Subsections

• 3.6.1 Mixed-Mode Simulation
• 3.6.2 The Nodal Approach and Modified Nodal Approach
• 3.6.3 Two-Level Newton and Full Newton Methods
• 3.6.4 Iteration Schemes
• 3.6.5 The Mixed-Mode AC Capabilities

3.6 Small-Signal Capabilities for Mixed-Mode Device/Circuit Simulations

All presented small-signal features are not only provided for single-mode simulations, but also for the mixed-mode of MINIMOS-NT. After a short introduction, the mixed-mode AC capabilities of MINIMOS-NT are discussed.

3.6.1 Mixed-Mode Simulation

Traditional device simulation has considered the behavior of isolated device structures under artificial boundary conditions (single-mode). To gain additional insight into the performance of devices under realistic dynamic boundary conditions imposed by a circuit, mixed-mode simulations have proven to be invaluable [81]. The main advantages of mixed-mode simulations are [231]:

• A calibrated device simulator can be directly employed for circuit simulations: No subsequent and often expensive parameter/model extraction is necessary. Thus, in time-to-market considerations results of many different devices are available at significantly earlier times.
• It is common practice to create optimization loops consisting of process and device simulators. Controlled by various kinds of optimizers, device figures of merit (for example, cut-off frequency $f_\mathrm{t}$) trigger process variations in order to be improved. By switching the device simulator into the mixed-mode, also circuit figures of merit can be optimization targets.

The major drawback in comparison to compact model approaches is the significant performance difference, since much larger equation systems have to be assembled and solved. However, the compact models can only be applied after the cumbersome extraction of the various parameters of the respective models. For example, the BSIM model [48] for short-channel MOS transistors provides over 300 parameters for calibration purposes, the VBIC95 model [143] for bipolar junction transistors offers about 30.

A physical circuit consists of an interconnection of circuit elements. Two well-known different aspects have to be considered when developing a mathematical model for a circuit. First, the circuit equations must satisfy Kirchhoff's topological laws:

• Kirchhoff's current law: the algebraic sum of currents leaving a circuit node must be zero at every instant of time (provided there is no charge on the nodes).
• Kirchhoff's voltage law: the algebraic sum of voltages around a circuit loop must be zero at every instant of time.

Second, each circuit element has to satisfy its branch relation which will be called a constitutive relation in the following. There are current-defined branches where the branch current is given in terms of circuit and device parameters, and voltage-defined branches where the branch voltage is given in terms of circuit and device parameters. Devices with $N$ terminals can be described using $N \cdot (N - 1)/2$ branch relations. It is not necessary to include all branch currents and voltages into the vector of unknowns. It is possible to also include charges and fluxes. The wide choice of possible unknown quantities leads to a variety of equation formulations that are available. From the number of published methods, the nodal approach and the tableau approach [46] are the most important. Whereas the latter is the most general approach allowing also simulation of many idealized theoretical circuit elements, it has several inherent disadvantages (for example, ill-conditioned system matrices). Since one main objective is to solve realistic devices, the nodal approach perfectly suits the needs.

3.6.2 The Nodal Approach and Modified Nodal Approach

The independent variables of the nodal approach are the node voltages of each circuit node to a reference node which can be chosen arbitrarily. Kirchhoff's current law is applied to each node other except the reference node in the circuit such that the summation of the currents leaving the node is zero. Thus, in matrix representation, the admittance matrix of the circuit is assembled, which consists of $N-1$ independent equations for a circuit of $N$ nodes.

The admittance matrix can be assembled by taking all contributions of each element into account. The various admittance matrices of the circuit elements can simply be superpositioned to yield the complete circuit admittance matrix. Current sources contribute to the current source vector on the right-hand-side of the equation system [81]. All contributions are commonly referred to as stamps as they can be directly stamped into the equation system without considering the rest of the circuit.

For circuits containing conductances and current sources only, the condition of the resulting system matrix is very good, because the nodal approach produces diagonally dominant matrices which are well suited for iterative solution procedures. Two additional devices can be modeled, namely a voltage controlled current source and the gyrator [204]. However, these devices destroy the diagonal dominance of the circuit admittance matrix.

One disadvantage of the nodal approach is the inadequate treatment of voltage sources. Ideal voltage sources and current controlled elements cannot be modeled with this approach. However, a very large class of integrated circuits can be accommodated by adding a provision for grounded sources. The modified nodal approach [99] overcomes these shortcomings by introducing branch currents as independent variables, which are available to formulate the device constitutive relations.

Since it is not difficult to implement, the modified nodal approach enjoys large popularity. However, the numerically well-behaved system matrix obtained by the nodal approach is distorted by those additional equations, and some additional measures (see Chapter 4) have to be taken. Furthermore, the additional equations can even produce zero diagonal entries which are avoided by exchanging the rows of the admittance matrix [147].

3.6.3 Two-Level Newton and Full Newton Methods

Several efforts dealing with circuit simulation using distributed devices have been introduced [175,144]. Most publications deal with the coupling of device simulators to SPICE [168,147]. This results in a two-level Newton algorithm since the device and circuit equations are handled subsequently. Each solution of the circuit equations gives new operating conditions for the distributed devices. After creating a new input-deck the device simulator is then invoked to calculate the resulting currents and the derivatives of these currents with respect to the contact voltages [81].

The alternative approach is called full Newton algorithm as it combines the device and circuit equations in one single equation system. This equation system is then solved applying a Newton method. In contrast to the two-level Newton algorithm where the device and circuit unknowns are solved in a decoupled manner, here the complete set of unknowns is solved simultaneously. In MINIMOS-NT the capability to solve circuit equations was added to the simulator kernel. This allows to assemble the circuit and the device equations into one system matrix which results in a real full Newton method. Since the contact currents are solution variables, derivatives of the contact currents in respect to the contact voltages need not be calculated explicitly.

Although the full Newton algorithm seems to be more effective as the complete set of capabilities is part of the simulator, the two-level Newton algorithm has particular parallelization advantages. Whereas the relatively quick circuit simulation can be done on one host, the simulations of all devices can be distributed over a network in a straightforward way. In contrast, the full Newton algorithm is restricted to parallelization strategies regarding the solution of the large linear equation system (see Chapter 5).

3.6.4 Iteration Schemes

The need for iteration schemes arises from the fact that when solving very complex coupled equation systems, the solution can often not be obtained from the available initial-guess as the region of attraction for the Newton scheme would be too small. Since the equations are split into their terms, the flexible equation assembly can simply neglect some of the contributions. Thus, it is possible to apply iteration schemes as described in [58,59], where for example some of the derivatives are neglected (see also Section 2.3.1).

Hence, the problem can be split into different levels of complexity with each of them using the previous level as an initial-guess to further refine the solution by applying more complicated models. This procedure is called iteration scheme. MINIMOS-NT provides an interface so that iteration schemes can be arbitrarily programmed with several additional options making use of the features provided by the input-deck [81]. An iteration scheme consists of arbitrarily nested iteration blocks. Each block can have subblocks which will be evaluated recursively.

For mixed-mode simulations an iteration scheme consisting of two blocks has been created. In the first block, specified node voltages are kept constant in order to obtain a converged solution for the distributed devices. This block is similar to single-mode device simulation. In the second block, the fixed voltages are set free in order to start the full circuit simulation. This procedure can be further improved by providing a previously obtained solution in an initial file.

3.6.5 The Mixed-Mode AC Capabilities

The AC features are activated the very same way than for the single-mode. However, circuits have to be extended by complex-valued sources. The sources are basically responsible for the setup of the mixed-mode AC mode. The mixed-mode of MINIMOS-NT can be used as pure circuit simulator with compact models only. In order to demonstrate the complex-valued sources, a parallel resonant circuit and a band rejection filter are simulated. For the simulation setup, see Appendix A.8. The output curves of both examples are shown in Figure 3.14.

Figure 3.14: Results of mixed-mode AC simulations with compact models only: the resonant circuit on the left side and the band rejection filter on the right side.

MINIMOS-NT provides a feature to calculate the complete admittance, scattering, and capacitance matrix of a circuit. In contrast to the single-mode, the user has to specify the output nodes which should act as small-signal terminals of the circuit. This is performed by including voltage sources. In Appendix A.10, an example setup for the simulation of a heterojunction bipolar transistor is shown. The H-, Z-, and ABCD-parameter sets provided for the single-mode are bound to the parasitic circuit, so they are not provided for the mixed-mode, because the parasitic elements can be included in the fully simulated circuit.