In order to derive the complex-valued small-signal system based on the
approach, the equations (2.21), (2.22), and (2.23)
can be symbolically written as [90]:
To obtain the linearized version of (2.117), (2.118), and (2.119), a Taylor series expansion is performed, which is generally defined for a function with several unknowns as follows [29]:
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(2.128) |
By applying (2.129) for the functions of equations (2.123),
(2.124), and (2.125) with the functions (2.120),
(2.121), and (2.122), the following approximation is derived:
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(2.130) |
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(2.131) |
If this approximation is substituted into equations (2.123), (2.124), and (2.125), the resulting equation system reads
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(2.132) |
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(2.133) |
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(2.134) |
According to equations (2.99), (2.100), and (2.101), the
steady-state solutions are equal to zero. This linear equation system can be
written in the following matrix notation [127], where the subscript
emphasizes that all derivatives are evaluated at the steady-state
operating point:
The real-valued part of the system matrix equals the Jacobian matrix as shown in (2.108). For that reason, the assembly of this part of the matrix can be performed in exactly the same way as for steady-state analysis. The complex-valued contributions are then added by the transient models. As a consequence, the real-valued part can be stored during a frequency stepping since only the complex-valued part is modified.
As already discussed in Section 2.3.1, the exact Jacobian matrix
can be replaced by a simpler matrix. Since the solution of the
small-signal system does not involve an iterative process but is based on the
linearization of the device, it is absolutely necessary to take all derivatives
into account. If the simulator offers the possibility to skip several entries
(for example iteration schemes, see Section 3.6.4), it has to be
ensured that the complex-valued system matrices contain all necessary
entries. The same problem occurs while iteratively solving the complex-valued
equation system. The preconditioner has to be configured in such a way that no
entries are removed (see Section 5.2.5).
In contrast to the Newton procedure, the right-hand-side vector is
mostly zero. The small-signal Neumann boundary conditions are the
same as during steady-state analysis. The frequency-independent boundary
conditions for and
are zero, because the derivatives vanish in the
Taylor series expansion of the contact control function.
Real- or complex-valued Dirichlet boundary conditions for
can be used to excite the system.
After the complete complex-valued linear system for a small-signal analysis is assembled, the following parts can be identified:
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(2.137) |
The small-signal system for the energy-transport transport model reads as follows:
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(2.138) |
With matrix
from (2.139), the small-signal system for the six moments
transport models is eventually given by
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(2.140) |
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(2.141) |
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(2.142) |
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(2.143) |