1.5 \mathrm{V}$, the orbit is unstable. The interpolation error in shown with (black -).