2.4 Small-Signal Simulation

Small-signal device simulation is used to extract the relationship between small sinusoidal terminal voltages and currents which are superimposed upon an already calculated steady-state operating point. This relationship depends on the DC operating point and on the frequency. The amplitude of the superimposed signal is considered to be small as long as harmonics are not generated within the device.

A small-signal simulation mode can be based on several approaches, some of them which will be shortly discussed in this section in accordance with the well-known overview from [127]. Whereas many of these approaches are based in the time domain and can thus use a transient simulation mode, the ${\textrm{S}^3\textrm{A}}$ approach (Sinusoidal Steady-State Analysis) is directly applied in the frequency domain.

In Figure 2.4 a comparison between approaches based in the transient and frequency domain is shown. The time derivatives are usually discretized by a backward Euler discretization, and thus a high number of steps has to be performed to achieve sufficient accuracy. For that reason the time consumption is usually reduced by extracting an equivalent circuit using the information of only one frequency.

**Figure 2.4:** Comparison of transient and frequency-domain-based approaches [233]. The dashed rectangles of the $\textrm {S}^3\textrm {A}$ approach symbolize complex-valued equation systems, the other real-valued ones.
$\includegraphics[width=6cm,angle=0]{figures/acdc_mod2.eps}$

Fourier decomposition techniques were one of the first choices to characterize AC device behavior [127]. The entries of the admittance matrix are obtained after the Fourier transformation of transient current and voltage responses. This allows to employ a transient simulation mode followed by transformation algorithm, for example a fast Fourier transformation. The technique is rigorous and universally applicable, but requires much computational resources, as a high number of time steps is required in order to achieve sufficient accuracy in the time and frequency domain.

An alternative are the so-called incremental charge partitioning heuristics. An entry of the capacitance matrix is obtained by $C_{ij} = \ensuremath{\Delta\,Q_i}/\ensuremath{\Delta\,V_j}$ . $\Delta\,Q_i$ is the incremental charge at the contact

. Whereas the results can be accurate and computationally inexpensive, this approach cannot be applied in a general-purpose device simulator. For example the gate capacitances of a MOSFET can be readily computed, since the transient current is solely a displacement current and the integral can be evaluated. The incremental charge is then simply the incremental charge induced at the gate by the voltage perturbation. The charge partitioning technique is heuristic and cannot be generally applied. For specific problems good results can be obtained with small computational resources.

2.4.1 Equivalent Circuits

In order to perform small and large signal simulations, equivalent circuits are frequently extracted and applied. The advantages of these circuits can be summarized as follows: like the compact models used in circuit simulations, they can be evaluated very efficiently. In addition, the values for the circuit elements can be optimized in order to deliver a nearly perfect match with the measurement data used for the calibration. The extraction procedure and the limitation for predefined operating conditions can be regarded as the disadvantages of this approach.

The small-signal modeling of a GaAs heterojunction bipolar transistor is often based on the linear hybrid $\Pi$ model [74,187]. The applied model is extended by the separation of inner and outer base resistance and of the base-collector capacitances. An alternative model is the T-model as discussed in [130]. In Figure 2.5 a standard $\Pi$ -type small-signal equivalent circuit of a HEMT [169] (left) and a T-type eight-element small-signal equivalent circuit of an HBT are shown [160]. Although this approach can be very efficient, inaccurate compact models can endanger the quality of the results.

**Figure 2.5:** These figures [160] show a standard $\Pi$ -type small-signal equivalent circuit of a HEMT (left) and a T-type eight-element small-signal equivalent circuit of an HBT. The dashed rectangles denote the intrinsic devices, the terminal resistances can be additionally included in the simulation.
$\includegraphics[width=0.98\linewidth ]{figures/eqcircuits.eps}$

2.4.2 Sinusoidal Steady-State Analysis

The most rigorous small-signal simulation mode is based on the sinusoidal steady-state analysis ( ${\textrm{S}^3\textrm{A}}$ ) approach, which is well-established in the device simulation area [127,90,77,218,111,214,216]. In contrast to the alternative approaches, transient analysis is not employed. The approach is rigorously correct and can be applied with small computational effort. Its results are very accurate due to the formal linearization of the device. In addition, the analysis is based directly in the frequency domain. Harmonics cannot be considered, because the device is linearized by a Taylor series expansion terminated after the linear term. Two properties can be identified why ${\textrm{S}^3\textrm{A}}$ is both an extremely accurate and very efficient approach: